research papers in mathematics

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Journals High quality journals covering a broad range of mathematical disciplines.

Current issue · All issues

Notices of the American Mathematical Society ISSN 1088-9477 (online) ISSN 0002-9920 (print) MCQ: 0.45

Bulletin of the American Mathematical Society ISSN 1088-9485 (online) ISSN 0273-0979 (print) MCQ: 0.47

All issues : 2009 - Present

Abstracts of Papers Presented to the American Mathematical Society ISSN 2689-4831 (online) ISSN 0192-5857 (print) MCQ: 0.00

MCQ Info The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period.

Current volume · All volumes

Communications of the American Mathematical Society ISSN 2692-3688 MCQ: 0.47

Journal of the American Mathematical Society ISSN 1088-6834 (online) ISSN 0894-0347 (print) MCQ: 4.79

Representation Theory ISSN 1088-4165 MCQ: 0.7

Proceedings of the American Mathematical Society ISSN 1088-6826 (online) ISSN 0002-9939 (print) MCQ: 0.85

Proceedings of the American Mathematical Society Series B ISSN 2330-1511 MCQ: 0.84

Mathematics of Computation ISSN 1088-6842 (online) ISSN 0025-5718 (print) MCQ: 1.98

Conformal Geometry and Dynamics ISSN 1088-4173 MCQ: 0.5

Memoirs Home

Memoirs of the American Mathematical Society ISSN 1947-6221 (online) ISSN 0065-9266 (print) MCQ: 0.51

Transactions of the American Mathematical Society ISSN 1088-6850 (online) ISSN 0002-9947 (print) MCQ: 1.43

Transactions of the American Mathematical Society Series B ISSN 2330-0000 MCQ: 1.79

All volumes

Electronic Research Announcements ISSN 1079-6762 MCQ: 0.00

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St. Petersburg Mathematical Journal ISSN 1547-7371 (online) ISSN 1061-0022 (print) MCQ: 0.54

Transactions of the Moscow Mathematical Society ISSN 1547-738X (online) ISSN 0077-1554 (print) MCQ: 0.51

Sugaku Expositions ISSN 2473-585X (online) ISSN 0898-9583 (print) MCQ: 0.10

Annales Scientifiques de l'École Normale Supérieure ISSN: 1088-4173 MCQ: 2.09

Astérisque ISSN: 0303-1179 MCQ: 0.45

Bulletin de la Société Mathématique de France ISSN 0037-9484 MCQ: 0.70

Theory of Probability and Mathematical Statistics ISSN 1547-7363 (online) ISSN 0094-9000 (print) MCQ: 0.12

Journal of Algebraic Geometry ISSN 1534-7486 (online) ISSN 1056-3911 (Print) MCQ: 1.37

ISSN 0379-4024 (print) MCQ: 0.60

Quarterly of Applied Mathematics ISSN 1552-4485 (online) ISSN 0033-569X (print) MCQ: 0.60

Moscow Mathematical Journal 1609-3321 (print) MCQ: 0.61

Mémoires de la Société Mathématique de France ISSN 0249-633X MCQ: 1.83

Journal of the Ramanujan Mathematical Society ISSN 0970-1249 MCQ: 0.24

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Home > Mathematics > MATHSTUDENT

Mathematics, Department of

Department of mathematics: dissertations, theses, and student research.

Prefix-Rewriting: The Falsification by Fellow Traveler Property and Practical Computation , Ash DeClerk

Positioning Undergraduate Learning Assistants in Instruction: A Case Study of the LA Role in Active Learning Mathematics Classrooms at the University of Nebraska-Lincoln , Rachel Funk

Classroom Social Support: A Multiple Phenomenological Case Study of Mathematics Graduate Teaching Assistants’ Decision Making in the Classroom , Brittany Johnson

On the Superabundance of Singular Varieties in Positive Characteristic , Jake Kettinger

Intrinsic Tame Filling Functions and Other Refinements of Diameter Functions , Andrew Quaisley

Partitions of R^n with Maximal Seclusion and their Applications to Reproducible Computation , Jason Vander Woude

Gordian Distance and Complete Alexander Neighbors , Ana Wright

Extremal Problems in Graph Saturation and Covering , Adam Volk

Free Semigroupoid Algebras from Categories of Paths , Juliana Bukoski

Frobenius and Homological Dimensions of Complexes , Taran Funk

N-Fold Matrix Factorizations , Eric Hopkins

Free Complexes over the Exterior Algebra with Small Homology , Erica Hopkins

Gauge-Invariant Uniqueness and Reductions of Ordered Groups , Robert Huben

Results on Nonorientable Surfaces for Knots and 2-knots , Vincent Longo

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids , George Nasr

Bootstrap Percolation on Random Geometric Graphs , Alyssa Whittemore

Hadamard Well-Posedness for two Nonlinear Structure Acoustic Models , Andrew Becklin

Optimal Allocation of Two Resources in Annual Plants , David McMorris

Spectral Properties of a Non-Compact Operator in Ecology , Matthew Reichenbach

Exploring Pedagogical Empathy of Mathematics Graduate Student Instructors , Karina Uhing

