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What is Research: Definition, Methods, Types & Examples

The search for knowledge is closely linked to the object of study; that is, to the reconstruction of the facts that will provide an explanation to an observed event and that at first sight can be considered as a problem. It is very human to seek answers and satisfy our curiosity. Let’s talk about research.

Content Index

What is Research?

What are the characteristics of research.

Comparative analysis chart

Qualitative methods

Quantitative methods, 8 tips for conducting accurate research.

Research is the careful consideration of study regarding a particular concern or research problem using scientific methods. According to the American sociologist Earl Robert Babbie, “research is a systematic inquiry to describe, explain, predict, and control the observed phenomenon. It involves inductive and deductive methods.”

Inductive methods analyze an observed event, while deductive methods verify the observed event. Inductive approaches are associated with qualitative research , and deductive methods are more commonly associated with quantitative analysis .

Research is conducted with a purpose to:

Identify potential and new customers
Understand existing customers
Set pragmatic goals
Develop productive market strategies
Address business challenges
Put together a business expansion plan
Identify new business opportunities
Good research follows a systematic approach to capture accurate data. Researchers need to practice ethics and a code of conduct while making observations or drawing conclusions.
The analysis is based on logical reasoning and involves both inductive and deductive methods.
Real-time data and knowledge is derived from actual observations in natural settings.
There is an in-depth analysis of all data collected so that there are no anomalies associated with it.
It creates a path for generating new questions. Existing data helps create more research opportunities.
It is analytical and uses all the available data so that there is no ambiguity in inference.
Accuracy is one of the most critical aspects of research. The information must be accurate and correct. For example, laboratories provide a controlled environment to collect data. Accuracy is measured in the instruments used, the calibrations of instruments or tools, and the experiment’s final result.

What is the purpose of research?

There are three main purposes:

Exploratory: As the name suggests, researchers conduct exploratory studies to explore a group of questions. The answers and analytics may not offer a conclusion to the perceived problem. It is undertaken to handle new problem areas that haven’t been explored before. This exploratory data analysis process lays the foundation for more conclusive data collection and analysis.

LEARN ABOUT: Descriptive Analysis

Descriptive: It focuses on expanding knowledge on current issues through a process of data collection. Descriptive research describe the behavior of a sample population. Only one variable is required to conduct the study. The three primary purposes of descriptive studies are describing, explaining, and validating the findings. For example, a study conducted to know if top-level management leaders in the 21st century possess the moral right to receive a considerable sum of money from the company profit.

LEARN ABOUT: Best Data Collection Tools

Explanatory: Causal research or explanatory research is conducted to understand the impact of specific changes in existing standard procedures. Running experiments is the most popular form. For example, a study that is conducted to understand the effect of rebranding on customer loyalty.

Here is a comparative analysis chart for a better understanding:

It begins by asking the right questions and choosing an appropriate method to investigate the problem. After collecting answers to your questions, you can analyze the findings or observations to draw reasonable conclusions.

When it comes to customers and market studies, the more thorough your questions, the better the analysis. You get essential insights into brand perception and product needs by thoroughly collecting customer data through surveys and questionnaires . You can use this data to make smart decisions about your marketing strategies to position your business effectively.

To make sense of your study and get insights faster, it helps to use a research repository as a single source of truth in your organization and manage your research data in one centralized data repository .

Types of research methods and Examples

Research methods are broadly classified as Qualitative and Quantitative .

Both methods have distinctive properties and data collection methods.

Qualitative research is a method that collects data using conversational methods, usually open-ended questions . The responses collected are essentially non-numerical. This method helps a researcher understand what participants think and why they think in a particular way.

Types of qualitative methods include:

One-to-one Interview
Focus Groups
Ethnographic studies
Text Analysis

Quantitative methods deal with numbers and measurable forms . It uses a systematic way of investigating events or data. It answers questions to justify relationships with measurable variables to either explain, predict, or control a phenomenon.

Types of quantitative methods include:

Survey research
Descriptive research
Correlational research

LEARN MORE: Descriptive Research vs Correlational Research

Remember, it is only valuable and useful when it is valid, accurate, and reliable. Incorrect results can lead to customer churn and a decrease in sales.

It is essential to ensure that your data is:

Valid – founded, logical, rigorous, and impartial.
Accurate – free of errors and including required details.
Reliable – other people who investigate in the same way can produce similar results.
Timely – current and collected within an appropriate time frame.
Complete – includes all the data you need to support your business decisions.

Gather insights

Identify the main trends and issues, opportunities, and problems you observe. Write a sentence describing each one.
Keep track of the frequency with which each of the main findings appears.
Make a list of your findings from the most common to the least common.
Evaluate a list of the strengths, weaknesses, opportunities, and threats identified in a SWOT analysis .
Prepare conclusions and recommendations about your study.
Act on your strategies
Look for gaps in the information, and consider doing additional inquiry if necessary
Plan to review the results and consider efficient methods to analyze and interpret results.

Review your goals before making any conclusions about your study. Remember how the process you have completed and the data you have gathered help answer your questions. Ask yourself if what your analysis revealed facilitates the identification of your conclusions and recommendations.

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Research Basics

What Is Research?
Types of Research
Secondary Research | Literature Review
Developing Your Topic
Primary vs. Secondary Sources
Evaluating Sources
Responsible Conduct of Research
Additional Help

Research is formalized curiosity. It is poking and prying with a purpose. - Zora Neale Hurston

A good working definition of research might be:

Research is the deliberate, purposeful, and systematic gathering of data, information, facts, and/or opinions for the advancement of personal, societal, or overall human knowledge.

Based on this definition, we all do research all the time. Most of this research is casual research. Asking friends what they think of different restaurants, looking up reviews of various products online, learning more about celebrities; these are all research.

Formal research includes the type of research most people think of when they hear the term “research”: scientists in white coats working in a fully equipped laboratory. But formal research is a much broader category that just this. Most people will never do laboratory research after graduating from college, but almost everybody will have to do some sort of formal research at some point in their careers.

So What Do We Mean By “Formal Research?”

Casual research is inward facing: it’s done to satisfy our own curiosity or meet our own needs, whether that’s choosing a reliable car or figuring out what to watch on TV. Formal research is outward facing. While it may satisfy our own curiosity, it’s primarily intended to be shared in order to achieve some purpose. That purpose could be anything: finding a cure for cancer, securing funding for a new business, improving some process at your workplace, proving the latest theory in quantum physics, or even just getting a good grade in your Humanities 200 class.

What sets formal research apart from casual research is the documentation of where you gathered your information from. This is done in the form of “citations” and “bibliographies.” Citing sources is covered in the section "Citing Your Sources."

Formal research also follows certain common patterns depending on what the research is trying to show or prove. These are covered in the section “Types of Research.”

Next: Types of Research >>
Last Updated: Dec 21, 2023 3:49 PM
URL: https://guides.library.iit.edu/research_basics

Module 1: Introduction: What is Research?

Learning Objectives

By the end of this module, you will be able to:

Explain how the scientific method is used to develop new knowledge
Describe why it is important to follow a research plan

The Scientific Method consists of observing the world around you and creating a hypothesis about relationships in the world. A hypothesis is an informed and educated prediction or explanation about something. Part of the research process involves testing the hypothesis , and then examining the results of these tests as they relate to both the hypothesis and the world around you. When a researcher forms a hypothesis, this acts like a map through the research study. It tells the researcher which factors are important to study and how they might be related to each other or caused by a manipulation that the researcher introduces (e.g. a program, treatment or change in the environment). With this map, the researcher can interpret the information he/she collects and can make sound conclusions about the results.

Research can be done with human beings, animals, plants, other organisms and inorganic matter. When research is done with human beings and animals, it must follow specific rules about the treatment of humans and animals that have been created by the U.S. Federal Government. This ensures that humans and animals are treated with dignity and respect, and that the research causes minimal harm.

No matter what topic is being studied, the value of the research depends on how well it is designed and done. Therefore, one of the most important considerations in doing good research is to follow the design or plan that is developed by an experienced researcher who is called the Principal Investigator (PI). The PI is in charge of all aspects of the research and creates what is called a protocol (the research plan) that all people doing the research must follow. By doing so, the PI and the public can be sure that the results of the research are real and useful to other scientists.

Module 1: Discussion Questions

How is a hypothesis like a road map?
Who is ultimately responsible for the design and conduct of a research study?
How does following the research protocol contribute to informing public health practices?

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USC Libraries
Research Guides

Organizing Your Social Sciences Research Paper

4. The Introduction
Purpose of Guide
Design Flaws to Avoid
Independent and Dependent Variables
Glossary of Research Terms
Reading Research Effectively
Narrowing a Topic Idea
Broadening a Topic Idea
Extending the Timeliness of a Topic Idea
Academic Writing Style
Choosing a Title
Making an Outline
Paragraph Development
Research Process Video Series
Executive Summary
The C.A.R.S. Model
Background Information
The Research Problem/Question
Theoretical Framework
Citation Tracking
Content Alert Services
Evaluating Sources
Primary Sources
Secondary Sources
Tiertiary Sources
Scholarly vs. Popular Publications
Qualitative Methods
Quantitative Methods
Insiderness
Using Non-Textual Elements
Limitations of the Study
Common Grammar Mistakes
Writing Concisely
Avoiding Plagiarism
Footnotes or Endnotes?
Further Readings
Generative AI and Writing
USC Libraries Tutorials and Other Guides
Bibliography

The introduction leads the reader from a general subject area to a particular topic of inquiry. It establishes the scope, context, and significance of the research being conducted by summarizing current understanding and background information about the topic, stating the purpose of the work in the form of the research problem supported by a hypothesis or a set of questions, explaining briefly the methodological approach used to examine the research problem, highlighting the potential outcomes your study can reveal, and outlining the remaining structure and organization of the paper.

Key Elements of the Research Proposal. Prepared under the direction of the Superintendent and by the 2010 Curriculum Design and Writing Team. Baltimore County Public Schools.

Importance of a Good Introduction

Think of the introduction as a mental road map that must answer for the reader these four questions:

What was I studying?
Why was this topic important to investigate?
What did we know about this topic before I did this study?
How will this study advance new knowledge or new ways of understanding?

According to Reyes, there are three overarching goals of a good introduction: 1) ensure that you summarize prior studies about the topic in a manner that lays a foundation for understanding the research problem; 2) explain how your study specifically addresses gaps in the literature, insufficient consideration of the topic, or other deficiency in the literature; and, 3) note the broader theoretical, empirical, and/or policy contributions and implications of your research.

A well-written introduction is important because, quite simply, you never get a second chance to make a good first impression. The opening paragraphs of your paper will provide your readers with their initial impressions about the logic of your argument, your writing style, the overall quality of your research, and, ultimately, the validity of your findings and conclusions. A vague, disorganized, or error-filled introduction will create a negative impression, whereas, a concise, engaging, and well-written introduction will lead your readers to think highly of your analytical skills, your writing style, and your research approach. All introductions should conclude with a brief paragraph that describes the organization of the rest of the paper.

Hirano, Eliana. “Research Article Introductions in English for Specific Purposes: A Comparison between Brazilian, Portuguese, and English.” English for Specific Purposes 28 (October 2009): 240-250; Samraj, B. “Introductions in Research Articles: Variations Across Disciplines.” English for Specific Purposes 21 (2002): 1–17; Introductions. The Writing Center. University of North Carolina; “Writing Introductions.” In Good Essay Writing: A Social Sciences Guide. Peter Redman. 4th edition. (London: Sage, 2011), pp. 63-70; Reyes, Victoria. Demystifying the Journal Article. Inside Higher Education.

Structure and Writing Style

I. Structure and Approach

The introduction is the broad beginning of the paper that answers three important questions for the reader:

What is this?
Why should I read it?
What do you want me to think about / consider doing / react to?

Think of the structure of the introduction as an inverted triangle of information that lays a foundation for understanding the research problem. Organize the information so as to present the more general aspects of the topic early in the introduction, then narrow your analysis to more specific topical information that provides context, finally arriving at your research problem and the rationale for studying it [often written as a series of key questions to be addressed or framed as a hypothesis or set of assumptions to be tested] and, whenever possible, a description of the potential outcomes your study can reveal.

These are general phases associated with writing an introduction: 1. Establish an area to research by:

Highlighting the importance of the topic, and/or
Making general statements about the topic, and/or
Presenting an overview on current research on the subject.

2. Identify a research niche by:

Opposing an existing assumption, and/or
Revealing a gap in existing research, and/or
Formulating a research question or problem, and/or
Continuing a disciplinary tradition.

3. Place your research within the research niche by:

Stating the intent of your study,
Outlining the key characteristics of your study,
Describing important results, and
Giving a brief overview of the structure of the paper.

NOTE: It is often useful to review the introduction late in the writing process. This is appropriate because outcomes are unknown until you've completed the study. After you complete writing the body of the paper, go back and review introductory descriptions of the structure of the paper, the method of data gathering, the reporting and analysis of results, and the conclusion. Reviewing and, if necessary, rewriting the introduction ensures that it correctly matches the overall structure of your final paper.

II. Delimitations of the Study

Delimitations refer to those characteristics that limit the scope and define the conceptual boundaries of your research . This is determined by the conscious exclusionary and inclusionary decisions you make about how to investigate the research problem. In other words, not only should you tell the reader what it is you are studying and why, but you must also acknowledge why you rejected alternative approaches that could have been used to examine the topic.

Obviously, the first limiting step was the choice of research problem itself. However, implicit are other, related problems that could have been chosen but were rejected. These should be noted in the conclusion of your introduction. For example, a delimitating statement could read, "Although many factors can be understood to impact the likelihood young people will vote, this study will focus on socioeconomic factors related to the need to work full-time while in school." The point is not to document every possible delimiting factor, but to highlight why previously researched issues related to the topic were not addressed.

Examples of delimitating choices would be:

The key aims and objectives of your study,
The research questions that you address,
The variables of interest [i.e., the various factors and features of the phenomenon being studied],
The method(s) of investigation,
The time period your study covers, and
Any relevant alternative theoretical frameworks that could have been adopted.

Review each of these decisions. Not only do you clearly establish what you intend to accomplish in your research, but you should also include a declaration of what the study does not intend to cover. In the latter case, your exclusionary decisions should be based upon criteria understood as, "not interesting"; "not directly relevant"; “too problematic because..."; "not feasible," and the like. Make this reasoning explicit!

NOTE: Delimitations refer to the initial choices made about the broader, overall design of your study and should not be confused with documenting the limitations of your study discovered after the research has been completed.

ANOTHER NOTE : Do not view delimitating statements as admitting to an inherent failing or shortcoming in your research. They are an accepted element of academic writing intended to keep the reader focused on the research problem by explicitly defining the conceptual boundaries and scope of your study. It addresses any critical questions in the reader's mind of, "Why the hell didn't the author examine this?"

III. The Narrative Flow

Issues to keep in mind that will help the narrative flow in your introduction :

Your introduction should clearly identify the subject area of interest . A simple strategy to follow is to use key words from your title in the first few sentences of the introduction. This will help focus the introduction on the topic at the appropriate level and ensures that you get to the subject matter quickly without losing focus, or discussing information that is too general.
Establish context by providing a brief and balanced review of the pertinent published literature that is available on the subject. The key is to summarize for the reader what is known about the specific research problem before you did your analysis. This part of your introduction should not represent a comprehensive literature review--that comes next. It consists of a general review of the important, foundational research literature [with citations] that establishes a foundation for understanding key elements of the research problem. See the drop-down menu under this tab for " Background Information " regarding types of contexts.
Clearly state the hypothesis that you investigated . When you are first learning to write in this format it is okay, and actually preferable, to use a past statement like, "The purpose of this study was to...." or "We investigated three possible mechanisms to explain the...."
Why did you choose this kind of research study or design? Provide a clear statement of the rationale for your approach to the problem studied. This will usually follow your statement of purpose in the last paragraph of the introduction.

IV. Engaging the Reader

A research problem in the social sciences can come across as dry and uninteresting to anyone unfamiliar with the topic . Therefore, one of the goals of your introduction is to make readers want to read your paper. Here are several strategies you can use to grab the reader's attention:

Open with a compelling story . Almost all research problems in the social sciences, no matter how obscure or esoteric , are really about the lives of people. Telling a story that humanizes an issue can help illuminate the significance of the problem and help the reader empathize with those affected by the condition being studied.
Include a strong quotation or a vivid, perhaps unexpected, anecdote . During your review of the literature, make note of any quotes or anecdotes that grab your attention because they can used in your introduction to highlight the research problem in a captivating way.
Pose a provocative or thought-provoking question . Your research problem should be framed by a set of questions to be addressed or hypotheses to be tested. However, a provocative question can be presented in the beginning of your introduction that challenges an existing assumption or compels the reader to consider an alternative viewpoint that helps establish the significance of your study.
Describe a puzzling scenario or incongruity . This involves highlighting an interesting quandary concerning the research problem or describing contradictory findings from prior studies about a topic. Posing what is essentially an unresolved intellectual riddle about the problem can engage the reader's interest in the study.
Cite a stirring example or case study that illustrates why the research problem is important . Draw upon the findings of others to demonstrate the significance of the problem and to describe how your study builds upon or offers alternatives ways of investigating this prior research.

NOTE: It is important that you choose only one of the suggested strategies for engaging your readers. This avoids giving an impression that your paper is more flash than substance and does not distract from the substance of your study.

Freedman, Leora and Jerry Plotnick. Introductions and Conclusions. University College Writing Centre. University of Toronto; Introduction. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Introductions. The Writing Center. University of North Carolina; Introductions. The Writer’s Handbook. Writing Center. University of Wisconsin, Madison; Introductions, Body Paragraphs, and Conclusions for an Argument Paper. The Writing Lab and The OWL. Purdue University; “Writing Introductions.” In Good Essay Writing: A Social Sciences Guide . Peter Redman. 4th edition. (London: Sage, 2011), pp. 63-70; Resources for Writers: Introduction Strategies. Program in Writing and Humanistic Studies. Massachusetts Institute of Technology; Sharpling, Gerald. Writing an Introduction. Centre for Applied Linguistics, University of Warwick; Samraj, B. “Introductions in Research Articles: Variations Across Disciplines.” English for Specific Purposes 21 (2002): 1–17; Swales, John and Christine B. Feak. Academic Writing for Graduate Students: Essential Skills and Tasks . 2nd edition. Ann Arbor, MI: University of Michigan Press, 2004 ; Writing Your Introduction. Department of English Writing Guide. George Mason University.

Writing Tip

Avoid the "Dictionary" Introduction

Giving the dictionary definition of words related to the research problem may appear appropriate because it is important to define specific terminology that readers may be unfamiliar with. However, anyone can look a word up in the dictionary and a general dictionary is not a particularly authoritative source because it doesn't take into account the context of your topic and doesn't offer particularly detailed information. Also, placed in the context of a particular discipline, a term or concept may have a different meaning than what is found in a general dictionary. If you feel that you must seek out an authoritative definition, use a subject specific dictionary or encyclopedia [e.g., if you are a sociology student, search for dictionaries of sociology]. A good database for obtaining definitive definitions of concepts or terms is Credo Reference .

Saba, Robert. The College Research Paper. Florida International University; Introductions. The Writing Center. University of North Carolina.

Another Writing Tip

When Do I Begin?

A common question asked at the start of any paper is, "Where should I begin?" An equally important question to ask yourself is, "When do I begin?" Research problems in the social sciences rarely rest in isolation from history. Therefore, it is important to lay a foundation for understanding the historical context underpinning the research problem. However, this information should be brief and succinct and begin at a point in time that illustrates the study's overall importance. For example, a study that investigates coffee cultivation and export in West Africa as a key stimulus for local economic growth needs to describe the beginning of exporting coffee in the region and establishing why economic growth is important. You do not need to give a long historical explanation about coffee exports in Africa. If a research problem requires a substantial exploration of the historical context, do this in the literature review section. In your introduction, make note of this as part of the "roadmap" [see below] that you use to describe the organization of your paper.

Introductions. The Writing Center. University of North Carolina; “Writing Introductions.” In Good Essay Writing: A Social Sciences Guide . Peter Redman. 4th edition. (London: Sage, 2011), pp. 63-70.

Yet Another Writing Tip

Always End with a Roadmap

The final paragraph or sentences of your introduction should forecast your main arguments and conclusions and provide a brief description of the rest of the paper [the "roadmap"] that let's the reader know where you are going and what to expect. A roadmap is important because it helps the reader place the research problem within the context of their own perspectives about the topic. In addition, concluding your introduction with an explicit roadmap tells the reader that you have a clear understanding of the structural purpose of your paper. In this way, the roadmap acts as a type of promise to yourself and to your readers that you will follow a consistent and coherent approach to addressing the topic of inquiry. Refer to it often to help keep your writing focused and organized.

Cassuto, Leonard. “On the Dissertation: How to Write the Introduction.” The Chronicle of Higher Education , May 28, 2018; Radich, Michael. A Student's Guide to Writing in East Asian Studies . (Cambridge, MA: Harvard University Writing n. d.), pp. 35-37.

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Research Paper Introduction – Writing Guide and Examples

Table of Contents

Research Paper Introduction

Research paper introduction is the first section of a research paper that provides an overview of the study, its purpose, and the research question (s) or hypothesis (es) being investigated. It typically includes background information about the topic, a review of previous research in the field, and a statement of the research objectives. The introduction is intended to provide the reader with a clear understanding of the research problem, why it is important, and how the study will contribute to existing knowledge in the field. It also sets the tone for the rest of the paper and helps to establish the author’s credibility and expertise on the subject.

How to Write Research Paper Introduction

Writing an introduction for a research paper can be challenging because it sets the tone for the entire paper. Here are some steps to follow to help you write an effective research paper introduction:

Start with a hook : Begin your introduction with an attention-grabbing statement, a question, or a surprising fact that will make the reader interested in reading further.
Provide background information: After the hook, provide background information on the topic. This information should give the reader a general idea of what the topic is about and why it is important.
State the research problem: Clearly state the research problem or question that the paper addresses. This should be done in a concise and straightforward manner.
State the research objectives: After stating the research problem, clearly state the research objectives. This will give the reader an idea of what the paper aims to achieve.
Provide a brief overview of the paper: At the end of the introduction, provide a brief overview of the paper. This should include a summary of the main points that will be discussed in the paper.
Revise and refine: Finally, revise and refine your introduction to ensure that it is clear, concise, and engaging.

Structure of Research Paper Introduction

The following is a typical structure for a research paper introduction:

Background Information: This section provides an overview of the topic of the research paper, including relevant background information and any previous research that has been done on the topic. It helps to give the reader a sense of the context for the study.
Problem Statement: This section identifies the specific problem or issue that the research paper is addressing. It should be clear and concise, and it should articulate the gap in knowledge that the study aims to fill.
Research Question/Hypothesis : This section states the research question or hypothesis that the study aims to answer. It should be specific and focused, and it should clearly connect to the problem statement.
Significance of the Study: This section explains why the research is important and what the potential implications of the study are. It should highlight the contribution that the research makes to the field.
Methodology: This section describes the research methods that were used to conduct the study. It should be detailed enough to allow the reader to understand how the study was conducted and to evaluate the validity of the results.
Organization of the Paper : This section provides a brief overview of the structure of the research paper. It should give the reader a sense of what to expect in each section of the paper.

Research Paper Introduction Examples

Research Paper Introduction Examples could be:

Example 1: In recent years, the use of artificial intelligence (AI) has become increasingly prevalent in various industries, including healthcare. AI algorithms are being developed to assist with medical diagnoses, treatment recommendations, and patient monitoring. However, as the use of AI in healthcare grows, ethical concerns regarding privacy, bias, and accountability have emerged. This paper aims to explore the ethical implications of AI in healthcare and propose recommendations for addressing these concerns.

Example 2: Climate change is one of the most pressing issues facing our planet today. The increasing concentration of greenhouse gases in the atmosphere has resulted in rising temperatures, changing weather patterns, and other environmental impacts. In this paper, we will review the scientific evidence on climate change, discuss the potential consequences of inaction, and propose solutions for mitigating its effects.

Example 3: The rise of social media has transformed the way we communicate and interact with each other. While social media platforms offer many benefits, including increased connectivity and access to information, they also present numerous challenges. In this paper, we will examine the impact of social media on mental health, privacy, and democracy, and propose solutions for addressing these issues.

Example 4: The use of renewable energy sources has become increasingly important in the face of climate change and environmental degradation. While renewable energy technologies offer many benefits, including reduced greenhouse gas emissions and energy independence, they also present numerous challenges. In this paper, we will assess the current state of renewable energy technology, discuss the economic and political barriers to its adoption, and propose solutions for promoting the widespread use of renewable energy.

Purpose of Research Paper Introduction

The introduction section of a research paper serves several important purposes, including:

Providing context: The introduction should give readers a general understanding of the topic, including its background, significance, and relevance to the field.
Presenting the research question or problem: The introduction should clearly state the research question or problem that the paper aims to address. This helps readers understand the purpose of the study and what the author hopes to accomplish.
Reviewing the literature: The introduction should summarize the current state of knowledge on the topic, highlighting the gaps and limitations in existing research. This shows readers why the study is important and necessary.
Outlining the scope and objectives of the study: The introduction should describe the scope and objectives of the study, including what aspects of the topic will be covered, what data will be collected, and what methods will be used.
Previewing the main findings and conclusions : The introduction should provide a brief overview of the main findings and conclusions that the study will present. This helps readers anticipate what they can expect to learn from the paper.

When to Write Research Paper Introduction

The introduction of a research paper is typically written after the research has been conducted and the data has been analyzed. This is because the introduction should provide an overview of the research problem, the purpose of the study, and the research questions or hypotheses that will be investigated.

Once you have a clear understanding of the research problem and the questions that you want to explore, you can begin to write the introduction. It’s important to keep in mind that the introduction should be written in a way that engages the reader and provides a clear rationale for the study. It should also provide context for the research by reviewing relevant literature and explaining how the study fits into the larger field of research.

Advantages of Research Paper Introduction

The introduction of a research paper has several advantages, including:

Establishing the purpose of the research: The introduction provides an overview of the research problem, question, or hypothesis, and the objectives of the study. This helps to clarify the purpose of the research and provide a roadmap for the reader to follow.
Providing background information: The introduction also provides background information on the topic, including a review of relevant literature and research. This helps the reader understand the context of the study and how it fits into the broader field of research.
Demonstrating the significance of the research: The introduction also explains why the research is important and relevant. This helps the reader understand the value of the study and why it is worth reading.
Setting expectations: The introduction sets the tone for the rest of the paper and prepares the reader for what is to come. This helps the reader understand what to expect and how to approach the paper.
Grabbing the reader’s attention: A well-written introduction can grab the reader’s attention and make them interested in reading further. This is important because it can help to keep the reader engaged and motivated to read the rest of the paper.
Creating a strong first impression: The introduction is the first part of the research paper that the reader will see, and it can create a strong first impression. A well-written introduction can make the reader more likely to take the research seriously and view it as credible.
Establishing the author’s credibility: The introduction can also establish the author’s credibility as a researcher. By providing a clear and thorough overview of the research problem and relevant literature, the author can demonstrate their expertise and knowledge in the field.
Providing a structure for the paper: The introduction can also provide a structure for the rest of the paper. By outlining the main sections and sub-sections of the paper, the introduction can help the reader navigate the paper and find the information they are looking for.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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Organizing Academic Research Papers: 4. The Introduction

Purpose of Guide
Design Flaws to Avoid
Glossary of Research Terms
Narrowing a Topic Idea
Broadening a Topic Idea
Extending the Timeliness of a Topic Idea
Academic Writing Style
Choosing a Title
Making an Outline
Paragraph Development
Executive Summary
Background Information
The Research Problem/Question
Theoretical Framework
Citation Tracking
Content Alert Services
Evaluating Sources
Primary Sources
Secondary Sources
Tertiary Sources
What Is Scholarly vs. Popular?
Qualitative Methods
Quantitative Methods
Using Non-Textual Elements
Limitations of the Study
Common Grammar Mistakes
Avoiding Plagiarism
Footnotes or Endnotes?
Further Readings
Annotated Bibliography
Dealing with Nervousness
Using Visual Aids
Grading Someone Else's Paper
How to Manage Group Projects
Multiple Book Review Essay
Reviewing Collected Essays
About Informed Consent
Writing Field Notes
Writing a Policy Memo
Writing a Research Proposal
Acknowledgements

The introduction serves the purpose of leading the reader from a general subject area to a particular field of research. It establishes the context of the research being conducted by summarizing current understanding and background information about the topic, stating the purpose of the work in the form of the hypothesis, question, or research problem, briefly explaining your rationale, methodological approach, highlighting the potential outcomes your study can reveal, and describing the remaining structure of the paper.

Key Elements of the Research Proposal. Prepared under the direction of the Superintendent and by the 2010 Curriculum Design and Writing Team. Baltimore County Public Schools.

Importance of a Good Introduction

Think of the introduction as a mental road map that must answer for the reader these four questions:

What was I studying?
Why was this topic important to investigate?
What did we know about this topic before I did this study?
How will this study advance our knowledge?

A well-written introduction is important because, quite simply, you never get a second chance to make a good first impression. The opening paragraph of your paper will provide your readers with their initial impressions about the logic of your argument, your writing style, the overall quality of your research, and, ultimately, the validity of your findings and conclusions. A vague, disorganized, or error-filled introduction will create a negative impression, whereas, a concise, engaging, and well-written introduction will start your readers off thinking highly of your analytical skills, your writing style, and your research approach.

Introductions . The Writing Center. University of North Carolina.

Structure and Writing Style

I. Structure and Approach

The introduction is the broad beginning of the paper that answers three important questions for the reader:

What is this?
Why am I reading it?
What do you want me to think about / consider doing / react to?

Think of the structure of the introduction as an inverted triangle of information. Organize the information so as to present the more general aspects of the topic early in the introduction, then narrow toward the more specific topical information that provides context, finally arriving at your statement of purpose and rationale and, whenever possible, the potential outcomes your study can reveal.

These are general phases associated with writing an introduction:

Highlighting the importance of the topic, and/or
Making general statements about the topic, and/or
Presenting an overview on current research on the subject.
Opposing an existing assumption, and/or
Revealing a gap in existing research, and/or
Formulating a research question or problem, and/or
Continuing a disciplinary tradition.
Stating the intent of your study,
Outlining the key characteristics of your study,
Describing important results, and
Giving a brief overview of the structure of the paper.

NOTE: Even though the introduction is the first main section of a research paper, it is often useful to finish the introduction very late in the writing process because the structure of the paper, the reporting and analysis of results, and the conclusion will have been completed and it ensures that your introduction matches the overall structure of your paper.

