85 results
Search Results
2. On the size of subsets of $$\mathbb{F}_p^n$$ without p distinct elements summing to zero
- Author
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Lisa Sauermann
- Subjects
Mathematics - Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Zero (complex analysis) ,Lattice (group) ,0102 computer and information sciences ,Infinity ,01 natural sciences ,Upper and lower bounds ,Prime (order theory) ,Combinatorics ,Integer ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Maximum size ,Combinatorics (math.CO) ,Number Theory (math.NT) ,0101 mathematics ,Constant (mathematics) ,media_common ,Mathematics - Abstract
Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages
- Published
- 2021
3. On the pair correlations of powers of real numbers
- Author
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Christoph Aistleitner and Simon Baker
- Subjects
11K06, 11K60 ,General Mathematics ,Modulo ,FOS: Physical sciences ,0102 computer and information sciences ,Lebesgue integration ,01 natural sciences ,Combinatorics ,symbols.namesake ,Pair correlation ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematical Physics ,Real number ,Mathematics ,Sequence ,Mathematics - Number Theory ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,010201 computation theory & mathematics ,symbols ,Martingale (probability theory) ,Mathematics - Probability - Abstract
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics
- Published
- 2021
4. Finite groups, 2-generation and the uniform domination number
- Author
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Scott Harper and Timothy C. Burness
- Subjects
Domination analysis ,Group (mathematics) ,General Mathematics ,Group Theory (math.GR) ,Conjugate element ,Combinatorics ,Conjugacy class ,Symmetric group ,Dominating set ,Simple group ,FOS: Mathematics ,Classification of finite simple groups ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G$ be a finite $2$-generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_1, \ldots, x_k$, there exists $y \in G$ such that $G = \langle x_i, y\rangle$ for all $i$. The more restrictive notion of uniform spread, denoted $u(G)$, requires $y$ to be chosen from a fixed conjugacy class of $G$, and a theorem of Breuer, Guralnick and Kantor states that $u(G) \geqslant 2$ for every non-abelian finite simple group $G$. For any group with $u(G) \geqslant 1$, we define the uniform domination number $\gamma_u(G)$ of $G$ to be the minimal size of a subset $S$ of conjugate elements such that for each nontrivial $x \in G$ there exists $y \in S$ with $G = \langle x, y \rangle$ (in this situation, we say that $S$ is a uniform dominating set for $G$). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on $\gamma_u(G)$ for all simple groups $G$. In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups $G$ with $\gamma_u(G)=2$, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for $G$. We also establish new results concerning the $2$-generation of soluble and symmetric groups, and we present several open problems., Comment: 60 pages; to appear in Israel Journal of Mathematics
- Published
- 2020
5. Tangent categories of algebras over operads
- Author
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Joost Nuiten, Matan Prasma, Yonatan Harpaz, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and Harpaz, Yonatan
- Subjects
Model category ,General Mathematics ,Parameterized complexity ,[MATH] Mathematics [math] ,[MATH.MATH-AT] Mathematics [math]/Algebraic Topology [math.AT] ,0102 computer and information sciences ,Mathematics::Algebraic Topology ,01 natural sciences ,Spectrum (topology) ,Combinatorics ,Mathematics::Category Theory ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Cotangent complex ,Mathematics - Algebraic Topology ,[MATH]Mathematics [math] ,0101 mathematics ,Algebra over a field ,Mathematics ,010102 general mathematics ,Tangent ,55P42, 18G55, 18D50 ,16. Peace & justice ,Cohomology ,010201 computation theory & mathematics ,[MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT] - Abstract
Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper., Comment: The section concerning stabilization of model categories was separated into an independent paper, appearing now as arXiv:1802.08031
- Published
- 2019
6. Optimal quantization for the Cantor distribution generated by infinite similutudes
- Author
-
Mrinal Kanti Roychowdhury
- Subjects
General Mathematics ,Quantization (signal processing) ,010102 general mathematics ,Dynamical Systems (math.DS) ,0102 computer and information sciences ,01 natural sciences ,Probability vector ,Combinatorics ,Cantor set ,60Exx, 28A80, 94A34 ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Cantor distribution ,Borel probability measure ,Mathematics - Abstract
Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that $$P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}$$ , where for each j ∈ ℕ and x ∈ ℝ, $$S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}$$ . Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector $$(\frac{1}{2},\frac{1}{{{2^2}}},...)$$ , for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.