Trisections of Flat Surface Bundles over Surfaces , Marla Williams

Operator algebras generated by left invertibles , Derek DeSantis

Admissibility of C*-Covers and Crossed Products of Operator Algebras , Mitchell A. Hamidi

Unbounded Derivations of C*-algebras and the Heisenberg Commutation Relation , Lara M. Ismert

Individual Based Model to Simulate the Evolution of Insecticide Resistance , William B. Jamieson

The Derived Category of a Locally Complete Intersection Ring , Joshua Pollitz

Sequential Differences in Nabla Fractional Calculus , Ariel Setniker

The T 3 , T 4 -conjecture for links , Katie Tucker

Design and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks , Allison Beemer

Graphs with few spanning substructures , Jessica De Silva

Fractional Difference Operators and Related Boundary Value Problems , Scott C. Gensler

Green's Functions and Lyapunov Inequalities for Nabla Caputo Boundary Value Problems , Areeba Ikram

On the well-posedness and global boundary controllability of a nonlinear beam model , Jessie Jamieson

Resolutions of Finite Length Modules over Complete Intersections , Seth Lindokken

On Coding for Partial Erasure Channels , Carolyn Mayer

High Cognitive Demand Examples in Precalculus: Examining the Work and Knowledge Entailed in Enactment , Erica R. Miller

A Tensor's Torsion , Neil Steinburg

Properties and Convergence of State-based Laplacians , Kelsey Wells

The Existence of Solutions for a Nonlinear, Fractional Self-Adjoint Difference Equation , Kevin Ahrendt

Ideal Containments under Flat Extensions and Interpolation on Linear Systems in P 2 , Solomon Akesseh

Stable Cohomology Of Local Rings And Castelnuovo-Mumford Regularity Of Graded Modules , Luigi Ferraro

Languages, geodesics, and HNN extensions , Maranda Franke

Antichains and Diameters of Set Systems , Brent McKain

Rigidity of the Frobenius, Matlis Reflexivity, and Minimal Flat Resolutions , Douglas J. Dailey

Management of Invasive Species using Optimal Control Theory , Christina J. Edholm

Cohen-Macaulay Dimension for Coherent Rings , Rebecca Egg

Adian inverse semigroups , Muhammad Inam

Homological characterizations of quasi-complete intersections , Jason M. Lutz

Bridge spectra of cables of 2-bridge knots , Nicholas John Owad

Applications of Discrete Mathematics for Understanding Dynamics of Synapses and Networks in Neuroscience , Caitlyn Parmelee

A Caputo Boundary Value Problem in Nabla Fractional Calculus , Julia St. Goar

Stable local cohomology and cosupport , Peder Thompson

Graph centers, hypergraph degree sequences, and induced-saturation , Sarah Lynne Behrens

Knörrer Periodicity and Bott Periodicity , Michael K. Brown

The Strict Higher Grothendieck Integral , Scott W. Dyer

Invariant Basis Number and Basis Types for C*-Algebras , Philip M. Gipson

Bioinformatic Game Theory and Its Application to Cluster Multi-domain Proteins , Brittney Keel

Extremal Results for the Number of Matchings and Independent Sets , Lauren Keough

Crosscap Number: Handcuff Graphs and Unknotting Number , Anne Kerian

Tame Filling Functions and Closure Properties , Anisah Nu'Man

Analysis of Neuronal Sequences Using Pairwise Biases , Zachary Roth

Systems of parameters and the Cohen-Macaulay property , Katharine Shultis

Local and Nonlocal Models in Thin-Plate and Bridge Dynamics , Jeremy Trageser

Betti sequences over local rings and connected sums of Gorenstein rings , Zheng Yang

Boundary Value Problems of Nabla Fractional Difference Equations , Abigail M. Brackins

Results on edge-colored graphs and pancyclicity , James Carraher

An Applied Functional and Numerical Analysis of a 3-D Fluid-Structure Interactive PDE , Thomas J. Clark

Algebraic Properties of Ext-Modules over Complete Intersections , Jason Hardin

Combinatorial and Algebraic Coding Techniques for Flash Memory Storage , Kathryn A. Haymaker

Well-posedness and stability of a semilinear Mindlin-Timoshenko plate model , Pei Pei

The Neural Ring: Using Algebraic Geometry to Analyze Neural Codes , Nora Youngs

Development and Application of Difference and Fractional Calculus on Discrete Time Scales , Tanner J. Auch

BOUNDARY VALUE PROBLEMS FOR DISCRETE FRACTIONAL EQUATIONS , Pushp R. Awasthi

Embedding and Nonembedding Results for R. Thompson's Group V and Related Groups , Nathan Corwin

Periodic modules over Gorenstein local rings , Amanda Croll

Results on Containments and Resurgences, with a Focus on Ideals of Points in the Plane , Annika Denkert

REGULARITY FOR SOLUTIONS TO PARABOLIC SYSTEMS AND NONLOCAL MINIMIZATION PROBLEMS , Joe Geisbauer

Decompositions of Betti Diagrams , Courtney Gibbons

Symbolic Powers of Ideals in k [ P N ] , Michael Janssen

Closure and homological properties of (auto)stackable groups , Ashley Johnson

Random search models of foraging behavior: theory, simulation, and observation. , Ben C. Nolting