II. Delimitations of the Study

Delimitations refer to those characteristics that limit the scope and define the conceptual boundaries of your study . This is determined by the conscious exclusionary and inclusionary decisions you make about how to investigate the research problem. In other words, not only should you tell the reader what it is you are studying and why, but you must also acknowledge why you rejected alternative approaches that could have been used to examine the research problem.

Examples of delimitating choices would be:

The key aims and objectives of your study,
The research questions that you address,
The variables of interest [i.e., the various factors and features of the phenomenon being studied],
The method(s) of investigation, and
Any relevant alternative theoretical frameworks that could have been adopted.

Review each of these decisions. You need to not only clearly establish what you intend to accomplish, but to also include a declaration of what the study does not intend to cover. In the latter case, your exclusionary decisions should be based upon criteria stated as, "not interesting"; "not directly relevant"; “too problematic because..."; "not feasible," and the like. Make this reasoning explicit!

III. The Narrative Flow

Issues to keep in mind that will help the narrative flow in your introduction :

Your introduction should clearly identify the subject area of interest . A simple strategy to follow is to use key words from your title in the first few sentences of the introduction. This will help focus the introduction on the topic at the appropriate level and ensures that you get to the primary subject matter quickly without losing focus, or discussing information that is too general.
Establish context by providing a brief and balanced review of the pertinent published literature that is available on the subject. The key is to summarize for the reader what is known about the specific research problem before you did your analysis. This part of your introduction should not represent a comprehensive literature review but consists of a general review of the important, foundational research literature (with citations) that lays a foundation for understanding key elements of the research problem. See the drop-down tab for "Background Information" for types of contexts.
Clearly state the hypothesis that you investigated . When you are first learning to write in this format it is okay, and actually preferable, to use a past statement like, "The purpose of this study was to...." or "We investigated three possible mechanisms to explain the...."
Why did you choose this kind of research study or design? Provide a clear statement of the rationale for your approach to the problem studied. This will usually follow your statement of purpose in the last paragraph of the introduction.

IV. Engaging the Reader

The overarching goal of your introduction is to make your readers want to read your paper. The introduction should grab your reader's attention. Strategies for doing this can be to:

Open with a compelling story,
Include a strong quotation or a vivid, perhaps unexpected anecdote,
Pose a provocative or thought-provoking question,
Describe a puzzling scenario or incongruity, or
Cite a stirring example or case study that illustrates why the research problem is important.

NOTE: Only choose one strategy for engaging your readers; avoid giving an impression that your paper is more flash than substance.

Writing Tip

Avoid the "Dictionary" Introduction

Giving the dictionary definition of words related to the research problem may appear appropriate because it is important to define specific words or phrases with which readers may be unfamiliar. However, anyone can look a word up in the dictionary and a general dictionary is not a particularly authoritative source. It doesn't take into account the context of your topic and doesn't offer particularly detailed information. Also, placed in the context of a particular discipline, a term may have a different meaning than what is found in a general dictionary. If you feel that you must seek out an authoritative definition, try to find one that is from subject specific dictionaries or encyclopedias [e.g., if you are a sociology student, search for dictionaries of sociology].

Saba, Robert. The College Research Paper . Florida International University; Introductions . The Writing Center. University of North Carolina.

Another Writing Tip

When Do I Begin?

A common question asked at the start of any paper is, "where should I begin?" An equally important question to ask yourself is, "When do I begin?" Research problems in the social sciences rarely rest in isolation from the history of the issue being investigated. It is, therefore, important to lay a foundation for understanding the historical context underpinning the research problem. However, this information should be brief and succinct and begin at a point in time that best informs the reader of study's overall importance. For example, a study about coffee cultivation and export in West Africa as a key stimulus for local economic growth needs to describe the beginning of exporting coffee in the region and establishing why economic growth is important. You do not need to give a long historical explanation about coffee exportation in Africa. If a research problem demands a substantial exploration of historical context, do this in the literature review section; note in the introduction as part of your "roadmap" [see below] that you covering this in the literature review.

Yet Another Writing Tip

Always End with a Roadmap

The final paragraph or sentences of your introduction should forecast your main arguments and conclusions and provide a description of the rest of the paper [a "roadmap"] that let's the reader know where you are going and what to expect.

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Knowledge Base
Dissertation

How to Write a Thesis or Dissertation Introduction

Published on September 7, 2022 by Tegan George and Shona McCombes. Revised on November 21, 2023.

The introduction is the first section of your thesis or dissertation , appearing right after the table of contents . Your introduction draws your reader in, setting the stage for your research with a clear focus, purpose, and direction on a relevant topic .

Your introduction should include:

Your topic, in context: what does your reader need to know to understand your thesis dissertation?
Your focus and scope: what specific aspect of the topic will you address?
The relevance of your research: how does your work fit into existing studies on your topic?
Your questions and objectives: what does your research aim to find out, and how?
An overview of your structure: what does each section contribute to the overall aim?

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How to start your introduction, topic and context, focus and scope, relevance and importance, questions and objectives, overview of the structure, thesis introduction example, introduction checklist, other interesting articles, frequently asked questions about introductions.

Although your introduction kicks off your dissertation, it doesn’t have to be the first thing you write — in fact, it’s often one of the very last parts to be completed (just before your abstract ).

It’s a good idea to write a rough draft of your introduction as you begin your research, to help guide you. If you wrote a research proposal , consider using this as a template, as it contains many of the same elements. However, be sure to revise your introduction throughout the writing process, making sure it matches the content of your ensuing sections.

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Begin by introducing your dissertation topic and giving any necessary background information. It’s important to contextualize your research and generate interest. Aim to show why your topic is timely or important. You may want to mention a relevant news item, academic debate, or practical problem.

After a brief introduction to your general area of interest, narrow your focus and define the scope of your research.

You can narrow this down in many ways, such as by:

Geographical area
Time period
Demographics or communities
Themes or aspects of the topic

It’s essential to share your motivation for doing this research, as well as how it relates to existing work on your topic. Further, you should also mention what new insights you expect it will contribute.

Start by giving a brief overview of the current state of research. You should definitely cite the most relevant literature, but remember that you will conduct a more in-depth survey of relevant sources in the literature review section, so there’s no need to go too in-depth in the introduction.

Depending on your field, the importance of your research might focus on its practical application (e.g., in policy or management) or on advancing scholarly understanding of the topic (e.g., by developing theories or adding new empirical data). In many cases, it will do both.

Ultimately, your introduction should explain how your thesis or dissertation:

Helps solve a practical or theoretical problem
Addresses a gap in the literature
Builds on existing research
Proposes a new understanding of your topic

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Perhaps the most important part of your introduction is your questions and objectives, as it sets up the expectations for the rest of your thesis or dissertation. How you formulate your research questions and research objectives will depend on your discipline, topic, and focus, but you should always clearly state the central aim of your research.

If your research aims to test hypotheses , you can formulate them here. Your introduction is also a good place for a conceptual framework that suggests relationships between variables .

Conduct surveys to collect data on students’ levels of knowledge, understanding, and positive/negative perceptions of government policy.
Determine whether attitudes to climate policy are associated with variables such as age, gender, region, and social class.
Conduct interviews to gain qualitative insights into students’ perspectives and actions in relation to climate policy.

To help guide your reader, end your introduction with an outline of the structure of the thesis or dissertation to follow. Share a brief summary of each chapter, clearly showing how each contributes to your central aims. However, be careful to keep this overview concise: 1-2 sentences should be enough.

I. Introduction

Human language consists of a set of vowels and consonants which are combined to form words. During the speech production process, thoughts are converted into spoken utterances to convey a message. The appropriate words and their meanings are selected in the mental lexicon (Dell & Burger, 1997). This pre-verbal message is then grammatically coded, during which a syntactic representation of the utterance is built.

Speech, language, and voice disorders affect the vocal cords, nerves, muscles, and brain structures, which result in a distorted language reception or speech production (Sataloff & Hawkshaw, 2014). The symptoms vary from adding superfluous words and taking pauses to hoarseness of the voice, depending on the type of disorder (Dodd, 2005). However, distortions of the speech may also occur as a result of a disease that seems unrelated to speech, such as multiple sclerosis or chronic obstructive pulmonary disease.

This study aims to determine which acoustic parameters are suitable for the automatic detection of exacerbations in patients suffering from chronic obstructive pulmonary disease (COPD) by investigating which aspects of speech differ between COPD patients and healthy speakers and which aspects differ between COPD patients in exacerbation and stable COPD patients.

Checklist: Introduction

I have introduced my research topic in an engaging way.

I have provided necessary context to help the reader understand my topic.

I have clearly specified the focus of my research.

I have shown the relevance and importance of the dissertation topic .

I have clearly stated the problem or question that my research addresses.

I have outlined the specific objectives of the research .

I have provided an overview of the dissertation’s structure .

You've written a strong introduction for your thesis or dissertation. Use the other checklists to continue improving your dissertation.

If you want to know more about AI for academic writing, AI tools, or research bias, make sure to check out some of our other articles with explanations and examples or go directly to our tools!

Research bias

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The introduction of a research paper includes several key elements:

A hook to catch the reader’s interest
Relevant background on the topic
Details of your research problem

and your problem statement

A thesis statement or research question
Sometimes an overview of the paper

Don’t feel that you have to write the introduction first. The introduction is often one of the last parts of the research paper you’ll write, along with the conclusion.

This is because it can be easier to introduce your paper once you’ve already written the body ; you may not have the clearest idea of your arguments until you’ve written them, and things can change during the writing process .

Research objectives describe what you intend your research project to accomplish.

They summarize the approach and purpose of the project and help to focus your research.

Your objectives should appear in the introduction of your research paper , at the end of your problem statement .

Scope of research is determined at the beginning of your research process , prior to the data collection stage. Sometimes called “scope of study,” your scope delineates what will and will not be covered in your project. It helps you focus your work and your time, ensuring that you’ll be able to achieve your goals and outcomes.

Defining a scope can be very useful in any research project, from a research proposal to a thesis or dissertation . A scope is needed for all types of research: quantitative , qualitative , and mixed methods .

To define your scope of research, consider the following:

Budget constraints or any specifics of grant funding
Your proposed timeline and duration
Specifics about your population of study, your proposed sample size , and the research methodology you’ll pursue
Any inclusion and exclusion criteria
Any anticipated control , extraneous , or confounding variables that could bias your research if not accounted for properly.

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George, T. & McCombes, S. (2023, November 21). How to Write a Thesis or Dissertation Introduction. Scribbr. Retrieved April 9, 2024, from https://www.scribbr.com/dissertation/introduction-structure/

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What Is Research Methodology? A Plain-Language Explanation & Definition (With Examples)

By Derek Jansen (MBA) and Kerryn Warren (PhD) | June 2020 (Last updated April 2023)

If you’re new to formal academic research, it’s quite likely that you’re feeling a little overwhelmed by all the technical lingo that gets thrown around. And who could blame you – “research methodology”, “research methods”, “sampling strategies”… it all seems never-ending!

In this post, we’ll demystify the landscape with plain-language explanations and loads of examples (including easy-to-follow videos), so that you can approach your dissertation, thesis or research project with confidence. Let’s get started.

Research Methodology 101

What exactly research methodology means
What qualitative , quantitative and mixed methods are
What sampling strategy is
What data collection methods are
What data analysis methods are
How to choose your research methodology
Example of a research methodology

What is research methodology?

Research methodology simply refers to the practical “how” of a research study. More specifically, it’s about how a researcher systematically designs a study to ensure valid and reliable results that address the research aims, objectives and research questions . Specifically, how the researcher went about deciding:

What type of data to collect (e.g., qualitative or quantitative data )
Who to collect it from (i.e., the sampling strategy )
How to collect it (i.e., the data collection method )
How to analyse it (i.e., the data analysis methods )

Within any formal piece of academic research (be it a dissertation, thesis or journal article), you’ll find a research methodology chapter or section which covers the aspects mentioned above. Importantly, a good methodology chapter explains not just what methodological choices were made, but also explains why they were made. In other words, the methodology chapter should justify the design choices, by showing that the chosen methods and techniques are the best fit for the research aims, objectives and research questions.

So, it’s the same as research design?

Not quite. As we mentioned, research methodology refers to the collection of practical decisions regarding what data you’ll collect, from who, how you’ll collect it and how you’ll analyse it. Research design, on the other hand, is more about the overall strategy you’ll adopt in your study. For example, whether you’ll use an experimental design in which you manipulate one variable while controlling others. You can learn more about research design and the various design types here .

Need a helping hand?

What are qualitative, quantitative and mixed-methods?

Qualitative, quantitative and mixed-methods are different types of methodological approaches, distinguished by their focus on words , numbers or both . This is a bit of an oversimplification, but its a good starting point for understanding.

Let’s take a closer look.

Qualitative research refers to research which focuses on collecting and analysing words (written or spoken) and textual or visual data, whereas quantitative research focuses on measurement and testing using numerical data . Qualitative analysis can also focus on other “softer” data points, such as body language or visual elements.

It’s quite common for a qualitative methodology to be used when the research aims and research questions are exploratory in nature. For example, a qualitative methodology might be used to understand peoples’ perceptions about an event that took place, or a political candidate running for president.

Contrasted to this, a quantitative methodology is typically used when the research aims and research questions are confirmatory in nature. For example, a quantitative methodology might be used to measure the relationship between two variables (e.g. personality type and likelihood to commit a crime) or to test a set of hypotheses .

As you’ve probably guessed, the mixed-method methodology attempts to combine the best of both qualitative and quantitative methodologies to integrate perspectives and create a rich picture. If you’d like to learn more about these three methodological approaches, be sure to watch our explainer video below.

What is sampling strategy?

Simply put, sampling is about deciding who (or where) you’re going to collect your data from . Why does this matter? Well, generally it’s not possible to collect data from every single person in your group of interest (this is called the “population”), so you’ll need to engage a smaller portion of that group that’s accessible and manageable (this is called the “sample”).

How you go about selecting the sample (i.e., your sampling strategy) will have a major impact on your study. There are many different sampling methods you can choose from, but the two overarching categories are probability sampling and non-probability sampling .

Probability sampling involves using a completely random sample from the group of people you’re interested in. This is comparable to throwing the names all potential participants into a hat, shaking it up, and picking out the “winners”. By using a completely random sample, you’ll minimise the risk of selection bias and the results of your study will be more generalisable to the entire population.

Non-probability sampling , on the other hand, doesn’t use a random sample . For example, it might involve using a convenience sample, which means you’d only interview or survey people that you have access to (perhaps your friends, family or work colleagues), rather than a truly random sample. With non-probability sampling, the results are typically not generalisable .

To learn more about sampling methods, be sure to check out the video below.

What are data collection methods?

As the name suggests, data collection methods simply refers to the way in which you go about collecting the data for your study. Some of the most common data collection methods include:

Interviews (which can be unstructured, semi-structured or structured)
Focus groups and group interviews
Surveys (online or physical surveys)
Observations (watching and recording activities)
Biophysical measurements (e.g., blood pressure, heart rate, etc.)
Documents and records (e.g., financial reports, court records, etc.)

The choice of which data collection method to use depends on your overall research aims and research questions , as well as practicalities and resource constraints. For example, if your research is exploratory in nature, qualitative methods such as interviews and focus groups would likely be a good fit. Conversely, if your research aims to measure specific variables or test hypotheses, large-scale surveys that produce large volumes of numerical data would likely be a better fit.

What are data analysis methods?

Data analysis methods refer to the methods and techniques that you’ll use to make sense of your data. These can be grouped according to whether the research is qualitative (words-based) or quantitative (numbers-based).

Popular data analysis methods in qualitative research include:

Qualitative content analysis
Thematic analysis
Discourse analysis
Narrative analysis
Interpretative phenomenological analysis (IPA)
Visual analysis (of photographs, videos, art, etc.)

Qualitative data analysis all begins with data coding , after which an analysis method is applied. In some cases, more than one analysis method is used, depending on the research aims and research questions . In the video below, we explore some common qualitative analysis methods, along with practical examples.

Moving on to the quantitative side of things, popular data analysis methods in this type of research include:

Descriptive statistics (e.g. means, medians, modes )
Inferential statistics (e.g. correlation, regression, structural equation modelling)

Again, the choice of which data collection method to use depends on your overall research aims and objectives , as well as practicalities and resource constraints. In the video below, we explain some core concepts central to quantitative analysis.

How do I choose a research methodology?

As you’ve probably picked up by now, your research aims and objectives have a major influence on the research methodology . So, the starting point for developing your research methodology is to take a step back and look at the big picture of your research, before you make methodology decisions. The first question you need to ask yourself is whether your research is exploratory or confirmatory in nature.

If your research aims and objectives are primarily exploratory in nature, your research will likely be qualitative and therefore you might consider qualitative data collection methods (e.g. interviews) and analysis methods (e.g. qualitative content analysis).

Conversely, if your research aims and objective are looking to measure or test something (i.e. they’re confirmatory), then your research will quite likely be quantitative in nature, and you might consider quantitative data collection methods (e.g. surveys) and analyses (e.g. statistical analysis).

Designing your research and working out your methodology is a large topic, which we cover extensively on the blog . For now, however, the key takeaway is that you should always start with your research aims, objectives and research questions (the golden thread). Every methodological choice you make needs align with those three components.

Example of a research methodology chapter

In the video below, we provide a detailed walkthrough of a research methodology from an actual dissertation, as well as an overview of our free methodology template .

Psst… there’s more (for free)

This post is part of our dissertation mini-course, which covers everything you need to get started with your dissertation, thesis or research project.

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199 Comments

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I am writing a APA Format paper . I using questionnaire with 120 STDs teacher for my participant. Can you write me mthology for this research. Send it through email sent. Just need a sample as an example please. My topic is ” impacts of overcrowding on students learning

Thanks for your comment.

We can’t write your methodology for you. If you’re looking for samples, you should be able to find some sample methodologies on Google. Alternatively, you can download some previous dissertations from a dissertation directory and have a look at the methodology chapters therein.

All the best with your research.

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MLA Jansen, Derek, and Kerryn Warren. “What (Exactly) Is Research Methodology?” Grad Coach, June 2021, gradcoach.com/what-is-research-methodology/.

APA Jansen, D., & Warren, K. (2021, June). What (Exactly) Is Research Methodology? Grad Coach. https://gradcoach.com/what-is-research-methodology/

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Education Scholarship in Healthcare pp 13–23 Cite as

Introduction to Education Research

Sharon K. Park 3 ,
Khanh-Van Le-Bucklin 4 &
Julie Youm 4
First Online: 29 November 2023

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Educators rely on the discovery of new knowledge of teaching practices and frameworks to improve and evolve education for trainees. An important consideration that should be made when embarking on a career conducting education research is finding a scholarship niche. An education researcher can then develop the conceptual framework that describes the state of knowledge, realize gaps in understanding of the phenomenon or problem, and develop an outline for the methodological underpinnings of the research project. In response to Ernest Boyer’s seminal report, Priorities of the Professoriate , research was conducted about the criteria and decision processes for grants and publications. Six standards known as the Glassick’s criteria provide a tangible measure by which educators can assess the quality and structure of their education research—clear goals, adequate preparation, appropriate methods, significant results, effective presentation, and reflective critique. Ultimately, the promise of education research is to realize advances and innovation for learners that are informed by evidence-based knowledge and practices.

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Boyer EL. Scholarship reconsidered: priorities of the professoriate. Princeton: Carnegie Foundation for the Advancement of Teaching; 1990.

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Park, S.K., Le-Bucklin, KV., Youm, J. (2023). Introduction to Education Research. In: Fitzgerald, A.S., Bosch, G. (eds) Education Scholarship in Healthcare. Springer, Cham. https://doi.org/10.1007/978-3-031-38534-6_2

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The importance of crafting a good introduction to scholarly research: strategies for creating an effective and impactful opening statement

Mohsen tavakol.

1 Medical Education Centre, School of Medicine, The University of Nottingham, UK

David O'Brien

Introduction.

The introduction section is arguably one of the most critical elements of a written piece of research work, often setting the tone for the remainder of any dissertation or research article. The primary purpose of an introduction is to provide the reader with a clear understanding of the research question, in addition to the scope, rationale, aims and objectives of the study. This ensures the reader can more easily comprehend the context of the research, which will consequently help them better interpret and evaluate the study results. One could liken an introduction to a trailer for a movie, where the plot of the film (the research topic) is introduced by setting the scene (outlining the significance of the topic) and enticing you to watch the full movie (understanding the research and its importance).

Despite this, our experience suggests that students frequently pay insufficient attention to the introduction section of their dissertation or omit elements which we consider essential to address. This editorial aims to help researchers appreciate the importance of a comprehensive dissertation introduction in medical education research and learn how to effectively manage this key section of their work. Although it focuses purely on the introduction section of a written research submission, readers interested in learning more about the other primary steps of the research process are encouraged to read AMEE Guide No. 90 1 , 2 textbooks on research methods and both consult and seek constructive feedback from colleagues with expertise in research methods and writing for publication.

Here we aim to provide the reader with a simple structure of how best to construct the introduction for a dissertation and recommend that this should typically include the following essential components and principles.

Background to the research topic

The purpose of providing background information in an introduction is to supply the context and other essential information concerning the research topic, and thus allow the reader to understand the significance of the specific research question and where it sits within the broader field of study. This aids the reader to better understand how the research question contributes to the existing body of knowledge and why it is, necessary to investigate this specific aspect further. For example, suppose the study concerns the effectiveness of simulation-based training in medical education. In this case, the broader field of the study may include relevant areas such as medical simulation, medical education research, health care education, standardised patients, simulation-based training, and curriculum development based on simulation training. After providing the reader with an understanding of the context and relevance of the topic of interest, the researcher must then establish a theoretical or conceptual framework. This underpins the study topic in order that the reader can understand how any research questions and objectives are formulated. It is important to distinguish between these two frameworks. A theoretical framework describes the rationale for applying a particular theory to provide support and structure for the topic being studied. In the absence of an applicable theory, a conceptual framework substantiates the significance of a particular problem, context or phenomenon within a specific area of the study by illustrating its relevance and connection to research topic. 3 A conceptual framework highlights the importance of a research topic by showing how it relates to the larger body of knowledge in a particular field. Here is an example to demonstrate the use of a theoretical framework in a research context.

When considering Social Cognitive Theory (SCT), one of the key constructs is self-efficacy, as described by Albert Bandura, 4 and refers to the belief that a person has it within their own ability to accomplish a specific task successfully. This is not related to what a person does, but more how they perceive their ability to use these skills. So, based on this construct of self-efficacy, a researcher may formulate a research hypothesis; that examiners with higher self-efficacy in OSCEs will demonstrate improved performance in subsequent exams compared to those with lower self-efficacy. Now the researcher is in a position to identify the fundamental concepts of the research, i.e., self-efficacy (personal factors), examiner performance (behavioural factors) and examination conditions and examiner scaffolding support (environmental factors). Identifying key concepts helps the researcher find the relationship between these, and develop appropriate research questions, e.g., 1) How does an examiner's self-efficacy in OSCEs affect their ability to assess students in subsequent exams? 2) How does the support provided to examiners and exam conditions influence the link between self-efficacy and examiner performance in OSCEs? 3) Do examiners with high self-efficacy provide fairer scores than those with low self-efficacy in OSCEs? By having a theoretical framework, researchers can establish a foundation for their research and provide a clear picture of the relationship between the key concepts involved in the study. Researchers must also provide any conceptual and operational definitions for key concepts or variables that will be used in the study. Clearly defining key concepts and variables in the background section of a dissertation can also help establish the significance of the research question and its relevance to the broader field of study. As the name implies, a conceptual definition refers to a variable's meaning in a conceptual, abstract, or theoretical sense. Conceptual definitions are often used to describe concepts which cannot be directly measured, such as active learning, rote learning, inter-professional learning, inter-professional education, or constructs such as clinical performance. Conversely, operational definitions define the steps researchers must take in order to collect data to measure a phenomenon or concept. 5 For example, clinical performance can be considered a conceptual construct but may also be defined operationally as the ability of students to pass 12 out of 16 stations of an OSCE. The researcher having already pre-specified specific the criteria for classifying students as pass/fail in order to determine the ability of students to perform clinically. This operational definition provides a clear method for evaluating and measuring student ability, which can then be used to give feedback and guide further learning or to establish clear expectations for students and provide a basis for evaluating and assessing their performance. In general, it can be beneficial for medical education programs to define aspects such as clinical performance operationally in this way in rather than conceptually, especially if there is a need to ensure that students meet a required standard of competence and are prepared for the demands of real-world clinical practice. These definitions can also then be used to establish the methods and criteria by which the variables of the study will subsequently be measured or altered.

Citing the existing literature to support the research aim

A literature review is the process of critically evaluating existing research and utilising it to inform and guide the research proposal under investigation. Taking this approach enables researchers to ensure that their research is not only grounded in, but also contributes meaningfully to, any existing knowledge as a whole. Critically reviewing the literature provides evidence and justification for any research and is essential when formulating a hypothesis, question, or study objectives. In addition, and perhaps most importantly, it helps identify any gaps or inconsistencies in the existing knowledge base. Determining the knowledge gap is critical in justifying the necessity for our research and advancing knowledge. A comprehensive literature review also helps establish the theoretical or conceptual frameworks to ground any subsequent research, providing researchers with guidance and direction on how best to conduct their future studies. Understanding from the literature what has worked previously and what may pose challenges or limitations assists researchers when exploring the best methods and techniques for answering new research questions. To clarify, consider a hypothetical study in which researchers wish to examine the effectiveness of a specific educational intervention in medical students to improve patient safety. Based on the existing literature, let's assume that researchers learned that most studies had only focused on short-term outcomes rather than long-term ones. The long-term effects of any intervention in medical students on patient safety therefore remain uncertain. Researchers may therefore wish to consider conducting longitudinal studies months after interventions have been carried out, rather than simply repeating research based on short-term outcomes, in order to address the current knowledge gap. A review of existing literature may highlight hitherto previously unconsidered logistical difficulties in conducting longitudinal studies in this area that the researcher may need to be aware of.

Stating the significance of the research

More than simply reporting the existing research, one of the key objectives in any literature review is to summarise and synthesise existing research on the intended topic in order to analyse the significance of the research in question. In this process, diverse ideas can be merged to form fresh new perspectives. Any gaps, limitations, or controversies in medical education can be identified, and potential future benefits and implications of the proposed research explained to the reader. Based on any potential impact or perceived importance, the introduction provides an excellent opportunity for the researcher to affirm the significance of the research study and why it should be conducted.

By way of an example, the significance of a study concerning feedback given to examiners for Objective Structured Clinical Examinations (OSCEs) is used to illustrate this point further. The potential significance of this research lies in improving the validity and reliability of OSCE scores in medical education. As a result of reviewing different types of feedback given to examiners, the research may assist in identifying the most effective strategies for improving the quality of OSCEs in medical education. By providing new insights into how feedback can improve the reliability and validity of OSCE results, the research could also contribute to the broader knowledge of assessment in general. This may result in the development of more accurate and robust medical education assessments, which in turn may potentially enhance delivery of healthcare and improve patient outcomes and safety. It may also address the current challenges and gaps in medical education assessment by providing evidence-based approaches for improving OSCE quality.

Formulating Research Questions and Objectives

Researchers formulate research questions and objectives based on the topic they are seeking to address. As noted previously, these will have already been derived as a result of a comprehensive literature review of any existing knowledge and based on a theoretical or conceptual framework. Furthermore, in medical education, the literature review provides researchers with the opportunity to formulate new research questions or research objectives to address any gaps or limitations in the existing literature and add something new to the current body of knowledge. Research questions and objectives should be stated clearly, being both specific, and measurable. These should then guide the subsequent selection of appropriate research methods, data collection and any subsequent analytical process. Clear, focused, and rigorous research questions and objectives will ensure the study is well-designed and make a valuable contribution to the existing body of knowledge.

Qualitative research questions should be open-ended and exploratory rather than focused on a specific hypothesis or proposition. It is common for qualitative studies to focus on understanding how and why certain phenomena occur, rather than simply describing what has occurred. These should be formulated to elicit rich, detailed, and context-specific data that can provide insights into the experiences, perspectives, and meanings of the participants. In contrast, quantitative research questions are more specific and are designed to test a particular hypothesis or relationship. In medical education, it is imperative to emphasise the importance of both qualitative and quantitative research questions when it comes to generating new knowledge. Combining both quantitative and qualitative research methods (mixed methods) can be particularly powerful in providing a more comprehensive understanding of any phenomena under study. Assume again that we are examining the effectiveness of feedback on the performance of medical students and adopt a mixed-methods approach using a combination of qualitative and quantitative research methods. A quantitative research question may be, what is the impact of feedback on the performance of medical students as measured by OSCE mark? How the experience of receiving feedback on performance contributes to the future professional development of medical students is a more qualitative research question. This combination of quantitative and qualitative research questions will provide an in depth understanding of the effectiveness of feedback on medical student performance. It is important to note that in qualitative research methods particularly, there can be a wide variety of research question types. For example, grounded theory researchers may ask so-called "process questions", such as 'how do students interpret and use the feedback they are given?' Phenomenologists, on the other hand, are concerned with lived experience of research subjects and frequently ask questions looking to understand the "meaning" of any such experience, often aiming to attribute feelings to this experience, for example, ‘how do students feel when they receive feedback?’ Ethnographers look to understand how culture contributes to an experience, and may ask more "descriptive questions" 5 for example, ‘how does the culture within a specific medical school affect students receiving feedback on their performance?’

For ease of reference, the key points we recommended are considered in any dissertation introduction are summarised below:

1. Set the context for the research

2. Establish a theoretical or conceptual framework to support your study

3. Define key variables both conceptually and theoretically

4. Critically appraise relevant papers during the literature review

5. Review previous studies to identify and define the knowledge gap by assessing what has already been studied and what areas remain unexplored

6. Clearly articulate the rationale behind your study, emphasising its importance in the intended field

7. Clearly define your research objectives, questions, and hypotheses

Conclusions

Whilst crafting a research introduction may seem a challenging and time-consuming task, it is well worth the effort to convey your research clearly and engage potential readers. Providing sufficient background information on the research topic, conducting a comprehensive review of the existing research, determining the knowledge gap, understanding any limitations or controversies in the topic of interest, before then exploring any theoretical or conceptual frameworks to develop the research concepts, research questions and methodology are fundamental steps. Articulating any conceptual and operational definitions of key concepts and clearly defining any key terms, including explanations of how these will be used in the study is also paramount to a good introduction. It is essential to clearly present the rationale behind the research and why this is significant, clarifying what it adds to the existing body of knowledge in medical education and exploring any potential future implications. Lastly, it is vital to ensure that any research questions are clearly stated and are open-ended and exploratory in the case of qualitative studies, or specific and measurable in the case of quantitative studies.