- Published
- 2019
7. Counting non-uniform lattices
- Author
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Mikhail Belolipetsky and Alexander Lubotzky
- Subjects
Conjecture ,Mathematics - Number Theory ,22E40 (Primary), 11N45, 20G30 (Secondary) ,Rank (linear algebra) ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Conjugacy class ,010201 computation theory & mathematics ,Log-log plot ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Constant (mathematics) ,Mathematics - Group Theory ,Mathematics - Abstract
In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices., Comment: 23 pages, revised following referee's comments. Dedicated to Aner Shalev on his 60th birthday. This paper is related to our previous work arXiv:0905.1841 with which it shares some preliminaries
- Published
- 2019
8. Homotopical Morita theory for corings
- Author
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Alexander Berglund and Kathryn Hess
- Subjects
Pure mathematics ,General Mathematics ,Coalgebra ,Mathematics::Algebraic Topology ,01 natural sciences ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,16T15, 55U35 (Primary), 18G55, 55U15 (Secondary) ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Category Theory (math.CT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Algebra over a field ,Descent (mathematics) ,Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Monoidal category ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Coring ,Rings and Algebras (math.RA) ,Morita therapy ,010307 mathematical physics - Abstract
A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring., Comment: 46 pages
- Published
- 2018
9. Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems
- Author
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Gerhard Larcher, Mark Lewko, and Christoph Aistleitner
- Subjects
General Mathematics ,FOS: Physical sciences ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Mathematical Physics ,Mathematics ,Sequence ,Mathematics - Number Theory ,Lebesgue measure ,Probability (math.PR) ,010102 general mathematics ,Zero (complex analysis) ,Mathematical Physics (math-ph) ,Function (mathematics) ,16. Peace & justice ,Distribution (mathematics) ,Mathematics - Classical Analysis and ODEs ,010201 computation theory & mathematics ,Hausdorff dimension ,Metric (mathematics) ,11K55, 11B30, 11B13, 11J54, 11J71, 11K60 ,Mathematics - Probability ,Energy (signal processing) - Abstract
For a sequence of integers $\{a(x)\}_{x \geq 1}$ we show that the distribution of the pair correlations of the fractional parts of $\{ \langle \alpha a(x) \rangle \}_{x \geq 1}$ is asymptotically Poissonian for almost all $\alpha$ if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of $\alpha$ such that $\{\langle \alpha x^d \rangle\}$ fails to have Poissonian pair correlation is at most $\frac{d+2}{d+3} < 1$. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least $\frac{2}{d+1}$. An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved., Comment: 22 pages. Version 2 contains an appendix by Jean Bourgain. Version 3 with corrections suggested by the referee. The paper will be published in the Israel Journal of Mathematics
- Published
- 2017
10. PFA and guessing models
- Author
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Nam Trang
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Supercompact cardinal ,03E45, 03E55, 03E47 ,Mathematics::General Topology ,Mathematics - Logic ,01 natural sciences ,Mathematics::Logic ,Variation (linguistics) ,Consistency (statistics) ,Hull ,0103 physical sciences ,FOS: Mathematics ,Proper forcing axiom ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Logic (math.LO) ,Mathematics - Abstract
This paper explores the consistency strength of The Proper Forcing Axiom ($\textsf{PFA}$) and the theory (T) which involves a variation of the Viale-Wei$\ss$ guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper implies that the theory "$\sf{AD}$$_\mathbb{R} + \Theta$ is regular" is consistent relative to (T) and to $\textsf{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\textsf{PFA}$, which is roughly "$\sf{AD}$$_\mathbb{R} + \textsf{DC}$".
- Published
- 2016
11. Homological smoothness and deformations of generalized Weyl algebras
- Author
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Liyu Liu
- Subjects
Polynomial ,Pure mathematics ,Weyl algebra ,Smoothness (probability theory) ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,K-Theory and Homology (math.KT) ,Mathematics - Rings and Algebras ,16. Peace & justice ,Quantitative Biology::Genomics ,01 natural sciences ,Algebra ,Rings and Algebras (math.RA) ,Mathematics - K-Theory and Homology ,0103 physical sciences ,Converse ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
It is an immediate conclusion from Bavula's papers \cite{Bavula:GWA-def}, \cite{Bavula:GWA-tensor-product} that if a generalized Weyl algebra $A=\kk[z;\lambda,\eta,\varphi(z)]$ is homologically smooth, then the polynomial $\varphi(z)$ has no multiple roots. We prove in this paper that the converse is also true. Moreover, formal deformations of $A$ are studied when $\kk$ is of characteristic zero., Comment: Final version, 36 pages
- Published
- 2015
12. Uncountably many permutation stable groups
- Author
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Levit, Arie and Lubotzky, Alexander
- Subjects
General Mathematics ,FOS: Mathematics ,Group Theory (math.GR) ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,Mathematics - Group Theory - Abstract
In a 1937 paper B.H. Neumann constructed an uncountable family of $2$-generated groups. We prove that all of his groups are permutation stable by analyzing the structure of their invariant random subgroups.
