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1Geometric Combinatorics - Making Undergraduate Research Work
Speaker: Francis Su
“Geometric Combinatorics - Making Undergraduate Research Work” Metadata:
- Title: ➤ Geometric Combinatorics - Making Undergraduate Research Work
- Language: English
“Geometric Combinatorics - Making Undergraduate Research Work” Subjects and Themes:
- Subjects: Mathematics - lecture
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- Internet Archive ID: lecture10589
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2Let’s Flying By Wing - Mathematical Combinatorics & Smarandache Multi-Spaces (new Extended Version) [Chinese]
By Linfan Mao
Speaker: Francis Su
“Let’s Flying By Wing - Mathematical Combinatorics & Smarandache Multi-Spaces (new Extended Version) [Chinese]” Metadata:
- Title: ➤ Let’s Flying By Wing - Mathematical Combinatorics & Smarandache Multi-Spaces (new Extended Version) [Chinese]
- Author: Linfan Mao
- Language: English
Edition Identifiers:
- Internet Archive ID: multimedia-larga-9781599732145
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3Graphs And Combinatorics 1995: Vol 11 Table Of Contents
Graphs and Combinatorics 1995: Volume 11 , Issue CONTENTS. Digitized from IA1514110-05 . Previous issue: sim_graphs-and-combinatorics_1994_10_4 . Next issue: sim_graphs-and-combinatorics_1995_11_1 .
“Graphs And Combinatorics 1995: Vol 11 Table Of Contents” Metadata:
- Title: ➤ Graphs And Combinatorics 1995: Vol 11 Table Of Contents
- Language: English
“Graphs And Combinatorics 1995: Vol 11 Table Of Contents” Subjects and Themes:
- Subjects: ➤ Mathematics - Mathematics--Computer Applications - Scholarly Journals - microfilm
Edition Identifiers:
- Internet Archive ID: ➤ sim_graphs-and-combinatorics_1995_11_contents
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4Introductory Combinatorics
By Bogart, Kenneth P
Graphs and Combinatorics 1995: Volume 11 , Issue CONTENTS. Digitized from IA1514110-05 . Previous issue: sim_graphs-and-combinatorics_1994_10_4 . Next issue: sim_graphs-and-combinatorics_1995_11_1 .
“Introductory Combinatorics” Metadata:
- Title: Introductory Combinatorics
- Author: Bogart, Kenneth P
- Language: English
“Introductory Combinatorics” Subjects and Themes:
- Subjects: ➤ Combinatorial analysis - Computer science -- Mathematics
Edition Identifiers:
- Internet Archive ID: introductorycomb0000boga
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5International Journal Of Mathematical Combinatorics (IJMC). Volume 1/2009
By Linfan Mao (Editor in Chief)
The International J. Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-Euclidean geometry and topology and their applications to other sciences.
“International Journal Of Mathematical Combinatorics (IJMC). Volume 1/2009” Metadata:
- Title: ➤ International Journal Of Mathematical Combinatorics (IJMC). Volume 1/2009
- Author: Linfan Mao (Editor in Chief)
- Language: English
“International Journal Of Mathematical Combinatorics (IJMC). Volume 1/2009” Subjects and Themes:
- Subjects: Smarandache geometries - Smarandache curves
Edition Identifiers:
- Internet Archive ID: ijmc-1-2009
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6International Journal Of Mathematical Combinatorics (IJMC). Volume 3/2009
By Linfan Mao
The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-Euclidean geometry and topology and their applications to other sciences.
“International Journal Of Mathematical Combinatorics (IJMC). Volume 3/2009” Metadata:
- Title: ➤ International Journal Of Mathematical Combinatorics (IJMC). Volume 3/2009
- Author: Linfan Mao
- Language: English
“International Journal Of Mathematical Combinatorics (IJMC). Volume 3/2009” Subjects and Themes:
- Subjects: ➤ Smarandache multi-spaces - Smarandache geometries - Differential Geometry - Smarandache curves - Combinatorial designs - Combinatorial enumeration - Low Dimensional Topology - Differential Topology
Edition Identifiers:
- Internet Archive ID: ijmc-3-2009
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7Combinatorics Of Rational Functions And Poincare-Birkhoff-Witt Expansions Of The Canonical U(n-)-valued Differential Form
By R. Rimanyi, L. Stevens and A. Varchenko
We study the canonical U(n-)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie albegras A_r, B_r, C_r, D_r in a conviniently chosen Poincare-Birkhoff-Witt basis. As a byproduct we obtain results on the combinatorics of rational functions, namely non-trivial identities are proved between certain rational functions with partial symmetries.