We feel that observing these basic principles and adhering to these few simple steps will hopefully set the stage for a highly successful piece of research and will certainly go some way to achieving a favourable editorial outcome for possible subsequent publication of the work.

Conflict of Interest

The authors declare that they have no conflict of interest.

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How to Write a Research Paper Introduction (with Examples)

The research paper introduction section, along with the Title and Abstract, can be considered the face of any research paper. The following article is intended to guide you in organizing and writing the research paper introduction for a quality academic article or dissertation.

The research paper introduction aims to present the topic to the reader. A study will only be accepted for publishing if you can ascertain that the available literature cannot answer your research question. So it is important to ensure that you have read important studies on that particular topic, especially those within the last five to ten years, and that they are properly referenced in this section. 1 What should be included in the research paper introduction is decided by what you want to tell readers about the reason behind the research and how you plan to fill the knowledge gap. The best research paper introduction provides a systemic review of existing work and demonstrates additional work that needs to be done. It needs to be brief, captivating, and well-referenced; a well-drafted research paper introduction will help the researcher win half the battle.

The introduction for a research paper is where you set up your topic and approach for the reader. It has several key goals:

Present your research topic
Capture reader interest
Summarize existing research
Position your own approach
Define your specific research problem and problem statement
Highlight the novelty and contributions of the study
Give an overview of the paper’s structure

The research paper introduction can vary in size and structure depending on whether your paper presents the results of original empirical research or is a review paper. Some research paper introduction examples are only half a page while others are a few pages long. In many cases, the introduction will be shorter than all of the other sections of your paper; its length depends on the size of your paper as a whole.

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What is the introduction for a research paper, why is the introduction important in a research paper, craft a compelling introduction section with paperpal. try now, 1. introduce the research topic:, 2. determine a research niche:, 3. place your research within the research niche:, craft accurate research paper introductions with paperpal. start writing now, frequently asked questions on research paper introduction, key points to remember.

The introduction in a research paper is placed at the beginning to guide the reader from a broad subject area to the specific topic that your research addresses. They present the following information to the reader

Scope: The topic covered in the research paper
Context: Background of your topic
Importance: Why your research matters in that particular area of research and the industry problem that can be targeted

The research paper introduction conveys a lot of information and can be considered an essential roadmap for the rest of your paper. A good introduction for a research paper is important for the following reasons:

It stimulates your reader’s interest: A good introduction section can make your readers want to read your paper by capturing their interest. It informs the reader what they are going to learn and helps determine if the topic is of interest to them.
It helps the reader understand the research background: Without a clear introduction, your readers may feel confused and even struggle when reading your paper. A good research paper introduction will prepare them for the in-depth research to come. It provides you the opportunity to engage with the readers and demonstrate your knowledge and authority on the specific topic.
It explains why your research paper is worth reading: Your introduction can convey a lot of information to your readers. It introduces the topic, why the topic is important, and how you plan to proceed with your research.
It helps guide the reader through the rest of the paper: The research paper introduction gives the reader a sense of the nature of the information that will support your arguments and the general organization of the paragraphs that will follow. It offers an overview of what to expect when reading the main body of your paper.

What are the parts of introduction in the research?

A good research paper introduction section should comprise three main elements: 2

What is known: This sets the stage for your research. It informs the readers of what is known on the subject.
What is lacking: This is aimed at justifying the reason for carrying out your research. This could involve investigating a new concept or method or building upon previous research.
What you aim to do: This part briefly states the objectives of your research and its major contributions. Your detailed hypothesis will also form a part of this section.

How to write a research paper introduction?

The first step in writing the research paper introduction is to inform the reader what your topic is and why it’s interesting or important. This is generally accomplished with a strong opening statement. The second step involves establishing the kinds of research that have been done and ending with limitations or gaps in the research that you intend to address. Finally, the research paper introduction clarifies how your own research fits in and what problem it addresses. If your research involved testing hypotheses, these should be stated along with your research question. The hypothesis should be presented in the past tense since it will have been tested by the time you are writing the research paper introduction.

The following key points, with examples, can guide you when writing the research paper introduction section:

Highlight the importance of the research field or topic
Describe the background of the topic
Present an overview of current research on the topic

Example: The inclusion of experiential and competency-based learning has benefitted electronics engineering education. Industry partnerships provide an excellent alternative for students wanting to engage in solving real-world challenges. Industry-academia participation has grown in recent years due to the need for skilled engineers with practical training and specialized expertise. However, from the educational perspective, many activities are needed to incorporate sustainable development goals into the university curricula and consolidate learning innovation in universities.

Reveal a gap in existing research or oppose an existing assumption
Formulate the research question

Example: There have been plausible efforts to integrate educational activities in higher education electronics engineering programs. However, very few studies have considered using educational research methods for performance evaluation of competency-based higher engineering education, with a focus on technical and or transversal skills. To remedy the current need for evaluating competencies in STEM fields and providing sustainable development goals in engineering education, in this study, a comparison was drawn between study groups without and with industry partners.

State the purpose of your study
Highlight the key characteristics of your study
Describe important results
Highlight the novelty of the study.
Offer a brief overview of the structure of the paper.

Example: The study evaluates the main competency needed in the applied electronics course, which is a fundamental core subject for many electronics engineering undergraduate programs. We compared two groups, without and with an industrial partner, that offered real-world projects to solve during the semester. This comparison can help determine significant differences in both groups in terms of developing subject competency and achieving sustainable development goals.

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The purpose of the research paper introduction is to introduce the reader to the problem definition, justify the need for the study, and describe the main theme of the study. The aim is to gain the reader’s attention by providing them with necessary background information and establishing the main purpose and direction of the research.

The length of the research paper introduction can vary across journals and disciplines. While there are no strict word limits for writing the research paper introduction, an ideal length would be one page, with a maximum of 400 words over 1-4 paragraphs. Generally, it is one of the shorter sections of the paper as the reader is assumed to have at least a reasonable knowledge about the topic. 2 For example, for a study evaluating the role of building design in ensuring fire safety, there is no need to discuss definitions and nature of fire in the introduction; you could start by commenting upon the existing practices for fire safety and how your study will add to the existing knowledge and practice.

When deciding what to include in the research paper introduction, the rest of the paper should also be considered. The aim is to introduce the reader smoothly to the topic and facilitate an easy read without much dependency on external sources. 3 Below is a list of elements you can include to prepare a research paper introduction outline and follow it when you are writing the research paper introduction. Topic introduction: This can include key definitions and a brief history of the topic. Research context and background: Offer the readers some general information and then narrow it down to specific aspects. Details of the research you conducted: A brief literature review can be included to support your arguments or line of thought. Rationale for the study: This establishes the relevance of your study and establishes its importance. Importance of your research: The main contributions are highlighted to help establish the novelty of your study Research hypothesis: Introduce your research question and propose an expected outcome. Organization of the paper: Include a short paragraph of 3-4 sentences that highlights your plan for the entire paper

Cite only works that are most relevant to your topic; as a general rule, you can include one to three. Note that readers want to see evidence of original thinking. So it is better to avoid using too many references as it does not leave much room for your personal standpoint to shine through. Citations in your research paper introduction support the key points, and the number of citations depend on the subject matter and the point discussed. If the research paper introduction is too long or overflowing with citations, it is better to cite a few review articles rather than the individual articles summarized in the review. A good point to remember when citing research papers in the introduction section is to include at least one-third of the references in the introduction.

The literature review plays a significant role in the research paper introduction section. A good literature review accomplishes the following: Introduces the topic – Establishes the study’s significance – Provides an overview of the relevant literature – Provides context for the study using literature – Identifies knowledge gaps However, remember to avoid making the following mistakes when writing a research paper introduction: Do not use studies from the literature review to aggressively support your research Avoid direct quoting Do not allow literature review to be the focus of this section. Instead, the literature review should only aid in setting a foundation for the manuscript.

Remember the following key points for writing a good research paper introduction: 4

Avoid stuffing too much general information: Avoid including what an average reader would know and include only that information related to the problem being addressed in the research paper introduction. For example, when describing a comparative study of non-traditional methods for mechanical design optimization, information related to the traditional methods and differences between traditional and non-traditional methods would not be relevant. In this case, the introduction for the research paper should begin with the state-of-the-art non-traditional methods and methods to evaluate the efficiency of newly developed algorithms.
Avoid packing too many references: Cite only the required works in your research paper introduction. The other works can be included in the discussion section to strengthen your findings.
Avoid extensive criticism of previous studies: Avoid being overly critical of earlier studies while setting the rationale for your study. A better place for this would be the Discussion section, where you can highlight the advantages of your method.
Avoid describing conclusions of the study: When writing a research paper introduction remember not to include the findings of your study. The aim is to let the readers know what question is being answered. The actual answer should only be given in the Results and Discussion section.

To summarize, the research paper introduction section should be brief yet informative. It should convince the reader the need to conduct the study and motivate him to read further. If you’re feeling stuck or unsure, choose trusted AI academic writing assistants like Paperpal to effortlessly craft your research paper introduction and other sections of your research article.

1. Jawaid, S. A., & Jawaid, M. (2019). How to write introduction and discussion. Saudi Journal of Anaesthesia, 13(Suppl 1), S18.

2. Dewan, P., & Gupta, P. (2016). Writing the title, abstract and introduction: Looks matter!. Indian pediatrics, 53, 235-241.

3. Cetin, S., & Hackam, D. J. (2005). An approach to the writing of a scientific Manuscript1. Journal of Surgical Research, 128(2), 165-167.

4. Bavdekar, S. B. (2015). Writing introduction: Laying the foundations of a research paper. Journal of the Association of Physicians of India, 63(7), 44-6.

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Guide on How to Write a Research Paper Introduction

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Inhaltsverzeichnis

1 Research Paper Introduction – Definition
3 Research Paper Introduction: Structure
4 The Do’s and Don’ts
5 Research Paper Introduction: Example
6 In a nutshell

Research Paper Introduction – Definition

The research paper introduction arrests the reader’s attention from a general perspective to one specific area of a study. It outlines a summary of the research being conducted by condensing current understanding and background information about the topic, presenting the importance of the research in the form of a hypothesis, research questions , or research problem. It also outlines the methodological approach touching the likely outcomes that your study can reveal, and describing the remaining structure of the research paper.

Research paper introduction in academic writing is widely used in the presentation of a thesis and academic work. This article highlights the best ways to go about writing a captivating introduction to help you fine-tune your writing skills at the introductory level.

What is the purpose of a research paper introduction?

It establishes the depth, the context, and the importance of the research by summarizing and bringing the reader’s attention to your thesis. The research topic should be clear from the get-go. The introduction needs to draw in the reader whilst summarizing for them what it is that they’re about to read.

How do you start a research paper introduction?

You start the introduction of the research paper by presenting what your research paper is about. You’ll need some great sentence starters and transition words because your introduction needs to be well written in order to envoke the reader’s interest. Don’t forget to create some context and inform the reader about the research you have carried out.

How do you write a research paper introduction?

Draft your introduction on a piece of paper and edit it extensively before you add it to the final copy of your research paper. Be sure to refer to the research paper outline that you created before you started writing. Your sentences should be short and precise. It’s also important that you do not oversell your ideas at this point- remember that you’re still trying to draw the reader in.

What do you include in a research paper introduction?

You should highlight the key aspects of your thesis. It’s important that your thesis statement is placed towards the end of your research paper introduction. You are essentially briefly introducing the reader to concepts that they will come across in your research work.

How do you write a research paper introduction to a scientific research paper?

The information included in a scientific research paper introduction is very similar to what you would include in any other research paper . However, the overall structure of a scientific research paper is a bit different as you’ll need to include sections like ‘materials’ and ‘scientific processes’. Your introduction to a scientific research paper should highlight sufficient background information on the experiment that you did, making it easy for readers to understand and evaluate your research work.

What is the rationale in the research paper introduction?

The rationale for research is the highlight of why your research topic is worthy of the study and experimentation and how it adds value to already existing research works. You will probably need to bury yourself in books, do your research in the library and undertake descriptive research for your specific field. You need to become an expert in your chosen field and you should know exactly what you are contributing to the academic community with your research.

Tip: Read about the different parts of a research paper for a full rundown of which parts go where.

Research Paper Introduction: Structure

The structure of a research paper introduction should contain the main goal and the objective of the research. It should be a concise but enlightening outline of the soon-after context. Here you are required to state your rationale or reasons why you want to major into a particular subject or instead what problems you seek to solve in the subject matter.

Therefore, you need comprehensible argumentation to emphasize the importance of your research topic to your reader. In addition, you want to excite the readers curiosity for the subject. Below you will find the prime points to create a convincing research paper introduction.

The Do’s and Don’ts

One of the things that should be evident throughout your research paper introduction is honesty to your readers. This will go a long way in establishing a piece of research work that can be relied on by other students and researchers in the future. You will also not find it hard explaining the rest of the research paper to the panellists.

• Your research paper introduction should be short, accurate and precise. Don´t tell stories at the introductory level of your research.

• Pick-point the ideas you want to talk about and the methodologies that you have derived from the course work for you to solve the hindrances that you encountered on the ground.

• Refer to diverse research paper introduction works and make sure to look for up-to-date researches for your thesis.

• Provide tangible shreds of evidence and supporting arguments to blueprint your findings, and at least prove the fact that what you are presenting is well researched as well as authentic.

• Find it worth to include relevant terms, may it be scientific or mathematical or even theological.

• Always remember to proofread your work.

• Scrutinize your research paper introduction before presentation for reliability and present it with utmost logic to show how it supports your research and not a mere throwing in of figures.

• Do not try explaining ideas that do not answer your research questions. This is a mere waste of time and will not lead to any new conclusion about your research paper introduction work.

• Do not write a lengthy research paper introduction. What will you write in the rest of the paper if you tell it all here?

• Do not state incomplete reasons for carrying out the research. You want to be as convincing as possible in your research paper introduction.

• Do not exceed the stated word limit. It brings about the fact that you do not know what you are talking about, instead, you present yourself as a bluff.

• Do not plagiarize your research paper introduction, just like any other portion of your research work. Check this before any submissions. Make sure all the hypothetical findings are genuine and unique.

Research Paper Introduction: Example

In a nutshell

Research paper introduction introduces the core topic to your thesis.
The introduction explains where you are coming from concerning your research. Therefore make your research paper introduction precise.
The research paper introduction should be short, concise, and accurate.
Your research paper introduction should highlight the rationale of your research, which is the support of the worthiness of your study and research experiments.
A research paper introduction should be free from plagiarism.

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Open access
Published: 10 July 2023

The evolution of Big Data in neuroscience and neurology

Laura Dipietro 1 ,
Paola Gonzalez-Mego 2 ,
Ciro Ramos-Estebanez 3 ,
Lauren Hana Zukowski 4 ,
Rahul Mikkilineni 4 ,
Richard Jarrett Rushmore 5 &
Timothy Wagner 1 , 6

Journal of Big Data volume 10 , Article number: 116 ( 2023 ) Cite this article

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This article has been updated

Neurological diseases are on the rise worldwide, leading to increased healthcare costs and diminished quality of life in patients. In recent years, Big Data has started to transform the fields of Neuroscience and Neurology. Scientists and clinicians are collaborating in global alliances, combining diverse datasets on a massive scale, and solving complex computational problems that demand the utilization of increasingly powerful computational resources. This Big Data revolution is opening new avenues for developing innovative treatments for neurological diseases. Our paper surveys Big Data’s impact on neurological patient care, as exemplified through work done in a comprehensive selection of areas, including Connectomics, Alzheimer’s Disease, Stroke, Depression, Parkinson’s Disease, Pain, and Addiction (e.g., Opioid Use Disorder). We present an overview of research and the methodologies utilizing Big Data in each area, as well as their current limitations and technical challenges. Despite the potential benefits, the full potential of Big Data in these fields currently remains unrealized. We close with recommendations for future research aimed at optimizing the use of Big Data in Neuroscience and Neurology for improved patient outcomes.

Introduction

The field of Neuroscience was formalized in 1965 when the “Neuroscience Research Program” was established at the Massachusetts Institute of Technology with the objective of bringing together several varied disciplines including molecular biology, biophysics, and psychology to study the complexity of brain and behavior [ 1 ]. The methods employed by the group were largely data driven, with a foundation based on the integration of multiple unique data sets across numerous disciplines. As Neuroscience has advanced as a field, appreciation of the nervous system’s complexity has grown with the acquisition and analysis of larger and more complex datasets. Today, many Neuroscience subfields are implementing Big Data approaches, such as Computational Neuroscience [ 2 ], Neuroelectrophysiology [ 3 , 4 , 5 , 6 ], and Connectomics [ 7 ] to elucidate the structure and function of the brain. Modern Neuroscience technology allows for the acquisition of massive, heterogeneous data sets whose analysis requires a new set of computational tools and resources for managing computationally intensive problems [ 7 , 8 , 9 ]. Studies have advanced from small labs using a single outcome measure to large teams using multifaceted data (e.g., combined imaging, behavioral, and genetics data) collected across multiple international sites via numerous technologies and analyzed with high-performance computational methods and Artificial Intelligence (AI) algorithms. These Big Data approaches are being used to characterize the intricate structural and functional morphology of healthy nervous systems, and to describe and treat neurological disorders.

Jean-Martin Charcot (1825–1893), considered the father of Neurology, was a pioneering figure in utilizing a scientific, data-driven approach to innovate neurological treatments [ 10 ]. For example, in the study of multiple sclerosis (MS), once considered a general "nervous disorder" [ 10 ], Charcot's approach integrated multiple facets of anatomical and clinical data to delineate MS as a distinct disease. By connecting pathoanatomical data with behavioral and functional data, Charcot's work ultimately transformed our understanding and treatment of MS. Furthermore, Charcot’s use of medical photographs in his practice was an early instance of incorporating ‘imaging’ data in Neurology and Psychiatry [ 11 ]. Today, Neuroimaging, spurred on by new technologies, computational methods, and data types, is at the forefront of Big Data in Neurology [ 9 , 12 ]—see Fig. 1 . Current neurology initiatives commonly use large, highly heterogeneous datasets (e.g., neuroimaging, genetic testing, or clinical assessments from 1000s to 100,000s patients [ 13 , 14 , 15 , 16 , 17 , 18 ]) and acquire data with increasing velocity (e.g., using wearable sensors [ 6 ]) and technologies adapted from other Big Data fields (e.g., automatized clinical note assessment [ 19 ], social media-based infoveillance applications [ 16 , 20 ]). Similar to how Big Data has spurred on Neuroscience, the exponentially growing size, variety, and collection speed of datasets combined with the need to investigate their correlations is revolutionizing Neurology and patient care (see Fig. 1 ).

Evolution of data types [ 21 ]. The evolution of Data types in the development of Computational Neuroscience can be traced from Golgi and Ramón y Cajal’s structural data descriptions of the neuron in the nineteenth century [ 22 ]; to Hodgkin, Huxley, and Ecceles’s biophysical data characterization of the “all-or-none” action potential during the early to mid-twentieth century [ 23 ]; to McCulloch and Pitts’ work on the use of ‘the "all-or-none" character of nervous activity’ to model neural networks descriptive of fundamentals of nervous system [ 24 ]. Similarly, Connectomics’ Data evolution [ 25 ] can be traced from Galen’s early dissection studies [ 26 ], to Wernicke’s and Broca’s postulations on structure and function [ 27 ], to imaging of the nervous system [ 28 , 29 ], and brain atlases (e.g., Brodmann, Talairach) and databases [ 30 , 31 ] into the Big Data field that is today as characterized by the Human Connectome Project [ 32 ] and massive whole brain connectome models [ 7 , 33 ]. Behavioral Neuroscience and Neurology can be tracked from early brain injury studies [ 34 ] to stimulation and surgical studies [ 35 , 36 ], to Big Data assessments in cognition and behavior [ 37 ]. All these fields are prime examples of the transformative impact of the Big Data revolution on Neuroscience and Neurology sub-fields

This paper examines the evolving impact of Big Data in Neuroscience and Neurology, with a focus on treating neurological disorders. We critically evaluate available solutions and limitations, propose methods to overcome these limitations, and highlight potential innovations that will shape the fields' future.

Problem definition

According to the United States (US) National Institutes of Health (NIH), neurological disorders affect ~ 50 M/yr. people in the US, with a total annual cost of hundreds of billions of dollars [ 38 ]. Globally, neurological disorders are responsible for the highest incidence of disability and rank as the second leading cause of death [ 39 ]. These numbers are expected to grow over time as the global population ages. The need for new and innovative treatments is of critical and growing importance given the tremendous personal and societal impact of diseases of the nervous system and brain.

Big Data holds great potential for advancing the understanding of neurological diseases and the development of new treatments. To comprehend how such advancements can occur and have been occurring, it is important to appreciate how this type of research is enabled, not only through methods classically used in clinical research in Neurology such as clinical trials but also via advancing Neuroscience research.

This paper aims to review how Big Data is currently used and transforming the fields of Neuroscience and Neurology to advance the treatment of neurological disorders. Our intent is not merely to survey the most prominent research in each area, but to give the reader a historical perspective on how key areas moved from an earlier Small Data phase to the current Big Data phase. For applications in Neurology, while numerous clinical areas are evolving with Big Data and exemplified herein (e.g., Depression, Stroke, Alzheimer’s Disease (AD)), we highlight its impact on Parkinson’s Disease (PD), Substance Use Disorders (SUD), and Pain to provide a varied, yet manageable, review of the impact of Big Data on patient care. To balance brevity and completeness, we summarize a fair amount of general information in tabular form and limit our narrative to exemplify the Big Data trajectories of Neurology and Neuroscience. Additionally, in surveying this literature, we have identified a common limitation; specifically, the conventional application of Big Data, as characterized by the 5 V’s (see Fig. 2 ), is often unevenly or insufficiently applied in Neurology and Neuroscience. The lack of standardization for the Big Data in studies across Neurology and Neuroscience as well as field-specific and study-specific differences in application limit the reach of Big Data for improving patient treatments. We will examine the reasons that contribute to any mismatch and areas where past studies have not reached their potential. Finally, we identify the limitations of current Big Data approaches and discuss possible solutions and opportunities for future research.

The 5 V’s. While the 5 V’s of Big Data (“Volume, Variety, Velocity, Veracity, and Value”) are clearly found in certain fields (e.g., social media) there are many "Big Data" Neuroscience and Neurology projects where categories are not explored or are underexplored. Many self-described “Big Data” studies are limited to Volume and/or Variety. Furthermore, most “Big Data” clinical trial speeds move at the variable pace of patient recruitment which can pale in comparison to the speeds of Big Data Velocity in the finance and social media spaces. “Big Data” acquisition and processing times are also sporadically detailed in the fields. Finally, there is not an accepted definition of data Veracity as it pertains to healthcare (e.g., error, bias, incompleteness, inconsistency) and Veracity can be assessed on multiple levels (e.g., from data harmonization techniques to limitations in experimental methods used in studies)

Our paper differs from other Big Data review papers in Neuroscience and/or Neurology (e.g., [ 12 ], [ 40 , 41 , 42 , 43 ]) as it specifically examines the crucial role of Big Data in transforming the clinical treatment of neurological disorders. We go beyond previous papers that have focused on specific subfields (such as network data (e.g., [ 44 ]), neuroimaging (e.g., [ 12 ]), stroke (e.g., [ 45 ]), or technical methodologies related to data processing (e.g., [ 46 , 47 ]) and/or sharing (e.g., [ 48 , 49 ]). Furthermore, our review spans a broad range of treatments, from traditional pharmacotherapy to neuromodulation and personalized therapy guided by Big Data methods. This approach allows for a comparison of the evolving impact of Big Data across Neurology sub-specialties, such as Pain versus PD. Additionally, we take a cross-disciplinary approach to analyze applications in both Neuroscience and Neurology, synthesizing and categorizing available resources to facilitate insights between neuroscientists and neurologists. Finally, our study appraises the present implementation of the Big Data definition within the fields of Neuroscience and Neurology. Overall, we differentiate ourselves in terms of scope, breadth, and interdisciplinary analysis.

Existing solutions

Big Data use in Neuroscience and Neurology has matured as a result of national and multi-national projects [ 40 , 41 , 42 , 43 ]. In the early to mid-2000’s, several governments started national initiatives aimed at understanding brain function, such as the NIH Brain Initiative in the US [ 50 ], the Brain Project in Europe [ 51 , 52 ], and the Brain Mapping by Integrated Neurotechnologies for Disease Studies (Brain/MINDS) project in Japan [ 53 ]. Although not always without controversy [ 40 , 51 , 52 ], many initiatives soon became global and involved increasingly larger groups of scientists and institutions focused on collecting and analyzing voluminous data including neuroimaging, genetic, biospecimen, and/or clinical assessments to unlock the secrets of the nervous system (the reader is referred to Table 1 and Additional file 1 : Table S1 for exemplary projects or reviews [ 40 , 41 , 42 , 43 ]). These projects spurred the creation of open-access databases and resource depositories (the reader is referred to Table 2 and Additional file 1 : Table S2 for exemplary databases or reviews [ 41 , 42 ]). The specific features of the collected data sets, such as large volume, high heterogeneity/variety, and inconsistencies across sites/missing data, necessitated the development of ad-hoc resources, procedures, and standards for data collection and processing. Moreover, these datasets created the need for hardware and software for data-intensive computing, such as supercomputers and machine learning techniques, which were not conventionally used in Neuroscience and Neurology [ 54 , 55 , 56 , 57 , 58 ]. Most significantly, the Big Data revolution is improving our understanding and treatment of neurological diseases, see Tables 3 – 6 and Additional file 1 : Tables S3-S6.

National projects and big data foundations: Connectomes, neuroimaging, and genetics

The human brain contains ~ 100 billion neurons connected via ~ 10 14 synapses, through which electrochemical data is transmitted [ 59 ]. Neurons are organized into discrete regions or nuclei and connect in precise and specific ways to neurons in other regions; the aggregated connections between all neurons in an individual comprises their connectome. The connectome is a term coined by Sporns et al. designed to be analogous to the genome; like the genome, the connectome is a large and complex dataset characterized by tremendous interindividual variability [ 60 ]. Connectomes, at the level of the individual or as aggregate data from many individuals, have the potential to produce a better understanding of how brains are wired as well as to unravel the “basic network causes of brain diseases” for prevention and treatment [ 60 , 61 , 62 , 63 ]. Major investments in human connectome studies in health and disease came in ~ 2009, when the NIH Blueprint for Neuroscience Research launched the Blueprint Grand Challenges to catalyze research. As part of this initiative, the Human Connectome Project (HCP) was launched to chart human brain connectivity, with two research consortia awarded approximately $40 M. The Wu-Minn-Ox consortium sought to map the brain connectivity (structural and functional) of 1200 healthy young adults and investigate the associations between behavior, lifestyle, and neuroimaging outcomes. The MGH-UCLA (Massachusetts General Hospital-University of California Los Angeles) consortium aimed to build a specialized magnetic resonance imager optimized for measuring connectome data. The Brain Activity Map (BAM) Project was later conceived during the 2011 London workshop “Opportunities at the Interface of Neuroscience and Nanoscience.” The BAM group proposed the initiation of a technology-building research program to investigate brain activity from every neuron within a neural circuit. Recordings of neurons would be carried out with timescales over which behavioral outputs or mental states occur [ 64 , 65 ]. Following up on this idea, in 2013, the NIH BRAIN Initiative was initiated by the Obama administration, to “accelerate the development and application of new technologies that will enable researchers to produce dynamic pictures of the brain that show how individual brain cells and complex neural circuits interact at the speed of thought”. Other countries and consortia generated their own initiatives, such as the European Human Brain Project, the Japan Brain/MINDS project, Alzheimer’s Disease Neuroimaging Initiative (ADNI), Enhancing Neuroimaging Genetics through Meta-analysis (ENIGMA), and the China Brain Project. These projects aimed to explore brain structure and function, with the goal of guiding the development of new treatments for neurological diseases. The scale of these endeavors, and the insights they generated into the nervous system, were made possible by the collection and analysis of Big Data (see Table 1 ). Below, we succinctly exemplify ways in which Big Data is transforming Neuroscience and Neurology through the HCP (and similar initiatives), ADNI, and ENIGMA projects.

Ways in which Big Data is transforming Neuroscience and Neurology are exemplified through advancements in elucidating the connectome (see for example Table 3 and Additional file 1 : Table S3). Early studies in organisms such as the nematode C. elegans used electron microscopy (EM) to image all 302 neurons and 5000 connections of the animal [ 66 ], while analyses on animals with larger nervous systems collated neuroanatomical tracer studies to extract partial cerebral cortex connectivity matrices, e.g., cat [ 67 ] and macaque monkey [ 68 , 69 ]. More recently, advancements in imaging and automation techniques, including EM and two-photon (2P) fluorescence microscopy, have enabled the creation of more complete maps of the nervous system in zebrafish and drosophila [ 7 , 33 , 70 , 71 ]. Despite the diminutive size of their nervous systems, the amount of data is enormous. Scheffer and colleagues generated a connectome for portion of the central brain of the fruit fly “encompassing 25,000 neurons and 20 million chemical synapses” [ 7 ]. This effort required “numerous machine-learning algorithms and over 50 person-years of proofreading effort over ≈2 calendar years” processing > 20 TB of raw data into a 26 MB connectivity graph, “roughly a million fold reduction in data size” (note, a review of the specific computational techniques is outside this paper’s scope, see [ 7 , 33 , 58 , 70 , 71 ] for more examples). Thus, connectomes can be delineated in simple animal models; however, without automation and the capacity to acquire Big Data of this type, such a precise reconstruction could not be accomplished. Extending this detailed analysis to the human brain will be a larger challenge, as evidenced by the stark contrast between the 25,000 neurons analyzed in the above work and the 100 billion neurons and ~ 10 14 synapses present in the human brain.