- Published
- 2022
13. Derived categories of skew quadric hypersurfaces
- Author
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Ueyama, Kenta
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Probability ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,General Mathematics ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Mathematics - Rings and Algebras ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
The existence of a full strong exceptional sequence in the derived category of a smooth quadric hypersurface was proved by Kapranov. In this paper, we present a skew generalization of this result. Namely, we show that if $S$ is a standard graded $(\pm 1)$-skew polynomial algebra in $n$ variables with $n \geq 3$ and $f = x_1^2+\cdots +x_n^2 \in S$, then the derived category $\operatorname{\mathsf{D^b}}(\operatorname{\mathsf{qgr}} S/(f))$ of the noncommutative scheme $\operatorname{\mathsf{qgr}} S/(f)$ has a full strong exceptional sequence. The length of this sequence is given by $n-2+2^r$ where $r$ is the nullity of a certain matrix over $\mathbb F_2$. As an application, by studying the endomorphism algebra of this sequence, we obtain the classification of $\operatorname{\mathsf{D^b}}(\operatorname{\mathsf{qgr}} S/(f))$ for $n=3, 4$., Comment: 23 pages
- Published
- 2022
14. Representability of matroids by c-arrangements is undecidable
- Author
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Kühne, Lukas and Yashfe, Geva
- Subjects
Mathematics::Combinatorics ,General Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,05B35, 52B40, 14N20, 52C35, 20F10, 03D40 - Abstract
For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank functions of $c$-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a $c$ such that a given matroid has a $c$-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain network coding problems are also undecidable. In the proof, we introduce a generalized Dowling geometry to encode an instance of the uniform word problem for finite groups in matroids of rank three. The $c$-arrangement condition gives rise to some difficulties and their resolution is the main part of the paper., Comment: Improved exposition and added application to network coding
- Published
- 2022
15. Images of multilinear polynomials on n × n upper triangular matrices over infinite fields
- Author
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Gargate, Ivan Gonzales and de Mello, Thiago Castilho
- Subjects
Rings and Algebras (math.RA) ,General Mathematics ,FOS: Mathematics ,16R10 ,Mathematics - Rings and Algebras - Abstract
In this paper we prove that the image of multilinear polynomials evaluated on the algebra $UT_n(K)$ of $n\times n$ upper triangular matrices over an infinite field $K$ equals $J^r$, a power of its Jacobson ideal $J=J(UT_n(K))$. In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for $UT_n(K)$ is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree $r$ in terms of its coefficients. It turns out that the image of a multilinear polynomial $f$ on $UT_n(K)$ is $J^r$ if and only if $f$ has commutator degree $r$., To appear in Israel Journal of Mathematics
- Published
- 2022
16. Amalgamation functors and boundary properties in simple theories
- Author
-
Alexei Kolesnikov, Byunghan Kim, and John Goodrick
- Subjects
Higher-dimensional algebra ,Functor ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Context (language use) ,Elementary abelian group ,Mathematics - Logic ,0102 computer and information sciences ,16. Peace & justice ,01 natural sciences ,Algebra ,03C45 ,010201 computation theory & mathematics ,Simple (abstract algebra) ,FOS: Mathematics ,Stable theory ,0101 mathematics ,Abelian group ,Logic (math.LO) ,Mathematics - Abstract
This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various "dimensions" n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context., 33 pages, 1 figure
- Published
- 2012
17. Strongly dense free subgroups of semisimple algebraic groups
- Author
-
Ben Green, Emmanuel Breuillard, Terence Tao, and Robert M. Guralnick
- Subjects
General Mathematics ,Simple Lie group ,010102 general mathematics ,20G40, 20N99 ,Group Theory (math.GR) ,010103 numerical & computational mathematics ,Reductive group ,Lattice (discrete subgroup) ,01 natural sciences ,Representation theory ,Semisimple algebraic group ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Group Theory ,Group of Lie type ,Locally finite group ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Group Theory ,Group theory ,Mathematics - Abstract
We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type., 27 pages, no figures, submitted, Israel J. Math. It turns out that there is one specific family of algebraic group - namely Sp(4) in characteristic 3 - which our methods are unable to resolve, and we have adjusted the paper to exclude this case from the results
- Published
- 2012
18. Finite groups have even more conjugacy classes
- Author
-
Thomas Michael Keller
- Subjects
Finite group ,Group (mathematics) ,General Mathematics ,Group Theory (math.GR) ,Combinatorics ,Conjugacy class ,Character table ,Symmetric group ,FOS: Mathematics ,Order (group theory) ,20E45 ,Algebra over a field ,Mathematics - Group Theory ,Mathematics - Abstract
In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper., 13 pages
- Published
- 2011
19. Multi-secant lemma
- Author
-
Mina Teicher, J. Y. Kaminski, and A. Kanel-Belov
- Subjects
Combinatorics ,Discrete mathematics ,Mathematics - Algebraic Geometry ,Lemma (mathematics) ,Generalization ,General Mathematics ,FOS: Mathematics ,Equidimensional ,Algebraic Geometry (math.AG) ,Projective variety ,Mathematics - Abstract
We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations. Let $X$ be an equidimensional projective variety of dimension $d$. For a given $k \leq d + 1$, we are interested in the study of the variety of $k$-secants. The classical trisecant lemma just considers the case where $k = 3$ while elsewhere the case $k = d + 2$ is considered. Secants of order from $4$ to $d + 1$ provide service for our main result. In this paper, we prove that if the variety of $k$-secants ($k \leq d + 1$) satisfies the three following conditions: (i) trough every point in $X$, passes at least one $k$-secant, (ii) the variety of $k$-secant satisfies a strong connectivity property that we defined in the sequel, (iii) every $k$-secant is also a ($k+1$)-secant, then the variety $X$ can be embedded into $P^{d+1}$. The new assumption, introduced here, that we called strong connectivity is essential because a naive generalization that does not incorporate this assumption fails as we show in some example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption., Comment: arXiv admin note: text overlap with arXiv:0712.3878
- Published
- 2010
20. Slice monogenic functions
- Author
-
Irene Sabadini, Daniele C. Struppa, and Fabrizio Colombo
- Subjects
Pure mathematics ,Multivector ,Mathematics - Complex Variables ,General Mathematics ,Algebra of physical space ,Universal geometric algebra ,Clifford algebra ,Tensor algebra ,30G35 ,Algebra ,Geometric algebra ,Classification of Clifford algebras ,FOS: Mathematics ,Cellular algebra ,Complex Variables (math.CV) ,Mathematics - Abstract
In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series., Comment: to appear in Israel Journal of Mathematics
- Published
- 2009
21. A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra
- Author
-
Sandro Mattarei
- Subjects
Mathematics - Number Theory ,Divisor ,General Mathematics ,Order (ring theory) ,Mathematics - Rings and Algebras ,law.invention ,Combinatorics ,Invertible matrix ,Rings and Algebras (math.RA) ,law ,Lie algebra ,FOS: Mathematics ,Number Theory (math.NT) ,Algebra over a field ,17B50, 17B40, 12C15, 20C15 ,Mathematics - Abstract
A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method., 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Math
- Published
- 2009
22. Zero-sum problems for abelian p-groups and covers of the integers by residue classes
- Author
-
Zhi-Wei Sun
- Subjects
Discrete mathematics ,Mathematics - Number Theory ,General Mathematics ,Existential quantification ,Elementary abelian group ,11B75, 05A05, 05C07, 05E99, 11B25, 11C08, 11D68, 20D60 ,Combinatorics ,Integer ,FOS: Mathematics ,Mathematics - Combinatorics ,Number Theory (math.NT) ,Combinatorics (math.CO) ,Abelian group ,Prime power ,Mathematics - Abstract
Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.