Geometric Study of the Category of Matrix Factorizations , Xuan Yu

BOUNDARY VALUE PROBLEMS FOR DISCRETE FRACTIONAL EQUATIONS , Khulud Alyousef

PRIME IDEALS IN TWO-DIMENSIONAL NOETHERIAN DOMAINS AND FIBER PRODUCTS AND CONNECTED SUMS , Ela Celikbas

Modeling and Mathematical Analysis of Plant Models in Ecology , Eric A. Eager

An Analysis of Nonlocal Boundary Value Problems of Fractional and Integer Order , Christopher Steven Goodrich

SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SUPERCRITICAL SOURCES , Yanqiu Guo

Commutative Rings Graded by Abelian Groups , Brian P. Johnson

The Weak Discrepancy and Linear Extension Diameter of Grids and Other Posets , Katherine Victoria Johnson

Combinatorics Using Computational Methods , Derrick Stolee

On the Betti Number of Differential Modules , Justin DeVries

On Morrey Spaces in the Calculus of Variations , Kyle Fey

Formalizing Categorical and Algebraic Constructions in Operator Theory , William Benjamin Grilliette

The Theory of Discrete Fractional Calculus: Development and Application , Michael T. Holm

Covariant Representations of C*-dynamical systems Involving Compact Groups , Firuz Kamalov

Homology of Artinian Modules Over Commutative Noetherian Rings , Micah J. Leamer

Annihilators of Local Cohomology Modules , Laura Lynch

Groups and Semigroups Generated by Automata , David McCune

Hilbert-Samuel and Hilbert-Kunz Functions of Zero-Dimensional Ideals , Lori A. McDonnell

On a Family of Generalized Wiener Spaces and Applications , Ian Pierce

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How to Effectively Write a Mathematics Research Paper

Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition .

Structure and Style

Formal Exposition

The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.

Informal Exposition

Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper . Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.

Conventions of Mathematics

Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.

Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.

Theorems and Equations

Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.

Math Symbols and Notations

Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources .

Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.

LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

Good Writing and Logical Constructions

Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

• Concept of a math paper
• Title, acknowledgment, and list of authors
• Introduction
• Body of the work
• Conclusion, appendix, and references
• Publication of a math paper
• Preprint archive
• Choice of the journal, submission
• Publication

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.

Good advice. For me, writing an essay on mathematics was very difficult. I did not have enough time and knowledge to write a quality essay. I worked a lot in the library and read many articles on the Internet. I studied information about essay writing. But I couldn’t finish the essay in full. I had to look for professional writers on the subject of mathematics. He helped me finish a few paragraphs. The work was delivered on time and on an excellent assessment.

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Writing math research papers: a guide for students and instructors.

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Robert Gerver

• Table of Contents

Writing Math Research Papers  is primarily a guide for high school students that describes how to write aand present mathematics research papers. But it’s really much more than that: it’s a systematic presentation of a philosophy that writing about math helps many students to understand it, and a practical method to move students from the relatively passive role of someone doing what is assigned to them, to creative thinkers and published writers who contribute to the mathematical literature.

As experienced writers know, the actual writing is not the half of it. William Zinsser once taught a writing class at the New School for Social Research which involved no writing at all: students talked through their ideas in class and through that process discovered the real story which could be written from their tangle of experiences, hopes and dreams. The actual writing was secondary, once they understood how to find the story and organize it.

Gerver, an experienced high school mathematics teacher, takes a similar approach. The primary audience is high school students who want to prepare formal papers or presentations, for contests or for a “math day” at their high school. But the discovery, research and organizational processes involved in writing an original paper, as opposed to rehashing information from a reference book, can help any student learn and understand math, and the experience will be useful even if the paper is never written.

Gerver leads students through a discovery process beginning with examining their own knowledge of mathematics and reviewing the basics of problem solving. The “math annotation” project follows next, in which students organize their class notes for one topic for presentation to their peers, resulting in a product similar to a section of a textbook or handbook, complete with illustrations and the necessary background and review material. Practical advice about finding a topic, developing it by keeping a research journal, and creating a final product, either a research paper or oral presentation, follows.

Writing Math Research Papers  is directed primarily to students, and could be assigned as a supplementary textbook for high school mathematics classes. It will also be useful to teachers who incorporate writing into their classes or who serve as mentors to the math club, and for student teachers in similar situations. An appendix for teachers includes practical advice about helping students through the research and writing process, organizing consultations, and grading the student papers and presentations. Excerpts from student research papers are included as well, and more materials are available from the web site www.keypress.com/wmrp .

Robert Gerver, PhD, is a mathematics instructor at North Shore High School in New York. He received the Presidential Award for Excellence in Mathematical Teaching in 1988 and the Tandy Prize and Chevron Best Practices Award in Education in 1997. He has been publishing mathematics. Dr. Gerver has written eleven mathematics textbooks and numerous articles, and holds two U.S. patents for educational devices.

