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Articles 1 - 30 of 59
Full-Text Articles in Entire DC Network
Lagrange’S Study Of Wilson’S Theorem, Carl Lienert
Lagrange’S Study Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of The Converse Of Wilson’S Theorem, Carl Lienert
Lagrange’S Proof Of The Converse Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of Wilson’S Theorem—And More!, Carl Lienert
Lagrange’S Proof Of Wilson’S Theorem—And More!, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert
Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
A Discrete Morse Approach For Computing Homotopy Types: An Exploration Of The Morse, Generalized Morse, Matching, And Independence Complexes, Connor Donovan
A Discrete Morse Approach For Computing Homotopy Types: An Exploration Of The Morse, Generalized Morse, Matching, And Independence Complexes, Connor Donovan
Mathematics Honors Papers
In this thesis, we study possible homotopy types of four families of simplicial complexes–the Morse complex, the generalized Morse complex, the matching complex, and the independence complex–using discrete Morse theory. Given a simplicial complex, K, we can construct its Morse complex from all possible discrete gradient vector fields on K. A similar construction will allow us to build the generalized Morse complex while considering edges and vertices will allow us to construct the matching complex and independence complex. In Chapter 3, we use the Cluster Lemma and the notion of star clusters to apply matchings to families of Morse, generalized …
Understanding Compactness Through Primary Sources: Early Work Uniform Continuity To The Heine-Borel Theorem, Naveen Somasunderam
Understanding Compactness Through Primary Sources: Early Work Uniform Continuity To The Heine-Borel Theorem, Naveen Somasunderam
Analysis
No abstract provided.
Towards The Homotopy Type Of The Morse Complex, Connor Donovan
Towards The Homotopy Type Of The Morse Complex, Connor Donovan
Mathematics Summer Fellows
Mathematicians have long been interested in studying the properties of simplicial complexes. In 1998, Robin Forman developed gradient vector fields as a tool to study these complexes. Having gradient vector fields to study these simplicial complexes, in 2005, Chari and Joswig discovered the Morse complex, a complex consisting of all gradient vector fields on a fixed complex. Although the Morse complex has been studied since 2005, there is little information regarding its homotopy type for different simplicial complexes. Pursuing our curiosity of the topic, we extend a result by Ayala et. al., stating that the pure Morse complex of a …
The French Connection: Borda, Condorcet And The Mathematics Of Voting Theory, Janet Heine Barnett
The French Connection: Borda, Condorcet And The Mathematics Of Voting Theory, Janet Heine Barnett
General Education and Liberal Studies
No abstract provided.
Stitching Dedekind Cuts To Construct The Real Numbers, Michael P. Saclolo
Stitching Dedekind Cuts To Construct The Real Numbers, Michael P. Saclolo
Analysis
No abstract provided.
Cross-Cultural Comparisons: The Art Of Computing The Greatest Common Divisor, Mary K. Flagg
Cross-Cultural Comparisons: The Art Of Computing The Greatest Common Divisor, Mary K. Flagg
Number Theory
No abstract provided.
Investigations Into Bolzano's Bounded Set Theorem, Dave Ruch
Investigations Into Bolzano's Bounded Set Theorem, Dave Ruch
Analysis
No abstract provided.
Investigations Into D'Alembert's Definition Of Limit (Real Analysis Version), Dave Ruch
Investigations Into D'Alembert's Definition Of Limit (Real Analysis Version), Dave Ruch
Analysis
No abstract provided.
The Mobius Function And Mobius Inversion, Carl Lienert
The Mobius Function And Mobius Inversion, Carl Lienert
Number Theory
No abstract provided.
Merge Trees In Discrete Morse Theory, Benjamin Johnson
Merge Trees In Discrete Morse Theory, Benjamin Johnson
Mathematics Summer Fellows
The field of topological data analysis seeks to use techniques in topology to study large data sets. The hope is that rather than single quantities that summarize the data, such as mean or standard deviation, information about the data can be learned by studying the overall ``shape” of the data. One way to summarize this data is through a merge tree. Merge trees can be thought of as keeping track of certain clusters of data and determining when they merge together. In this paper, we will study merge trees induced by a discrete Morse function on a tree. Under a …
Euler’S Calculation Of The Sum Of The Reciprocals Of Squares, Kenneth M. Monks
Euler’S Calculation Of The Sum Of The Reciprocals Of Squares, Kenneth M. Monks
Calculus
No abstract provided.
Connectedness- Its Evolution And Applications, Nicholas A. Scoville
Connectedness- Its Evolution And Applications, Nicholas A. Scoville
Topology
No abstract provided.
Greatest Common Divisor: Algorithm And Proof, Mary K. Flagg
Greatest Common Divisor: Algorithm And Proof, Mary K. Flagg
Number Theory
No abstract provided.
How To Calculate Pi: Buffon's Needle (Non-Calculus Version), Dominic Klyve
How To Calculate Pi: Buffon's Needle (Non-Calculus Version), Dominic Klyve
Pre-calculus and Trigonometry
No abstract provided.
The Origin Of The Prime Number Theorem, Dominic Klyve
The Origin Of The Prime Number Theorem, Dominic Klyve
Number Theory
No abstract provided.
Dual Perspectives On Desargues' Theorem, Carl Lienert
Otto Holder's Formal Christening Of The Quotient Group Concept, Janet Heine Barnett
Otto Holder's Formal Christening Of The Quotient Group Concept, Janet Heine Barnett
Abstract Algebra
No abstract provided.
Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu
Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu
Mathematics Summer Fellows
We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. We conclude with an example illustrating our construction.
Nearness Without Distance, Nicholas A. Scoville
Determining The Determinant, Danny Otero
From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville
From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville
Topology
No abstract provided.
The Roots Of Early Group Theory In The Works Of Lagrange, Janet Heine Barnett
The Roots Of Early Group Theory In The Works Of Lagrange, Janet Heine Barnett
Abstract Algebra
No abstract provided.
The Pell Equation In India, Toke Knudsen, Keith Jones
The Pell Equation In India, Toke Knudsen, Keith Jones
Number Theory
No abstract provided.
Generating Pythagorean Triples: A Gnomonic Exploration, Janet Heine Barnett
Generating Pythagorean Triples: A Gnomonic Exploration, Janet Heine Barnett
Number Theory
No abstract provided.
Gaussian Guesswork: Infinite Sequences And The Arithmetic-Geometric Mean, Janet Heine Barnett
Gaussian Guesswork: Infinite Sequences And The Arithmetic-Geometric Mean, Janet Heine Barnett
Calculus
No abstract provided.
Rigorous Debates Over Debatable Rigor: Monster Functions In Introductory Analysis, Janet Heine Barnett
Rigorous Debates Over Debatable Rigor: Monster Functions In Introductory Analysis, Janet Heine Barnett
Analysis
No abstract provided.