- Published
- 2009
23. Free group representations from vector-valued multiplicative functions. III
- Author
-
Kuhn, MG, Saliani, S, Steger, T, Kuhn, M, Saliani, S, and Steger, T
- Subjects
Irreducible unitary representation ,Boundary realization ,General Mathematics ,Free group ,FOS: Mathematics ,Duplicity ,Representation Theory (math.RT) ,Oddity ,Mathematics - Representation Theory - Abstract
Let $\pi$ be an irreducible unitary representation of a finitely generated nonabelian free group $\Gamma$; suppose $\pi$ is weakly contained in the regular representation. In 2001 the first and third authors conjectured that such a representation must be either odd or monotonous or duplicitous. In 2004 they introduced the class of multiplicative representations: this is a large class of representations obtained by looking at the action of $\Gamma$ on its Cayley graph. In the second paper of this series we showed that some of the multiplicative representations were monotonous. Here we show that all the other multiplicative representations are either odd or duplicitous. The conjecture is therefore established for multiplicative representations.
- Published
- 2023
24. Complexes of graph homomorphisms
- Author
-
Eric Babson and Dmitry N. Kozlov
- Subjects
General Mathematics ,0102 computer and information sciences ,Mathematics::Algebraic Topology ,01 natural sciences ,Combinatorics ,Windmill graph ,Computer Science::Discrete Mathematics ,Graph power ,Mathematics::Category Theory ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Combinatorics ,Graph homomorphism ,Mathematics - Algebraic Topology ,0101 mathematics ,Mathematics ,Discrete mathematics ,Mathematics::Combinatorics ,Moser spindle ,010102 general mathematics ,Voltage graph ,Butterfly graph ,05C15 ,55P91, 55S35, 57M15 ,010201 computation theory & mathematics ,Friendship graph ,Topological graph theory ,Combinatorics (math.CO) - Abstract
$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lov\'asz to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$ is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have $\varpi_1^k(\thom(K_m,G))\neq 0$, then $\chi(G)\geq k+m$; here $Z_2$-action is induced by the swapping of two vertices in $K_m$, and $\varpi_1$ is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of $Hom(G,H)$ induces a homotopy equivalence. It then follows that $Hom(F,K_n)$ is homotopy equivalent to a direct product of $(n-2)$-dimensional spheres, while $Hom(\bar{F},K_n)$ is homotopy equivalent to a wedge of spheres, where $F$ is an arbitrary forest and $\bar{F}$ is its complement., Comment: This is the first part of the series of papers containing the complete proofs of the results announced in "Topological obstructions to graph colorings". This is the final version which is to appear in Israel J. Math., it has an updated list of references and new remarks on latest developments
- Published
- 2006
25. A twisted Laurent series ring that is a noncrossed product
- Author
-
Timo Hanke
- Subjects
Pure mathematics ,Ring (mathematics) ,General Mathematics ,Laurent series ,Outer automorphism group ,Order (ring theory) ,16S35 (Primary) 16K20 16W60 11Y40 ,Mathematics - Rings and Algebras ,Cyclotomic field ,Rings and Algebras (math.RA) ,Product (mathematics) ,FOS: Mathematics ,Division algebra ,Hasse norm theorem ,Mathematics - Abstract
The striking results on noncrossed products were their existence (Amitsur) and the determination of Q(t) and Q((t)) as their smallest possible centres (Brussel). This paper gives the first fully explicit noncrossed product example over Q((t)). As a consequence, the use of deep number theoretic theorems (local-global principles such as the Hasse norm theorem and density theorems) in order to prove existence is eliminated. Instead, the example can be verified by direct calculations. The noncrossed product proof is short and elementary., Comment: 4 pages (A4), updated version of published paper, the sign error in the definition of the element \pi has been corrected
- Published
- 2005
26. Integral morphisms and log blow-ups
- Author
-
Kato, Fumiharu
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Primary: 14A99, Secondary: 14E05 ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
This paper is a revision of the author's old preprint "Exactness, integrality, and log modifications". We will prove that any quasi-compact morphism of fs log schemes can be modified locally on the base to an integral morphism by base change by fs log blow-ups., Comment: 9 pages, accepted and to appear in Israel Journal of Mathematics
- Published
- 2021
27. Eta invariants of Dirac operators on circle bundles over riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces
- Author
-
Liviu I. Nicolaescu
- Subjects
Mathematics - Differential Geometry ,General Mathematics ,Computation ,Riemann surface ,Dirac (software) ,Mathematical analysis ,Collapse (topology) ,Type (model theory) ,Moduli space ,symbols.namesake ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,Bounding overwatch ,FOS: Mathematics ,symbols ,58G10, 58G18, 53C21, 53B21 ,Adiabatic process ,Mathematics::Symplectic Geometry ,Mathematics ,Mathematical physics - Abstract