Sarah Boslaugh, ( [email protected] ) is a Performance Review Analyst for BJC HealthCare and an Adjunct Instructor in the Washington University School of Medicine, both in St. Louis, MO. Her books include An Intermediate Guide to SPSS Programming: Using Syntax for Data Management  (Sage, 2004), Secondary Data Sources for Public Health: A Practical Guide (Cambridge, 2007), and Statistics in a Nutshell (O'Reilly, forthcoming), and she is Editor-in-Chief of The Encyclopedia of Epidemiology (Sage, forthcoming).

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Trefethen's list of 13 classic papers in applied mathematics

• J.W. Cooley and J.W. Tukey, " An algorithm for the machine calculation of complex Fourier series ," Math. Comp. , 19 (1965), pp. 297–301.
• R. Courant, K. Friedrichs, and H. Lewy, " On the partial difference equations of mathematical physics ," IBM J. Res. Develop. , 11 (1967), pp. 215–234.
• A.S. Householder, " Unitary triangularization of a nonsymmetric matrix ," J. Assoc. Comput. Mach. , 5 (1958), pp. 339–342.
• C.F. Curtiss and J.O. Hirschfelder, " Integration of stiff equations ," Proc. Nat. Acad. Sci. U.S.A. , 38 (1952), pp. 235–243.
• C. de Boor, " On calculating with B -splines ," J. Approximation Theory , 6 (1972), pp. 50–62.
• R. Courant, " Variational methods for the solution of problems of equilibrium and vibrations ," Bull. Amer. Math. Soc. , 49 (1943), pp. 1–23.
• G. Golub and W. Kahan, " Calculating the singular values and pseudo-inverse of a matrix ," J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. , 2 (1965), pp. 205–224.
• A. Brandt, " Multi-level adaptive solutions to boundary-value problems ," Math. Comp. , 31 (1977), no. 138, pp. 333–390.
• M.R. Hestenes and E. Stiefel, " Methods of conjugate gradients for solving linear systems ," J. Research Nat. Bur. Standards , 49 (1952), pp. 409–436.
• R. Fletcher and M.J.D. Powell, "A rapidly convergent descent method for minimization," Comput. J. , 6 (1963/1964), pp. 163–168.
• G. Wanner, E. Hairer, and S.P. Nørsett, " Order stars and stability theorems ," BIT , 18 (1978), no. 4, pp. 475–489.
• N. Karmarkar, "A new polynomial-time algorithm for linear programming," Combinatorica , 4 (1984), no. 4, pp. 373–395.
• L. Greengard and V. Rokhlin, " A fast algorithm for particle simulations ," J. Comput. Phys. , 73 (1987), no. 2, pp. 325–348.

Origin of this list

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Title: evolutionary game theory: the mathematics of evolution and collective behaviours.

Abstract: This brief discusses evolutionary game theory as a powerful and unified mathematical tool to study evolution of collective behaviours. It summarises some of my recent research directions using evolutionary game theory methods, which include i) the analysis of statistical properties of the number of (stable) equilibria in a random evolutionary game, and ii) the modelling of safety behaviours' evolution and the risk posed by advanced Artificial Intelligence technologies in a technology development race. Finally, it includes an outlook and some suggestions for future researchers.

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Mathematics Research Paper Topics

See our list of mathematics research paper topics . Mathematics is the science that deals with the measurement, properties, and relationships of quantities, as expressed in either numbers or symbols. For example, a farmer might decide to fence in a field and plant oats there. He would have to use mathematics to measure the size of the field, to calculate the amount of fencing needed for the field, to determine how much seed he would have to buy, and to compute the cost of that seed. Mathematics is an essential part of every aspect of life—from determining the correct tip to leave for a waiter to calculating the speed of a space probe as it leaves Earth’s atmosphere.

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• Boolean algebra
• Chaos theory
• Complex numbers
• Correlation
• Fraction, common
• Game theory
• Graphs and graphing
• Imaginary number
• Multiplication
• Natural numbers
• Number theory
• Numeration systems
• Probability theory
• Proof (mathematics)
• Pythagorean theorem
• Trigonometry

Mathematics undoubtedly began as an entirely practical activity— measuring fields, determining the volume of liquids, counting out coins, and the like. During the golden era of Greek science, between about the sixth and third centuries B.C., however, mathematicians introduced a new concept to their study of numbers. They began to realize that numbers could be considered as abstract concepts. The number 2, for example, did not necessarily have to mean 2 cows, 2 coins, 2 women, or 2 ships. It could also represent the idea of “two-ness.” Modern mathematics, then, deals both with problems involving specific, concrete, and practical number concepts (25,000 trucks, for example) and with properties of numbers themselves, separate from any practical meaning they may have (the square root of 2 is 1.4142135, for example).

Fields of Mathematics

Mathematics can be subdivided into a number of special categories, each of which can be further subdivided. Probably the oldest branch of mathematics is arithmetic, the study of numbers themselves. Some of the most fascinating questions in modern mathematics involve number theory. For example, how many prime numbers are there? (A prime number is a number that can be divided only by 1 and itself.) That question has fascinated mathematicians for hundreds of years. It doesn’t have any particular practical significance, but it’s an intriguing brainteaser in number theory.