At present, the study of the human connectome has principally relied on clinical neuroimaging methods, including Diffusion Tensor Imaging (DTI) and Magnetic Resonance Imaging (MRI), to generate anatomical connectomes, and on neuroimaging techniques such as functional MRI (fMRI), to generate functional connectomes [ 9 , 12 ]. For example, in what might be considered a “Small Data” step, P. van den Heuvel and Sporns, demonstrated “rich-club” organization in the human brain (“tendency for high-degree nodes to be more densely connected among themselves than nodes of a lower degree, providing important information on the higher-level topology of the brain”) via DTI and simulation studies based on imaging from 21 subjects focused on 12 brain regions [ 72 ]. This type of work has quickly become “Big Data” science, as exemplified by Bethlehem et al.’s study of “Brain charts for the human lifespan” which was based on 123,984 aggregated MRI scans, “across more than 100 primary studies, from 101,457 human participants between 115 days post-conception and 100 years of age” [ 13 ]. The study provides instrumental evidence towards neuroimaging phenotypes and developmental trajectories via MRI imaging. Human connectome studies are also characterized by highly heterogeneous datasets, owing to the use of multimodal imaging, which are often integrated with clinical and/or biospecimen datasets. For example, studies conducted under the HCP [ 32 ] have implemented structural MRI (sMRI), task fMRI (tfMRI), resting-state fMRI (rs-fMRI), and diffusion MRI (dMRI) imaging modalities, with subsets undergoing Magnetoencephalography (MEG) and Electroencephalography (EEG). These studies usually involve hundreds to thousands of subjects, such as the Healthy Adult and HCP Lifespan Studies [ 73 ]. While the above connectome studies have primarily focused on anatomical, functional, and behavioral questions, connectome studies are used across the biological sciences (e.g., study evolution by comparing mouse, non-human primates, and human connectomes [ 74 ]) and as an aid in assessing and treating neuropathologies (as will be elaborated on further below).

In the same period that the NIH was launching its Neuroscience Blueprint Program (2005), it also helped launch the ADNI in collaboration with industry and non-profit organizations. The primary objectives of ADNI are to develop “biomarkers for early detection” and monitoring of AD; support “intervention, prevention, and treatment” through early diagnostics; and share data worldwide [ 75 , 76 , 77 ]. Its Informatics Core [ 78 ], which was established for data integration, analysis, and dissemination, was hosted at University of Southern California, and highlights the Big Data underpinnings of ADNI ( https://adni.loni.usc.edu ). ADNI was originally designed to last 5 years with bi-annual data collection of cognition; brain structural and metabolic changes via Positron Emission Technology (PET) and MRIs; genetic data; “and biochemical changes in blood, cerebrospinal fluid (CSF), and urine in a cohort of 200 elderly control subjects, 400 Mild Cognitive Impairment patients, and 200 mild AD patients" [ 75 , 76 , 79 ]. The project is currently in its fourth iteration, ADNI4, with funding through 2027 [ 80 , 81 ]. To date, ADNI has enrolled > 2000 participants who undergo continuing longitudinal assessments. The ADNI study has paved the way for the diagnosis of AD through the usage of biomarker tests such as amyloid PET scans and lumbar punctures for CSF, and demonstrated that ~ 25% of people in their mid-70’s has a very early stage of AD (“preclinical AD”), which would have previously gone undetected. These results have helped encourage prevention and early treatment as the most effective approach to the disease.

During the same period that major investments were beginning in connectome projects (2009), the ENIGMA Consortium was established [ 82 , 83 ]. It was founded with the initial aim of combining neuroimaging and genetic data to determine genotype–phenotype brain relationships. As of 2022, the consortium included > 2000 scientists hailing from 45 countries and collaborating across more than 50 working groups [ 82 ]. These efforts helped spur on many discoveries, including genome-wide variants associated with human brain imaging phenotypes (see, the 60 + center large-scale study with > 30,000 subjects that provided evidence of the genetic impact on hippocampal volume [ 84 , 85 ], whose reduction is possibly a risk factor for developing AD). The group has also conducted large scale MRI studies in multiple pathologies and showed imaging-based abnormalities or structural changes [ 82 , 83 ] in numerous conditions, such as major depressive disorder (MDD) [ 86 ] and bipolar disorder [ 87 ]. Other genetics/imaging-based initiatives have made parallel advancements, such as the genome-wide association studies of UK Biobank [ 88 , 89 , 90 ], Japan’s Brain/MINDS work [ 53 ], and the Brainstorm Consortium [ 91 ]. For example, the Brainstorm Consortium assessed “25 brain disorders from genome-wide association studies of 265,218 patients and 784,643 control participants and assessed their relationship to 17 phenotypes from 1,191,588 individuals.” Ultimately, Big Data-based genetic and imaging assessments have permeated the Neurology space, significantly impacting patient care through enhanced diagnostics and prognostics, as will be discussed further below.

From discovery research to improved neurological disease treatment

The explosive development of studies spurred on by these national projects with growing size, variety, and speed of data, combined with the development of new technologies and analytics, has provoked a paradigm shift in our understanding of brain changes through lifespan and disease [ 7 , 92 , 93 , 94 , 95 , 96 ], leading to changes in the investigation and treatment development for neurological diseases and profoundly impacting the field of Neurology. Over the past decade, such impact has occurred in multiple ways. First, Big Data has opened the opportunity to analyze combined large, incomplete, disorganized, and heterogenous datasets [ 97 ], which may yield more impactful results as compared to clean curated, small datasets (with all their external validity questions and additional limitations). Second, Big Data studies have improved our basic understanding (i.e., mechanisms of disease) of numerous neurological conditions. Third, Big Data has aided diagnosis improvement (including phenotyping) and subsequently refined the determination of a presumptive prognosis. Fourth, Big Data has enhanced treatment monitoring, which further aids treatment outcome prediction. Fifth, Big Data studies have recently started to change clinical research methodology and design and thus directly impact the development of novel therapies. In the remainder of this section, we will elaborate on the aforementioned topics, followed by the presentation of particular case studies in select areas of Neurology.

Opportunities and improved understanding

As introduced above, Big Data solutions have impacted our understanding of the fundamentals of brain sciences and disease, such as brain structure and function (e.g., HCP) and the genetic basis of disease (e.g., ENIGMA). Advancements in connectome and genetics studies, along with improved analytics, have advanced our understanding of brain changes throughout the lifespan and supported hypotheses linking abnormal connectomes to many neurological diseases [ 13 , 72 , 92 , 98 ]. Studies have consistently shown that architecture and properties of functional brain networks (which can be quantified in many ways, e.g., with graph theoretical approaches [ 94 ]) correlate with individual cognitive performance and dynamically change through development, aging, and neurological disease states including neurodegenerative diseases, autism, schizophrenia, and cancer (see, e.g., [ 92 , 93 , 95 , 96 ]). Beyond genetics and connectomes, Big Data methods are used in vast ways in brain research and the understanding of diseases, such as from brain electrophysiology [ 99 ], brain blood-flow [ 100 ], brain material properties [ 101 ], perceptual processing [ 102 , 103 ], and motor control [ 104 ].

Diagnostics/prognostics/monitoring

Big Data methods are also increasing in prevalence in diagnostics and prognostics. For example, the US Veterans Administration recently reported on the genetic basis of depression based on analysis from > 1.2 M individuals, identifying 178 genomic risk loci, and confirming it in a large independent cohort (n > 1.3 M) [ 105 ]. Subsequent to the European Union (EU) neuGRID and neuGRID4You projects, Munir et. al. used fuzzy logic methods to derive a single “Alzheimer’s Disease Identification Number” for tracking disease severity [ 106 ]. Eshaghi et. al. identified MS subtypes via MRI Data and unsupervised machine learning [ 107 ] and Mitelpunkt et al. used multimodal data from the ADNI registry to identify dementia subtypes [ 108 ]. Big Data methods have also been used to identify common clinical risk factors for disease, such as gender, age, and geographic location for stroke [ 109 ] (and/or its genetic risk factors [ 110 ]). Big Data approaches to predict response to treatment are also increasing in frequency. For example, for depression, therapy choice often involves identifying subtypes of patients based on co-occurring symptoms or clinical history, but these variables are often not sufficient for Precision Medicine (i.e., predict unique patient response to specific treatment) nor even at times to differentiate patients from healthy controls [ 17 , 111 ]. Noteworthy progress has been made in depression research, such as successful prediction of treatment response using connectome gradient dysfunction and gene expression [ 18 ], through resting state connectivity markers of Transcranial Magnetic Stimulation (TMS) response [ 17 ], and via a sertraline-response EEG signature [ 111 ]. As another example, the Italian I-GRAINE registry is being developed as a source of clinical, biological, and epidemiologic Big Data on migraine used to address therapeutic response rates and efficiencies in treatment [ 112 ].

Additionally, Big Data approaches of combining high volumes of varied data at high velocities are offering the potential for new "real-time" biomarkers [ 113 ]. For instance, data collected with wearable sensors has been increasingly used in clinical studies to monitor patient behavior at home or in real-world settings. While the classic example is the use of EEG for epilepsy [ 114 ], numerous other embodiments can be found in the literature. For example, another developing approach is utilizing smartphone data to evaluate daily changes in symptom severity and sensitivity to medication in PD patients [ 115 ]. This approach has led to a memory test and simple finger tapping and to track the status of study participants [ 116 ]. Collectively, these examples highlight Big Data’s potential for facilitating participatory Precision Medicine (i.e., tailored to each patient) in trials and clinical practice (which is covered in more detail in Sect. “ Proposed Solutions ”).

Evolving evaluation methods

The way in which new potential neurological therapies are being developed is also changing. Traditionally, Randomized Controlled Trials (RCTs) evaluate the safety and efficacy of potential new treatments. In an RCT the treatment group is compared to a control or placebo group, in terms of outcome measures, at predefined observation points. While RCTs are the gold standard for developing new treatments, they have several limitations [ 117 ], which can include high cost, lengthy completion times, limited generalizability of results, and restricted observations (e.g., made at a limited number of predefined time points in a protocol (e.g., baseline, end of treatment)). Thereby, clinical practice is currently limited by RCT and evidence-based medicine interpretations and limitations [ 118 ], which are largely responsible for a predominant physician’s responsive mindset. A wealth of recent manuscripts on Big Data analysis facilitates a potential solution for individual patient behavior prediction and proactive Precision Medicine management [ 119 ] by augmenting and extending RCT design [ 117 ]. Standardization and automation of procedures using Big Data make entering and extracting data easier and could reduce the effort and cost of running an RCT. They can also be used to formulate hypotheses fueled by large, preliminary observational studies and/or carry out virtual trials. For example, Peter et al. showed how Big Data could be used to move from basic scientific discovery to translation to patients in a non-linear fashion [ 120 ]. Given the potential pathophysiological connection between PD and inflammatory bowel disease (IBD), they evaluated the incidence of PD in IBD patients and investigated whether anti-tumor necrosis factor (anti-TNF) treatment for IBD affected the risk of developing PD. Rather than a traditional RCT, they ran a virtual repurposing trial using data from 170 million people in two large administrative claims databases. The study observed a 28% higher incidence rate of PD in IBD patients than in unaffected matched controls. In IBD patients, anti-TNF treatment resulted in 78% reduction in the rate of PD incidence relative to patients that did not receive the treatment [ 120 , 121 ]. A similar approach was reported by Slade et al. They conducted experiments on rats to investigate the effects of Attention Deficit Hyperactivity Disorder (ADHD) medication (type and timing) on the “rats’ propensity to exhibit addiction-like behavior”, which led to the hypothesis that initiating ADHD medication in adolescence “may increase the risk for SUD in adulthood”. To test this hypothesis in humans, rather than running a traditional RCT, they used healthcare Big Data from a large claim database and, indeed, found that “temporal features of ADHD medication prescribing”, not subject demographics, predicted SUD development in adolescents on ADHD medication [ 122 ]. A hybrid approach was used in the study by Yu et al. [ 123 ]. Their study examined the potential of vitamin K2 (VK2) to reduce the risk of PD, given its anti-inflammatory properties and inflammation's role in PD pathogenesis. Initially, Yu et al. assessed 93 PD patients and 95 controls and determined that the former group had lower serum VK2 levels compared to the healthy controls. To confirm the connection between PD and inflammation, the study then analyzed data from a large public database, which revealed that PD patients exhibit dysregulated inflammatory responses and coagulation cascades that correlate with decreased VK2 levels [ 123 ].

Even though these pioneering studies demonstrate potential ways in which Big Data can be used to perform virtual RCT trials, several challenges remain. The processing pipeline of Big Data, from collection to analysis, has still to be refined. Moreover, it is still undetermined how regulatory bodies will ultimately utilize this type of data. In the US, the Food and Drug Administration (FDA) has acknowledged the future potential of “Big Data” approaches, such as using data that could be gathered from Electronic Health Records (EHRs), pharmacy dispensing, and payor records, to help evaluate the safety and efficacy of therapeutics [ 124 ]. Furthermore, the FDA has begun the exploration and use of High-Performance Computing (HPC) to internally tackle Big Data problems [ 125 ] and concluded that Big Data methodologies could broaden “the range of investigations that can be performed in silico” and potentially improve “confidence in devices and drug regulatory decisions using novel evidence obtained through efficient big data processing”. The FDA is also employing Big Data based on Real World Evidence (RWE), such as with their Sentinel Innovation Center, which will implement data science advances (e.g., machine learning, natural language processing) to expand EHR data use for medical product surveillance [ 126 , 127 ]. Lastly, the exploration of crowdsourcing of data acquisition and analysis is an area still to be explored and outside the scope of this review [ 128 ].

Big Data case studies in neurology

To provide the reader with a sample of existing Big Data solutions for improving patient care (beyond those surveyed above), we focus on three separate disorders, PD, SUD, and Pain. While Big Data has positively impacted numerous other neuropathologies (e.g., [ 129 , 130 , 131 , 132 ]), we have chosen these three disorders due to their significant societal impact and their representation of varying stages of maturity in the application of Big Data to Neurology. Finally, we exemplify Big Data’s foreseeable role in therapeutic technology via brain stimulation, which is used in the aforementioned disorders and is particularly suitable for Precision Medicine.

After AD, PD is the second most prevalent neurodegenerative disorder [ 133 , 134 , 135 ]. About 10,000 million people live with PD worldwide, with ~ 1 million cases in the US. The loss of dopamine-producing neurons leads to symptoms such as tremor, rigidity, bradykinesia, and postural instability [ 136 ]. Traditional treatments include levodopa, physical therapy, and neuromodulation (including Deep Brain Stimulation (DBS) and Noninvasive Brain Stimulation (NIBS) [ 36 , 137 , 138 ].

The increasing significance of Big Data in both PD research and patient care can be measured by the rising number of published papers over the past decade (Fig. 3 ). Several national initiatives have been aimed at building public databases to facilitate research. For example, the Michael J. Fox Foundation’s Parkinson’s Progression Markers Initiative (PPMI) gathers data from about 50 sites in several nations including the US, Europe, Israel, and Australia with the objective of identifying potential biomarkers of disease progression [ 139 , 140 ]. A major area of research involving Big Data analytics focuses on PD’s risk factors, particularly through genetic data analysis. The goal is to enhance our comprehension of the causes of the disease and develop preventive treatments. The meta-analysis of PD genome-wide association studies by Nalls et al. illustrates this approach, which involved the examination of “7,893,274 variants” among “13,708 cases and 95,282 controls”. The findings revealed and confirmed “28 independent risk variants” for PD “across 24 loci” [ 141 ]. Patient phenotyping for treatment outcome prediction is another research area that utilizes Big Data analytics. Wong et al.’s paper discusses this approach, reviewing the use of structural and functional connectivity studies to enhance the efficacy of DBS treatment for PD and other neurological diseases [ 142 ]. An emerging area of patient assessment is wearable sensors and/or apps for potential real-time monitoring of symptoms and response to treatment [ 143 ]. A major project in this area is the iPrognosis mobile app, which was funded by the EU Research Programme Horizon 2020 and aimed at accelerating PD diagnosis and developing strategies to help improve and maintain the quality of life of PD patients via capturing data during user interaction with smart devices, including smartphones and smartwatches [ 144 ]. Similar to other diseases, PD analysis is also being conducted via social media (e.g., [ 16 , 145 ]) and EHR [ 146 , 147 ] analyses. See Table 4 and Additional file 1 : Table S4 or review articles in [ 148 , 149 , 150 , 151 , 152 , 153 , 154 ] for further examples of Big Data research in PD.

Cumulative number of papers on Big Data over time for different areas, as per Pubmed. The panels illustrate when Big Data started to impact the area and allow a comparison across areas As graphs were simply created by using the keywords “Big Data” AND “area”, with "area" being “Parkinson’s Disease”, “Addiction”, etc. as opposed to using multiple keywords that may be used to describe each field, actual numbers are likely to be underestimated

SUD and Opioid Use Disorder (OUD)

The economic and social burden associated with SUDs is enormous. OUD is the leading cause of overdoses due to substance abuse disorders, where death rates have drastically increased, with over 68,000 people in 2020 [ 155 ]. The US economic cost of OUD alone and fatal opioid overdoses was $471 billion and $550 billion, respectively, in 2017 [ 156 ]. Treatments focus on replacement (e.g., nicotine and opioid replacement) and abstinence and are often combined with self-help groups or psychotherapy [ 157 , 158 ].

Like PD, the increasing impact of Big Data in SUD and OUD research and patients care can be measured by the increased number of papers published in Pubmed over the past decade (Fig. 3 ). Several national initiatives have been aimed at building public databases to facilitate SUD research. For example, since 2009, the ENIGMA project includes a working group specifically focused on addiction, which has gathered genetic, epigenetic, and/or imaging data from 1000’s of SUD subjects from 33 sites as of 2020 [ 37 ]. As part of this research, Mackey et al. have been investigating the association between dependence and regional brain volumes, both substance-specific and general [ 159 ]. Similarly, studies implementing data sets from the UK BioBank and 23andMe (representing > 140,000 subjects) have been used for developing the Alcohol Use Disorder Identification Test (AUDIT) to identify the genetic basis of alcohol consumption and alcohol use disorder [ 160 ]. Big Data is also being used to devise strategies for retaining patients on medication for OUD, as roughly 50% of persons discontinue OUD therapy within a year [ 158 ]. The Veterans Health Administration is spearheading such an initiative based on data (including clinical, insurance claim, imaging, and genetic data) from > 9 M veterans [ 158 ]. Social media is also emerging as a method to monitor substance abuse and related behaviors. For example, Cuomo et al. reported on the results of an analysis of geo-localized Big Data collected in 2015 via 10 M tweets from Twitter regressed with Indiana State Department of Health data on non-fatal opioid-related hospitalizations and new “HIV cases from the US Centers for Disease Control and Prevention" to examine the transition from "opioid prescription abuse to heroin injection and HIV transmission risk” [ 161 ]. Leveraging Big Data from online content is likely to aid public health practitioners in monitoring SUD. Table 5 and Additional file 1 : Table S5 summarize Big Data research in SUD and OUD.

Chronic pain is a widespread condition that affects a significant portion of the global population, with an estimated 20% of adults suffering from it and 10% newly diagnosed each year [ 162 ]. In the US, this condition is most prevalent and affects over 50 million adults. The most common pain locations are the back, hip, knee, or foot [ 163 ], which are chiefly due to neural entrapment syndromes (e.g., Carpal Tunnel Syndrome (CTS)), peripheral neuropathy (such as from diabetes), or unknown causes (such as non-specific chronic Lower Back Pain (LBP)). Pain treatment remains challenging and includes physical therapy, pharmacological and neuromodulation approaches [ 164 ]. As in other areas of Neurology, the Big Data revolution has been impacting pain research and management strategies. As reviewed by Zaslansky et al., multiple databases have been created to monitor pain, for example the international acute pain registry PAIN OUT, established in 2009 with EU funds, to improve the management of surgeries [ 165 , 166 ]. Besides risk factors [ 167 ], such as those based on genetic data (e.g., see [ 168 , 169 ]), pain studies using Big Data mainly focus on management of symptoms and improving therapy outcomes. Large-scale studies aimed at comparing different treatments [ 170 , 171 ] or at identifying phenotypes in order to classify and diagnose patients (see for example [ 172 ]) are particularly common. Table 6 and Additional file 1 : Table S6 summarize Big Data research in Pain, while Fig. 3 shows the increasing number of published papers in the field.

Example of Big Data impact on treatments and diagnostics-brain stimulation

In the last twenty years, neurostimulation methods have seen a substantial rise in application for neurological disease treatment [ 36 , 138 , 173 ]. Among the most used approaches are invasive techniques like DBS [ 173 , 174 , 175 , 176 ], which utilize implanted devices to apply electrical currents directly into the neural tissue and modulate neural activity. Noninvasive techniques, on the other hand, like those applied transcranially, offer stimulation without the risks associated with surgical procedures (such as bleeding or infection) [ 36 ]. Both invasive and noninvasive approaches have been used for psychiatric and neurological disorders treatments, including those for depression, PD, addiction, and pain. While High Performance Computing has been used in the field for some time (see Fig. 4 ), Big Data applications have just recently started to be explored in brain stimulation. For example, structural and functional connectome studies have yielded new insights into the potential targets for stimulation, in the quest to enhance stimulation effectiveness. Although DTI has optimized the definition of targets for DBS and noninvasive stimulation technologies since mid-2000 [ 177 , 178 , 179 ], Big Data and advances in computational methods have enabled new venues for DTI to further improve stimulation, which have enhanced clinical results. For example, in 2017, Horn et al. utilized structural and functional connectivity data of open-source connectome databases (including healthy subjects connectome from the Brain Genomics Superstruct Project, the HCP, and PD connectome from the PPMI) to build a computational model to predict outcomes following subthalamic nucleus modulation with DBS in PD. As a result, Big Data allowed the identification of a distinct pattern of functional and structural connectivity, which independently accurately predicted DBS response. Additionally, the findings held external validity as connectivity profiles obtained from one cohort were able to predict clinical outcomes in a separate DBS center’s independent cohort. This work also demonstrated the prospective use of Big Data in Precision Medicine by illustrating how connectivity profiles can be utilized to predict individual patient outcomes [ 180 ]. For a more comprehensive review of application of functional connectome studies to DBS, the reader is referred to [ 142 ], where Wong et al. discuss application of structural and functional connectivity to phenotyping of patients undergoing DBS treatment and prediction of DBS treatment response. Big Data is also expected to augment current efforts in the pursuit of genetic markers to optimize DBS in PD (e.g., [ 148 , 181 , 182 ]).

High Performance Computing solutions for modeling brain stimulation dosing have been explored for well over a decade. The above figure is adapted from [ 183 ], where Sinusoidal Steady State Solutions of the electromagnetic fields during TMS and DBS were determined from MRI derived Finite Element Models based on frequency specific tissue electromagnetic properties of head and brain tissue. The sinusoidal steady state solutions were then transformed into the time domain to rebuild the transient solution for the stimulation dose in the targeted brain tissues. These solutions were then coupled with single cell conductance-based models of human motor neurons to explore the electrophysiological response to stimulation. Today, high resolution patient specific models are being developed (see below), implementing more complicated biophysical modeling (e.g., coupled electromechanical field models) and are being explored as part of large heterogenous data sets (e.g., clinical, imaging, and movement kinematic) to optimize/tune therapy

Compared to DBS, studies on NIBS have been sparser. However, the use of Big Data methodologies has facilitated the improvement and standardization of established TMS techniques (i.e., single and paired pulse), which had large inter-subject variability, by identifying factors that affect responses to this stimulation in a multicentric sample [ 184 ]. A similar paradigm was followed to characterize theta-burst stimulation [ 185 ]. Regarding disease, a large multisite TMS study (n = 1188), showed that resting state connectivity in limbic and frontostriatal networks can be used for neurophysiological subtype classification in depression. Moreover, individual connectivity evaluations predicted TMS therapy responsiveness better than isolated symptomatology in a subset of patients (n = 154) [ 17 ].

Proposed solutions

As reviewed above, Big Data has been improving the care of patients with neurological diseases in multiple ways. It has elevated the value of diverse and often incomplete data sources, enhanced data sharing and multicentric studies, streamlined multidisciplinary collaboration, and improved the understanding of neurological disease (diagnosis, prognosis, optimizing current treatment, and helping develop novel therapies). Nevertheless, existing methodologies suffer from several limitations, which have prevented the full realization of Big Data’s potential in Neuroscience and Neurology. Below, we discuss the limitations of current approaches and propose possible solutions.

Full exploitation of available resources

Many Neuroscience and Neurology purported “Big Data” studies do not fully implement the classic 3 V's (i.e., “Volume, Variety, and Velocity”) or 5 V’s (i.e., “Volume, Variety, Velocity, Veracity and Value”) and/or are characterized by the high heterogeneity in which the V’s can be interpreted. For example, in “Big Data” Neuroscience and Neurology studies, Volume sometimes refers to studies with hundreds of thousands of patients’ multidimensional datasets and other times to studies with 10's of patients’ unidimensional datasets. Value, a characteristic of Big Data typically defined in financial terms in other Big Data fields, is not usually considered in Big Data studies in Neuroscience and Neurology. In this paper, across studies and databases, we adopted a measure of clinical or preclinical Value where financial information was not given (see Tables 2 – 6 and Additional file 1 : Tables S2–S6). Data Veracity is not standardized in Neuroscience or Neurology and thus, we focused our analysis on both typical data Veracity measures and potential experimental sources of error in the data sets from studies that we reviewed above. In terms of Variety, few clinical studies make use of large multimodal data sets and even fewer are acquired and processed at a rapid Velocity. Data Velocity information is sparsely reported throughout the literature, but its clear reporting would enable a better understanding and refinement of methodologies through the research community.

While these limitations may be simply labeled as semantics, we believe that these deficits often result in Big Data analytics being underexploited, which limits the potential impact of a study and possibly increases its cost. Thus, aligning studies in Neuroscience and Neurology to the V’s represents an opportunity to leverage the knowledge, technology, analytics, and principles established in fields that have been using Big Data more extensively, thereby improving the Big Data studies in Neurology and Neuroscience. Identifying whether a study is suitable for using Big Data approaches makes it easier to choose the best tools for the study and exploit the plethora of resources (databases, software, models, data management strategies) that are already available (part of which we have reviewed herein, see for example Tables 1 – 2 and Additional file 1 : Tables S1, S2).

Tools for data harmonization

The overall lack of tools for data harmonization (particularly for multimodal datasets used in clinical research and care) is a significant issue of current Big Data studies. Creation of methods for sharing data and open-access databases has been a priority of Big Data initiatives since their inception. Data sharing is required by many funding agencies and scientific journals, and publicly available repositories have been established. While these repositories have become more common and organized (see Sect. “ Existing Solutions ”), there has been less emphasis on the development of tools for quality control, standardization of data acquisition, visualization, pre-processing, and analysis. With the proliferation of initiatives promoting data sharing and pooling of existing resources, the need for better tools in these areas is becoming increasingly urgent. Despite efforts made by the US Department of Health and Human Service to establish standardized libraries of outcome measures in various areas, such as Depression [ 186 , 187 ], and by the NIH that has spearheaded Clinical Trials Network (CTN)-recommended Common Data Elements (CDEs) for use in RCTs and EHRs [ 188 ], more work is needed to ensure data harmonization across not only clinical endpoints but also across all data types that typically comprise Big Data in Neuroscience and Neurology. For example, in neuroimaging, quality control of acquired images is a long-standing problem. Traditionally, this is performed visually, but in Big Data sets, large volumes make this approach exceedingly expensive and impractical. Thus, methods for automatic quality control have become in high demand [ 189 ]. Quality control issues are compounded in collaborative datasets, where variability may stem from multiple sources. In multisite studies, a typical source of variability arises from the use of different MRI scanners (i.e., from different manufacturers, with different field strengths or hardware drifts [ 190 , 191 ]). Variability can also arise from data pre-processing techniques and pipelines. For example, the pre-processing pipeline of MRI data involves a variety of steps (such as correcting field inhomogeneity and motion, segmentation, and registration) and continues to undergo refinement through algorithm development, ultimately affecting reproducibility/Veracity of study results. As an additional example, while working on data harmonization methods in genome-wide association studies Chen et. al. have noted similar problems where an “aggregation of controls from multiple sources is challenging due to batch effects, difficulty in identifying genotyping errors and the use of different genotyping platforms” [ 192 ].

Some progress towards harmonization of data and analysis procedures [ 193 ] has been enabled by the availability of free software packages that incorporate widely accepted sets of best practices, see, e.g., Statistical Parametric Mapping (SPM), FreeSurfer, FMRIB Software Library (FSL), Analysis of Functional NeuroImages (AFNI), or their combination (such as Fusion of Neuroimaging Processing (FuNP) [ 194 ]). In addition, open-access pre-processed datasets have been made available (see Table 2 and Additional file 1 : Table S2); for example, the Preprocessed Connectome Project has been systematically pre-processing the data from the International Neuroimaging Data-sharing Initiative and 1000 Functional Connectomes Project [ 195 , 196 ] or GWAS Central (Genome-wide association study Central) which “provides a centralized compilation of summary level findings from genetic association studies” [ 197 ]. As another example, EU-funded NeuGRID and neuGRID4You projects included a set of analysis tools and services for neuroimaging analysis [ 106 ]. Development of software like Combat (which was initially created to eliminate batch effects in genomic data [ 198 ] and subsequently adapted to handle DTI, cortical thickness measurements [ 199 ], and functional connectivity matrices [ 200 ]) can also help researchers harmonize data from various types of study, regardless of whether they are analyzing newly collected or retrospective data gathered with older standards. For more detailed discussions on efforts to address data harmonization challenges in neuroimaging, the reader is directed to the review papers of Li et al. [ 12 ], Pinto et al. [ 201 ], and Jovicich et al. [ 202 ]. In clinical studies using data different from neuroimaging (and/or biospecimen sources), standardization of clinical assessments and measures of outcome across multiple sites has also proven to be challenging. For example, as shown by the ENIGMA study group, multi-center addiction studies face notable methodological challenges due to the heterogeneity of measurements for substance consumption in the context of genomic studies [ 203 ].

Developing tools to harmonize datasets across different sources and data types (e.g., based on machine learning [ 191 ]) for Neurology-based clinical studies might allow researchers to exploit Big Data to their full potential. Tools for complex data visualization and interactive manipulation are also needed to allow researchers from different backgrounds to fully understand the significance of their data [ 204 ]. For studies that are in the design phase, identifying whether tools for data harmonization are available or developing such tools in an early phase of the study will allow researchers to enhance the Veracity, and ultimately the impact of the study, while cutting costs.

New technologies for augmented study design and patient data collection

Traditional clinical studies are associated with several recognized limitations. However, a few recent Big Data studies have shown potential in mitigating some of these limitations.