We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces on any 4-manifold bounding unions of circle bundles. This belated paper should be regarded as the analytical backbone of dg-ga/9711006. There, we indicated only what changes are needed to extend the methods of the present paper to Seifert fibrations and we focused only to topological and number theoretic aspects related to Froyshov invariants, Latex 2.09, 57 pages, 6 figures
- Published
- 1999
28. Groups whose prime graph on class sizes has a cut vertex
- Author
-
Silvio Dolfi, Lucia Sanus, Emanuele Pacifici, and Víctor Sotomayor
- Subjects
Class (set theory) ,Finite group ,General Mathematics ,Prime number ,Group Theory (math.GR) ,Vertex (geometry) ,Set (abstract data type) ,Combinatorics ,Conjugacy class ,Simple (abstract algebra) ,Prime graph ,FOS: Mathematics ,20E45 ,Finite groups, Conjugacy classes, Prime graph ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\Delta(G)$ has a cut vertex.
- Published
- 2021
29. Weak (1,1) estimates for multiple operator integrals and generalized absolute value functions
- Author
-
Fedor Sukochev, Dmitriy Zanin, and Martijn Caspers
- Subjects
Current (mathematics) ,General Mathematics ,Mathematics - Operator Algebras ,Order (ring theory) ,Absolute value ,Interval (mathematics) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Combinatorics ,Bounded function ,FOS: Mathematics ,Ideal (ring theory) ,Divided differences ,Operator Algebras (math.OA) ,Trace class ,Mathematics - Abstract
Consider the generalized absolute value function defined by \[ a(t) = \vert t \vert t^{n-1}, \qquad t \in \mathbb{R}, n \in \mathbb{N}_{\geq 1}. \] Further, consider the $n$-th order divided difference function $a^{[n]}: \mathbb{R}^{n+1} \rightarrow \mathbb{C}$ and let $1 < p_1, \ldots, p_n < \infty$ be such that $\sum_{l=1}^n p_l^{-1} = 1$. Let $\mathcal{S}_{p_l}$ denote the Schatten-von Neumann ideals and let $\mathcal{S}_{1,\infty}$ denote the weak trace class ideal. We show that for any $(n+1)$-tuple ${\bf A}$ of bounded self-adjoint operators the multiple operator integral $T_{a^{[n]}}^{\bf A}$ maps $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1, \infty}$ boundedly with uniform bound in ${\bf A}$. The same is true for the class of $C^{n+1}$-functions that outside the interval $[-1, 1]$ equal $a$. In [CLPST16] it was proved that for a function $f$ in this class such boundedness of $T^{ {\bf A} }_{f^{[n]}}$ from $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1}$ may fail, resolving a problem by V. Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences., to appear in Israel Journal of Mathematics
- Published
- 2021
30. The normalized cyclomatic quotient associated with presentations of finitely generated groups
- Author
-
Amnon Rosenmann
- Subjects
Discrete mathematics ,Fundamental group ,Presentation of a group ,Cayley graph ,General Mathematics ,Amenable group ,Cyclic group ,Group Theory (math.GR) ,Combinatorics ,Free product ,FOS: Mathematics ,Mathematics - Group Theory ,Quotient ,Bass–Serre theory ,Mathematics - Abstract
Given the Cayley graph of a finitely generated group $G$, with respect to a presentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of vertices of the subgraphs gives rise to the definition of the normalized cyclomatic quotient $\Xi (G^{\alpha})$. The asymptotic behavior of this quotient is similar to the asymptotic behavior of the quotient of the cardinality of the boundary of the subgraph by the cardinality of the subgraph. Using Følner's criterion for amenability one gets that $\Xi (G^{\alpha})$ vanishes for infinite groups if and only if they are amenable. When $G$ is finite then $\Xi (G^{\alpha})=1/|G|$, where $|G|$'> is the cardinality of $G$, and when $G$ is non-amenable then $1-n\leq\Xi (G^{\alpha})\le 0$, with $\Xi (G^{\alpha})=1-n$ if and only if $G$ is free of rank $n$. Thus we see that on special cases $\Xi (G^{\alpha})$ takes the values of the Euler characteristic of $G$. Most of the paper is concerned with formulae for the value of $\Xi (G^{\alpha})$ with respect to that of subgroups and factor groups, and with respect to the decomposition of the group into direct product and free product. Some of the formulae and bounds we get for $\Xi (G^{\alpha})$ are similar to those given for the spectral radius of symmetric random walks on the graph of $G^{\alpha}$, but this is not always the case. In the last section of the paper we define and touch very briefly the balanced cyclomatic quotient, which is defined on concentric balls in the graph and is related to the growth of $G$., Comment: LaTex, 23 pages, no figures
- Published
- 1997
31. Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices
- Author
-
Vishesh Jain
- Subjects
General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Zero (complex analysis) ,0102 computer and information sciences ,01 natural sciences ,Square matrix ,Independent vector ,Combinatorics ,Singular value ,010201 computation theory & mathematics ,Simple (abstract algebra) ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Algebra over a field ,Computer Science::Data Structures and Algorithms ,Random matrix ,Mathematics - Probability ,Mathematics - Abstract
An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $\eta \geq 0$, $\Pr(s_n(M_n) \leq \eta) \lesssim n^{C}\eta + \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a `truly combinatorial' setting., Comment: 28 pages; comments welcome!