Geometry, a second branch of mathematics, deals with shapes and spatial relationships. It also was established very early in human history because of its obvious connection with practical problems. Anyone who wants to know the distance around a circle, square, or triangle, or the space contained within a cube or a sphere has to use the techniques of geometry.

Algebra was established as mathematicians recognized the fact that real numbers (such as 4 and 5.35) can be represented by letters. It became a way of generalizing specific numerical problems to more general situations.

Analytic geometry was founded in the early 1600s as mathematicians learned to combine algebra and geometry. Analytic geometry uses algebraic equations to represent geometric figures and is, therefore, a way of using one field of mathematics to analyze problems in a second field of mathematics.

Over time, the methods used in analytic geometry were generalized to other fields of mathematics. That general approach is now referred to as analysis, a large and growing subdivision of mathematics. One of the most powerful forms of analysis—calculus—was created almost simultaneously in the early 1700s by English physicist and mathematician Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716). Calculus is a method for analyzing changing systems, such as the changes that take place as a planet, star, or space probe moves across the sky.

Statistics is a field of mathematics that grew in significance throughout the twentieth century. During that time, scientists gradually came to realize that most of the physical phenomena they study can be expressed not in terms of certainty (“A always causes B”), but in terms of probability (“A is likely to cause B with a probability of XX%”). In order to analyze these phenomena, then, they needed to use statistics, the field of mathematics that analyzes the probability with which certain events will occur.

Each field of mathematics can be further subdivided into more specific specialties. For example, topology is the study of figures that are twisted into all kinds of bizarre shapes. It examines the properties of those figures that are retained after they have been deformed.

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Spatial ability in STEM learning

Spatial Reasoning in Context: Bridging Cognitive and Educational Perspectives of Spatial-Mathematics Relations

• 1 University of Canberra, Australia

The final, formatted version of the article will be published soon.

Spatial reasoning is ingrained in daily life, such as when locating our keys or parking our car. At a broad level, spatial reasoning describes the ability to mentally represent and transform objects and their relations. Spatial reasoning is comprised of distinct, yet related, spatial skills, most of which have strong links with mathematics achievement. Subsequently, understanding the ways spatial reasoning connects with mathematics has the potential to support achievement in school. However, current research practices have failed to translate into practical outcomes for students. To date, research has often focused on decontextualized spatial skills, measured by psychometric tests, to generalize about broader models of spatial reasoning. However, spatial reasoning goes beyond test performance. In this theoretical review, I have sought to find the points of connection between the fields of cognitive psychology, often based in the lab, and mathematics education, situated within classrooms, and discussed ways to connect this currently siloed work for greater impact on classroom practice. The paper addresses the emergence of spatial research from its historical roots in intelligence testing and the influence these conceptualizations have had on contemporary methodologies. It goes on to discuss how these research traditions may be limiting our ability to understand the mechanisms linking spatial reasoning and mathematics. The paper argues for a broader view of research problems and methodologies in spatial cognition research to facilitate the translation of research to meaningful contexts in pedagogy and learning.

Keywords: spatial reasoning, mathematics education, Spatial skills, spatial cognition, classroom practice

Received: 26 Sep 2023; Accepted: 29 Nov 2023.

Copyright: © 2023 Harris. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Danielle Harris, University of Canberra, Canberra, Australia

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Research Paper Topics on Mathematics

Research papers in mathematics usually have a set of basic requirements that apply to all students. As a rule, they relate to the format of the research work and the requirements for it. Students need to adhere to a specific font, spacing, and writing format. It is necessary to have a title page of the research paper and the project, as well as the correct text formatting with recommendations and pagination.

Students must adhere to a clear heading hierarchy in research work and adhere to the correct abbreviations and formulas in the design of the entire project. As a rule, research work associated with the use of certain drawings, tables, diagrams, and graphs. Also, there may be charts which must also be marked following the requirements.

Students must reveal the topic given to them and bring clear arguments in favor of their statements. It should be a full-fledged mathematics research topic that fully reveals the theme and provides the reader with real evidence of opinion. It's important to choose an interesting math research topic.

The main task for each student is to relate everything they plan to write with one Topic. It will allow the teacher not to waste time looking for information and get the opportunity to see holistic work that fully meets all the established requirements. Students should also check the practical part as mathematics is an exact science that does not tolerate assumptions. So, let's take a look at the math research topics for middle schools.

Algebra & Algebraic Geometry

Algebra and algebraic geometry are the oldest branches of this science. This section arose as a result of the need to solve arithmetic problems of the same type. Now, this is a huge branch of science, which is extremely important because of its ability to make accurate calculations and definitions in various areas of our life. Choose any research topic for mathematics that suits you well.

• Formality in Deformation Theory
• The World of P-adic Numbers
• Toric Geometry and Mirror Symmetry
• Algebraic Geometry and Singularity Theory
• P-adic Dynamics and Applications of P-adic Numbers in Other Sciences
• Algebraic Geometry New Trends: The height Functions in Modern Science
• Theory of Height Functions & in Math

Algebraic Topology

This section of typology studies topological spaces using the juxtaposition of algebraic objects. At the moment, it is one of the more popular offshoots that uses homotopy groups and homomorphism. Any research topic in mathematics is a chance to create a good paper. This area of mathematics is popular thanks to several scientific studies. Here is a list of paper themes, but you can also find math education research topics by yourself.