First, traditional clinical studies, particularly RCTs which serve as the standard in clinical trials, are often expensive and inefficient. The integration of Big Data, particularly in the form of diverse data types or multicenter trials, can further amplify these issues and lead to exponential increases in costs. Thus, there is a pressing need for tools that can optimize resources and contain expenses. Virtual trials are a promising but underutilized approach that can potentially enhance study design and address cost-related challenges. To achieve this, health economics methods could be used to compare different scenarios, such as recruitment strategies or inclusion criteria, and select the most effective one prior to initiating an actual clinical study. These methods can also assign quantitative values to data sets or methods [ 205 ]. For studies testing interventions, virtual experiments that use simulations can be performed. For example, in the area of brain stimulation, virtual DBS is being explored [ 206 ] to supplement existing study design. Similarly, for NIBS, our group and others are building biophysics-based models that can be used to personalize interventions [ 58 ].

Second, traditional clinical studies, including RCTs, often suffer from limited data and limited generalizability of conclusions. Collected data is often too limited to fully account for highly multidimensional and heterogenous neurological conditions. PD is an example of this, where patients’ clinical presentation, progression and response to different treatment strategies can vary significantly, even within a single day [ 153 ]. Limited external validity due to discrepancies between the study design (patient inclusion criteria) and real-world clinical scenarios, as well as limited generalizability of findings to different time points beyond those assessed during the study are other known limitations. Relaxing study criteria and increasing timepoints could provide more data, but often at the expense of increased patient burden and study cost. Mobile applications can potentially help overcome some of these limitations while offering other advantages. For example, by allowing a relatively close monitoring of patients mobile applications may help capture features of symptoms not easily observable during hospital visits. This richer dataset could be used to design algorithms for patient classification/phenotyping or medication tuning. However, data collected via mobile technology is often limited to questionnaires or by the type of data that can be collected with sensors that can be embedded in mobile/wearable devices (typically accelerometers in motor disorders studies). Leveraging Big Data in this context would require the development of technology to monitor patients outside the time and space constraints of a traditional clinical study/RCT (e.g., home, or other unstructured environments); such technology should be sufficiently inexpensive to be useful at scale, while still providing reliable and clinically valuable data. Other related approaches include additional nontraditional data sources, such as information gathered from Payer Databases, EHR, or social media particular to a disease and treatment to support conventional findings. For example, the FDA is poised to pursue Big Data approaches to continue to assess products through their life cycle to "fill knowledge gaps and inform FDA regulatory decision-making" [ 207 ].

Finally, clinical studies might be subject to bias due to important clinical information being missing. This is particularly true for studies that rely on databases for billing or claim purposes, part of which we have reviewed herein, as they use data which were not collected primarily for research (see Additional file 1 : Tables S4–S6). A possible way to overcome this limitation is to more directly couple payer data with clinical data and correlating the results. This approach is still mostly theoretical: modern patient tracking systems like Epic are beginning to offer billing code data within the EHR, but the system was not designed for population-based analysis. Ideally, information such as payer data can be used for exploration purposes and results of the analysis can guide the design of more rigorous studies aimed at testing specific clinical hypotheses.

Tools for facilitating interdisciplinary research

As the use of Big Data continues to expand across various fields, there is a growing need for better tools that can facilitate collaborations among professionals with different backgrounds. A project that exemplifies this need is the American Heart Association (AHA) Precision Medicine Platform [ 208 ]. This platform aims to "realize precision cardiovascular and stroke medicine" by merging large, varying datasets and providing analytical tools and tutorials for clinicians and researchers. Despite the strong technological and community-based support of this platform, major challenges related to scalability, security, privacy, and ease of use have prevented it from being integrated into mainstream medicine, subsequently obstructing its full exploitation.

Creating tools to visualize and interactively manipulate multidimensional data (e.g., borrowing from fields such as virtual or augmented reality that already use these tools [ 209 ]) might help overcome this type of issue.

Future directions

We have identified current limitations in the application of Big Data to Neuroscience and Neurology and have proposed general solutions to overcome them. One area where the limitations in Big Data, as currently defined and implemented, could be addressed, and make a major impact is in the development of personalized therapies and Precision Medicine. In this field, the acceleration Big Data could enable has not yet occurred [ 210 ]. Unlike a traditional one-size-fits-all approach, Precision Medicine seeks to optimize patient care based on individual patient characteristics, including genetic makeup, environmental factors, and lifestyle. This approach can help in preventing, diagnosing, or treating diseases. Precision oncology has been a driver of Precision Medicine for approximately two decades [ 211 ] and exploited availability of big, multi-omics data to develop data-driven approaches to predict risk of developing a disease, help diagnosis, identify patient phenotypes, and identify new therapeutic targets. In Neurology, availability of large neuroimaging, connectivity, and genetics datasets has opened the possibility for data-driven approaches in Precision Medicine. However, these approaches have not yet been fully integrated with clinical decision making and personalized care. Diagnosis and treatment are still often guided by only clinical symptoms. Currently, there are no widely used platforms, systems, or projects that analytically combine personalized data, either to generate personalized treatment plans or assist physicians with diagnostics. However, the AHA Precision Medicine Platform [ 208 ] aims to address this gap by providing a means to supplement treatment plans with personalized analytics. Despite the strong technological and community-based support of this platform, integration of the software into mainstream medicine has been challenging, as discussed above (see SubSect. “ Future Directions ” in Sect. “ Proposed Solutions ").

As a potential way to acquire large real-time multimodal data sets for use in personalized care in the movement disorder, pain, and rehabilitation spaces we have been developing an Integrated Motion Analysis Suite (IMAS), which combines motion capture technology, inertial sensors (gyroscope/accelerometers), and force sensors to assess patient movement kinematics from multiple body joints as well as kinetics. The hardware system for movement kinematic and kinetic data capture is underpinned with an AI driven computational system with algorithms for data reduction, modeling, and prediction of clinical scales, prognostic potential for motor recovery (e.g., in the case of injury such as stroke), and response to treatment. Ultimately, the low-cost hardware package is coupled to computational packages to holistically aid clinicians in motor symptom assessments. The system is currently being investigated as part of a stroke study [ 212 ] and supporting other studies in the movement disorder [ 213 ] and Chronic Pain [ 214 , 215 ] spaces. As for the Big Data component, the system has been designed for different data streams and systems to be networked and interconnected. As a result, data such as multiple patients’ kinematic/kinetic, imaging, EHR, payer database, and clinical data can be longitudinally assessed and analyzed to develop a continually improving model of patient disease progression. This approach also serves as a method to personalize and optimize therapy delivery and/or predict response to therapy (see below).

Our group is also developing a new form of NIBS, electrosonic stimulation (ESStim™) [ 138 ], and testing it in multiple areas (e.g., diabetic neuropathic pain [ 215 ], LBP, CTS pain [ 214 ], PD [ 138 ], and OUD [ 216 ]). While the RCTs that are being conducted for the device are based on classic safety and efficacy endpoints, several of our studies are also focused on developing models of stimulation efficacy through combined imaging data, clinical data, kinematic data, and/or patient specific biophysical models of stimulation dose at the targeted brain sites to identify best responders to therapy (e.g., in PD, OUD, and Pain). These computational models are being developed with the goal of not only identifying the best responders but as a future means to personalize therapy based on the unique characteristics of the individual patients [ 58 ] and multimodal disease models. It is further planned that the IMAS system, with its Big Data backbone, will be integrated with the ESStim™ system to further aid in personalizing patient stimulation dose in certain indications (e.g., PD, CTS pain).

Finally, our group is working on developing a trial optimization tool based on health economics modeling (e.g., Cost Effective Analysis (CEA)) [ 205 , 217 ]. The software we are generating allows for a virtual trial design and the predicting of the cost effectiveness of the trial. We anticipate that the software could also be implemented to quantify data set values in health economic terms or used to quantify non-traditional data for use in RCT design or assessment (e.g., for the OUD patient population CEA methodologies could be used to quantify the impact of stigma on the patient, caregiver, or society with traditional (e.g., biospecimen) and non-traditional data sets (e.g., EHR, social media)). Ultimately, we see all these systems being combined into a personalized treatment suite, based on a Big Data infrastructure, whereby the multimodal data sets (e.g., imaging, biophysical field-tissue interaction models, clinical, and biospecimen data) are coupled rapidly to personalize brain stimulation-based treatments in diverse and expansive patient cohorts (see Fig. 5 ).

Schematic of our suite under development for delivering personalized treatments based on a Big Data infrastructure, whereby multimodal data sets (e.g., imaging, biophysical field-tissue interaction models, clinical, biospecimen data) can be coupled to deliver personalized brain stimulation-based treatments in a diverse and expansive patient cohort. Each integrated step can be computationally intensive (e.g., see Fig. 4 for simplified dosing example for exemplary electromagnetic brain stimulation devices)

Elaboration

The Section “ Existing Solutions ” has reviewed the influence of Big Data on Neuroscience and Neurology, specifically in the context of advancing treatments for neurological diseases. Our analysis spans the last few decades and includes a diverse selection of cutting-edge projects in Neuroscience and Neurology that illustrate the continuing shift towards a Big Data-driven paradigm; also, it reveals that certain areas of neurological treatment development have not fully embraced the potential of the Big Data revolution, as demonstrated through our comprehensive review of clinical literature in Sect. “ Proposed Solutions ”.

One sign of this gap is that there are differences between the definition of Big Data and the use the 3 V's or 5 V’s across studies that are considered “Big Data” studies in Neuroscience and Neurology literature. Several definitions can be found in the literature from these fields. For example, van den Heuvel et al. noted that the term “Big Data” includes many data types, such as “observational study data, large datasets, technology-generated outcomes (e.g., from wearable sensors), passively collected data, and machine-learning generated algorithms” [ 153 ]; Muller-Wirtz and Volk stated that “Big Data can be defined as Extremely large datasets to be analyzed computationally to reveal patterns, trends, and associations, especially relating to human behavior and interactions” [ 166 ]; and Eckardt et al. referred to Big Data science as the “application of mathematical techniques to large data sets to infer probabilities for prediction and find novel patterns to enable data driven decisions” [ 218 ]. Other definitions also include the techniques required for data analysis. For example, van den Heuvel et al. stated that “these information assets (characterized by high Volume, Velocity, and Variety) require specific technology and analytical methods for its transformation into Value” [ 153 ]; and according to Banik and Bandyopadhyay, the term “Big Data encompassed massive data sets having large, more varied, and complex structure with the difficulties of storing, analyzing, and visualizing for further processes or results” [ 219 ]. Thus, what constitutes Big Data in Neuroscience and Neurology is not established nor always aligned with the definition of Big Data outside of these fields.

In addition, in the fields of Neuroscience and Neurology, often some V’s are incompletely considered or even dismissed. At present, Neuroscience study data from “Big Data” studies are often just big and sometimes multimodal, and Neurology studies with "Big Data" are often characterized by small multimodal datasets. Incorporating all the V’s into studies might spur innovation. The area of research focused on OUD treatments is a particularly salient example. Adding “Volume” to OUD studies by integrating OUD patient databases, as it has been done for other diseases, could lead to better use of Big Data techniques and ultimately help understand the underlying disease and develop new treatments (e.g., see the work of Slade et. al. discussed above [ 122 ]). Similarly, adding “Velocity” to OUD studies by developing technology for increasing dataflow (e.g., integrating clinical data collected during hospital visits with home monitoring signals collected with mobile apps) might lead to using Big Data techniques for uncovering data patterns that could ultimately translate into development of new, personalized OUD treatments. In this vein, Variety in OUD studies could significantly add to the clinical toolbox of caregivers or researchers developing new technologies. For example, infovelliance of social media combined with machine learning algorithms, such as those developed for use during the COVID Pandemic [ 220 ], could be used to assess the stigma associated with potential treatment options for OUD patients, and quantify potential methods to lower patient treatment hesitancy. As for data Veracity, additional metrics of veracity could be garnered from clinical data sets to further assessment of the internal and external validity of trial results. For example, in OUD, Big Data sets could be used to assess the validity of self-reported opioid use, such as data gathered from drug diaries, in reference to other components of the Data Set (e.g., social media presence, sleep patterns, biospecimens, etc.). Finally, while we characterized Value herein as direct or indirect in terms of clinical utility, one could assign economic value to the Neuroscience and Neurology data sets through health economics methods. For example, in the OUD patient population, CEA or cost benefit analysis methodologies could be used to quantify the value of the data in health economics terms and guide policy makers in the design of studies or programs for aiding OUD treatment.

Finally, the rapid growth of Big Data in Neuroscience and Neurology has brought to the forefront ethical considerations that must be addressed [ 221 , 222 ]. For example, a perennial concern is data security and how to best manage patient confidentiality [ 223 ]. In the US, current laws and regulations require that SUD treatment information be kept separate from patient’s EHR, which can limit Big Data approaches for improving OUD treatment [ 158 ]. The cost versus benefit of making the information more accessible poses ethical challenges as there are risks to trying to acquire such sensitive protected health information (PHI). As of November 28, 2022, the US Health and Human Services Department, through the Office for Civil Rights (OCR) and the Substance Abuse and Mental Health Services Administration (SAMHSA) put forth proposed modifications to rules and has requested public comments on the issue [ 224 ]. Ultimately, as the use of Big Data in the treatment of neurological patients progresses, such challenges will need to be addressed in a manner which provides the most benefit to the patient with minimal risks [ 225 , 226 ].

This paper has provided a comprehensive analysis of how Big Data has influenced Neuroscience and Neurology, with an emphasis on the clinical treatment of a broad sample of neurological disorders. It has highlighted emerging trends, identified limitations of current approaches, and proposed possible methodologies to overcome these limitations. Such a comprehensive review can foster further innovation by enabling readers to identify unmet needs and fill them with a Mendeleyevization-based approach; to compare how different (but related) areas have been advancing and assess whether a solution from an area can be applied to another (Cross-disciplinarization); or to use Big Data to enhance traditional solutions to a problem (Implantation) [ 227 ]. This paper has also tackled the issue of the application of the classic 5 V’s or 3 V’s definitions of Big Data in Neuroscience and Neurology, an aspect that has been overlooked in previous literature. Review of the literature under this perspective has contributed to highlight the limitations of current Big Data studies which, as a result, rarely take advantage of AI methods typical of Big Data analytics. This can significantly impact treatment of neurological disorders, which are highly heterogeneous in both symptom presentation and etiology, and would benefit significantly from the application of these methods. At the same time, assessing the missing V’s of Big Data can provide the basis to improve study design. In light of our findings, we recommend that future research should focus on the following areas:

Augment and standardize the way the 5 V’s are currently defined and implemented , since not all "Big Data" studies are truly "Big Data" studies.

Encourage collaborative, multi-center studies : especially in clinical research, adding Volume might help overcome the limitations of classical RCTs (e.g., type II error).

Leverage new technologies for real-time data collection : for diseases characterized by time-varying patterns of symptoms, higher data Velocity such as implemented in home monitoring or wearables might help personalize treatments and/or improve treatment effectiveness.

Diversify data types collected in the clinic and/or home : as data Variety can help uncover patterns in patients subtypes or treatment responses.

Enforce protocols for data harmonization to improve Veracity.

Consider each V in terms of Value and identify ways to categorize and increase Value out of a study, since adding V’s might amplify study costs (and not all data is preclinically or clinically meaningful).

Funding agencies should encourage initiatives aimed at educating junior and established scientists on the methods, tools, and resources that Big Data challenges require.

It often happens that when new methods/techniques/technologies are developed or simply get the attention of researchers in a field, that field changes trajectory. In Neuroscience and Neurology, the use of Big Data has been an evolving trend, as evident from our review of over 300 papers and 120 databases. We discussed how Big Data is altering the course of these fields by leveraging computational tools to develop innovative treatments for neurological diseases, a major global health concern. While our analysis has identified significant advancements made in the fields, we also note that the use of Big Data remains fragmented. Nevertheless, we view this as an opportunity for progress in these rapidly developing fields, which can ultimately benefit patients with improved diagnosis and treatment options.

Availability of data and materials

Data sharing is not applicable to this survey article as no primary research datasets were generated during the survey (further, all data survey material is included in the manuscript and/or Additional file 1 ).

Change history

28 july 2023.

The clean version of ESM has been updated.

Abbreviations

Artificial Intelligence

Multiple Sclerosis

United States

National Institutes of Health

Volume, Variety, Velocity, Veracity, and Value

Alzheimer’s Disease

Parkinson’s Disease

Substance Use Disorder

Brain Mapping by Integrated Neurotechnologies for Disease Studies

Human Connectome Project

Massachusetts General Hospital

University of California Los Angeles

Brain Activity Map Project

Alzheimer’s Disease Neuroimaging Initiative

Enhancing Neuroimaging Genetics through Meta-Analysis

Electron Microscopy

Two-photon Fluorescence Microscopy

Magnetic Resonance Imaging

Diffusion Tensor Imaging

Functional Magnetic Resonance Imaging

Resting State Magnetic Resonance Imaging

Task Functional Magnetic Resonance Imaging

Diffusion Magnetic Resonance Imaging

Magnetoencephalography

Electroencephalography

Positron Emission Technology

Cerebrospinal Fluid

Major Depressive Disorder

Transcranial Magnetic Stimulation

Randomized Controlled Trial

Inflammatory Bowel Disease

Anti-Tumor Necrosis Factor

Attention Deficit Hyperactivity Disorder

Food and Drug Administration

Electronic Health Records

High Performance Computing

Real World Evidence

Deep Brain Stimulation

Non-Invasive Brain Stimulation

Parkinson’s Progression Markers Initiative

European Union

Opioid Use Disorder

Alcohol Use Disorder Identification Test

Carpal Tunnel Syndrome

Lower Back Pain

Volume, Variety, and Velocity

Clinical Trials Network

Common Data Elements

Statistical Parametric Mapping

Analysis of Functional NeuroImages

FMRIB Software Library (FSL)

Fusion of Neuroimaging Processing

Genome-Wide Association Study

A grid-based e-Infrastructure for neuroimaging research

American Heart Association

Integrated Motion Analysis Suite

Electrosonic Stimulation

Cost Effective Analysis

Protected Health Information

Office for Civil Rights

Substance Abuse and Mental Health Services Administration

Switzerland

United Kingdom

South Korea

Healthy and Pathology

Chronic Back Pain

Fibromyalgia

Irritable Bowel Syndrome

Neurodegenerative Disease

Cerebral Palsy

Computed Tomography

Single-Photon Emission Computerized Tomography

Second Capture

Spinal Muscular Atrophy

Structural Magnetic Resonance Imaging

Alzheimer’s Disease and Related Dementias

Electro-Corticography

Event Related Potential

Intracranial Electroencephalography

Electromyography

Central Nervous System

Autism Spectrum Disorder

Arterial Spin Labeling

In Situ Hybridization

Intensive Care Unit

National Science Foundation

Fixed Studies

Fixed (Updates Anticipated)

Open/Closed To Uploads

Pre-Clinical

Pre-Clinical and Clinical

Mushroom Body

Data verified through automated analytical process (AI, statistical methods)

Manual Verification

Dependent on Methodological Limitations

Mobile App Realtime Dependent

Social Media Dependent

Hospital upload Dependent

Spinal Cord Stimulation

Osteoarthritis

Interstitial Cystitis

Bladder Pain Syndrome

Visual Analog Scale

National Institute of Aging

National Institute of Diabetes and Digestive and Kidney Diseases

National Institute of Neurological Disorders and Stroke

National Institute of Arthritis and Musculoskeletal and Skin Diseases

National Institute on Drug Abuse

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LD crafted the idea for the manuscript and wrote the initial manuscript. TW, PGM, JR, LHZ, and CR contributed to multiple sections of the manuscript. RM contributed to ethics component. LD, TW, JR, PGM, LHZ, and RM helped compile and/or review the tabular material. TW, LHZ, and LD generated the graphics. All authors read and approved the final manuscript.

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Additional file 1.

: Table S1 . Sample of national projects that spurred on the big data revolution. Table S2 . Sample of neurology and neuroscience databases. Table S3 . Sample of connectome studies and evolving big data use. Table S4 . Sample of PD "Big Data" studies. Table S5 . Sample of SUD and OUD "Big Data" studies. Table S6 . Sample of pain "Big Data" studies.

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Dipietro, L., Gonzalez-Mego, P., Ramos-Estebanez, C. et al. The evolution of Big Data in neuroscience and neurology. J Big Data 10 , 116 (2023). https://doi.org/10.1186/s40537-023-00751-2

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1 introduction, 2 a primer on the local |$h^{*}$| -polynomial of a lattice polytope, 3 definition, basic properties, and examples of thin polytopes, 4 classification of thin polytopes in dimension |$3$|, 5 thin gorenstein polytopes and gorenstein joins, 6 characterization of thin gorenstein polytopes, acknowledgments.

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Thin Polytopes: Lattice Polytopes With Vanishing Local h * -Polynomial

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Christopher Borger, Andreas Kretschmer, Benjamin Nill, Thin Polytopes: Lattice Polytopes With Vanishing Local h * -Polynomial, International Mathematics Research Notices , Volume 2024, Issue 7, April 2024, Pages 5619–5657, https://doi.org/10.1093/imrn/rnad231

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In this paper, we study the novel notion of thin polytopes: lattice polytopes whose local |$h^{*}$| -polynomials vanish. The local |$h^{*}$| -polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov, and Mavlyutov; Schepers; and Katz and Stapledon. The study of thin simplices was originally proposed by Gelfand, Kapranov, and Zelevinsky, where in this case the local |$h^{*}$| -polynomial simply equals its so-called box polynomial. Our main results are the complete classification of thin polytopes up to dimension 3 and the characterization of thinness for Gorenstein polytopes. The paper also includes an introduction to the local |$h^{*}$| -polynomial with a survey of previous results.

In this paper, we propose to investigate thin polytopes : lattice polytopes with vanishing local |$h^{*}$| -polynomials. Local |$h^{*}$| -polynomials are also called |$\ell ^{*}$| -polynomials or |$\tilde{S}$| -polynomials. In the case of lattice simplices, they equal the so-called box polynomial, see Example 2.15 . Thin simplices were first defined in the context of regular |$A$| -determinants and |$A$| -discriminants by Gelfand, Kapranov, and Zelevinsky [ 24 , 11.4.B] as those lattice simplices whose Newton numbers are zero, see Remark 3.2 . As has been noted in [ 24 ], “a classification of thin lattice simplices seems to be an interesting problem in the geometry of numbers.” In this paper, we extend this endeavor to thin lattice polytopes, which we throughout refer to for simplicity as thin polytopes. Our main results are a complete classification of thin polytopes up to dimension |$3$| (Theorem 4.3 ) and a characterization of thin Gorenstein polytopes in any dimension (Theorem 6.3 ). The latter relies crucially on a recent non-negativity result by Katz and Stapledon [ 34 , Theorem 6.1]. As a consequence, we solve the original problem of [ 24 ] in these two cases and answer questions posed by Borisov, Schepers, and the last named author that came up in the investigation of stringy |$E$| -polynomials of Gorenstein polytopes.

We also hope that this paper leads to renewed interest in the study of the local |$h^{*}$| -polynomial as a fundamental invariant of a lattice polytope with many fruitful connections as pioneered in the work of Stanley [ 48 ]; Karu [ 32 ]; Batyrev, Borisov, and Mavlyutov [ 7 , 13 ]; Schepers [ 42 , 43 ]; and Katz and Stapledon [ 34 ].

Let us give an overview of this paper. In Section 2 , we give a comprehensive survey on the local |$h^{*}$| -polynomial of a lattice polytope. In Section 3 , we define thin polytopes, present the main examples, and discuss several open questions (e.g., Question 3.16 ). Section 4 contains the complete classification of three-dimensional thin polytopes. In particular, we prove that three-dimensional lattice simplices are thin if and only if they are lattice pyramids (Corollary 4.10 ). Section 6.1 presents the characterization of thin Gorenstein polytopes (Theorem 6.3 ). In particular, we deduce that thin Gorenstein polytopes have lattice width |$1$| (Corollary 6.7 ), being thin is invariant under the duality of Gorenstein polytopes (Corollary 6.12 ) and show that Gorenstein simplices are thin if and only if they are lattice pyramids (Corollary 6.15 ). For these results, we study in Section 5 the behavior of local |$h^{*}$| -polynomials under joins, particularly for Gorenstein polytopes.

As the local |$h^{*}$| -polynomial is still not as well known in Ehrhart theory as the |$h^{*}$| -polynomial and also has been studied with different names and notations, we will give a slightly more thorough account on previous research than strictly necessary for the mere purpose of the results of this paper.

2.1 Toric |$g$| - and |$h$| -polynomials of lower Eulerian posets

In [ 47 ], Stanley generalized the notion of |$h$| -vectors of simplicial complexes and simplicial polytopes significantly. For this, let us recall some basic terminology.

The dual of a finite poset |$\mathcal{P}$| is denoted |$\mathcal{P}^{\ast }$|⁠ . A finite poset |$\mathcal{P}$| is locally graded if every inclusion-maximal chain in every interval |$[x,y]$| has the same length |$r(x,y)$|⁠ . The rank |$\operatorname{rk}(\mathcal{P})$| is the length of the longest chain in |$\mathcal{P}$|⁠ . If in addition there exists a rank function |$\rho : \mathcal{P} \rightarrow{\mathbb{Z}}$|⁠ , that is, |$r(x,y) = \rho (y) - \rho (x)$| for every interval |$[x,y]$|⁠ , then |$\mathcal{P}$| is called ranked . If |$\mathcal{P}$| is ranked and every interval |$[x,y]$| with |$x \neq y$| has the same number of even rank and odd rank elements, then |$\mathcal{P}$| is locally Eulerian . If |$\mathcal{P}$| is locally Eulerian and contains a minimal element |$\hat{0}$|⁠ , then it is called lower Eulerian . If it also contains a maximum |$\hat{1}$|⁠ , then |$\mathcal{P}$| is called Eulerian . In the presence of a minimum |$\hat{0}$| in a ranked poset |$\mathcal{P}$|⁠ , we will always assume that the rank function satisfies |$\rho (\hat{0}) = 0$|⁠ .

Here is the definition of the |$g$| -polynomial and the |$h$| -polynomial for lower Eulerian posets according to Stanley [ 47 ].

Definition 2.2. Let |$\mathcal{P}$| be a lower Eulerian poset with rank function |$\rho $| and rank |$d$|⁠ . We define the |$g$| -polynomial |$g_{\mathcal{P}}(t)$| and the |$h$| -polynomial |$h_{\mathcal{P}}(t)$| recursively by introducing a third polynomial |$f_{\mathcal{P}}(t)$| as an intermediate step. Let $$\begin{gather*} f_{\emptyset}(t) = g_{\emptyset}(t) = h_{\emptyset}(t) = 1 \end{gather*}$$ and if |$\mathcal{P}\not =\emptyset $|⁠ , we set $$\begin{gather*} f_{\mathcal{P}}(t) = \sum_{x \in \mathcal{P}} (t-1)^{d - \rho(x)} g_{[\hat{0},x)}(t) \end{gather*}$$ and define for |$f_{\mathcal{P}}(t)=\sum _{i=0}^{d}f_{i} t^{i}$|⁠ , $$\begin{gather*} g_{\mathcal{P}}(t) = \sum_{i=0}^{\lfloor d/2 \rfloor} (f_{i}-f_{i-1}) t^{i}, \ \textrm{and}\ \\ h_{\mathcal{P}}(t) = \sum_{i=0}^{d} f_{d-i} t^{i}. \end{gather*}$$

Hence, |$h_{\mathcal{P}}(t)$| is a polynomial with constant term |$1$| of degree |$\leq d$|⁠ , and |$g_{\mathcal{P}}(t)$| is a polynomial with degree |$\le d/2$|⁠ .

If for |$x \in \mathcal{P}$|⁠ , the interval |$[\hat{0},x]$| is boolean, then |$g_{[\hat{0},x)}(t) = 1$|⁠ , see [ 47 , Prop. 2.1].

Remark 2.4. Let us recall the situation of simplicial complexes |$\Delta $| (see [ 47 ]), where the previous definition of the |$h$| -polynomial agrees with the usual one. For this, we identify |$\Delta $| with its face poset that is a lower Eulerian poset with minimum |$\emptyset \in \Delta $|⁠ . Throughout, we use the convention that |$\dim (\emptyset ) = -1$|⁠ . It follows from Remark 2.3 that |$g_{[\emptyset ,\sigma )}(t) = 1$| for all faces |$\sigma $| of |$\Delta $|⁠ . If |$\Delta $| has dimension |$d-1$|⁠ , we get |$f_{\Delta }(t) = \sum _{i=0}^{d} f_{i-1} (t-1)^{d-i}$|⁠ , where |$f_{j}$| denotes the number of faces of |$\Delta $| of dimension |$j$|⁠ . Hence, this implies $$\begin{align*} &h_{\Delta}(t) = \sum_{\sigma \in \Delta} t^{\dim(\sigma)+1} (1-t)^{d-1-\dim(\sigma)},\end{align*}$$ which indeed equals the usual |$h$| -polynomial of |$\Delta $|⁠ , and where its coefficients form the usual |$h$| -vector of |$\Delta $| [ 47 , page 199]. For instance, if |$\Delta $| is the boundary complex of a |$d$| -dimensional simplex, then |$h_{\Delta }(t)=1+t+\cdots +t^{d}$|⁠ . Let us also give one example to illustrate that the previous formula for |$h_{\Delta }(t)$| fails in the non-simplicial situation. Let |$P$| be the pyramid over the square. In this case, the |$h$| -polynomial of the boundary complex of |$P$| equals |$1+2t+2t^{2}+t^{3}$|⁠ , while the previous formula would give |$1+2t+t^{2}+t^{3}$|⁠ . Note that the |$h$| -polynomial is palindromic while the latter expression is not.

Stanley proved in [ 47 , Theorem 2.4] the following combinatorial palindromicity result generalizing the Dehn–Sommerville equations for face numbers of simplicial polytopes.

Let |$\hat{\mathcal{P}}$| be an Eulerian poset and |$\mathcal{P} := \hat{\mathcal{P}} \setminus \hat{1}$| with |$\operatorname{rk}(\mathcal{P}) = d$|⁠ . Then the |$h$| -polynomial |$h_{\mathcal{P}}(t) = \sum _{i=0}^{d} h_{i} t^{i}$| is palindromic of degree |$d$|⁠ , that is, |$h_{i} = h_{d-i}$| for all |$i=0, \ldots , d$|⁠ .

In particular, we have |$f_{\mathcal{P}}(t) = h_{\mathcal{P}}(t)$| in this case.