- Published
- 2021
32. Lagrangians of hypergraphs II: When colex is best
- Author
-
Natasha Morrison, Shoham Letzter, Vytautas Gruslys, and Apollo - University of Cambridge Repository
- Subjects
Hypergraph ,Mathematics::Combinatorics ,Conjecture ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,05C65, 05C35 ,010201 computation theory & mathematics ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Order (group theory) ,Combinatorics (math.CO) ,0101 mathematics ,Algebra over a field ,Lagrangian ,Counterexample ,Mathematics ,Initial segment - Abstract
A well-known conjecture of Frankl and F\"{u}redi from 1989 states that an initial segment of colex of has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov and a counterexample to an old conjecture of Ahlswede and Katona., Comment: We split our original paper (arXiv:1807.00793v2) into two parts. The first part can be found in arXiv:1807.00793. This is the second part, which consists of 18 pages, including a two-page appendix
- Published
- 2021
33. Hall algebras and graphs of Hecke operators for elliptic curves
- Author
-
Roberto Alvarenga
- Subjects
Pure mathematics ,Structure constants ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,TEORIA DOS NÚMEROS ,Automorphic form ,Elliptic function ,Field (mathematics) ,0102 computer and information sciences ,01 natural sciences ,Mathematics - Algebraic Geometry ,Elliptic curve ,Operator (computer programming) ,Hall algebra ,010201 computation theory & mathematics ,FOS: Mathematics ,Number Theory (math.NT) ,Representation Theory (math.RT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Hecke operator ,Mathematics - Abstract
The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section., Comment: 38 pages, comments are welcome - minor typos were corrected in the second version
- Published
- 2020
34. Detecting structural properties of finite groups by the sum of element orders
- Author
-
Marius Tărnăuceanu
- Subjects
Finite group ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,0102 computer and information sciences ,Function (mathematics) ,01 natural sciences ,Mathematics::Group Theory ,Nilpotent ,010201 computation theory & mathematics ,FOS: Mathematics ,0101 mathematics ,Abelian group ,Algebra over a field ,Element (category theory) ,Mathematics - Group Theory ,Mathematics - Abstract
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
- Published
- 2020
35. The dimension of solution sets to systems of equations in algebraic groups
- Author
-
Anton A. Klyachko and Maria A. Ryabtseva
- Subjects
Finite group ,Pure mathematics ,Rank (linear algebra) ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Mathematics - Algebraic Geometry ,010201 computation theory & mathematics ,Algebraic group ,FOS: Mathematics ,Order (group theory) ,Finitely generated group ,0101 mathematics ,Algebraic number ,Mathematics - Group Theory ,Algebraic Geometry (math.AG) ,Irreducible component ,Mathematics - Abstract
The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th unknown in $i$-th equation is less than the number unknowns. We obtain analogues of this and similar facts for algebraic groups. In particular, our results imply that the dimension of each irreducible component of the variety of homomorphisms from a finitely generated group with infinite abelianisation into an algebraic group $G$ is at least $\dim G$., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.2: An open question added, misprints corrected. V.3: misprints corrected
- Published
- 2020
36. The topological entropy of endomorphisms of Lie groups
- Author
-
Mauro Patrão
- Subjects
Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Lie group ,Torus ,Dynamical Systems (math.DS) ,Group Theory (math.GR) ,0102 computer and information sciences ,Topological entropy ,01 natural sciences ,010201 computation theory & mathematics ,Variational principle ,FOS: Mathematics ,Ergodic theory ,Entropy (information theory) ,Maximal torus ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
In this paper, we determine the topological entropy $h(\phi)$ of a continuous endomorphism $\phi$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(\phi)$ associated to a left invariant distance $d$ just provides an upper bound to $h(\phi)$, which is characterized by the so called variational principle. We prove that \[ h\left(\phi\right) = h\left(\phi|_{T(G_\phi)}\right) \] where $G_\phi$ is the maximal connected subgroup of $G$ such that $\phi(G_\phi) = G_\phi$, and $T(G_\phi)$ is the maximal torus in the center of $G_\phi$. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.