• Main Calculus Functions and How to Use it in Real Life
• The Calculus of Functors and Applications
• How to Use the Algebraic Models for Spaces in Real Life
• New Wave of Automorphisms in Math
• The Asymptotic Properties of Polynomials in Higher Dimensions
• Moduli Spaces Geometry: Main Features
• The Lefschetz Properties and Main Parameters
• The Group Theory in Mathematics: Main Benefits
• Automorphisms of Manifolds and Related Options
• Modern Trends in Algebraic Topology and How to Use it Properly

Analysis & PDEs

This list of topics can be especially interesting for those who want to develop methods for solving equations of mathematical physics. Let's proceed to researchable topics in mathematics. In particular, students can create their work based on the approximation of the differential operator and use numerical methods to solve the tasks and open the topic of scientific research.

• Regularity Theory for Linear, Semilinear, and Fully Nonlinear Elliptic and Parabolic Equation
• New Theory for Linear, Semilinear, and Nonlinear Functions
• Qualitative Properties of Solutions to Reaction-diffusion Equations
• The Analysis of Problems in Mathematical Physics and Mathematical Modeling
• Delay Equations Formulation of Structured Population Dynamics
• Fractional differential equations and Fractals
• Mathematical Modeling of Ecological Systems and Epidemic Systems
• Delay Differential Equations is One Best Research Area in Mathematics.

According to many historians, the foundations of geometry were laid by the ancient Greeks, who took over from the Egyptians some mathematical calculations and general principles. This section received such a name in 1822. Nevertheless, all the foundations and important stages of this science branch were laid many centuries ago. Here are the applied mathematics research topics.

• The Bending Active Approach In Lightweight Design
• The Need Of Geometry In Orthodontics
• Archimedes Theory of a Circle ABCD and a Triangle K
• Applications of Toric Geometry to Geometric Representation Theory
• Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations
• Geometric Constructions of Mapping Cones in the Fukaya Category
• Euclidean and Non-Euclidean Geometries. Going Beyond the Space, We Used to know

Mathematical Logic & Foundations

This branch of mathematics may be of interest to those who want to study the nature of mathematical proof in general and various mathematical judgments as a branch of informal logic. In many respects, the foundation of this science was laid by Aristotle. Nevertheless, mathematical logic is developing in our time. Therefore, this is a good basis for scientific work.

• Deductive Parsing of Visual Languages
• Reduction of the Yablo paradox
• Gödel's: Minds are not Machines
• Ways to use the Mathematical logic in data science
• Applied Propositional Logic and digital circuit design
• Category Theory and Model Theory of Constructive Systems
• Constructive Mathematics and Point-free Methods in Topology and Analysis
• Ways to Use Martin-Löf type Theory and Univalent Foundations in Real Life

Number Theory

This is one of the popular branches of mathematics, consisting of many areas such as the Euclidean algorithm, continued fractions, Diophantine equations, and farm theory. At the moment, this industry has a lot of interesting topics to write a research paper. That's why you can find interesting mathematical topics.

• Writing Pi As the Sum of Arctangents With Recurring Linear Sequences, the Golden Mean and Lucas Numbers
• Binary Options Winning Formula: Make Consistent Wins Every Time
• A Theory of Numbers and New Operations
• The Twin Prime Conjecture in a Number Theory
• Pythagorean Triplets (Alternative Approach, Algebraic Operations, Dual of Given Triplets, and New Observations)
• A New Theory of Numbers: The Latest Discoveries in Mathematics
• Alternative Formula for the Series of Consecutive m-Squares under Alternating Signs
• A Clever Way To Factor One Less A Perfect Square
• How to use Number Theory to fight with gender stereotypes
• Goldbachs Twin Prime Conjecture

Probability & Statistics

By choosing this direction of mathematics, you can write a lot of interesting materials on modeling and data distribution and the study of two-dimensional numerical data. This is a particularly popular area of mathematics that will allow you to choose interesting topics for yourself with experiments and proofs. Let's check the research topics for mathematic paper.

• Highlighting Probability Issues in Simulated Annealing and Tabu Search
• Estimation of the Probability of Non-Response in Sampling Surveys Using Kernel Density Estimation Methods
• Importance of Statistical Tools and Methods in Data Science
• A Statistical Approach on Experimental Study for Determining Switching Frequency of Retro Reflector Sensor Using PLC
• The Statistics Analysis on Gender: Ain't ia Woman
• Quantitative Data Management, Statistical Analysis, and Graphics Using Stata
• The Data Science and Modern Statistics: the Symbiotic Connection
• Sei Shonagon's Pillow Book as an Example of Probability & Statistics Processes
• The Weibull Length Biased Exponential Distribution: Statistical Properties and Applications
• Univariate and Bivariate Transformations

Representation Theory

This branch of mathematics is especially interesting due to algebraic structures and linear transformations of vector spaces. Students can use Number Theory and Differential Geometry to describe certain nuances of Fourier theory or use topological groups to prove new theories. Here you can choose an undergraduate math research topic.