We emphasize that in the situation of Theorem 2.5 it is important to distinguish between the |$g$| - and |$h$| -polynomials of |$\mathcal{P}$| and |$\hat{\mathcal{P}}$|⁠ . Indeed, |$g_{\hat{\mathcal{P}}}(t) = 0$| and |$h_{\hat{\mathcal{P}}}(t) = g_{\mathcal{P}}(t)$|⁠ . Unfortunately, in this regard, the different notations employed in the literature can be confusing. Our notation follows that of Stanley while Katz and Stapledon in [ 34 ] write |$g(\hat{\mathcal{P}};t)$| for our |$g_{\mathcal{P}}(t)$| but also use |$h(\mathcal{P};t)$| for our |$h_{\mathcal{P}}(t)$|⁠ . In Borisov and Mavylutov [ 13 ], as well as in [ 7 , 42 ], our |$g_{\mathcal{P}}(t)$| and |$h_{\mathcal{P}}(t)$| would be |$g_{\hat{\mathcal{P}}^{\ast }}(t)$| and |$h_{\hat{\mathcal{P}}^{\ast }}(t)$|⁠ .

Let us give the definition of |$h$| - and |$g$| -polynomials of polytopes.

For |$P$| a polytope we define its (toric) |$h$| -polynomial |$h_{P}(t)$|⁠ , and its (toric) |$g$| -polynomial |$g_{P}(t)$| as the |$h$| -, resp., |$g$| -polynomial of the face lattice |$[\emptyset , P)$| of proper faces of |$P$|⁠ . Note that |$g_{P}(t) = h_{[\emptyset ,P]}(t)$|⁠ , see Remark 2.6 .

Note that by Remark 2.3 , we have |$g_{P}(t)=1$| if |$P$| is a simplex.

Theorem 2.8. Let |$P$| be a polytope of dimension |$d$|⁠ . Then |$h_{P}(t)=\sum _{i=0}^{d} h_{i} t^{i}$| is a palindromic polynomial with positive integer coefficients that form a unimodal sequence, that is, $$\begin{align*}& 1 = h_{0} \leq h_{1} \leq \cdots \leq h_{\lfloor \frac{d}{2} \rfloor}. \end{align*}$$ Equivalently, |$g_{P}(t)$| has non-negative coefficients.

Palindromicity follows from Theorem 2.5 . For rational polytopes |$P$|⁠ , nonnegativity follows from the interpretation of the coefficients of |$h_{P}(t)$| as the dimensions of the even intersection cohomology groups of the toric variety associated with |$P$| and the unimodality property follows from the hard Lefschetz theorem [ 47 , Theorem 3.1, Corollary 3.2]. The non-rational case has been treated by Karu in [ 31 ].

Let us mention the following less well-known duality property of |$g$| -polynomials that will be of importance in Section 6.1 . This is a result by Kalai, published in [ 14 , Theorem 4.5] as a consequence of the main result in that paper by Braden. Here, |$\mathcal{P}^{*}$| denotes the dual poset of a poset |$\mathcal{P}$|⁠ .

Let |$P$| be a polytope. Then |$\deg (g_{[\emptyset , P)}) = \deg (g_{(\emptyset ,P]^{\ast }})$|⁠ .

In other words, if |$Q$| is any polytope that is combinatorially dual to |$P$|⁠ , then |$\deg (g_{P}) = \deg (g_{Q})$|⁠ .

2.2 (Relative) local |$h$| -polynomials of polyhedral subdivisions

We give the definition of the local |$h$| -polynomial (and its generalized relative version) of a polyhedral subdivision |$\Delta $| of a polytope |$P$|⁠ , following [ 48 ] and [ 34 ] (the relative version was introduced independently in [ 2 ] and [ 41 ]). Here, we define the link of a face |$\sigma \in \Delta $| as |$\operatorname{link}(\Delta ,\sigma ):= \{\rho \in \Delta \,:\, \sigma \subseteq \rho \}$|⁠ . We view |$\operatorname{link}(\Delta ,\sigma )$| as a lower Eulerian poset with minimum |$\sigma $|⁠ .

We call |$\ell _{\Delta } (t):= \ell _{\Delta ,\emptyset }$| the local |$h$| -polynomial of |$\Delta $|⁠ .

We suppress |$P$| in this notation as it equals |$|\Delta | = \bigcup _{\sigma \in \Delta } \sigma $|⁠ , the support of |$\Delta $|⁠ . We remark that the same definition of the local |$h$| -polynomial can be extended to the so-called strong formal subdivisions of Eulerian posets, see [ 34 , Definition 4.1].

|$\ell _{\Delta ,\sigma }(t)$| has nonnegative integer coefficients.

|$\ell _{\Delta ,\sigma }(t)$| is palindromic, that is, |$\ell _{i} = \ell _{d-\dim (\sigma )-i}$| for |$i=0, \ldots , d-\dim (\sigma )$|⁠ .

If |$\Delta $| is regular, then the coefficients of |$\ell _{\Delta ,\sigma }(t)$| form a unimodal sequence.

Let us give the references. (2): For the local |$h$| -polynomial this is a special case of [ 48 , Corollary 7.7], for the relative local |$h$| -polynomial see [ 34 , Corollary 4.5]. (1) and (3): For |$\Delta $| a rational polyhedral subdivision this has been proven in [ 48 , Theorem 7.9], respectively, [ 34 , Theorem 6.1] using the decomposition theorem (cf. [ 11 , 17 , 18 ]). As pointed out in [ 34 , Remark 6.6], the only missing ingredient to drop the rationality hypothesis was the relative hard Lefschetz theorem for the intersection cohomology of fans, which was subsequently proven in [ 33 ].

The following decomposition theorem was one of the main motivations of Stanley for the notion of local |$h$| -vectors. This is proven in [ 48 , Theorem 7.8], and the general version in [ 34 ] (see, e.g., second equation in proof of Lemma 6.4). To stress the analogy to Theorem 2.17 , we state the equality also using the |$h$| -polynomial.

Proposition 2.12. Let |$P$| be a polytope of dimension |$d$|⁠ , |$\Delta $| a polyhedral subdivision of |$P$|⁠ , and |$\sigma \in \Delta $|⁠ . Then $$\begin{align*} &h_{\operatorname{link}(\Delta,\sigma)}(t) = \sum_{\sigma \subseteq F \le P} \ell_{\Delta_{F},\sigma}(t) g_{[F,P)}(t) = \sum_{\sigma \subseteq F \le P} \ell_{\Delta_{F},\sigma}(t) h_{[F,P]}(t).\end{align*}$$ In particular for |$\sigma =\emptyset $|⁠ , we get |$\ell _{\Delta }(t) \le h_{\Delta }(t)$| and |$g_{P}(t)= h_{[\emptyset ,P]} \le h_{\Delta }(t)$| coefficientwise.

As a consequence of the above nonnegativity results, Stanley and later Katz and Stapledon show that |$h$| -polynomials as well as relative local |$h$| -polynomials of polyhedral subdivisions are nonnegative and coefficientwise monotone under subdivision refinement [ 34 , Corollary 6.10].

2.3 The |$h^{*}$| -polynomial of a lattice polytope

Let us recall that two lattice polytopes |$P$| and |$Q$| (with respect to the lattice |$M$|⁠ ) are called isomorphic (or unimodularly equivalent ) if there is an affine lattice automorphism of |$M$| that maps the vertices of |$P$| to the vertices of |$Q$|⁠ . We say |$P$| is a unimodular simplex if |$P$| is isomorphic to the convex hull of an affine lattice basis of |$M$|⁠ . Now, |$P$| is a unimodular simplex if and only if |$h^{*}_{P}(t)=1$|⁠ , or equivalently, |$\deg (P)=0$|⁠ . For |$M={\mathbb{Z}}^{d}$|⁠ , let us also define the standard unimodular simplex |$\Delta _{d}:= \operatorname{conv}(0,e_{1}, \ldots , e_{d})$| for the standard lattice basis |$e_{1}, \ldots , e_{d}$|⁠ .

2.4 The local |$h^{*}$| -polynomial of a lattice polytope

Let us introduce the main player of this paper (see [ 48 , Example 7.13] and [ 34 , Def. 7.2]).

Definition 2.13. Let |$P$| be a lattice polytope. The local |$h^{\ast }$| -polynomial or |$\ell ^{\ast }$| -polynomial of |$P$| is defined as $$\begin{align*}& \ell^{\ast}_{P}(t) := \sum_{\emptyset \leq F \leq P} (-1)^{\dim(P)-\dim(F)} h^{\ast}_{F}(t) g_{(F,P]^{\ast}}(t). \end{align*}$$

Let us note that the local |$h^{*}$| -polynomial of the empty face equals |$1$|⁠ , while for a point it equals |$0$|⁠ . We also emphasize the analogy of Definition 2.13 with Definition 2.10 above. See also Subsection 2.5 for precise relationships between the |$h,\ell ,h^{\ast },\ell ^{\ast }$| -polynomials.

The local |$h^{*}$| -polynomial has been studied by Borisov, Batyrev, Mavlyutov, Schepers, and the third author under the name |$\tilde{S}$| -polynomial , see [ 13 , Definition 5.3]. It was used by Borisov and Mavlyutov to simplify the formulas for the stringy |$E$| -polynomial of Calabi–Yau complete intersections in Gorenstein toric Fano varieties originally described via the so-called |$B$| -polynomials [ 8 ]. We remark that the reader should be aware that in these papers in the definition of |$h$| - and |$g$| -polynomials the dual poset was used compared to the one given here.

For lattice simplices |$P$| (of dimension |$d>0$|⁠ ), the |$h^{*}$| - and |$\ell ^{*}$| -polynomial can be easily understood, as in this case the face posets are all Boolean. Let |$\Pi $| denote the half-open parallelepiped spanned by the vertices of |$P \times \{1\}$|⁠ . Then |$h^{*}_{P}(t)$| (resp., |$\ell ^{*}_{P}(t)$|⁠ ) enumerates the number of lattice points in |$\Pi $| (resp., in the interior of |$\Pi $|⁠ ). More precisely, we have |$h^{*}_{P}(t)=\sum _{i=0}^{d+1} h^{*}_{i} t^{i}$| and |$\ell ^{*}_{P}(t)=\sum _{i=0}^{d+1} \ell ^{*}_{i} t^{i}$|⁠ , where for |$i=0, \ldots , d+1$| the coefficient |$h^{*}_{i}$| (resp. |$\ell ^{*}_{i}$|⁠ ) equals the number of lattice points in |$\Pi $| (resp., in the interior of |$\Pi $|⁠ ) with last coordinate |$i$|⁠ . We refer to [ 7 , Prop. 4.6]. This polynomial |$\ell ^{*}_{P}(t)$| of a lattice simplex |$P$| is also often called box polynomial , cf. [ 15 , 25 , 46 ]. For instance, we have |$h^{*}_{P}(t)=1$| if and only if |$P$| is a unimodular simplex; in this case, |$\ell ^{*}_{P}(t)=0$|⁠ . Let us note that for |$h^{*}$| -polynomials this combinatorial interpretation of its coefficients can also be extended to arbitrary lattice polytopes, for example, by half-open decompositions [ 35 ]. On the other hand, there is not yet a combinatorial counting interpretation for the coefficients of the local |$h^{*}$| -polynomial of lattice polytopes known.

Let us summarize some of the basic properties of the local |$h^{*}$| -polynomial. Throughout, we use the convention that the degree of the zero-polynomial is zero.

|$\ell ^{*}_{P}(t)$| has nonnegative integer coefficients.

|$\ell ^{*}_{P}(t)$| is palindromic: we have |$\ell ^{*}_{i} = \ell ^{*}_{d+1-i}$| for |$i=1, \ldots , d$|⁠ .

If |$\ell ^{*}_{P}(t)$| does not vanish, then the degree of |$\ell ^{*}_{P}(t)$| equals at most the degree of |$h^{*}_{P}(t)$|⁠ , and its subdegree (i.e., the smallest |$i$| such that the |$i$| th coefficient of |$\ell ^{*}(t)$| is nonzero) is at least the codegree of |$P$|⁠ .

|$\ell ^{*}_{1} = \ell ^{*}_{d}$| equals the number of lattice points in the interior of |$P$|⁠ .

Let us give the corresponding references: (1) This was conjectured by Stanley [ 48 , Conj.7.14] and proven by Karu [ 32 ]. Using the |$\tilde{S}$| -notation for |$\ell ^{*}$| it also follows from its interpretation as the Hilbert function of a graded vector space by Borisov and Mavlyutov [ 13 , Prop. 5.5]. (2) This was observed in [ 13 , Remark 5.4]. (3) This follows directly, see also [ 42 , Cor.2.16(2)]. (4) For this observation, see [ 7 , Example 4.7].

In particular, the number |$\operatorname{int}_{\mathbb{Z}}(P)$| of interior lattice points completely determines the local |$h^{*}$| -polynomial up to dimension |$2$|⁠ . If |$d=0$|⁠ , then |$\ell ^{*}_{P}(t)=0$|⁠ ; if |$d=1$|⁠ , then |$\ell ^{*}_{P}(t) = \operatorname{int}_{\mathbb{Z}}(P) t$|⁠ ; and if |$d=2$|⁠ , then |$\ell ^{*}_{P}(t) = \operatorname{int}_{\mathbb{Z}}(P) t + \operatorname{int}_{\mathbb{Z}}(P) t^{2}$|⁠ .

2.5 Decomposing and relating the |$h,\ell ,h^{},\ell ^{}$| -polynomials

The following classical result by Betke and McMullen, generalized by Katz and Stapledon, explains the relation of |$h^{*}$| -polynomials to |$h$| -polynomials of a lattice subdivision (i.e., a polyhedral subdivision whose vertices are lattice points). Recall that a lattice triangulation is called unimodular if all its simplices are unimodular simplices.

Theorem 2.17. Let |$P$| be a lattice polytope with a lattice subdivision |$\Delta $|⁠ . Then the following holds: $$\begin{align*} &h^{*}_{P}(t) = \sum_{\sigma \in \Delta}\, \ell^{*}_{\sigma}(t)\, h_{\operatorname{link}(\Delta,\sigma)}(t).\end{align*}$$ In particular, we have |$h_{\Delta }(t) \le h^{*}_{P}(t)$| coefficientwise, where we have equality if and only if the local |$h^{*}$| -polynomial of every non-empty face of |$\Delta $| vanishes. If |$\Delta $| is a lattice triangulation, then this is equivalent to |$\Delta $| being a unimodular triangulation.

This is Lemma 7.12(3) of [ 34 ], generalizing [ 12 ]. We recall that the consequence follows from the nonnegativity of the occurring polynomials and the fact that the |$h$| -polynomials have constant term |$1$|⁠ . Second, the combinatorial description of the |$h^{*}$| - and |$\ell ^{*}$| -polynomial of a lattice simplex, Example 2.15 , implies that a lattice simplex |$S$| is a unimodular simplex if and only if |$\ell ^{*}_{\sigma }(t) = 0$| for all non-empty faces |$\sigma $| of |$S$|⁠ .

Let us note that this result was one motivation for Stanley to define local |$h$| -polynomials, as these allowed him to prove an analogous result in the combinatorial setting, namely, Proposition 2.12 above. And similar to that formula positively expressing the |$h$| -polynomial of a subdivision into local |$h$| -polynomials and toric |$h$| -polynomials, one can also decompose the |$h^{*}$| -polynomial of a lattice polytope positively into local |$h^{*}$| -polynomials and toric |$h$| -polynomials of its face poset.

Corollary 2.18. Let |$P$| be a lattice polytope. Then $$\begin{align*} &h^{*}_{P}(t) = \sum_{\emptyset \leq F \leq P} \ell^{\ast}_{F}(t) g_{[F,P)}(t)=\sum_{\emptyset \leq F \leq P} \ell^{\ast}_{F}(t) h_{[F,P]}(t).\end{align*}$$ In particular, |$\ell ^{*}_{P}(t) + g_{P}(t) = \ell ^{*}_{P}(t) + h_{[\emptyset ,P]}(t) \le h^{*}_{P}(t)$| coefficientwise.

A proof in greater generality is given in [ 43 , Prop. 2.9], see also [ 32 , Cor. 1.1] and [ 42 , Prop. 2.5]. We remark that Corollary 2.18 gives significance to thinking of the local |$h^{*}$| -polynomial as the “Ehrhart core” of the |$h^{*}$| -polynomial. This is most prominently clear in the case of lattice simplices, see Example 2.15 .

Now, just as the (generalized) Betke–McMullen formula transparently separates the lattice data (the |$\ell ^{*}$| -polynomials of the cells) and combinatorial data (the |$h$| -polynomials of the links of the cells) of the |$h^{*}$| -polynomial of the support of a lattice subdivision, the same can be done for the local |$h^{*}$| -polynomial. This was observed in [ 41 ], see also [ 34 , Lemma 7.12(4)].

Proposition 2.19. Let |$P$| be a lattice polytope with a lattice subdivision |$\Delta $|⁠ . Then the following holds: $$\begin{align*} &\ell^{*}_{P}(t) = \sum_{\sigma \in \Delta} \ell^{*}_{\sigma}(t)\; \ell_{\Delta,\sigma}(t).\end{align*}$$ In particular, |$\ell _{\Delta }(t) \le \ell ^{*}_{P}(t)$| coefficientwise, with equality if |$\Delta $| is a unimodular triangulation.

Here, we critically used the nonnegativity of the relative local |$h$| -polynomial, Theorem 2.11 (1), for the consequence. In particular, we get another proof of the nonnegativity of the local |$h^{*}$| -polynomial. Moreover, as already observed in [ 41 ], this implies that the unimodality of the |$\ell ^{*}$| -vector is an intrinsic obstruction for a lattice polytope to have a unimodular triangulation (apply Theorem 2.11 (3) with |$\sigma =\emptyset $|⁠ ). In fact, it is enough to have unimodality of the “local box polynomials” (this is Remark 7.23 in [ 34 ]). Such triangulations were called box unimodal in [ 44 ].

If |$P$| admits a regular triangulation such that the local |$h^{*}$| -polynomials of each cell have unimodal coefficients (e.g., the triangulation is unimodular), then its local |$h^{*}$| -polynomial has unimodal coefficients.

This was used in [ 25 ] to prove the unimodality of the local |$h^{*}$| -polynomial of |$s$| -lecture hall order polytopes.

For the purpose of this paper, the following innocent looking consequence of the nonnegativity of relative local |$h$| -polynomials is crucial.

Let |$P$| and |$P^{\prime}$| be lattice polytopes such that |$P^{\prime}$| is obtained from |$P$| by refining the lattice. Then |$h^{*}_{P}(t)\le h^{*}_{P^{\prime}}(t)$| and |$\ell ^{*}_{P}(t) \le \ell ^{*}_{P^{\prime}}(t)$| coefficientwise.

This follows from Theorem 2.17 , respectively, Proposition 2.19 , the explicit combinatorial description of the |$\ell ^{*}$| -polynomial of a lattice simplex, see Example 2.15 , and the nonnegativity of the |$h$| -polynomial, respectively, of the relative local |$h$| -polynomial.

We remark that for |$h^{*}$| -polynomials this lattice-monotonicity can also easily be seen combinatorially, for example, using half-open decompositions (see [ 10 ]). However, for local |$h^{*}$| -polynomials, there seems to be no such combinatorial argument known. This is also true for the next result. Note that by Stanley’s famous monotonicity result, the |$h^{*}$| -polynomial is coefficientwise monotone with respect to inclusion. However, this is not true for the local |$h^{*}$| -polynomial. Still, it holds when one considers subpolytopes that do not lie on the boundary.

Let |$P$| and |$Q$| be lattice polytopes such that |$\textrm{relint}(Q) \subseteq \operatorname{int}(P)$| (for instance, |$\dim (Q)=\dim (P)$|⁠ ). Then |$\ell ^{*}_{Q}(t) \le \ell ^{*}_{P}(t)$| coefficientwise.

Choose a lattice subdivision |$\Delta $| of |$P$| that contains |$Q$| as a cell. Then by the nonnegativity of the appearing polynomials, we see from Proposition 2.19 that |$\ell ^{*}_{Q}(t) \ell _{\Delta ,Q}(t) \le \ell ^{*}_{P}(t)$| coefficientwise. It remains to observe that since |$Q$| is in the relative interior of |$P$|⁠ , it follows directly from Definition 2.10 that |$\ell _{\Delta ,Q}(t) = h_{\operatorname{link}(\Delta ,Q)}(t)$| and hence has constant coefficient |$1$|⁠ .

Finally, let us just shortly mention that in the Katz–Stapledon paper [ 34 ], motivated by algebraic and tropical geometry, the |$h^{*}$| - and |$\ell ^{*}$| -polynomials are further refined to bivariate (and even trivariate) polynomials leading to the notion of |$h^{*}$| - and |$\ell ^{*}$| -diamonds. As we will use the following notation later for the computation of the local |$h^{*}$| -polynomial in dimension three, let us introduce it here.

Definition 2.23. Let |$P$| be a lattice polytope of dimension |$d$|⁠ . Then we define $$\begin{align*} &h^{*}_{P}(u,v):= \sum_{F \le P}v^{\dim(F)+1} \ell^{*}_{F}(uv^{-1}) g_{[F,P)}(uv).\end{align*}$$

Theorem 2.24. Let |$P$| be a lattice polytope of dimension |$d$| with |$\ell ^{*}_{P}(t)=\sum _{i=1}^{d} \ell ^{*}_{i} t^{*}$|⁠ . Then $$\begin{align*} &\ell^{*}_{1} \le \ell^{*}_{i} \quad \ \textrm{for}\ i=2, \ldots, d.\end{align*}$$

3.1 Main definition and known results from [ 24 ]

The following notion is the main focus of this paper.

A lattice polytope |$P$| is called thin if its local |$h^{*}$| -polynomial |$\ell ^{*}_{P}$| vanishes. By the nonnegativity of the coefficients, Theorem 2.16 (1), this is equivalent to |$\ell ^{*}_{P}(1)=0$|⁠ .

Let us note that lattice polytopes of dimension |$0$| as well as unimodular simplices are thin, see Example 2.15 . We remark that thin polytopes naturally appear in Theorem 2.17 .

Remark 3.2. Thin simplices were first investigated in [ 24 , 11-4-B] in the context of regular |$A$| -determinants and |$A$| -discriminants, more precisely, in the characterization of the so-called |$D$| -equivalence classes of regular triangulations of |$A$|⁠ . There a lattice simplex |$S$| was defined to be thin if its Newton number |$\nu (S)$| equals zero. Here, the Newton number is defined as follows: $$\begin{align}& \nu(S):= \sum_{\emptyset \le F \le S} (-1)^{\dim(S)-\dim(F)} \textrm{vol}_{\mathbb{Z}}(F) =0,\end{align}$$ (1) where, |$\textrm{vol}_{\mathbb{Z}}(\emptyset ):=1$| (also |$\textrm{vol}_{\mathbb{Z}}(F)=1$| if |$\dim (F)=0$|⁠ ). Recall from Example 2.15 that |$\textrm{vol}_{\mathbb{Z}}(F)=h^{*}_{F}(1)$| counts the number of lattice points in the half-open parallelepiped over |$F$|⁠ . Hence, by inclusion–exclusion, it is straightforward to deduce |$\nu (S)=\ell ^{*}_{S}(1)$|⁠ , the number of interior lattice points in the half-open parallelepiped over |$S$|⁠ . Thus, for lattice simplices, the definitions agree. Let us note that in [ 24 ] the nonnegativity of |$\nu (S)$| follows from quite deep algebro-geometric arguments, while it is combinatorially obvious from the interpretation of |$\ell ^{*}_{S}$| as the box polynomial of the lattice simplex |$S$|⁠ . The reader should also be warned that the expression in equation ( 1 ) may be negative for lattice polytopes . For instance, it equals |$-1$| for the 0/1-cube |$[0,1]^{3}$|⁠ .

Thin simplices were classified in [ 24 ] up to dimension |$2$|⁠ . Here, we can deduce the following statement directly from Theorem 2.16 (4). Let us define a lattice polytope to be hollow if it has no lattice points in its interior. Here, a |$0$| -dimensional lattice polytope is not hollow (but thin).

Thin polytopes of dimension |$> 0$| are hollow. The converse also holds in dimensions |$1$| and |$2$|⁠ .

In particular, |$\Delta _{1}$| is the only thin polytope of dimension |$1$|⁠ . Hollow polytopes in dimension |$2$| are well-known. They are either isomorphic to |$2 \Delta _{2}$| or have lattice width |$1$| (i.e., all vertices lie on two parallel hyperplanes of lattice distance one). Note that hollow three-dimensional lattice polytopes do not have to be thin, for example, |$2 \Delta _{3}$| and |$[0,1]^{3}$| are not thin.

One important construction for thin polytopes is to take lattice pyramids.

Definition 3.4. Let |$P \subset{\mathbb{R}}^{d}$| be a lattice polytope. Then $$\begin{align*} &\operatorname{conv}(P \times \{0\}, \{0\} \times \{1\}) \subset{\mathbb{R}}^{d} \times{\mathbb{R}}\end{align*}$$ is called the lattice pyramid over |$P$|⁠ . By convention, a lattice point is also considered a lattice pyramid.

It is well-known that the |$h^{*}$| -polynomial, and particularly the degree, does not change under taking lattice pyramids. The following result has already been observed in [ 24 ] for lattice simplices and in general in [ 7 ] for lattice polytopes.

Lattice pyramids over arbitrary lattice polytopes are thin.

Using this notation, we can state the classification of thin simplices up to dimension two as follows.

A lattice simplex of dimension at most |$\le 2$| is thin if and only if it is isomorphic to |$2 \Delta _{2}$| or it is a lattice pyramid.

In [ 24 ], thin triangulations were intensively studied. Recently, this notion has also been investigated by [ 19 ] where it was completely characterized up to dimension |$3$|⁠ . As Stanley observed at the end of Section 7 in [ 48 ], a thin triangulation may be defined by the vanishing of its local |$h$| -polynomial. Now, it follows from Proposition 2.19 that all lattice triangulations of thin polytopes are thin. This seems to be a quite strong combinatorial obstruction worth of further study.

By Corollary 2.21 , a thin polytope stays thin if the lattice is coarsened. We do not know of a purely combinatorial proof of this fact.

If a lattice polytope is contained in a thin polytope but not in its boundary, then it is also thin. This non-trivial fact follows from Corollary 2.22 .

Let us note that thin simplices turn up in [ 44 ] when studying conditions for unimodality of (local) |$h^{*}$| -polynomials in the context of box unimodal triangulations mentioned before Corollary 2.20 . Here, let us recall the following observation: if |$P$| admits a regular triangulation |$\Delta $| such that every non-empty face of |$\Delta $| is thin, then its local |$h^{*}$| -polynomial equals the local |$h$| -polynomial of |$\Delta $| and its |$h^{*}$| -polynomial equals the |$h$| -polynomial of |$\Delta $|⁠ , see Proposition 2.19 and Theorem 2.17 . Now, in [ 44 ], it is asked whether every IDP lattice polytope has a regular triangulation into lattice simplices that have vanishing or monomial |$\ell ^{*}$| -polynomial. The motivation was that the existence of a box unimodal triangulation of an IDP reflexive polytope implies unimodality of its |$h^{*}$| -polynomial. While the previous question is still open, a proof of the latter result using completely different methods was recently announced in [ 1 ].

3.2 Two classes of examples of thin polytopes

Let us describe two ways to get thin polytopes in higher dimensions.

The first observation is that lattice polytopes of small degree (in other words, “very hollow” lattice polytopes) are always thin.

We say, |$P$| is trivially thin if |$\dim (P) \ge 2 \deg (P)$|⁠ .

Trivially thin polytopes are thin.

A lattice polytope |$P$| is trivially thin if and only if |$\deg (P) < \operatorname{codeg}(P)$|⁠ . Now, the statement follows from Theorem 2.16 (3).

Typical examples of trivially thin polytopes are Cayley polytopes with many factors. We will talk about Cayley polytopes with two factors in much more detail later (see Definition 5.7 and Remark 5.8 ); however, let us already now give the definition of a Cayley polytope to make the previous statement precise. For this, we denote by a lattice projection |${\mathbb{R}}^{d} \to{\mathbb{R}}^{k}$| an affine-linear map mapping |${\mathbb{Z}}^{d}$| surjectively onto |${\mathbb{Z}}^{k}$|⁠ . If there is a lattice projection mapping a |$d$| -dimensional lattice polytope |$P$| onto a unimodular simplex |$\Delta _{k}$| with |$k \ge 1$|⁠ , then |$P$| is called a Cayley polytope with |$k+1$| factors (namely, the fibers of the vertices of |$\Delta _{k}$|⁠ ). One can easily deduce from [ 7 , Proposition 1.12] that |$P$| is trivially thin if |$k \ge d/2$|⁠ . An alternative way to view this is also the following. Take |$r$| lattice polytopes |$P_{0}, \ldots , P_{r-1}$| in |${\mathbb{R}}^{m}$|⁠ . Then the Cayley sum of |$P_{0}, \ldots , P_{r-1}$| is defined as the convex hull of |$P_{0} \times \{0\}$| and |$P_{i} \times \{e_{i}\}$| for |$i=1, \ldots , r-1$| in |${\mathbb{R}}^{m+r-1}$|⁠ . It is trivially thin if |$r \ge m+1$|⁠ . Note that a Cayley sum is a Cayley polytope, and every Cayley polytope is isomorphic to a Cayley sum.

A second way to get high-dimensional thin polytopes is to use free joins.

Definition 3.13. Let |$P \subset{\mathbb{R}}^{n}$| and |$Q \subset{\mathbb{R}}^{m}$| be lattice polytopes. We call $$\begin{align*} &P \circ_{\mathbb{Z}} Q:= \operatorname{conv}(P \times \{0\} \times \{0\}, \{0\} \times P \times \{1\}) \subset{\mathbb{R}}^{n} \times{\mathbb{R}}^{m} \times{\mathbb{R}},\end{align*}$$ the free join of |$P$| and |$Q$|⁠ .

For instance, the free join of |$[0,1]$| with itself is a unimodular |$3$| -simplex. Note that isomorphic factors lead to isomorphic free joins. From the Ehrhart-theoretic viewpoint the free join construction is important because of the following multiplicativity property, see [ 27 , Lemma 1.3] and [ 42 , Remark 4.6(5)].

Proposition 3.14. Let |$P \subset{\mathbb{R}}^{n}$| and |$Q \subset{\mathbb{R}}^{m}$| be lattice polytopes. Then $$\begin{align*} &h^{*}_{P \circ_{\mathbb{Z}} Q}(t) = h^{*}_{P}(t) \, h^{*}_{Q}(t),\; \ \textrm{and}\ \; \ell^{*}_{P \circ_{\mathbb{Z}} Q}(t) = \ell^{*}_{P}(t) \, \ell^{*}_{Q}(t).\end{align*}$$

The free join of two lattice polytopes is thin if and only if at least one of the two factors is thin.