- Published
- 2019
37. Ziegler spectra of serial rings
- Author
-
Lorna Gregory and Gena Puninski
- Subjects
Pure mathematics ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Mathematics - Logic ,Mathematics - Rings and Algebras ,0102 computer and information sciences ,Mathematics::Algebraic Topology ,01 natural sciences ,Mathematics::Logic ,Rings and Algebras (math.RA) ,010201 computation theory & mathematics ,Mathematics::Category Theory ,FOS: Mathematics ,Mathematics::Metric Geometry ,03C60, 16D10, 54B35 ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Logic (math.LO) ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper we prove that the Ziegler spectra of all serial rings are sober. We then use this proof to give a general framework for computing and understanding Ziegler spectra of uniserial rings up to topological indistinguishability. Finally, we illustrate this technique by computing the Ziegler spectra of all rank one uniserial domains up to topological indistinguishability., Comment: 30 pages
- Published
- 2019
38. Generators of semigroups on Banach spaces inducing holomorphic semiflows
- Author
-
Isabelle Chalendar and Wolfgang Arendt
- Subjects
Mathematics::Functional Analysis ,30D05, 47D03, 47B33 ,Pure mathematics ,Open unit ,Generator (category theory) ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Banach space ,0102 computer and information sciences ,Composition (combinatorics) ,01 natural sciences ,Domain (mathematical analysis) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010201 computation theory & mathematics ,FOS: Mathematics ,0101 mathematics ,Reciprocal ,Mathematics ,Analytic function - Abstract
Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication., 14 pages, Accepted, Israel Journal of Mathematics 2018
- Published
- 2018
39. Characters of Iwahori–Hecke algebras
- Author
-
Deke Zhao
- Subjects
Pure mathematics ,Generalization ,General Mathematics ,010102 general mathematics ,Character theory ,Duality (optimization) ,0102 computer and information sciences ,01 natural sciences ,Superalgebra ,010201 computation theory & mathematics ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Quantum ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper we prove a quantum generalization of Regev's theorems in (Israel. J. Math. 195 (2013), 31--35) by applying the Schur-Weyl duality between the quantum superalgebra and Iwahori-Hecke algebra. We also present an alternative proof of the quantized generalizations using the skew character theory of Iwahori-Hecke algebras., Final version: added a new section 5 to contain the algebraic proof of arXiv:1710.02778
- Published
- 2018
40. Topological dynamics and the complexity of strong types
- Author
-
Krzysztof Krupiński, Tomasz Rzepecki, and Anand Pillay
- Subjects
Pure mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Galois group ,Bohr compactification ,Mathematics::General Topology ,Mathematics - Logic ,03C45, 54H20, 03E15, 54H11 ,0102 computer and information sciences ,Type (model theory) ,01 natural sciences ,Mathematics::Logic ,010201 computation theory & mathematics ,Bounded function ,FOS: Mathematics ,Equivalence relation ,0101 mathematics ,Invariant (mathematics) ,Logic (math.LO) ,Trichotomy (mathematics) ,Mathematics - Abstract
We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g. we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over $\emptyset$ is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $\mathfrak{C}$ be a monster model of a countable theory, $p \in S(\emptyset)$, and $E$ be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on $p(\mathfrak{C})$. Then, exactly one of the following holds: (1) $E$ is relatively definable (on $p(\mathfrak{C})$), smooth, and has finitely many classes, (2) $E$ is not relatively definable, but it is type-definable, smooth, and has $2^{\aleph_0}$ classes, (3) $E$ is not type definable and not smooth, and has $2^{\aleph_0}$ classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups., 57 pages
- Published
- 2018
41. Universal inequalities in Ehrhart theory
- Author
-
Gabriele Balletti and Akihiro Higashitani
- Subjects
Discrete mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Polytope ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,010201 computation theory & mathematics ,52B20 (Primary), 52B12 (Secondary) ,Lattice (order) ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics ,media_common - Abstract
In this paper, we show the existence of universal inequalities for the $h^*$-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the $h^*$-polynomial which are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients $h^*_1$ and $h^*_2$ of the $h^*$-vector $(h^*_0,h^*_1,\ldots,h^*_d)$ of a lattice polytope of any degree satisfy Scott's inequality if $h^*_3=0$., Comment: 9 pages, 1 figure
- Published
- 2018
42. Dominated Pesin theory: convex sum of hyperbolic measures
- Author
-
Jairo Bochi, Katrin Gelfert, Christian Bonatti, Facultad de Matemáticas [Santiago de Chile], Pontificia Universidad Católica de Chile (UC), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Instituto de Matemática da Universidade Federal do Rio de Janeiro (IM / UFRJ), and Universidade Federal do Rio de Janeiro (UFRJ)
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,Convex set ,Regular polygon ,Dynamical Systems (math.DS) ,MSC: 37C29, 37C40, 37C50, 37D25, 37D30, 28A33 ,Mathematics::Geometric Topology ,01 natural sciences ,0103 physical sciences ,Converse ,FOS: Mathematics ,Ergodic theory ,Periodic orbits ,Convex combination ,010307 mathematical physics ,Homoclinic orbit ,Mathematics - Dynamical Systems ,37C29, 37C40, 37C50, 37D25, 37D30, 28A33 ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $\mu$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(\mu)$ of periodic orbits for the homoclinic relation, called its \emph{intersection class}. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index such as the dominated splitting is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption., Comment: final version, new co-author, to appear in: Israel Journal of Mathematics
- Published
- 2018
43. Fields of rationality of cusp forms
- Author
-
John Binder
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,Equidistribution theorem ,01 natural sciences ,Measure (mathematics) ,Upper and lower bounds ,Character (mathematics) ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we prove that for any totally real field $F$, weight $k$, and nebentypus character $\chi$, the proportion of Hilbert cusp forms over $F$ of weight $k$ and character $\chi$ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible $GL_2$ representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for $GL_2$; and third, a Plancherel equidsitribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin-Templier and mirrors work of Finis-Lapid-Mueller by introducing an explicit bound for certain families of orbital integrals., Comment: 41 pages
- Published
- 2017
44. Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles
- Author
-
Alexey Pokrovskiy, Noga Alon, and Benny Sudakov
- Subjects
Random graph ,Mathematics::Combinatorics ,Conjecture ,Distribution (number theory) ,05C38, 05C45, 05B15 ,General Mathematics ,010102 general mathematics ,ems ,Complete graph ,Rainbow ,0102 computer and information sciences ,Edge (geometry) ,01 natural sciences ,Hamiltonian path ,Combinatorics ,Set (abstract data type) ,symbols.namesake ,010201 computation theory & mathematics ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Abstract
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured $K_n$ formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties., 9 pages, 1 figure
- Published
- 2017
45. The number of Hamiltonian decompositions of regular graphs
- Author
-
Zur Luria, Roman Glebov, and Benny Sudakov
- Subjects
General Mathematics ,010102 general mathematics ,Complete graph ,Graph theory ,0102 computer and information sciences ,Disjoint sets ,01 natural sciences ,Hamiltonian path ,Graph ,Vertex (geometry) ,Combinatorics ,Hamiltonian decomposition ,symbols.namesake ,010201 computation theory & mathematics ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Partition (number theory) ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Abstract
A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kuhn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r (1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K n is $${n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}$$ .