• Computing Modular Forms for the Weil Representation
• Combinatorics of the Asymmetric Simple Exclusion Process
• The Representation Theory & Asymmetric Simple Exclusion
• The Weil Representation in Tropical Mathematics
• Camel and Cactus Test in Representation Theory
• Combinatorics and Computations in Tropical Mathematics
• Quaternionic Representation and Its Use in Real Life
• The Modern Trends in Representation Theory of Diffeomorphism Groups

How to Write a Research Paper on Mathematics?

As we wrote earlier, teachers are very scrupulous about the correct design and writing of research papers. And interesting math research paper topics are a must. First of all, you need to write a clear introduction and structure of all your work. It is worth noting that you need formal and informal exposure, as well as bringing evidence of your work. One of the sections there will be special recommendations that depend on a particular educational institution.

You should understand that any such work requires a clear technical design and the ghost of the necessary graphs, tables, formulas, and calculations. The most important task of such research papers is to provide clear and evidence of your point of view to argue for each point and paragraph of your work. This is the key aspect, without which you do not get good grades.

That is why you should turn to professionals if you are not confident in your abilities or want to save a little time. Our company can help you write a research paper in mathematics and accompany you until you receive your final grade. It's important to choose good research topics in mathematics education, and we are ready to help you.

An Inspiration Sources List:

• Maths At Home
• The Holy Grail for Any Student
• The Mathunion Website
• Yahoo Mathematics Page
• Computer Algebra Group
• MacTutor History of Math
• Mathematics on the Web

Research Papers on Mathematics

Mathematics is the study of the properties and measurements of quantities and sets, using symbols and numbers. People often prepare research papers on mathematics. Keeping this in mind, Researchomatic is providing a wide variety of research papers on this subject. These research papers are easily accessible by individuals and act as a guideline for them to prepare their own research papers on mathematics.

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Contrary to reports, OpenAI probably isn’t building humanity-threatening AI

Has OpenAI invented an AI technology with the potential to “threaten humanity”? From some of the recent headlines, you might be inclined to think so.

Reuters and The Information first reported last week that several OpenAI staff members had, in a letter to the AI startup’s board of directors, flagged the “prowess” and “potential danger” of an internal research project known as “Q*.” This AI project, according to the reporting, could solve certain math problems — albeit only at grade-school level — but had in the researchers’ opinion a chance of building toward an elusive technical breakthrough.

There’s now debate as to whether OpenAI’s board ever received such a letter — The Verge cites a source suggesting that it didn’t. But the framing of Q* aside, Q* in actuality might not be as monumental — or threatening — as it sounds. It might not even be new.

AI researchers on X (formerly Twitter), including Meta’s chief AI scientist Yann LeCun, were immediately skeptical that Q* was anything more than an extension of existing work at OpenAI — and other AI research labs besides. In a post on X, Rick Lamers, who writes the Substack newsletter Coding with Intelligence, pointed to an MIT guest lecture OpenAI co-founder John Schulman gave seven years ago during which he described a mathematical function called “Q*.”

Several researchers believe the “Q” in the name “Q*” refers to “Q-learning,” an AI technique that helps a model learn and improve at a particular task by taking — and being rewarded for — specific “correct” actions. Researchers say the asterisk, meanwhile, could be a reference to A*, an algorithm for checking the nodes that make up a graph and exploring the routes between these nodes.

Please ignore the deluge of complete nonsense about Q*. One of the main challenges to improve LLM reliability is to replace Auto-Regressive token prediction with planning. Pretty much every top lab (FAIR, DeepMind, OpenAI etc) is working on that and some have already published… — Yann LeCun (@ylecun) November 24, 2023

Both have been around a while.

Google DeepMind applied Q-learning to build an AI algorithm that could play Atari 2600 games at human level… in 2014. A* has its origins in an academic paper published in 1968. And researchers at UC Irvine several years ago explored improving A* with Q-learning — which might be exactly what OpenAI’s now pursuing.

Nathan Lambert, a research scientist at the Allen Institute for AI, told TechCrunch he believes that Q* is connected to approaches in AI “mostly [for] studying high school math problems” — not destroying humanity.

“OpenAI even shared work earlier this year improving the mathematical reasoning of language models with a technique called process reward models,” Lambert said, “but what remains to be seen is how better math abilities do anything other than make [OpenAI’s AI-powered chatbot] ChatGPT a better code assistant.”

Mark Riedl, a computer science professor at Georgia Tech, was similarly critical of Reuters’ and The Information’s reporting on Q* — and the broader media narrative around OpenAI and its quest toward artificial general intelligence (i.e. AI that can perform any task as well as a human can). Reuters, citing a source, implied that Q* could be a step toward artificial general intelligence (AGI). But researchers — including Riedl — dispute this.