“Combinatorics Of Rational Functions And Poincare-Birkhoff-Witt Expansions Of The Canonical U(n-)-valued Differential Form” Metadata:
- Title: ➤ Combinatorics Of Rational Functions And Poincare-Birkhoff-Witt Expansions Of The Canonical U(n-)-valued Differential Form
- Authors: R. RimanyiL. StevensA. Varchenko
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-math0407101
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8Journal Of Algebraic Combinatorics 1993: Vol 2 Table Of Contents
Journal of Algebraic Combinatorics 1993: Volume 2 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1992-12_1_4 . Next issue: sim_journal-of-algebraic-combinatorics_1993-03_2_1 .
“Journal Of Algebraic Combinatorics 1993: Vol 2 Table Of Contents” Metadata:
- Title: ➤ Journal Of Algebraic Combinatorics 1993: Vol 2 Table Of Contents
- Language: English
“Journal Of Algebraic Combinatorics 1993: Vol 2 Table Of Contents” Subjects and Themes:
- Subjects: ➤ Mathematical & Physical Sciences - Scholarly Journals - microfilm
Edition Identifiers:
- Internet Archive ID: ➤ sim_journal-of-algebraic-combinatorics_1993_2_contents
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9Journal Of Algebraic Combinatorics 1994: Vol 3 Table Of Contents
Journal of Algebraic Combinatorics 1994: Volume 3 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1993-11_2_4 . Next issue: sim_journal-of-algebraic-combinatorics_1994-01_3_1 .
“Journal Of Algebraic Combinatorics 1994: Vol 3 Table Of Contents” Metadata:
- Title: ➤ Journal Of Algebraic Combinatorics 1994: Vol 3 Table Of Contents
- Language: English
“Journal Of Algebraic Combinatorics 1994: Vol 3 Table Of Contents” Subjects and Themes:
- Subjects: ➤ Mathematical & Physical Sciences - Scholarly Journals - microfilm
Edition Identifiers:
- Internet Archive ID: ➤ sim_journal-of-algebraic-combinatorics_1994_3_contents
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10Unions Of Cubes In $\mathbb{R}^{n}$, Combinatorics In $\mathbb{Z}^{n}$ And The John-Nirenberg And John-Str\"omberg Inequalities
By Michael Cwikel
Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint "simple" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\cap W$ and $F\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by "simple" we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space $BMO$ of functions of bounded mean oscillation introduced by John and Nirenberg. The notion of a John-Str\"omberg pair is closely related to the above question, and the above mentioned result is obtained as a consequence of a general result about these pairs. We also present a number of additional results about these pairs.
“Unions Of Cubes In $\mathbb{R}^{n}$, Combinatorics In $\mathbb{Z}^{n}$ And The John-Nirenberg And John-Str\"omberg Inequalities” Metadata:
- Title: ➤ Unions Of Cubes In $\mathbb{R}^{n}$, Combinatorics In $\mathbb{Z}^{n}$ And The John-Nirenberg And John-Str\"omberg Inequalities
- Author: Michael Cwikel
“Unions Of Cubes In $\mathbb{R}^{n}$, Combinatorics In $\mathbb{Z}^{n}$ And The John-Nirenberg And John-Str\"omberg Inequalities” Subjects and Themes:
- Subjects: Metric Geometry - Functional Analysis - Mathematics
Edition Identifiers:
- Internet Archive ID: arxiv-1702.00602
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11The Geometry And Combinatorics Of Springer Fibers
By Julianna Tymoczko
This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.