As a lattice pyramid is the free join of a point (which is thin) and a lattice polytope, this generalizes Proposition 3.5 .

3.3 Are there other examples of thin polytopes?

It is not trivial to give examples of thin polytopes (such as Example 3.17 below) that do not fall in above described two classes. In order to formulate a natural question in this respect, let us recall two notions. First, a lattice polytope |$P$| is called spanning if every lattice point in its affine hull is an integer affine combination of the lattice points in |$P$|⁠ . Note that every lattice polytope becomes spanning after a possible coarsening of the ambient lattice (we refer to [ 29 ] for more background and results on spanning lattice polytopes). Second, let us call a lattice polytope |$P$| a join if there are two non-empty faces |$F$| and |$G$| of |$P$| such that the free join of |$F$| and |$G$| is affinely-isomorphic to |$P$|⁠ . Let us remark that if |$P$| is spanning and a join of |$F$| and |$G$| where every lattice point in |$P$| is contained in |$F$| or |$G$|⁠ , then it is automatically a free join.

Is every thin polytope trivially thin or a join?

Is every spanning thin polytope trivially thin or a free join?

Both questions are closely related but not directly. The reason is that the degree of the polytope can drop under coarsenings of the lattice, so a non-spanning thin but not trivially thin polytope could be trivially thin with respect to its spanning lattice. We also note that trivially thin polytopes are often not joins. For example the unit square |$[0,1]^{2}$| has degree |$1$| and is hence trivially thin, while triangles are the only polygons that are joins.

As the following example shows, the spanning hypothesis in the second part of Question 3.16 is indeed important. It is one of the apparently rare thin polytopes that are not trivially thin and not a free join.

Consider the |$4$| -simplex |$P = \operatorname{conv}(0, e_{1}, e_{2}, (1, 2, 4, 0), (2, 1, 0, 4)) \subseteq{\mathbb{R}}^{4}$|⁠ . The sublattice |$N$| of |${\mathbb{Z}}^{4}$| spanned by all lattice points of |$P$| has index |$2$| and the quotient |${\mathbb{Z}}^{4}/N \cong{\mathbb{Z}}/2{\mathbb{Z}}$| is generated by |$\overline{e_{3}} = \overline{e_{4}}$|⁠ . A computation in SageMath with backend Normaliz shows that |$P$| is thin and |$h_{P}^{\ast }(t) = t^{3} + 11t^{2} + 3t + 1$|⁠ , in particular |$\deg (P) = 3$|⁠ , so |$P$| is not trivially thin. A computation in Polymake shows that the lattice width of |$P$| is |$2$|⁠ , so that |$P$| is not a Cayley polytope, in particular not a free join. It can be checked that with respect to |$N$|⁠ , |$P$| is the lattice pyramid over a reflexive |$3$| -simplex of lattice volume |$8$|⁠ .

Question 3.16 should be understood as a guiding question for finding interesting high-dimensional thin polytopes. Let us discuss this problem in more detail as follows.

As being hollow is equivalent to |$\deg (P) < \dim (P)$|⁠ , it is evident that every hollow lattice polytope in dimension |$\le 2$| is trivially thin. Hence, by Proposition 3.3 every thin polytope in dimension |$\le 2$| is trivially thin. It will be proven in our first main result Theorem 4.3 that in dimension |$3$| all non-trivially thin polytopes are lattice pyramids. In particular, Question 3.16 has an affirmative answer in dimensions |$\le 3$|⁠ . Note that |$\operatorname{conv}(e_{1},e_{2},-e_{1}-e_{2}) \circ _{\mathbb{Z}} 2 \Delta _{2}$| is an example of a thin simplex in dimension |$5$| that is not trivially thin (it has degree |$3$|⁠ ), but is not a lattice pyramid, while being a free join with a (trivially) thin factor.

In higher dimensions, our second main result shows that non-trivially thin Gorenstein polytopes are the so-called Gorenstein joins (see Definition 5.11 ) with a trivially thin factor, so that Question 3.16 has an affirmative answer also in the Gorenstein case (see Corollary 6.4 ).

Computationally, we have verified that Question 3.16 has an affirmative answer for all |$4$| -dimensional lattice polytopes of lattice volume |$\leq 21$|⁠ , for all |$5$| -dimensional lattice simplices of lattice volume |$\leq 20$| and for all |$6$| -dimensional lattice simplices of lattice volume |$\leq 16$|⁠ . We provide some of the relevant data at [ 36 ].

3.4 Interesting thin empty simplices?

A lattice simplex is called empty if its vertices are its only lattice points. Among the hollow polytopes this is the class of lattice simplices that has been studied most intensively, see for example, [ 30 ] and the references therein. However, it turns out that there are no interesting thin empty simplices in dimension at most |$4$|⁠ . Let us give the easy reasoning. For this, we recall that the quotient group of a |$d$| -dimensional lattice simplex |$P \subset{\mathbb{R}}^{d}$| is defined as the quotient of |${\mathbb{Z}}^{d+1}$| by the subgroup generated by the vertices of |$P \times \{1\}$|⁠ .

Let |$P$| be a lattice simplex with cyclic quotient group. Then |$P$| is thin if and only if |$P$| is a lattice pyramid.

Let |$P \subset{\mathbb{R}}^{d}$| be |$d$| -dimensional. We denote by |$\Pi $| the half-open parallelepiped from Example 2.15 . Clearly, every element in the quotient group of |$P$| has a unique representative in |$\Pi \cap{\mathbb{Z}}^{d+1}$|⁠ . Let |$g \in \Pi \cap{\mathbb{Z}}^{d+1}$| be the representative of a generator of the quotient group of |$P$|⁠ . We assume that |$P$| is thin. Hence, there is a proper, non-empty subset |$V^{\prime}$| of the vertex set of |$S \times \{1\}$| such that |$g$| is a linear combination of vertices of |$V^{\prime}$|⁠ . In particular, this also holds for the representatives of all the elements in the quotient group of |$P$|⁠ . Now, it follows from [ 39 , Lemma 12] that |$P$| is a lattice pyramid.

It is well-known that all empty lattice simplices in dimension at most |$4$| have cyclic quotient group [ 5 ]. As also in higher dimensions most empty simplices constructed (but not all of them) have this property (see e.g., [ 22 ]), it seems to be a challenge to find examples of empty simplices that are thin but not simply lattice pyramids.

3.5 Are thin polytopes “flat”?

We observed above that all thin polytopes in dimension at most two have lattice width |$1$| except for |$2 \Delta _{2}$|⁠ . We leave it as an exercise to the reader to show that |$2 \Delta _{d}$| for even |$d$| is the only thin simplex among all lattice simplices of the form |$\operatorname{conv}(0,k_{1} e_{1}, \ldots , k_{d} e_{d}) \subset{\mathbb{R}}^{d}$| with |$k_{1}, \ldots , k_{d} \in{\mathbb{Z}}_{\ge 1}$| that are not lattice pyramids (i.e., |$k_{i}> 1$| for all |$i$|⁠ ). It will follow from our main results that thin polytopes in dimension three (Corollary 4.4 ) as well as thin Gorenstein polytopes in arbitrary dimension (Corollary 6.7 ) have lattice width |$1$|⁠ . In dimension four, Example 3.17 has lattice width |$2$|⁠ . As thin polytopes (of dimension |$> 0$|⁠ ) are hollow, in fixed dimension their lattice width is bounded. Now, our lack of “non-flat” examples motivates the following question.

Can one find (spanning) thin polytopes of arbitrarily large lattice width?

We expect that such examples with increasing lattice width should exist with increasing dimension. Note that if one assumes that Question 3.16 (2) has an affirmative answer, then for Question 3.19 it would be important to find the maximum width of trivially thin spanning polytopes |$P$|⁠ . However, it is a folklore open question, often called “the” Cayley conjecture (see [ 21 , 26 , 28 ]), that any lattice polytope with |$\dim (P)> 2 \deg (P)$| has lattice width |$1$|⁠ . Thus, assuming also that the Cayley conjecture holds essentially reduces the previous question to the study of spanning lattice polytopes with |$\dim (P) = 2 \deg (P)$|⁠ .

As observed above, three-dimensional lattice polytopes |$P$| that are lattice pyramids over polygons or have degree at most one are automatically thin. Our first main result, Theorem 4.3 , shows that in dimension three indeed all the thin polytopes are of this type.

Lattice polytopes of degree at most one are completely known in any dimension. For this, let us recall the following definition.

A Lawrence prism is a |$d$| -dimensional lattice polytope in |${\mathbb{R}}^{d}$| isomorphic to |$\operatorname{conv}(0,e_{1},\ldots , e_{d-1}, k_{0} e_{d}, e_{1}+k_{1} e_{d}, \ldots , e_{d-1}+k_{d-1} e_{d})$| with |$k_{0}, k_{1}, \ldots , k_{d-1} \in{\mathbb{Z}}_{\ge 1}$|⁠ .

The following result was proven in [ 6 ].

Any lattice polytope of degree |$1$| is either a lattice pyramid, a Lawrence prism or isomorphic to |$2 \Delta _{2}$|⁠ .

Here is the main result of this section.

|$P$| is a lattice pyramid over a lattice polygon, or

|$P$| is a Lawrence prism.

Every three-dimensional thin polytope has lattice width |$1$|⁠ .

The proof of Theorem 4.3 relies on two instances that seem to be exceptional to small dimensions. First, in dimension three, all the coefficients of the local |$h^{*}$| -polynomial can be explicitly determined.

Proposition 4.5. Let |$P \subseteq{\mathbb{R}}^{3}$| be a |$3$| -dimensional lattice polytope. Then $$\begin{align*}& \ell^{\ast}_{P}(t) = |\operatorname{int}_{{\mathbb{Z}}}(P)| (t + t^{3}) + \left(|\operatorname{int}_{{\mathbb{Z}}}(2P)| - 4 |\operatorname{int}_{{\mathbb{Z}}}(P)| - \sum_{F \leq P \textrm{ facet}} |\operatorname{int}_{{\mathbb{Z}}}(F)|\right) t^{2}. \end{align*}$$

Recall from Theorem 2.11 that |$\ell ^{\ast }_{1} = \ell ^{\ast }_{3} = h^{\ast }_{3} = |\operatorname{int}_{{\mathbb{Z}}}(P)|$|⁠ . Hence, we need only determine |$\ell ^{\ast }_{2}$|⁠ . From Stanley reciprocity, we deduce |$h^{\ast }_{2}=|\operatorname{int}_{{\mathbb{Z}}}(2P)| - 4 |\operatorname{int}_{{\mathbb{Z}}}(P)|$|⁠ . Now, in the notation of the |$h^{\ast }$| -diamond introduced in [ 34 ] (see Definition 2.23 ) we have |$h^{\ast }_{2} = h^{\ast }_{1,0}+h^{\ast }_{1,1}$|⁠ , where |$h^{\ast }_{1,1}=\ell ^{\ast }_{2}$| and |$h^{\ast }_{1,0} = \sum _{F \leq P \textrm{ facet}} |\operatorname{int}_{{\mathbb{Z}}}(F)|$| by [ 34 , Example 8.9]. This implies the statement.

Let us note that we get from the lower bound theorem of Katz–Stapledon, Theorem 2.24 , |$\ell ^{*}_{1} \le \ell ^{*}_{2}$|⁠ . This leads to the following non-obvious corollary. It would be very interesting to find a purely combinatorial proof.

Corollary 4.6. Let |$P \subseteq{\mathbb{R}}^{3}$| be a |$3$| -dimensional lattice polytope. Then $$\begin{align*} &|\operatorname{int}_{{\mathbb{Z}}}(2P)| \ge 5 \,|\operatorname{int}_{{\mathbb{Z}}}(P)| + \sum_{F \leq P \textrm{ facet}} |\operatorname{int}_{{\mathbb{Z}}}(F)|.\end{align*}$$

For our purposes, let us note the following numerical characterization of thinness in dimension three.

Corollary 4.7. Let |$P \subseteq{\mathbb{R}}^{3}$| be a |$3$| -dimensional lattice polytope. Then |$P$| is thin if and only if |$P$| is hollow and $$\begin{align*}& |\operatorname{int}_{{\mathbb{Z}}}(2P)| = \sum_{F \leq P \textrm{ facet}} |\operatorname{int}_{{\mathbb{Z}}}(F)|. \end{align*}$$

The second result that is not yet available in higher dimensions is a complete classification of hollow three-dimensional lattice polytopes.

|$P$| is contained in one of the |$12$| maximal hollow lattice polytopes classified in [ 3 ].

There is a lattice projection |${\mathbb{R}}^{3} \rightarrow{\mathbb{R}}^{1}$| mapping |$P$| onto |$\Delta _{1}$|⁠ .

There is a lattice projection |${\mathbb{R}}^{3} \rightarrow{\mathbb{R}}^{2}$| mapping |$P$| onto |$2 \Delta _{2}$|⁠ .

Before giving the proof of Theorem 4.3 let us also recall the following well-known formula for the mixed volume (e.g., [ 40 , Corollary 3.2]):

Lemma 4.9. Let |$P_{1}, P_{2} \subseteq{\mathbb{R}}^{2}$| be lattice polytopes. Then $$\begin{align*} \operatorname{MV}(P_{1}, P_{2}) = 1 &+ (-1)^{\dim(P_{1} + P_{2})}|\operatorname{int}_{{\mathbb{Z}}}(P_{1} + P_{2})| \\ &+(-1)^{\dim(P_{1}) - 1}|\operatorname{int}_{{\mathbb{Z}}}(P_{1})| + (-1)^{\dim(P_{2}) - 1}|\operatorname{int}_{{\mathbb{Z}}}(P_{2})|. \end{align*}$$

By Corollary 4.7 , |$P$| is hollow. We treat the three cases of Theorem 4.8 separately. A direct computation in Magma deals with case 1 , see [ 36 ].

If |$\dim (P_{1})=2$| and |$\dim (P_{2})=1$|⁠ , then Lemma 4.9 yields |$\operatorname{MV}(P_{1},P_{2}) = 1 + |\operatorname{int}_{{\mathbb{Z}}}(P_{2})|$|⁠ . On the other hand, |$\operatorname{MV}(P_{1},P_{2}) = V(\pi _{P_{2}}(P_{1}))(|\operatorname{int}_{{\mathbb{Z}}}(P_{2})|+1)$| by [ 45 , Theorem 5.3.1], where |$\pi _{P_{2}}$| is a lattice projection along the line segment |$P_{2}$| and |$V(\pi _{P_{2}}(P_{1}))$| denotes the lattice volume. Hence, |$V(\pi _{P_{2}}(P_{1})) = 1$| and therefore |$\pi _{P_{2}}(P_{1}) \cong \Delta _{1}$|⁠ . The lattice projection of |$P$| along |$P_{2}$| is then a lattice projection onto |$\Delta _{2}$|⁠ . Thus, |$\operatorname{codeg}(P) \ge 3$| and hence |$\deg (P) \leq 1$|⁠ , so either |$\deg (P) = 1$| or |$P \cong \Delta _{3}$| is a lattice pyramid.

The case |$\dim (P_{1}) = \dim (P_{2}) = 1$| is similar. Lemma 4.9 yields |$\operatorname{MV}(P_{1},P_{2})= 1 + |\operatorname{int}_{{\mathbb{Z}}}(P_{1})| + |\operatorname{int}_{{\mathbb{Z}}}(P_{2})|$|⁠ . On the other hand, again |$\operatorname{MV}(P_{1},P_{2})=V(\pi _{P_{2}}(P_{1})) (|\operatorname{int}_{{\mathbb{Z}}}(P_{2})|+1)$|⁠ . We may assume |$|\operatorname{int}_{{\mathbb{Z}}}(P_{1})| \leq |\operatorname{int}_{{\mathbb{Z}}}(P_{2})|$|⁠ . If |$V(\pi _{P_{2}}(P_{1})) \geq 2$| or |$V(\pi _{P_{2}}(P_{1})) = 0$|⁠ , we obtain a contradiction, so |$V(\pi _{P_{2}}(P_{1})) = 1$|⁠ . The same argument as above shows |$\deg (P) \leq 1$|⁠ .

Finally, if one of the |$P_{i}$| is zero-dimensional, then |$P$| is a lattice pyramid.

It is left to study case 3 , and we may assume |$P$| to be of lattice width at least |$2$| because width |$1$| is equivalent to |$P$| being a Cayley polytope that is precisely case 2 .

We distinguish several cases and always start by showing how, in each case, we can associate to each lattice point in the interior of a facet of |$P$|⁠ , in an injective way, a lattice point in the interior of |$2P$|⁠ . We then prove that there always exists an additional lattice point in the interior of |$2P$|⁠ , therefore showing that case 3 does not yield any new thin polytopes by Corollary 4.7 .

We may assume that |$P$| projects onto |$2 \Delta _{2}$| along the |$z$| -axis. As lattice projections map interior lattice points to interior lattice points, all interior lattice points of a facet of |$P$| are of the form |$x_{1}^{a}=(1,0,a)$|⁠ , |$x_{2}^{a}=(0,1,a)$|⁠ , or |$x_{3}^{a}=(1,1,a)$| for suitable |$a \in{\mathbb{Z}}$|⁠ . By fixing vertices |$v_{1}=(0,2,\alpha )$|⁠ , |$v_{2}=(2,0,\beta )$|⁠ , |$v_{3}=(0,0,\gamma )$| of |$P$|⁠ , we hence obtain points |$\frac{1}{2}(x_{i}^{a} + v_{i}) \in \operatorname{int}(P)$|⁠ , and therefore |$(x_{i}^{a} + v_{i}) \in \operatorname{int}_{{\mathbb{Z}}}(2P)$| for all |$a \in{\mathbb{Z}}$| such that |$x_{i}^{a}$| is an interior point of a facet of |$P$|⁠ . Then |$(x_{i}^{a}+v_{i}) \neq (x_{j}^{b}+v_{j})$| if |$i \neq j$| or |$a \neq b$|⁠ .

Now we show the existence of an additional interior lattice point. Indeed, |$P$| can have at most three facets containing interior lattice points, namely at most those facets, if there are such, that project to one of the three edges of |$2 \Delta _{2}$|⁠ .

Next, assume that |$P$| has exactly two facets containing interior lattice points, and say these are the facets opposite to |$v_{1}$| and |$v_{2}$|⁠ . We pick two such points |$(0,1,q)$|⁠ , |$(1,0,r)$|⁠ . Then we obtain as many interior lattice points in |$2P$| of the form |$x_{1}^{a}+v_{1}$| or |$x_{2}^{a}+v_{2}$| as there are points in the interiors of facets of |$P$|⁠ , and |$(1,1,q+r)$| is an additional interior lattice point of |$2P$|⁠ .

The only remaining case is the one where only one facet |$F$| of |$P$| contains interior lattice points. We may assume |$F$| is the facet that projects onto the edge |$[(0,0), (2,0)]$|⁠ . Then all interior lattice points of |$F$| project to |$(1,0)$| and are of the form |$x_{1}^{a}$| for |$a \in{\mathbb{Z}}$| ranging in a suitable interval. From this, we obtain |$|\operatorname{int}_{{\mathbb{Z}}}(F)|$| interior lattice points |$v_{1} + x_{1}^{a}$| of |$2P$|⁠ . Observe that all of them have second coordinate |$2$|⁠ . Therefore, it is enough to show that |$2P$| contains an interior lattice point with second coordinate |$1$|⁠ .

Now, we can deal with the remaining case |$|\operatorname{int}_{{\mathbb{Z}}}(F)| \in \{1,2\}$|⁠ . Let |$F^{\prime} \subseteq F$| run through the inclusion-minimal subpolygons of |$F$|⁠ , which contain the same interior lattice points as |$F$|⁠ . We will show that there are only two possibilities for |$F^{\prime}$|⁠ .

But only the last one does not contain three lattice points over |$(1,0)$|⁠ .

Again, only the last one does not contain three lattice points over |$(1,0)$|⁠ .

Lastly, for each of these two remaining polygons |$F^{\prime}$|⁠ , we may choose a lattice subpolytope |$P^{\prime}$| of |$P$| that is a pyramid of height |$2$| over |$F^{\prime}$|⁠ . We observe that for given |$F^{\prime}$| there are at most four non-isomorphic possibilities for |$P^{\prime}$| to consider as the first two coordinates of an apex in |${\mathbb{R}}^{2} \times \{2\}$| over a base polytope in |${\mathbb{R}}^{2} \times \{0\}$| may be chosen by a unimodular shearing to be in |$\{(0,0),(1,0),(0,1),(1,1)\}$|⁠ . Now, an explicit computation in SageMath shows that for all these at most eight cases we have |$|\operatorname{int}_{{\mathbb{Z}}}(2P^{\prime})|> |\operatorname{int}_{{\mathbb{Z}}}(F^{\prime})| = |\operatorname{int}_{{\mathbb{Z}}}(F)|$|⁠ , concluding the proof.

We can now answer the original question in [ 24 ] in dimension |$3$|⁠ .

A three-dimensional lattice simplex is thin if and only if it is a lattice pyramid.

This follows directly from Theorem 4.3 . The reader is cautioned not to jump to the conclusion that the same result may be true in higher dimensions. In dimension |$4$|⁠ , |$[-1,1] \circ _{\mathbb{Z}} 2 \Delta _{2}$| is an example of a (trivially) thin simplex that is not a lattice pyramid.

5.1 Gorenstein polytopes and their duals

A lattice polytope |$P$| is called Gorenstein if |$h^{*}_{P}$| is palindromic.

Definition 5.2. Let |$P \subset{\mathbb{R}}^{d}$| be a |$d$| -dimensional Gorenstein polytope. In this case, the dilate |$\operatorname{codeg}(P) \cdot P$| is a reflexive polytope (up to lattice translation), and we denote its unique interior lattice point by |$w$|⁠ . Then $$\begin{align*} &P^{\times}:= \{y \in ({\mathbb{R}}^{d+1})^{*} \;:\; \langle{y},{w}\rangle = 1 \ \textrm{and}\ \langle{y},{x}\rangle \ge 0 \;\forall\, x \in P \times \{1\}\}\end{align*}$$ is called the dual Gorenstein polytope of |$P$|⁠ .

Let |$P \subset{\mathbb{R}}^{d}$| be a |$d$| -dimensional Gorenstein polytope. Then |$P^{\times }$| is a Gorenstein polytope of the same dimension and degree as |$P$|⁠ , and it is combinatorially dual to |$P$|⁠ .

Note that, if a Gorenstein polytope is lower-dimensional, we consider, as usual, its ambient lattice in order to get its dual Gorenstein polytope.

If |$F$| is a face of |$P$|⁠ , we denote by |$F^{*}$| the dual face , that is, the corresponding face of |$P^{\times }$|⁠ .

Attention: it is important to distinguish the dual face |$F^{\ast }$| from |$F^{\times }$|⁠ , the latter being defined only if |$F$| is itself a Gorenstein polytope, which is not true in general. Even if this is the case, the two polytopes might have completely different dimensions (since the one definition is relative to |$P$| while the other one is intrinsic).

Local |$h^{*}$| -polynomials of Gorenstein polytopes (often called |$\tilde{S}$| -polynomials) allow to give an elegant formula for computing stringy |$E$| -polynomials of Calabi–Yau complete intersections in toric Gorenstein Fano varieties (we refer to [ 7 , 13 ]). In this context, several questions about stringy |$E$| -polynomials are still open, see [ 7 , 42 ]. Here, we make some progress in this direction by addressing the question when the local |$h^{*}$| -polynomial of a Gorenstein polytope vanishes. As one consequence of our main result, Theorem 6.3 , we will see that not only the degree of the |$h^{*}$| -polynomials of Gorenstein polytopes and their duals are the same but also of their |$\ell ^{*}$| -polynomials (Corollary 6.13 ).

5.2 Joins, Cayley polytopes, and Cayley joins

In the sequel, let us discuss some important notions of decomposing lattice polytopes that turn up naturally when studying Gorenstein polytopes (we refer to [ 7 , 42 ]). Let us first introduce a formal notation for a lattice polytope being a join (as already defined in Subsection 3.3 ).

Let |$P \subseteq{\mathbb{R}}^{d}$| be a polytope and |$F, G$| non-empty subsets of |$P$|⁠ . Then |$P$| is the join of |$F$| and |$G$| , written |$P = F \circ G$|⁠ , if |$P = \operatorname{conv}(F,G)$| and |$\dim (P) = \dim (F) + \dim (G) + 1$|⁠ .

Equivalently, |$P$| is affinely-isomorphic to the free join |$F \circ _{\mathbb{Z}} G$|⁠ . In particular, |$F$| and |$G$| are automatically faces of |$P$|⁠ .

Note that the join property is associative. Namely, given faces |$F,G,H$| of |$P$|⁠ , then |$P=F \circ (G \circ H)$|⁠ , respectively, |$P=(F \circ G) \circ H$|⁠ , are both equivalent to |$P=\operatorname{conv}(F,G,H)$| and |$\dim (P) = \dim (F) + \dim (G) + \dim (H) + 2$|⁠ .

Let us also give the formal notation for a lattice polytope being a Cayley polytope. We recall that the notion of Cayley polytopes and Cayley sums was already shortly mentioned and defined in Subsection 3.2 . Here, we will solely focus on the case of two factors. Note that if a Cayley polytope has more than two factors, it is still a Cayley polytope with two factors.

Let |$P \subseteq{\mathbb{R}}^{d}$| be a lattice polytope and |$F, G$| non-empty subsets of |$P$|⁠ . Then |$P$| is the Cayley polytope of (factors) |$F$| and |$G$|⁠ , written |$P = F * G$|⁠ , if |$P = \operatorname{conv}(F,G)$| and there exists an affine-linear map |${\mathbb{R}}^{d} \to{\mathbb{R}}$| mapping |${\mathbb{Z}}^{d} \to{\mathbb{Z}}$|⁠ , such that |$F \mapsto 0$| and |$G \mapsto 1$|⁠ . In other words, |$P$| is a Cayley polytope if and only if there is a lattice projection mapping |$P$| onto |$\Delta _{1}$|⁠ .

If |$P=F*G$|⁠ , then |$F$| and |$G$| are necessarily faces of |$P$|⁠ . Cayley polytopes can also be characterized as lattice polytopes with lattice width one. Cayley sums are explicit descriptions of Cayley polytopes.

Given lattice polytopes |$F$| and |$G$| in |${\mathbb{R}}^{d}$|⁠ , the convex hull of |$F \times \{0\}$| and |$G \times \{1\}$| is called the Cayley sum of |$F$| and |$G$|⁠ . Its dimension is one larger than the dimension of the Minkowski sum of |$F$| and |$G$|⁠ . If |$P=F*G$|⁠ , then |$P$| is isomorphic to the Cayley sum of |$F$| and |$G$|⁠ .

Cayley sums are important in the construction of high-dimensional Gorenstein polytopes, see for example, [ 7 , Theorem 2.6]. Note that the degree of a Cayley polytope is at most the dimension of the Minkowski sum of its factors, see Proposition [ 6 , Proposition 1.12].

Let |$P \subseteq{\mathbb{R}}^{d}$| be a full-dimensional lattice polytope and |$F,G \subseteq P$| faces. Then |$P$| is the Cayley join of |$F$| and |$G$|⁠ , written |$P = F \circ _{\operatorname{Cay}} G$|⁠ , if |$P=F \circ G$| and |$P=F * G$|⁠ .

Clearly, the notion of a Cayley join is more restrictive than that of a Cayley polytope (e.g., |$[0,1]^{2}$| is a Cayley polytope of two edges but not a Cayley join). The reader should be aware that Cayley polytopes and Cayley joins are in general not associative in the sense of Remark 5.6 , see Example 5.17 below.

Let us recall some properties of a Gorenstein polytope that is a join or Cayley join, see [ 42 , Lemma 4.8, Proposition 4.9].

If |$P=F \circ G$|⁠ , then |$P^{\times } = F^{*} \circ G^{*}$| with |$F$| and |$G^{*}$| (respectively, |$G$| and |$F^{*}$|⁠ ) being combinatorially dual to each other.

If |$P=F \circ _{\operatorname{Cay}} G$|⁠ , then |$F^{\ast }$| is a Gorenstein polytope with dual Gorenstein polytope |$(F^{\ast })^{\times }$|⁠ , which can be identified with the lattice polytope |$G$| with respect to a refined lattice.

5.3 Gorenstein joins

In [ 42 ], the following notion was defined.

Let |$F$| and |$G$| be faces of a Gorenstein polytope |$P$|⁠ . We say |$P$| is a Gorenstein join of |$F$| and |$G$|⁠ , denoted by |$P = F \circ _{\operatorname{Gor}} G$|⁠ , if |$P = F \circ _{\operatorname{Cay}} G$| and |$P^{\times } = F^{*} \circ _{\operatorname{Cay}} G^{*}$|⁠ . We call |$F$| and |$G$| the factors of the Gorenstein join.

We remark that Gorenstein joins do not have to be free joins, see [ 42 , Example 4.14]. The following result, a strengthening of Stanleys monotonicity theorem in the case of faces of Gorenstein polytopes, motivated the previous definition of a Gorenstein join and gives a direct enumerative criterion for its existence ([ 42 , Theorems 3.6+4.12]).

Let |$P$| be a Gorenstein polytope and |$F$| a non-empty proper face of |$P$|⁠ . Then |$\operatorname{codeg}(P) \le \operatorname{codeg}(F) + \operatorname{codeg}(F^{*})$| (equivalently, |$\deg (F) + \deg (F^{*}) \le \deg (P)$|⁠ ), with equality if and only if |$P$| is a Gorenstein join with factor |$F$|⁠ . In this case, |$F$| is a Gorenstein polytope.

Gorenstein polytopes that are not Gorenstein joins have been previously also called irreducible in [ 42 ]. As we see from the following result, it is not necessary to compute the dual Gorenstein polytope to check whether a Cayley join is a Gorenstein join.

Let |$P = F \circ _{\operatorname{Cay}} G$| be a Gorenstein polytope that is the Cayley join of two faces |$F, G \leq P$|⁠ . Then |$P = F \circ _{\operatorname{Gor}} G$| if and only if |$\operatorname{codeg}(P) = \operatorname{codeg}(F) + \operatorname{codeg}(G)$| (or equivalently, |$\deg (P) = \deg (F)+\deg (G)$|⁠ ).