People are saying this is Q* https://t.co/rI5dz0Maf7 Seems plausible, though it’s from May 2023 and no one lost their minds over this, nor should they. — Mark Riedl (@mark_riedl) November 25, 2023

“There’s no evidence that suggests that large language models [like ChatGPT] or any other technology under development at OpenAI are on a path to AGI or any of the doom scenarios,” Riedl told TechCrunch. “OpenAI itself has at best been a ‘fast follower,’ having taken existing ideas … and found ways to scale them up. While OpenAI hires top-rate researchers, much of what they’ve done can be done by researchers at other organizations. It could also be done if OpenAI researchers were at a different organization.

Riedl, like Lambert, didn’t guess at whether Q* might entail Q-learning or A*. But if it involved either — or a combination of the two — it’d be consistent with the current trends in AI research, he said.

“These are all ideas being actively pursued by other researchers across academia and industry, with dozens of papers on these topics in the last six months or more,” Riedl added. “It’s unlikely that researchers at OpenAI have had ideas that have not also been had by the substantial number of researchers also pursuing advances in AI.”

That’s not to suggest that Q* — which reportedly had the involvement of Ilya Sutskever, OpenAI’s chief scientist — might not move the needle forward.

Lamers asserts that, if Q* uses some of the techniques described in a paper published by OpenAI researchers in May, it could “significantly” increase the capabilities of language models. Based on the paper, OpenAI might’ve discovered a way to control the “reasoning chains” of language models, Lamers says — enabling them to guide models to follow more desirable and logically sound “paths” to reach outcomes.

“This would make it less likely that models follow ‘foreign to human thinking’ and spurious-patterns to reach malicious or wrong conclusions,” Lamers said. “I think this is actually a win for OpenAI in terms of alignment … Most AI researchers agree we need better ways to train these large models, such that they can more efficiently consume information.”

But whatever emerges of Q*, it — and the relatively simple math equations it solves — won’t spell doom for humanity.

• Main content

Everyone's talking about OpenAI's Q*. Here's what you need to know about the mysterious project.

• A mysterious new OpenAI model known as Q* has got the tech world talking.
• The model is said to have sparked concern at the startup that led to the resulting chaos.
• AI experts say the model could be a big step forward but is unlikely to end the world anytime soon.

As the dust settles on the chaos at OpenAI, we still don't know why CEO Sam Altman was fired — but reports have suggested it could be linked to a mysterious AI model .

The Information reported that a team led by OpenAI chief scientist Ilya Sutskever had made a breakthrough earlier this year, which allowed them to build a new model known as Q* (pronounced "Q star.") The outlet reported that the model could solve basic math problems.

Sources told Reuters that this model provoked an internal firestorm , with several staff members writing a letter to OpenAI's board warning that the new breakthrough could threaten humanity.

This warning was cited as one of the reasons that the board chose to fire Sam Altman, who returned as CEO on Wednesday after days of turmoil at the company, Reuters reported.

The ability to solve basic math problems may not sound that impressive, but AI experts told Business Insider it would represent a huge leap forward from existing models, which struggle to generalize outside the data they are trained on.

"If it has the ability to logically reason and reason about abstract concepts, which right now is what it really struggles with, that's a pretty tremendous leap," said Charles Higgins, a cofounder of the AI-training startup Tromero who's also a Ph.D. candidate in AI safety.

He added, "Maths is about symbolically reasoning — saying, for example, 'If X is bigger than Y and Y is bigger than Z, then X is bigger than Z.' Language models traditionally really struggle at that because they don't logically reason, they just have what are effectively intuitions."

Sophia Kalanovska, a fellow Tromero cofounder and Ph.D. candidate, told BI that Q*'s name implied it was a combination of two well-known AI techniques, Q-learning and A* search.

She said this suggested the new model could combine the deep-learning techniques that power ChatGPT with rules programmed by humans. It's an approach that could help fix the chatbot's hallucination problem .

"I think it's symbolically very important. On a practical level, I don't think it's going to end the world," Kalanovska said.

"I think the reason why people believe that Q* is going to lead to AGI is because, from what we've heard so far, it seems like it will combine the two sides of the brain and be capable of knowing some things out of experience, while still being able to reason about facts," she added, referring to artificial general intelligence.

"That is definitely a step closer to what we consider intelligence, and it is possible that it leads to the model being able to have new ideas, which is not the case with ChatGPT."

The inability to reason and develop new ideas, rather than just regurgitating information from within their training data, is seen as a huge limitation of existing models, even by the people building them .

Andrew Rogoyski, a director at the Surrey Institute for People-Centered AI, told BI that solving unseen problems was a key step toward creating AGI.

"In the case of math, we know existing AIs have been shown to be capable of undergraduate-level math but to struggle with anything more advanced," he said.

"However, if an AI can solve new, unseen problems, not just regurgitate or reshape existing knowledge, then this would be a big deal, even if the math is relatively simple," he added.

Not everyone was so enthused by the reported breakthrough. Gary Marcus, an AI expert and deep-learning critic , expressed doubts about Q*'s reported capabilities in a post on his Substack .

"If I had a nickel for every extrapolation like that—'today , it works for grade school students! next year, it will take over the world!'—I'd be Musk-level rich," wrote Marcus.

OpenAI did not immediately respond to a request for comment from Business Insider, made outside normal working hours.

Watch: Sam Altman moves to Microsoft after OpenAI fires him as CEO