“The Geometry And Combinatorics Of Springer Fibers” Metadata:
- Title: ➤ The Geometry And Combinatorics Of Springer Fibers
- Author: Julianna Tymoczko
“The Geometry And Combinatorics Of Springer Fibers” Subjects and Themes:
- Subjects: Representation Theory - Algebraic Geometry - Combinatorics - Mathematics
Edition Identifiers:
- Internet Archive ID: arxiv-1606.02760
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12Geometric Variational Problems Of Statistical Mechanics And Of Combinatorics
By Senya Shlosman
We present the geometric solutions of the various extremal problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape of a typical Young diagram and of a typical skyscraper.
“Geometric Variational Problems Of Statistical Mechanics And Of Combinatorics” Metadata:
- Title: ➤ Geometric Variational Problems Of Statistical Mechanics And Of Combinatorics
- Author: Senya Shlosman
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-math-ph0002035
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13The Rotor Model And Combinatorics
By M. T. Batchelor, J. de Gier and B. Nienhuis
We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the O($n=1$) model we find the number $A_{\rm V}(2m+1;3)$, which 3-enumerates vertically symmetric alternating sign matrices.
“The Rotor Model And Combinatorics” Metadata:
- Title: ➤ The Rotor Model And Combinatorics
- Authors: M. T. BatchelorJ. de GierB. Nienhuis
Edition Identifiers:
- Internet Archive ID: arxiv-math-ph0204002
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14Raise And Peel Models Of Fluctuating Interfaces And Combinatorics Of Pascal's Hexagon
By Pavel Pyatov
The raise and peel model of a one-dimensional fluctuating interface (model A) is extended by considering one source (model B) or two sources (model C) at the boundaries. The Hamiltonians describing the three processes have, in the thermodynamic limit, spectra given by conformal field theory. The probability of the different configurations in the stationary states of the three models are not only related but have interesting combinatorial properties. We show that by extending Pascal's triangle (which gives solutions to linear relations in terms of integer numbers), to an hexagon, one obtains integer solutions of bilinear relations. These solutions give not only the weights of the various configurations in the three models but also give an insight to the connections between the probability distributions in the stationary states of the three models. Interestingly enough, Pascal's hexagon also gives solutions to a Hirota's difference equation.
“Raise And Peel Models Of Fluctuating Interfaces And Combinatorics Of Pascal's Hexagon” Metadata:
- Title: ➤ Raise And Peel Models Of Fluctuating Interfaces And Combinatorics Of Pascal's Hexagon
- Author: Pavel Pyatov
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-math-ph0406025
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15Combinatorics Of Solitons In Noncritical String Theory
By Masafumi Fukuma and Shigeaki Yahikozawa
We study the combinatorics of solitons in $D
“Combinatorics Of Solitons In Noncritical String Theory” Metadata:
- Title: ➤ Combinatorics Of Solitons In Noncritical String Theory
- Authors: Masafumi FukumaShigeaki Yahikozawa
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-hep-th9610199
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16Combinatorics Of Geometrically Distributed Random Variables: Value And Position Of Large Left-to-right Maxima
By Helmut Prodinger
For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the r-th left-to-right maximum counted from the right, for fixed r and n->oo. This complements previous research where the analogous questions were considered for the r-th left-to-right maximum counted from the left.
“Combinatorics Of Geometrically Distributed Random Variables: Value And Position Of Large Left-to-right Maxima” Metadata:
- Title: ➤ Combinatorics Of Geometrically Distributed Random Variables: Value And Position Of Large Left-to-right Maxima
- Author: Helmut Prodinger
Edition Identifiers:
- Internet Archive ID: arxiv-math0009116
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17The Combinatorics Of The Garsia-Haiman Modules For Hook Shapes
By Ron M. Adin, Jeffrey B. Remmel and Yuval Roichman
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
“The Combinatorics Of The Garsia-Haiman Modules For Hook Shapes” Metadata:
- Title: ➤ The Combinatorics Of The Garsia-Haiman Modules For Hook Shapes
- Authors: Ron M. AdinJeffrey B. RemmelYuval Roichman
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-math0703308
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18Algorithmic Combinatorics On Partial Words
By Blanchet-Sadri, Francine
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
“Algorithmic Combinatorics On Partial Words” Metadata:
- Title: ➤ Algorithmic Combinatorics On Partial Words
- Author: Blanchet-Sadri, Francine
- Language: English
“Algorithmic Combinatorics On Partial Words” Subjects and Themes:
- Subjects: ➤ Computer algorithms - Computer science -- Mathematics - Combinatorial analysis - Algorithms - Algorithmes - Informatique -- Mathématiques - Analyse combinatoire - algorithms - COMPUTERS -- Programming -- Open Source - COMPUTERS -- Software Development & Engineering -- Tools - COMPUTERS -- Software Development & Engineering -- General
Edition Identifiers:
- Internet Archive ID: algorithmiccombi0000blan
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19Combinatorics
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
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20Mathematical Combinatorics. An International Book Series, Vol. 4, 2017
By Linfan Mao
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
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- Author: Linfan Mao
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21Mathematical Combinatorics. An International Book Series, Vol. 1, 2008
By Mao, Linfan.