- Published
- 2017
46. Gibbs u-states for the foliated geodesic flow and transverse invariant measures
- Author
-
Sébastien Alvarez
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,Lebesgue integration ,01 natural sciences ,Manifold ,symbols.namesake ,Harmonic function ,Unit tangent bundle ,0103 physical sciences ,FOS: Mathematics ,Foliation (geology) ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Invariant measure ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Probability measure ,Mathematics - Abstract
This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation with negatively curved leaves. On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when this foliated geodesic flow preserves a Gibbs su-state, i.e. a measure with Lebesgue disintegration both in the stable and unstable horospheres, then it has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. On the other hand we study in detail the projections of Gibbs u-states along the unit spheres tangent to the foliation. We show that they have Lebesgue disintegration in the leaves and that the local densities possess an integral representation analogue to the Poisson representation of harmonic functions., Comment: 50 pages, 13 figures, to appear in Israel Journal of Mathematics
- Published
- 2017
47. Polynomial identities for matrices over the Grassmann algebra
- Author
-
Péter E. Frenkel
- Subjects
Polynomial ,Degree (graph theory) ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Mathematics - Rings and Algebras ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Rings and Algebras (math.RA) ,010201 computation theory & mathematics ,FOS: Mathematics ,0101 mathematics ,Algebra over a field ,Exterior algebra ,Mathematics - Abstract
We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper., Comment: 9 pages
- Published
- 2017
48. The complexity of the topological conjugacy problem for Toeplitz subshifts
- Author
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Burak Kaya
- Subjects
Mathematics::Functional Analysis ,Class (set theory) ,Pure mathematics ,Mathematics::Dynamical Systems ,Relation (database) ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Logic ,Dynamical Systems (math.DS) ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,01 natural sciences ,Toeplitz matrix ,03 medical and health sciences ,0302 clinical medicine ,FOS: Mathematics ,03E15 (primary), 37B10 (secondary) ,030212 general & internal medicine ,Mathematics - Dynamical Systems ,0101 mathematics ,Algebra over a field ,Logic (math.LO) ,Topological conjugacy ,Computer Science::Formal Languages and Automata Theory ,Mathematics - Abstract
In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.
- Published
- 2017
49. Real zeroes of random polynomials, II. Descartes’ rule of signs and anti-concentration on the symmetric group
- Author
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Ken Söze
- Subjects
Discrete mathematics ,Degree (graph theory) ,Logarithm ,Series (mathematics) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,60-XX, 26C10 ,0102 computer and information sciences ,Expected value ,Random polynomials ,01 natural sciences ,Combinatorics ,010201 computation theory & mathematics ,Symmetric group ,Bounding overwatch ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Mathematics - Combinatorics ,Descartes' rule of signs ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Probability ,Mathematics - Abstract
In this sequel to Part-I, we present a different approach to bounding the expected number of real zeroes of random polynomials with real independent identically distributed coefficients or more generally, exchangeable coefficients. We show that the mean number of real zeroes does not grow faster than the logarithm of the degree. The main ingredients of our approach are Descartes' rule of signs and a new anti-concentration inequality for the symmetric group. This paper can be read independently of part-I in this series., Comment: 26 pages
- Published
- 2017
50. Combinatorial properties of Nil–Bohr sets
- Author
-
Jakub Konieczny
- Subjects
Conjecture ,Mathematics - Number Theory ,Relation (database) ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,01 natural sciences ,Bohr model ,Combinatorics ,Set (abstract data type) ,symbols.namesake ,Bounded function ,0103 physical sciences ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Number Theory (math.NT) ,Combinatorics (math.CO) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely $\mathrm{Nil}_d{-}\mathrm{Bohr}$ and $\mathrm{SG}_k$, as introduced by Host and Kra. We prove that any $\mathrm{Nil}_d{-}\mathrm{Bohr}_0$ set is necessarily $\mathrm{SG}_k$ where ${k}$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra., 25 pages
- Published
- 2017
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