By Theorem 5.12 , |$P = F \ast _{\operatorname{Gor}} G$| if and only if |$\operatorname{codeg}(F) + \operatorname{codeg}(F^{\ast }) = \operatorname{codeg}(P) =: r$|⁠ , and in this case by [ 42 , Theorem 4.12] |$\operatorname{codeg}(G) = \operatorname{codeg}(P) - \operatorname{codeg}(F) = \operatorname{codeg}(F^{\ast })$|⁠ . Conversely, assume |$\operatorname{codeg}(G) = \operatorname{codeg}(P) - \operatorname{codeg}(F)$|⁠ . By Theorem 5.12 , the inequality |$\operatorname{codeg}(F) + \operatorname{codeg}(F^{\ast }) \geq r$| always holds, so that by Theorem 5.12 again we only need to prove |$\operatorname{codeg}(F^{\ast }) \leq \operatorname{codeg}(G)$|⁠ . As |$P$| is the Cayley join of |$F$| and |$G$|⁠ , Proposition 5.10 yields that |$\operatorname{codeg}(F^{\ast }) = \operatorname{codeg}((F^{\ast })^{\times }) \leq \operatorname{codeg}(G)$| as the codegree can only decrease under refinements of the lattice.

As we will need it for the upcoming proofs, let us recall how to characterize Gorenstein polytope via cones. For more details, we refer to [ 7 ]. The cone over |$P$| is denoted by |$C_{P} \subseteq{\mathbb{R}}^{d+1}$| spanned by |$P \times \{1\} \subset{\mathbb{R}}^{d+1}$|⁠ . Any polyhedral cone in |${\mathbb{R}}^{d+1}$| that is unimodularly equivalent to some |$C_{P}$| is called a Gorenstein cone . Now, |$P$| is a Gorenstein polytope if and only if the dual cone |$C_{P}^{\vee } = \{y \in ({\mathbb{R}}^{d+1})^{*}: \langle y, x \rangle \ge 0 \;\forall \, x \in C_{P}\}$| is a Gorenstein cone. In this case, |$C_{P}^{\vee }$| is unimodularly equivalent to the cone over |$P^{\times }$|⁠ .

The following proposition contains a positive result regarding associativity of Gorenstein joins. In general, however, we do not expect associativity to hold.

Let |$P \subseteq{\mathbb{R}}^{d}$| be a |$d$| -dimensional Gorenstein polytope with faces |$F, G, H \leq P$| such that |$P = (F \ast _{\operatorname{Gor}} G) \ast _{\operatorname{Gor}} H$|⁠ . If |$F$| is a vertex or |$H$| is a vertex, then |$P = F \ast _{\operatorname{Gor}} (G \ast _{\operatorname{Gor}} H)$|⁠ . In particular, Gorenstein joins are associative for |$\dim (P) \leq 3$|⁠ .

By Lemma 5.13 , the faces |$\operatorname{conv}(F,G)$| and |$H$| of |$P$| are themselves Gorenstein polytopes with |$r:= \operatorname{codeg}(P) = \operatorname{codeg}(\operatorname{conv}(F,G)) + \operatorname{codeg}(H)$|⁠ . Applying the same result to the Gorenstein polytope |$\operatorname{conv}(F,G) = F \ast _{\operatorname{Gor}} G$|⁠ , we obtain that |$F$| and |$G$| are Gorenstein polytopes with |$\operatorname{codeg}(\operatorname{conv}(F,G)) = \operatorname{codeg}(F) + \operatorname{codeg}(G)$|⁠ . Hence, |$r = \operatorname{codeg}(F) + \operatorname{codeg}(G) + \operatorname{codeg}(H)$|⁠ . By Lemma 5.13 , it hence suffices to show |$P = F \ast _{\operatorname{Cay}} (G \ast _{\operatorname{Cay}} H)$|⁠ . That the join of |$G$| and |$H$| is a Cayley join is immediate from the assumption, so it is enough to show that the join of |$F$| and |$\operatorname{conv}(G,H)$| is a Cayley join.

Let now |$H$| be a vertex, so |$P$| is a lattice pyramid with vertex |$H$| and base |$\operatorname{conv}(F,G)$|⁠ . As |$F \circ _{\operatorname{Cay}} G$|⁠ , we may assume that |$\textrm{lin}(F,G)={\mathbb{R}}^{d-1} \times \{0\}$|⁠ , |$H = \{e_{d}\}$|⁠ , and there exists some |$u \in ({\mathbb{Z}}^{d-1} \times \{0\})^{*}$| such that |$\langle u, F \rangle = 0$| and |$\langle u, G \rangle = 1$|⁠ . Now, |$\langle u + e^{*}_{d}, F \rangle = 0$| and |$\langle u + e^{*}_{d}, \operatorname{conv}(G,H) \rangle = 1$|⁠ . In particular, the join of |$F$| and |$\operatorname{conv}(G,H)$| is a Cayley join.

Finally, if |$d = \dim (P) \leq 3$| and |$P$| is the join of |$F$|⁠ , |$G$| and |$H$|⁠ , then necessarily at least one of |$F$| and |$H$| is a vertex for dimension reasons, concluding the proof.

We observe that a Gorenstein polytope |$P$| is a lattice pyramid over a face |$F$| with apex a vertex |$v$| of |$P$| if and only if |$P$| is a Gorenstein join of |$F$| and |$v$|⁠ . Hence, the previous result has the following consequence.

If |$P$| is a Gorenstein join of two faces with one face a lattice pyramid, then |$P$| is also a lattice pyramid.

This result is already contained in the master’s thesis [ 38 ]. Let us give an example that shows that it fails for Cayley joins.

Consider |$F:= \operatorname{conv}(e_{1},e_{2}) \times \{0\}$| and |$G:= \operatorname{conv}(0,-e_{1}-e_{2}) \times \{1\}$| in |${\mathbb{R}}^{3}$|⁠ . Then its convex hull |$P$| is a tetrahedron that is a Gorenstein polytope of lattice volume |$2$| (with |$h^{*}_{P}(t)=1+t^{2}$| and |$\ell ^{*}_{P}(t)=t^{2}$|⁠ ). It is a Cayley join |$P = F \circ _{\operatorname{Cay}} G$| but not a Gorenstein join as |$\deg (P)=2\not =0=\deg (F)+\deg (G)$|⁠ . Note that |$F$| and |$G$| are lattice pyramids, but |$P$| is not. In particular, this example shows that the Cayley join property is not associative, and moreover, a Cayley join does not have to be thin if a factor of the Cayley join is thin.

5.4 Local |$h^{*}$| -polynomials of joins

Recall that |$h^{\ast }$| - and |$\ell ^{\ast }$| -polynomials are multiplicative with respect to free joins (Proposition 3.14 ). For general joins, one still gets inequalities.

Lemma 5.18. Let |$P$| be a lattice polytope, which is the join of two faces |$F$| and |$G$|⁠ . Then $$\begin{align*}& \ell^{\ast}_{F}(t) \cdot \ell^{\ast}_{G}(t) \leq \ell^{\ast}_{P}(t) \ \ \textrm{and}\ \ h^{\ast}_{F}(t) \cdot h^{\ast}_{G}(t) \leq h^{\ast}_{P}(t). \end{align*}$$ If moreover |$P$| is a Gorenstein polytope that is the Gorenstein join of |$F$| and |$G$|⁠ , then also $$\begin{align*}& \ell^{\ast}_{P^{\times}}(t) \leq \ell^{\ast}_{F^{\times}}(t) \cdot \ell^{\ast}_{G^{\times}}(t) \ \ \textrm{and}\ \ h^{\ast}_{P^{\times}}(t) \leq h^{\ast}_{F^{\times}}(t) \cdot h^{\ast}_{G^{\times}}(t). \end{align*}$$

We will use the notation of [ 42 ]. Let |$\overline{M} = {\mathbb{Z}}^{d+1}$|⁠ , and |$M(F)$| denote the sublattice of |$\overline{M}$| spanned by the lattice points in the linear hull of |$F \times \{1\}$|⁠ . Relative to the sublattice |$M(F) \oplus _{\mathbb{Z}} M(G)$| of |$\overline{M}$|⁠ , |$P$| becomes the free join of |$F$| and |$G$|⁠ . Recall that by Corollary 2.21 both the |$h^{\ast }$| -polynomial and the |$\ell ^{\ast }$| -polynomial are (weakly) monotonically increasing under refinements of the lattice. It hence follows from Proposition 3.14 that |$\ell ^{\ast }_{F}(t) \cdot \ell ^{\ast }_{G}(t) \leq \ell ^{\ast }_{P}(t)$| and |$h^{\ast }_{F}(t) \cdot h^{\ast }_{G}(t) \leq h^{\ast }_{P}(t)$|⁠ .

For the second claim, assume that |$P \subseteq{\mathbb{R}}^{d}$| is a full-dimensional Gorenstein polytope of codegree |$r$| with respect to the lattice |$M = {\mathbb{Z}}^{d} \subseteq{\mathbb{R}}^{d}$|⁠ . By assumption, |$P$| is the Gorenstein join of two faces |$F$| and |$G$|⁠ . By Theorem 5.12 and Lemma 5.13 , |$F$| and |$G$| are Gorenstein polytopes and |$\operatorname{codeg}(F) + \operatorname{codeg}(G) = r$|⁠ . For |$\overline{M} = {\mathbb{Z}}^{d+1}$|⁠ , we define the dual lattice |$\overline{N} := \textrm{Hom}_{\mathbb{Z}}(\overline{M},{\mathbb{Z}}) \subseteq ({\mathbb{R}}^{d+1})^{*}$|⁠ . By definition, as |$P$| is a Gorenstein polytope, |$C_{P}^{\vee }$| is a Gorenstein cone with respect to |$\overline{N}$|⁠ . Let |$n = e_{d+1}^{\ast } \in \overline{N}$| be the unique interior lattice point of |$C_{P}^{\vee }$| with |$P \times \{1\} = C_{P} \cap \{x \in{\mathbb{R}}^{d+1}: \langle n, x \rangle = 1\}$|⁠ . In the same way, we denote by |$m \in \overline{M}$| the unique interior lattice point of |$C_{P}$| such that |$P^{\times } = C_{P}^{\vee } \cap \{y \in ({\mathbb{R}}^{d+1})^{*}: \langle y, m \rangle = 1\}$|⁠ . Recall that |$\langle n, m \rangle = r$| and hence |$m = (p,r) \in M \oplus \mathbb{Z} = \overline{M}$| with |$p \in M$| the unique interior lattice point of the |$r$| -th dilate |$rP$| of |$P$|⁠ .

Now, let us consider the sublattice |$M(F) \oplus M(G) \subseteq \overline{M}$|⁠ . With respect to this coarser lattice, the polytope |$P$| is the free join of |$F$| and |$G$|⁠ , and this is clearly a Gorenstein polytope of codegree |$\operatorname{codeg}(F) + \operatorname{codeg}(G) = r$|⁠ . Hence, the |$r$| -th dilate |$rP$| contains a unique interior lattice point in the original as well as in the coarser lattice. These two points must therefore agree, so the unique interior lattice point |$m = (p,r) \in (rP) \times \{r\} \subseteq C_{P}$| with respect to the original lattice actually lies in |$M(F) \oplus M(G)$|⁠ . Let now |$\tilde{N} \subset ({\mathbb{R}}^{d+1})^{*}$| be the dual lattice of |$M(F) \oplus M(G)$|⁠ . Hence, |$P^{\times }$| is with respect to the finer lattice |$\tilde{N}$| the dual Gorenstein polytope of the Gorenstein polytope |$P$| considered with respect to the coarser lattice |$M(F) \oplus M(G)$|⁠ . By Proposition 5.10 , the Gorenstein dual of the free join of the Gorenstein polytopes |$F$| and |$G$|⁠ , is the free join of the Gorenstein duals |$F^{\times }$| and |$G^{\times }$|⁠ . Again, monotonicity and multiplicativity proves the second claim: |$\ell ^{\ast }_{P^{\times }}(t) \leq \ell ^{\ast }_{F^{\times }}(t) \cdot \ell ^{\ast }_{G^{\times }}(t)$| and |$h^{\ast }_{P^{\times }}(t) \leq h^{\ast }_{F^{\times }}(t) \cdot h^{\ast }_{G^{\times }}(t)$|⁠ , where |$P^{\times }$| is considered with respect to the original lattice |$\overline{N}$| again.

It follows in the situation of the second part of Lemma 5.18 from Proposition 5.10 and Remark 3.8 that |$\ell ^{\ast }_{F}(t) \leq \ell ^{\ast }_{(G^{\ast })^{\times }}(t)$| and |$\ell ^{\ast }_{G}(t) \leq \ell ^{\ast }_{(F^{\ast })^{\times }}(t)$|⁠ , because |$(G^{\ast })^{\times }$| is just the polytope |$F$| with a possibly finer lattice, and analogously for |$(F^{\ast })^{\times }$| and |$G$|⁠ . The same holds for the |$h^{\ast }$| -polynomial.

6.1 The main result

The following notion will occur naturally in the proof of Theorem 6.3 .

A lattice polytope |$P$| is called |$g$| -thin if |$\deg (g_{P}) = \deg (P)$|⁠ .

Let |$P$| denote the lattice pyramid over |$[-1,1]$|⁠ . Then |$P$| has dimension |$2$|⁠ , degree |$1$|⁠ , and |$\deg (g_{P})=0$| as it is a simplex. This is an example of a trivially thin Gorenstein polytope that is not |$g$| -thin. For another example, consider |$2 \Delta _{2}$|⁠ , which is a trivially thin (non-Gorenstein) simplex that is not |$g$| -thin. This example shows that a spanning thin polytope that is not a free join does not have to be |$g$| -thin (i.e., in Question 3.16 “trivially thin” cannot be strengthened by “g-thin”).

Here is our main result.

|$P$| is thin,

|$P$| is trivially thin or |$P = F \ast _{\operatorname{Gor}} G$| with at least one factor trivially thin,

|$P$| is |$g$| -thin or |$P = F \ast _{\operatorname{Gor}} G$| with |$\deg (\ell ^{*}_{F}) = \deg (F)$| and |$G$| |$g$| -thin.

Let us remark that if |$P$| is not thin, the last statement implies that |$\ell ^{*}_{P}(t)$| and |$h^{*}_{P}(t)$| have the same leading coefficient |$1$|⁠ , as |$h^{*}_{P}(t)$| is palindromic with constant coefficient |$1$|⁠ .

We have to leave it as an open question whether it is possible to strengthen in the previous result ‘Gorenstein join’ to ‘free join’. Let us note the following situation in which being a Gorenstein join (or even just a spanning join of faces) automatically implies being a free join.

Let |$P$| be a spanning Gorenstein polytope. Then |$P$| is thin if and only if it is trivially thin or a free join with a trivially thin factor (necessarily also a spanning Gorenstein polytope).

The proof of Theorem 6.3 relies critically on the decomposition of the |$h^{*}$| -polynomial into |$\ell ^{*}$| -polynomials and |$g$| -polynomials (Corollary 2.18 ), valid also for general lattice polytopes.

Let |$P$| be a lattice polytope with |$\deg (\ell ^{*}_{P}) < \deg (P)$|⁠ . Then |$P$| is |$g$| -thin or there exists a non-empty, proper face |$F$| of |$P$| with |$\deg (P)=\deg (\ell ^{*}_{F})+\deg (g_{[F,P)})$|⁠ .

Here, we recall |$\deg (\ell ^{*}_{\emptyset })=\deg (1)=0$| and |$\deg (g^{*}_{\emptyset })=\deg (1)=0$|⁠ .

By the nonnegativity of |$\ell ^{*}$| - and |$g$| -polynomials, Corollary 2.18 implies that there exists a face |$F$| of |$P$| with |$\deg (\ell ^{*}_{F})+\deg (g_{[F,P)}) = \deg (h^{*}_{P})$|⁠ . By our assumption, |$F\not =P$|⁠ . If |$F=\emptyset $|⁠ , then |$\deg (h^{*}_{P}) = \deg (g_{[\emptyset ,P)})$|⁠ , so |$P$| is |$g$| -thin.

Let us note the following observation:

Let |$P = F \ast _{\operatorname{Gor}} G$|⁠ . Then the Gorenstein polytopes |$F$|⁠ , |$G^{*}$|⁠ , |$F^{\times }$|⁠ , |$(G^{*})^{\times }$| have the same degree, dimension, and degree of their |$g$| -polynomials. In particular, if any of these Gorenstein polytopes are trivially thin (respectively, |$g$| -thin), then all of them are.

Proof. The Gorenstein property follows from Proposition 5.10 . It is well-known, cf. [ 7 ], that duality of Gorenstein polytopes keeps dimension and degree invariant. It follows from Theorem 2.9 that this is also true for the degree of the |$g$| -polynomial. By Proposition 5.10 , |$F$| and |$G^{*}$| are combinatorially dual to each other, hence, have the same dimension and by Theorem 2.9 the same degree of the |$g$| -polynomial. Finally, by Lemma 5.13 and Theorem 5.12 , $$\begin{align*} &\deg(P) - \deg(F) = \deg(G) = \deg(P) - \deg(G^{*}),\end{align*}$$ hence, |$\deg (F)=\deg (G^{*})$|⁠ .

The implication 3 |$\Rightarrow $| 2 is immediate.

2 |$\Rightarrow $| 1 : Let |$P = F \ast _{\operatorname{Gor}} G$| with |$F$| trivially thin. By Lemma 6.6 , it follows that |$(G^{\ast })^{\times }$| is trivially thin as well. Now, applying Lemma 5.18 to the factorization |$P^{\times } = F^{\ast } \ast _{\operatorname{Gor}} G^{\ast }$| yields |$\ell ^{\ast }_{P}(t) \leq \ell ^{\ast }_{(F^{\ast })^{\times }}(t) \cdot \ell ^{\ast }_{(G^{\ast })^{\times }}(t) = 0$|⁠ , so |$P$| is thin.

Every thin Gorenstein polytope (of dimension |$>0$|⁠ ) has lattice width |$1$|⁠ .

As Gorenstein joins are Cayley joins, it remains by Theorem 6.3 to show that a trivially thin Gorenstein polytope of dimension |$>0$| is a Cayley polytope. This is precisely the statement of Theorem 3.1 in [ 26 ].

Let us illustrate Theorem 6.3 by showing that all thin Gorenstein polytopes |$P$| of dimension |$d=3$| are lattice pyramids over Gorenstein polygons (without using Theorem 4.3 directly). Let us assume otherwise. If |$P$| is trivially thin, then |$\deg (P) \le 1$|⁠ , so by Theorem 4.2 |$P$| is a Lawrence prism. Palindromicity implies |$h^{*}_{P}(t)=1 + t$|⁠ , so lattice volume |$2$|⁠ , which is a contradiction because any three-dimensional Lawrence prism has at least lattice volume |$3$|⁠ . Hence, by Theorem 6.3 , |$P$| must be a lattice pyramid or a Gorenstein join of two Gorenstein intervals one of them being thin. As a thin interval is a unimodular simplex, thus a lattice pyramid, also |$P$| is a lattice pyramid by Corollary 5.16 .

6.2 Borisov’s proof of the degree of |$\ell ^{*}$| -polynomials of non-thin Gorenstein polytopes

Theorem 6.3 answers affirmatively Question 6.3(b) in [ 42 ] asking whether for Gorenstein polytopes having a non-vanishing |$\ell ^{*}$| -polynomial forces its degree to be maximal (i.e., equal to the degree of the |$h^{*}$| -polynomial). Lev Borisov has provided us with an alternative algebraic proof of this fact that we reproduce here. It uses the description of the local |$h^{*}$| -polynomial of a lattice polytope as a Hilbert series of a graded ideal given in [ 13 ].

Let |$P \subseteq{\mathbb{R}}^{d}$| be a Gorenstein polytope of codegree |$r$|⁠ . Then either |$P$| is thin or |$\ell ^{\ast }_{P}(t)$| starts with |$t^{r}$|⁠ . In this case, |$\ell ^{\ast }_{P}(t)$| has degree |$\deg (P)$| and leading coefficient |$1$|⁠ .

Let |$K \subseteq{\mathbb{Z}}^{d+1}$| be the lattice points in the Gorenstein cone over |$P \times \{1\}$|⁠ . Denote by |${\mathbb{C}}[K]$| the associated affine semi-group algebra with |$\mathbb{N}_{0}$| -grading given by the exponent of |$x_{d+1}$|⁠ , viewing |${\mathbb{C}}[K] \subseteq{\mathbb{C}}[x_{1}^{\pm 1}, \ldots , x_{d}^{\pm 1}, x_{d+1}]$|⁠ . As in [ 13 , Section 4], we let |$f \in{\mathbb{C}}[K]_{1}$| be non-degenerate and |$I \subseteq{\mathbb{C}}[K]$| the homogeneous ideal generated by the so-called logarithmic derivatives of |$f$|⁠ . Let moreover |$J \subseteq{\mathbb{C}}[K]$| be the homogeneous ideal generated by all lattice points in the relative interior |$K^{\circ }$| of |$K$|⁠ . Then Borisov and Mavlyutov define |$R_{1}(f,K)$| to be the image of |$J$| in the quotient ring |${\mathbb{C}}[K]/I$|⁠ , that is, |$R_{1}(f,K)$| is the homogeneous ideal |$(I+J)/I$| of |${\mathbb{C}}[K]/I$|⁠ . Now, by [ 13 , Proposition 5.5], |$\ell ^{\ast }_{P}(t)$| is the Hilbert series of |$R_{1}(f,K)$|⁠ . Moreover, as |$P$| is Gorenstein of codegree |$r$|⁠ , we have |$K^{\circ } = (p,r) + K$|⁠ , where |$p \in (rP) \cap{\mathbb{Z}}^{d}$| is the unique interior lattice point of |$rP$|⁠ . Therefore, |$R_{1}(f,K)$| is just the image of the principal ideal |$(x^{p} x_{d+1}^{r})$| in the quotient |${\mathbb{C}}[K]/I$|⁠ . Hence, |$R_{1}(f,K)$| is |$0$| if and only if |$x^{p} x_{d+1}^{r} \in I$|⁠ , and otherwise the lowest degree of its non-zero homogeneous components is |$r$| with |$R_{1}(f,K)_{r} = \langle x^{p} x_{d+1}^{r} \rangle $| of vector space dimension |$1$|⁠ . This proves the first claim, and the second follows from reciprocity, Theorem 2.16 (2).

In dimensions |$\leq 4$|⁠ , it is a consequence of the reciprocity of |$\ell ^{\ast }_{P}(t)$| that for any lattice polytope |$P$| either |$P$| is thin or |$\deg (\ell ^{\ast }_{P}) = \deg (P)$|⁠ . In higher dimensions, this property fails for non-Gorenstein lattice polytopes.

Consider the full-dimensional lattice simplex |$P \subseteq{\mathbb{R}}^{5}$| given as the convex hull |$P = \operatorname{conv}(0, e_{1}, e_{2}, e_{3}, (0,1,1,2,0), (5,3,3,2,6))$|⁠ . Then |$\ell _{P}^{\ast }(t) = 4t^{3}$| while |$h^{\ast }_{P}(t) = t^{4} + 5t^{3} + 4t^{2} + t + 1$|⁠ . In particular, |$P$| is not thin but |$\deg (\ell ^{\ast }_{P}) < \deg (h^{\ast }_{P}) = \deg (P)$|⁠ . This is the only such example among lattice simplices of dimension |$5$| with lattice volume |$\leq 15$|⁠ . It was found using the database [ 4 ]. The computations were performed in SageMath with backend Normaliz .

Consider the full-dimensional lattice simplex |$P \subseteq{\mathbb{R}}^{5}$| given as the convex hull |$P = \operatorname{conv}(0, e_{1}, e_{2}, (1, 1, 2, 0, 0), (3, 5, 6, 8, 0), (1, 1, 0, 0, 2))$|⁠ . Then |$\ell _{P}^{\ast }(t) = t^{3}$| while |$h^{\ast }_{P}(t) = 7t^{3} + 19t^{2} + 5t + 1$|⁠ . So |$\deg (\ell _{P}^{\ast }) = \deg (h_{P}^{\ast })$| but the leading coefficient of |$\ell _{P}^{\ast }$| is strictly smaller.

6.3 Thinness is invariant under duality

It was noted in Lemma 6.6 that being trivially thin as well as being |$g$| -thin is invariant under duality of Gorenstein polytopes. Let us explain how this allows us to deduce that also thinness has this beautiful duality property:

Let |$P$| be a Gorenstein polytope. Then |$P$| is thin if and only if |$P^{\times }$| is thin.

By Theorem 6.3 (ii), we may assume that |$P$| is a thin Gorenstein polytope such that |$P$| is a Gorenstein join of faces |$F$| and |$G$| with |$F$| being trivially thin. Hence, by Lemma 6.6 , we also have |$P^{\times } = F^{*} \ast _{\operatorname{Gor}} G^{*}$| with |$G^{*}$| being trivially thin. Again, by Theorem 6.3 , this implies that |$P^{\times }$| is thin.

Having such a direct proof answers a question of Lev Borisov, who communicated to us that this statement might also be proven using vertex algebra techniques.

In particular, as Theorem 6.3 implies that there are only two choices for the degree of the |$\ell ^{*}$| -polynomial of a Gorenstein polytope, we see that its degree is also invariant under duality (as it holds for the degrees of the |$h^{*}$| -polynomial and the |$g$| -polynomial).

Let |$P$| be a Gorenstein polytope. Then |$\deg (\ell ^{*}_{P}) = \deg (\ell ^{*}_{P^{\times }})$|⁠ .

The reader should be aware that the local |$h^{*}$| -polynomials of a Gorenstein polytope |$P$| and its dual |$P^{\times }$| may differ. For instance, for |$P=[-1,1]^{3}$|⁠ , we have |$\ell ^{*}_{P}(t) = t + 17 t^{2} + t^{3}$| and |$\ell ^{*}_{P^{\times }}(t)=t + 3 t^{2} + t^{3}$|⁠ .

6.4 Thin Gorenstein simplices

For the special case of Gorenstein simplices, we can answer the original question in [ 24 ] about classifying thin simplices.

Let |$P$| be a Gorenstein simplex. Then |$P$| is thin if and only if |$P$| is a lattice pyramid.

Let |$P$| be thin. If |$P$| is |$g$| -thin, then |$\deg (P)=\deg (g_{P})=0$| as |$P$| is a simplex. Hence, |$P$| is a unimodular simplex, in particular, a lattice pyramid.

Otherwise, Theorem 6.3 (iii) implies that there are faces |$F$| and |$G$| of |$P$| such that |$P=F \ast _{\operatorname{Gor}} G$| with |$G$| |$g$| -thin. As |$G$| is also a simplex, the previous consideration shows that |$G$| is a unimodular simplex, thus, a lattice pyramid. Hence, Corollary 5.16 implies that |$P$| is also a lattice pyramid.

This work was supported by the Research Training Group Mathematical Complexity Reduction funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [314838170, GRK 2297 MathCoRe].

Our deepest thanks go to Jan Schepers whose notes and insights during and after the collaboration with the third author on [ 42 ] contained in particular Lemma 6.5 and were the basis of the proof of Theorem 6.3 . We thank Lev Borisov for his comments, questions, and his kindness to share the proof of Proposition 6.9 . We are also grateful for Sam Payne and Liam Solus for their interest. We thank the anonymous referees for their careful reading and very useful suggestions.

Communicated by Prof. Dan-Virgil Voiculescu

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RESEARCH: MEANING, TYPES, SCOPE AND SIGNIFICANCE Structure 1.1 Introduction 1.2 Objectives 1.3 Meaning of Research 1.4 Definition of Research 1.5 Characteristics of Research 1.6 Types of Research 1.7 Methodology of Research 1.8 Formulation of Research Problem 1.9 Research Design 1.9.1 Meaning of Research Design
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Overview of the structure. To help guide your reader, end your introduction with an outline of the structure of the thesis or dissertation to follow. Share a brief summary of each chapter, clearly showing how each contributes to your central aims. However, be careful to keep this overview concise: 1-2 sentences should be enough.
What Is Research Methodology? Definition + Examples
As we mentioned, research methodology refers to the collection of practical decisions regarding what data you'll collect, from who, how you'll collect it and how you'll analyse it. Research design, on the other hand, is more about the overall strategy you'll adopt in your study. For example, whether you'll use an experimental design ...
Introduction to Education Research
Research is defined by the Oxford English Dictionary as "the systematic investigation into and study of materials and sources in order to establish facts and reach new conclusions." ... 'The research compass': an introduction to research in medical education: AMEE Guide no. 56. Med Teach. 2011;33(9):695-709. Article PubMed Google ...
The importance of crafting a good introduction to scholarly research
These definitions can also then be used to establish the methods and criteria by which the variables of the study will subsequently be measured or altered. ... Whilst crafting a research introduction may seem a challenging and time-consuming task, it is well worth the effort to convey your research clearly and engage potential readers. ...
How to Write a Research Paper Introduction (with Examples)
Define your specific research problem and problem statement. Highlight the novelty and contributions of the study. Give an overview of the paper's structure. The research paper introduction can vary in size and structure depending on whether your paper presents the results of original empirical research or is a review paper.
PDF UNIT 1 INTRODUCTION TO RESEARCH: Purpose, Nature and Scope ...
1.5.1 Exploratory or Formulative Research. Studies with a purpose of gaining familiarity with a phenomenon or to achieve new insights into it, often in order to formulate a more precise research problem or to develop hypotheses are known as Exploratory or Formulative research studies.
Guide on How to Write a Research Paper Introduction
Research Paper Introduction - Definition The research paper introduction arrests the reader's attention from a general perspective to one specific area of a study. It outlines a summary of the research being conducted by condensing current understanding and background information about the topic, presenting the importance of the research in ...
The evolution of Big Data in neuroscience and neurology
The field of Neuroscience was formalized in 1965 when the "Neuroscience Research Program" was established at the Massachusetts Institute of Technology with the objective of bringing together several varied disciplines including molecular biology, biophysics, and psychology to study the complexity of brain and behavior [].The methods employed by the group were largely data driven, with a ...
Thin Polytopes: Lattice Polytopes With Vanishing Local h*-Polynomial
Typical examples of trivially thin polytopes are Cayley polytopes with many factors. We will talk about Cayley polytopes with two factors in much more detail later (see Definition 5.7 and Remark 5.8); however, let us already now give the

What is Research: Definition, Methods, Types & Examples

What is Research?

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Research Paper Introduction

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What Is Research Methodology? A Plain-Language Explanation & Definition (With Examples)

Research Methodology 101

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Introduction to Education Research

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