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
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22MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) Vol. 3, 2009
By Mao, Linfan.
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
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- Title: ➤ MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) Vol. 3, 2009
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- Language: English
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23On Additive Combinatorics Of Permutations Of \mathbb{Z}_n
By L. Sunil Chandran, Deepak Rajendraprasad and Nitin Singh
Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we prove $s(n)\geq (n\phi(n))/2^k$ where $k$ is the number of distinct prime divisors of $n$. When $n$ is an odd prime we prove $s(n)\leq \frac{e^2}{\pi} n ((n-1)/e)^\frac{n-1}{2}$. For the second problem, we prove $2^{(n-1)/2}.(\frac{n-1}{2})!\leq t(n)\leq 2^k.(n-1)!/\phi(n)$ when $n$ is odd.
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- Authors: L. Sunil ChandranDeepak RajendraprasadNitin Singh
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- Subjects: Mathematics - Combinatorics
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24On The Combinatorics Of Smoothing
By Micah W. Chrisman
Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown how to use a modification of a theorem of Zulli together with a modification of the spectral theory of graphs to approach such problems systematically. We give an application to counting subdiagrams of pretzel knots which have one component after oriented and unoriented smoothings.
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- Author: Micah W. Chrisman
- Language: English
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25A Determinacy Approach To Borel Combinatorics
By Andrew Marks
We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.
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- Author: Andrew Marks
- Language: English
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26Combinatorics Of Geometrically Distributed Random Variables: Inversions And A Parameter Of Knuth
By Helmut Prodinger
For words of length $n$, generated by independent geometric random variables, we consider the mean and variance of the number of inversions and of a parameter of Knuth from permutation in situ. In this way, $q$--analogues for these parameters from the usual permutation model are obtained.
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- Author: Helmut Prodinger
- Language: English
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27Combinatorics Of The Free Baxter Algebra
By Marcelo Aguiar and Walter Moreira
We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schroeder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.
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- Authors: Marcelo AguiarWalter Moreira
- Language: English
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28The Jacobian Conjecture As A Problem In Combinatorics
By David Wright
The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools to prove the symmetric Jacobian Conjecture for the case $F=X-H$ with $H$ homogeneous and $JH^{3}=0$. Other special results are also derived. We pose a combinatorial statement which would give a complete proof the Jacobian Conjecture.
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- Author: David Wright
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29Cominuscule Tableau Combinatorics
By Hugh Thomas and Alexander Yong
We continue our study of "cominuscule tableau combinatorics" by generalizing constructions of M. Haiman, S. Fomin and M.-P. Sch\"{u}tzenberger. In particular, we extend the "dual equivalence" ideas of [Haiman, 1992] to reformulate the generalized Littlewood-Richardson rule for cominuscule G/P Schubert calculus from [Thomas-Yong, 2006]; this reformulation avoids certain arbitrary choices demanded by the original version. Also, we apply dual equivalence to give an alternative and independent proof of the jeu de taquin results of [Proctor, 2004] needed in our earlier work. Further, we extend Fomin's "growth diagram" description of jeu de taquin; the inherent symmetry of these diagrams leads to a generalization of Sch\"{u}tzenberger's "evacuation involution". We think of these results as characteristic of our viewpoint that the tableau combinatorics making the cohomology of Grassmannians attractive should have natural cominuscule generalizations. This principle incorporates the standard maxim that the Young tableau combinatorics of Schur polynomials should have "shifted analogues" for Schur $P-$ and $Q-$ polynomials (although this is often used with representation theory of the symmetric group, rather than geometry of flag manifolds, in mind).
