15 results
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2. On the correlation of increasing families.
- Author
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Kalai, Gil, Keller, Nathan, and Mossel, Elchanan
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STATISTICAL correlation , *MONOTONE operators , *OPERATOR theory , *COMBINATORICS , *MATHEMATICS - Abstract
The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand [19] established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
3. Rigidity of inversive distance circle packings revisited.
- Author
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Xu, Xu
- Subjects
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COMBINATORICS , *MATHEMATICAL analysis , *INFINITESIMAL geometry , *NONNEGATIVE matrices , *MATHEMATICS - Abstract
Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34] . They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [27] proved the global rigidity. In this paper, based on an observation of Zhou [37] , we prove this conjecture for inversive distance in ( − 1 , + ∞ ) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in ( − 1 , + ∞ ) . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
4. On solution-free sets of integers.
- Author
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Hancock, Robert and Treglown, Andrew
- Subjects
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INTEGERS , *SET theory , *LINEAR equations , *MATHEMATICS , *COMBINATORICS - Abstract
Given a linear equation L , a set A ⊆ [ n ] is L -free if A does not contain any ‘non-trivial’ solutions to L . In this paper we consider the following three general questions: (i) What is the size of the largest L -free subset of [ n ] ? (ii) How many L -free subsets of [ n ] are there? (iii) How many maximal L -free subsets of [ n ] are there? We completely resolve (i) in the case when L is the equation p x + q y = z for fixed p , q ∈ N where p ≥ 2 . Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L , thereby refining a special case of a result of Green (2005). We also give various bounds on the number of maximal L -free subsets of [ n ] for three-variable homogeneous linear equations L . For this, we make use of container and removal lemmas of Green (2005). [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
5. Limits of mappings.
- Author
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Hosseini, Lucas, Nešetřil, Jaroslav, and Ossona de Mendez, Patrice
- Subjects
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MATHEMATICAL mappings , *ORDERED algebraic structures , *MATHEMATICS theorems , *COMBINATORICS , *MATHEMATICS - Abstract
In this paper we consider a simple algebraic structure — sets with a single endofunction. We shall see that from the point of view of structural limits, even this simplest case is both interesting and difficult. Nevertheless we obtain the shape of limit objects in the full generality, and we prove the inverse theorem in the case of quantifier-free limits. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
6. On the generalised colouring numbers of graphs that exclude a fixed minor.
- Author
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van den Heuvel, Jan, de Mendez, Patrice Ossona, Quiroz, Daniel, Rabinovich, Roman, and Siebertz, Sebastian
- Subjects
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GRAPHIC methods , *COMBINATORICS , *POLYNOMIALS , *MATHEMATICS , *ALGORITHMS - Abstract
The generalised colouring numbers col r ( G ) and wcol r ( G ) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r -colouring number col r and a polynomial bound for the weak r -colouring number wcol r . In particular, we show that if G excludes K t as a minor, for some fixed t ≥ 4 , then col r ( G ) ≤ t − 1 2 ( 2 r + 1 ) and wcol r ( G ) ≤ r + t − 2 t − 2 ( t − 3 ) ( 2 r + 1 ) ∈ O ( r t − 1 ) . In the case of graphs G of bounded genus g , we improve the bounds to col r ( G ) ≤ ( 2 g + 3 ) ( 2 r + 1 ) (and even col r ( G ) ≤ 5 r + 1 if g = 0 , i.e. if G is planar) and wcol r ( G ) ≤ ( 2 g + r + 2 2 ) ( 2 r + 1 ) . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
7. The freeness of Ish arrangements.
- Author
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Abe, Takuro, Suyama, Daisuke, and Tsujie, Shuhei
- Subjects
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CATALAN numbers , *COMBINATORICS , *MATHEMATICS , *BINARY operations , *APPLIED mathematics - Abstract
The Ish arrangement was introduced by Armstrong to give a new interpretation of the q , t -Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
8. Smith normal form in combinatorics.
- Author
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Stanley, Richard P.
- Subjects
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COMBINATORICS , *MATHEMATICS , *ALGEBRAIC equations , *HYPERPLANES , *GEOMETRY - Abstract
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
9. Average radial integrability spaces of analytic functions
- Author
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Manuel D. Contreras, Luis Rodríguez-Piazza, Tanausú Aguilar-Hernández, Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI), and Universidad de Sevilla. Departamento de Análisis Matemático
- Subjects
Mathematics::Complex Variables ,30H20 (Primary), 47B33 (Primary), 47D06 (Primary), 46E15 (Secondary), 47G10 (Secondary) ,Holomorphic function ,Mixed norm spaces ,Hardy space ,Characterization (mathematics) ,Projection (linear algebra) ,Functional Analysis (math.FA) ,Separable space ,Mathematics - Functional Analysis ,Combinatorics ,symbols.namesake ,FOS: Mathematics ,symbols ,Radial integrability ,Bergman projection ,Unit (ring theory) ,Analysis ,Analytic function ,Mathematics - Abstract
In this paper we introduce the family of spaces $RM(p,q)$, $1\leq p,q\leq +\infty$. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when $p=\infty$) and Bergman spaces (when $p=q$). We characterize the inclusion between $RM(p_1,q_1)$ and $RM(p_2,q_2)$ depending on the parameters. For $1, Comment: 31 pages
- Published
- 2022
10. Friezes, weak friezes, and T-paths
- Author
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Ilke Canakci, Peter Jørgensen, and Mathematics
- Subjects
Cluster algebra ,Polygon dissection ,Frieze ,Mathematics::Combinatorics ,Applied Mathematics ,010102 general mathematics ,Frieze pattern ,Positivity ,01 natural sciences ,05B45, 05E15, 05E99, 51M20 ,Combinatorics ,Frieze group ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,010307 mathematical physics ,Generalised frieze pattern ,0101 mathematics ,Algebra over a field ,Computer Science::Data Structures and Algorithms ,Cluster expansion formula ,Semifield ,Mathematics - Abstract
Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued., 17 pages
- Published
- 2021
11. The (2,3)-generation of the finite unitary groups
- Author
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Marco Antonio Pellegrini and M.C. Tamburini Bellani
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Algebra and Number Theory ,Simple group ,010102 general mathematics ,Generation ,01 natural sciences ,Unitary state ,Combinatorics ,Unitary group ,Integer ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Prime power ,Settore MAT/02 - ALGEBRA ,Mathematics - Abstract
In this paper we prove that the unitary groups SU n ( q 2 ) are ( 2 , 3 ) -generated for any prime power q and any integer n ≥ 8 . By previous results this implies that, if n ≥ 3 , the groups SU n ( q 2 ) and PSU n ( q 2 ) are ( 2 , 3 ) -generated, except when ( n , q ) ∈ { ( 3 , 2 ) , ( 3 , 3 ) , ( 3 , 5 ) , ( 4 , 2 ) , ( 4 , 3 ) , ( 5 , 2 ) } .