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- Authors: Hugh ThomasAlexander Yong
- Language: English
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30Combinatorics Of Line Arrangements And Dynamics Of Polynomial Vector Fields
By Benoît Guerville-Ballé and Juan Viu-Sos
Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$--derivations. First, we give a dynamical interpretation of $\mathcal{D}(\mathcal{A})$ as the set of polynomial vector fields which posses $\mathcal{A}$ as invariant set. We characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree of polynomial vector fields fixing only a finite set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.
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- Authors: Benoît Guerville-BalléJuan Viu-Sos
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- Subjects: Mathematics - Dynamical Systems
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- Internet Archive ID: arxiv-1412.0137
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31Combinatorics And Genus Of Tropical Intersections And Ehrhart Theory
By Reinhard Steffens and Thorsten Theobald
Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in terms of mixed volumes. By establishing some aspects of a mixed version of Ehrhart theory we show that the genus of a tropical intersection curve equals the genus of a toric intersection curve corresponding to the same Newton polytopes.
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- Title: ➤ Combinatorics And Genus Of Tropical Intersections And Ehrhart Theory
- Authors: Reinhard SteffensThorsten Theobald
- Language: English
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32Heisenberg-Weyl Algebra Revisited: Combinatorics Of Words And Paths
By P. Blasiak, A. Horzela, G. H. E. Duchamp, K. A. Penson and A. I. Solomon
The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and applications.
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- Title: ➤ Heisenberg-Weyl Algebra Revisited: Combinatorics Of Words And Paths
- Authors: P. BlasiakA. HorzelaG. H. E. DuchampK. A. PensonA. I. Solomon
- Language: English
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- Internet Archive ID: arxiv-0904.1506
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33Combinatorics And Topology Of Straightening Maps II: Discontinuity
By Hiroyuki Inou
We continue the study of straightening maps for the family of polynomials of degree d>2. The notion of straightening map is originally introduced by Douady and Hubbard to study the self-similarity of the Mandelbrot set. As expected from their example of a cubic-like family with discontinuous straightening map, we prove that the straightening map is discontinuous unless it is of disjoint type (in this case, it is already known that the straightening map is continuous).
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- Title: ➤ Combinatorics And Topology Of Straightening Maps II: Discontinuity
- Author: Hiroyuki Inou
- Language: English
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- Internet Archive ID: arxiv-0903.4289
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34Abstract Integration, Combinatorics Of Trees And Differential Equations
By M. Gubinelli
This is a review paper on recent work about the connections between rough path theory, the Connes-Kreimer Hopf algebra on rooted trees and the analysis of finite and infinite dimensional differential equation. We try to explain and motivate the theory of rough paths introduced by T. Lyons in the context of differential equations in presence of irregular noises. We show how it is used in an abstract algebraic approach to the definition of integrals over paths which involves a cochain complex of finite increments. In the context of such abstract integration theories we outline a connection with the combinatorics of rooted trees. As interesting examples where these ideas apply we present two infinite dimensional dynamical systems: the Navier-Stokes equation and the Korteweg-de-Vries equation.
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- Title: ➤ Abstract Integration, Combinatorics Of Trees And Differential Equations
- Author: M. Gubinelli
- Language: English
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- Internet Archive ID: arxiv-0809.1821
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35Combinatorics Of Linked Systems Of Quartet Trees
By Emili Moan and Joseph Rusinko
We apply classical quartet techniques to the problem of phylogenetic decisiveness and find a value $k$ such that all collections of at least $k$ quartets are decisive. Moreover, we prove that this bound is optimal and give a lower-bound on the probability that a collection of quartets is decisive.
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- Title: ➤ Combinatorics Of Linked Systems Of Quartet Trees
- Authors: Emili MoanJoseph Rusinko
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- Subjects: Mathematics - Quantitative Methods - Quantitative Biology - Combinatorics
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- Internet Archive ID: arxiv-1405.2464
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36How Instanton Combinatorics Solves Painlevé VI, V And III's
By O. Gamayun, N. Iorgov and O. Lisovyy
We elaborate on a recently conjectured relation of Painlev\'e transcendents and 2D CFT. General solutions of Painlev\'e VI, V and III are expressed in terms of $c=1$ conformal blocks and their irregular limits, AGT-related to instanton partition functions in $\mathcal{N}=2$ supersymmetric gauge theories with $N_f=0,1,2,3,4$. Resulting combinatorial series representations of Painlev\'e functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the GUE, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.
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- Title: ➤ How Instanton Combinatorics Solves Painlevé VI, V And III's
- Authors: O. GamayunN. IorgovO. Lisovyy
- Language: English
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- Internet Archive ID: arxiv-1302.1832
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37Mathematical Combinatorics, International Book Series, Vol. 2, 2010
Papers on Crypto-Automorphism of the Buchsteiner Loops, Generalizations of Poly-Bernoulli Numbers and Polynomials, Open Alliance in Graphs, Forcing Weak Edge Detour Number of a Graph, New Families of Mean Graphs, Euler-Savary Formula for the Lorentzian Planar Homothetic Motions, and other topics.
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- Language: English
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- Subjects: ➤ Lorentzian Planar Homothetic Motions - Loops - Graph
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38Mathematical Combinatorics, International Book Series, Vol. 2, 2012
Papers on Non-Solvable Spaces of Linear Equation Systems, Roman Domination in Complementary Prism Graphs, On Pathos Total Semitotal and Entire Total Block Graph of a Tree, Distance Two Labeling of Generalized Cacti, Degree Splitting Graph on Graceful, Felicitous and Elegant Labeling, and other topics.
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39Combinatorics Of The Casselman-Shalika Formula In Type A
By Kyu-Hwan Lee, Philip Lombardo and Ben Salisbury
In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableaux model for the crystal and obtain the same coefficients using data from each individual tableaux; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a q-polynomial which exhibits many interesting properties. We use examples to illustrate these properties.
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- Title: ➤ Combinatorics Of The Casselman-Shalika Formula In Type A
- Authors: Kyu-Hwan LeePhilip LombardoBen Salisbury
- Language: English
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40Automatic Theorem-Proving In Combinatorics On Words
By Dane Henshall and Jeffrey Shallit
We describe a technique for mechanically proving certain kinds of theorems in combinatorics on words, using automata and a package for manipulating them. We illustrate our technique by solving, purely mechanically, an open problem of Currie and Saari on the lengths of unbordered factors in the Thue-Morse sequence.
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- Authors: Dane HenshallJeffrey Shallit
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41The Turán Number And Probabilistic Combinatorics
By Alan J. Aw
In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof of a classical theorem concerning the Tur\'{a}n number $T(n,k,l)$. Surprisingly, this proof cannot be found in existing literature.
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- Author: Alan J. Aw
- Language: English
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42A Theory Of Transformation Monoids: Combinatorics And Representation Theory
By Benjamin Steinberg
The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular, we obtain a generalization of a lemma of P. Neumann, from the theory of synchronizing groups, concerning the partition associated to a transformation of minimal rank.
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- Author: Benjamin Steinberg
- Language: English
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- Internet Archive ID: arxiv-1004.2982
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43Combinatorics Of Experimental Design
By Street, Anne Penfold
The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular, we obtain a generalization of a lemma of P. Neumann, from the theory of synchronizing groups, concerning the partition associated to a transformation of minimal rank.