- Published
- 2020
12. The Erdős-Ko-Rado theorem for the derangement graph of the projective general linear group acting on the projective space
- Author
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Pablo Spiga and Spiga, P
- Subjects
Conjecture ,010102 general mathematics ,Independent set ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Derangement graph ,Derangement ,Projective general linear group ,Computational Theory and Mathematics ,Hyperplane ,010201 computation theory & mathematics ,Erdős-Ko-Rado theorem ,Discrete Mathematics and Combinatorics ,Coset ,Projective space ,Projective linear group ,0101 mathematics ,Erdős–Ko–Rado theorem ,Mathematics - Abstract
In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL n + 1 ( q ) , in its natural action on the points of the n-dimensional projective space, is either a coset of the stabiliser of a point or a coset of the stabiliser of a hyperplane. This gives a positive solution to [15, Conjecture 2] .
- Published
- 2019
13. Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
- Author
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Victoria Lebed, Leandro Vendramin, Mathematics, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), CONICET, PICT-2014-1376, ICTP, Henri Lebesgue Center, University of Nantes, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
- Subjects
SHELF ,Algebraic structure ,CUBICAL HOMOLOGY ,Matemáticas ,General Mathematics ,Cellular homology ,Group cohomology ,BIRACK ,Yang-Baxter equation ,010103 numerical & computational mathematics ,16T25, 20N02, 55N35, 57M27 ,CYCLE SET ,QUANDLE ,01 natural sciences ,Matemática Pura ,Combinatorics ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,RACK ,0101 mathematics ,YANG–BAXTER EQUATION ,MSC2010: 16T25, 20N02, 55N35, 57M27 ,cubical homology ,Mathematics ,[MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT] ,quandle ,Yang–Baxter equation ,010102 general mathematics ,Degenerate energy levels ,extension ,rack ,birack ,K-Theory and Homology (math.KT) ,16. Peace & justice ,BRAIDED HOMOLOGY ,shelf ,cycle set ,Mathematics - K-Theory and Homology ,[MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA] ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,EXTENSION ,CIENCIAS NATURALES Y EXACTAS ,braided homology - Abstract
This paper deals with left non-degenerate set-theoretic solutions to the Yang-Baxter equation (=LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties., 39 pages, several figures. Accepted for publication in Advances in Mathematics
- Published
- 2017
14. Finite primitive permutation groups containing a permutation having at most four cycles
- Author
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Joy Morris, Simon Guest, Pablo Spiga, Cheryl E. Praeger, Guest, S, Morris, J, Praeger, C, and Spiga, P
- Subjects
Conjugacy classe ,Algebra and Number Theory ,Primitive permutation group ,Cycle structure ,010102 general mathematics ,Representation (systemics) ,Group Theory (math.GR) ,0102 computer and information sciences ,Disjoint sets ,Permutation group ,Fixed point ,01 natural sciences ,Combinatorics ,Permutation ,Conjugacy class ,010201 computation theory & mathematics ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
We classify the finite primitive groups containing a permutation with at most four cycles (including fixed points) in its disjoint cycle representation., Comment: 17 papers, 5 tables
- Published
- 2016
15. The limiting distribution of the product replacement algorithm for finitely generated prosoluble groups
- Author
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Andrea Lucchini, Eloisa Detomi, Marta Morigi, Detomi, Eloisa, Lucchini, Andrea, and Morigi, Marta
- Subjects
Discrete mathematics ,Finite group ,Uniform distribution (continuous) ,Algebra and Number Theory ,Distribution (number theory) ,Group (mathematics) ,010102 general mathematics ,Commutator subgroup ,Asymptotic distribution ,0102 computer and information sciences ,Generator ,01 natural sciences ,Prosoluble group ,Combinatorics ,Stallings theorem about ends of groups ,010201 computation theory & mathematics ,Bounded function ,0101 mathematics ,Algorithm ,Mathematics ,Probability - Abstract
Babai and Pak proved that the product replacement algorithm (a widely used heuristic algorithm intended to rapidly generate nearly uniformly distributed random elements in a finite group G) has a flaw. Indeed the projection of the uniform distribution on generating n-tuples onto the first component may not give the uniform distribution on elements of G. In this paper we examine the difference between this distribution and the uniform one, in the particular case when G is a finitely generated prosoluble group. Under some additional hypotheses (for example if the derived subgroup is pronilpotent), this bias is bounded away from 1. However even for soluble groups, the problem pointed out by Babai and Pak is unavoidable. We construct a sequence of finite soluble groups of derived length 4 for which the bias is close to 1.
- Published
- 2016
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