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- Title: ➤ Combinatorics Of Experimental Design
- Author: Street, Anne Penfold
- Language: English
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- Subjects: ➤ Experimental design - Combinatorial designs and configurations
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- Internet Archive ID: combinatoricsofe0000stre
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44Applied Combinatorics
By Mitchel T. Keller, William T. Trotter
Applied Combinatorics is an open-source textbook for a course covering the fundamental enumeration techniques (permutations, combinations, subsets, pigeon hole principle), recursion and mathematical induction, more advanced enumeration techniques (inclusion-exclusion, generating functions, recurrence relations, Polyá theory), discrete structures (graphs, digraphs, posets, interval orders), and discrete optimization (minimum weight spanning trees, shortest paths, network flows). There are also chapters introducing discrete probability, Ramsey theory, combinatorial applications of network flows, and a few other nuggets of discrete mathematics. Available at the Open Textbook Library: https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=339
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- Author: ➤ Mitchel T. Keller, William T. Trotter
- Language: English
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- Subjects: Combinatorics - Enumeration techniques - Mathematics - Discrete mathematics - Combinatorial mathematics
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45Short Formulas For Algebraic Covariant Derivative Curvature Tensors Via Algebraic Combinatorics
By Bernd Fiedler
We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S_r] and discrete Fourier transforms for symmetric groups S_r. For symbolic calculations we used the Mathematica packages Ricci and PERMS.
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- Author: Bernd Fiedler
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46A-walk-through-combinatorics
one of the best book for begineers
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47DTIC ADA187928: The Construction Of A Pedagogy To Promote Meaningful Learning Of Strategies And Tactics For Solving Problems Of Elementary Combinatorics.
By Defense Technical Information Center
The purpose of this study was to construct a new methodology for teaching combinatorics based on Doctor Ausubel's theory about meaningful learning. The key idea Ausubel's theory is that if learning has to be meaningful, then the learner has to have subsuming or anchoring concepts in his cognitive structure. Combinatorics has typically been one of those subjects the students have more difficulty in understanding. This phenomenon happens because previous presentations of combinatorics leave a great deal to be desired pedagogically, and do not place enough emphasis on visualization. As a result, students use to learn course materials in a rote manner, and find little motivation for such learning activities. A prescription has been found to remedy such pathology. A conceptual map, rather than typically organized hierarchy of concepts, has been developed. The conceptual map interrelates the main and subordinate concepts in a cyclical manner, in a repetitive way, in a gradual and smooth progress, to enable the reader to assimilate ideas meaningfully.
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- Subjects: ➤ DTIC Archive - Rodriguez-Pascual, Pedro - AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF SYSTEMS AND LOGISTICS - *PROBLEM SOLVING - *COMBINATORIAL ANALYSIS - *THEORY - ANCHORS - COGNITION - HIERARCHIES - LEARNING - METHODOLOGY - MOTIVATION - PATHOLOGY - PHYSICIANS - STRATEGY - STUDENTS - PERMUTATIONS - TEACHING METHODS - VARIATIONS - THESES
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48Combinatorics Of The Lipschitz Polytope
By J. Gordon and F. Petrov
Let $\rho$ be a metric on the set $X=\{1,2,\dots,n+1\}$. Consider the $n$-dimensional polytope of functions $f:X\rightarrow \mathbb{R}$, which satisfy the conditions $f(n+1)=0$, $|f(x)-f(y)|\leq \rho(x,y)$. The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by A. M. Vershik \cite{V}. We prove that for any "generic" metric the number of $(n-m)$-dimensional faces, $0\leq m\leq n$, equals $\binom{n+m}{m,m,n-m}=(n+m)!/m!m!(n-m)!$. This fact is intimately related to regular triangulations of the root polytope (the convex hull of the roots of $A_n$ root system). Also we get two-sided estimates for the logarithm of the number of Vershik classes of metrics: $n^3\log n$ from above and $n^2$ from below.
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- Title: ➤ Combinatorics Of The Lipschitz Polytope
- Authors: J. GordonF. Petrov
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- Subjects: Metric Geometry - Combinatorics - Mathematics
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- Internet Archive ID: arxiv-1608.06848
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49Q-combinatorics And Quantum Integrability
By A. Yu. Volkov
The idea that a Dynkin diagram can provide one of the `spatial' variables for an integrable difference-difference system is no news. I propose a `model' where the only variable is of this sort.
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- Author: A. Yu. Volkov
- Language: English
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50Asymptotics Of Lattice Walks Via Analytic Combinatorics In Several Variables
By Stephen Melczer and Mark C. Wilson
We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pad\'e-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.
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- Authors: Stephen MelczerMark C. Wilson
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- Subjects: Combinatorics - Mathematics
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- Internet Archive ID: arxiv-1511.02527
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