186 results
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2. Pointwise Boundary Differentiability for Fully Nonlinear Elliptic Equations
- Author
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Wu, Duan, Lian, Yuanyuan, and Zhang, Kai
- Subjects
Mathematics - Analysis of PDEs ,35B65, 35J25, 35J60, 35D40 ,General Mathematics ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this paper, we prove the pointwise boundary differentiability for viscosity solutions of fully nonlinear elliptic equations. This generalizes the previous related results for linear equations. The geometrical conditions in this paper are pointwise and more general than before. Moreover, our proofs are relatively simple.
- Published
- 2023
3. 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
4. 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
5. 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
6. 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
7. On connectivity of the facet graphs of simplicial complexes
- Author
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Ilan Newman and Yuri Rabinovich
- Subjects
Combinatorics ,Connected component ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Graph ,Mathematics - Abstract
The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of d-cycles, d-hypertrees and d-hypercuts are, respectively, (d +1)-, d-and (n − d − 1)-vertex-connected. It is also shown that the facet graph of a d-cycle cannot be split into more than s connected components by removing at most s vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.
- Published
- 2019
8. Optimal quantization for the Cantor distribution generated by infinite similutudes
- Author
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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
9. 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
10. Explicit sentences distinguishing Mcduff’s II1 factors
- Author
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Bradd Hart, Henry Towsner, and Isaac Goldbring
- Subjects
Pure mathematics ,Continuum (topology) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Upper and lower bounds ,Separable space ,Quantifier (logic) ,0103 physical sciences ,Pairwise comparison ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Arithmetic ,Mathematics - Abstract
Recently, Boutonnet, Chifan, and Ioana proved that McDuff’s examples of continuum many pairwise non-isomorphic separable II1 factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht–Fra¨isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.
- Published
- 2018
11. 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
12. 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
13. Uo-convergence and its applications to Cesàro means in Banach lattices
- Author
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Niushan Gao, Foivos Xanthos, and Vladimir G. Troitsky
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Net (mathematics) ,Mathematical proof ,01 natural sciences ,Wedge (geometry) ,010101 applied mathematics ,Alpha (programming language) ,Lattice (order) ,Convergence (routing) ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
A net (x α ) in a vector lattice X is said to uo-converge to x if $$\left| {{x_\alpha } - x} \right| \wedge u\xrightarrow{o}0$$ for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uoconvergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [27, 26]. In the second part, we use uo-convergence to study convergence of Cesaro means in Banach lattices. In particular, we establish an intrinsic version of Komlos’ Theorem, which extends the main results of [35, 16, 31] in a uniform way. We also develop a new and unified approach to Banach–Saks properties and Banach–Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [20, 24, 25].
- Published
- 2017
14. Sharp reversed Hardy–Littlewood–Sobolev inequality on R n
- Author
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Quốc Anh Ngô and Van Hoang Nguyen
- Subjects
010101 applied mathematics ,Combinatorics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Algebra over a field ,Type (model theory) ,01 natural sciences ,Mathematics ,Sobolev inequality - Abstract
This is the first in our series of papers that concerns Hardy–Littlewood–Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space R n, $$\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}$$ , for any non-negative functions f ∈ L p(R n), g ∈ L r(R n), and p, r ∈ (0, 1), λ > 0 such that 1/p+1/r −λ/n = 2. We will also explore some estimates for ln,p,r and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.
- Published
- 2017
15. 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
16. Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators
- Author
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Daniel Pellegrino, N. Albuquerque, Juan B. Seoane-Sepúlveda, and Frédéric Bayart
- Subjects
Multilinear map ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Probabilistic logic ,Banach space ,010103 numerical & computational mathematics ,Type inequality ,Type (model theory) ,Mathematical proof ,01 natural sciences ,Algebra ,0101 mathematics ,Hardy–Littlewood inequality ,Mathematics ,media_common - Abstract
In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust-Hilletype inequalities.
- Published
- 2015
17. Homological smoothness and deformations of generalized Weyl algebras
- Author
-
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
18. On the structure of the degrees of relative provability
- Author
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Uri Andrews, Mingzhong Cai, David Diamondstone, Joseph S. Miller, and Steffen Lempp
- Subjects
Discrete mathematics ,Operator (computer programming) ,Computable function ,General Mathematics ,Calculus ,Structure (category theory) ,Jump ,Consistency (knowledge bases) ,Extension (predicate logic) ,Inversion (discrete mathematics) ,Measure (mathematics) ,Mathematics - Abstract
We investigate the structure of the degrees of provability, which measure the proof-theoretic strength of statements asserting the totality of given computable functions. The degrees of provability can also be seen as an extension of the investigation of relative consistency statements for first-order arithmetic (which can be viewed as Π10 -statements, whereas statements of totality of computable functions are Π20 -statements); and the structure of the degrees of provability can be viewed as the Lindenbaum algebra of true Π20 -statements in first-order arithmetic. Our work continues and greatly expands the second author’s paper on this topic by answering a number of open questions from that paper, comparing three different notions of a jump operator and studying jump inversion as well as the corresponding high/low hierarchies, investigating the structure of true Π10 -statements as a substructure, and connecting the degrees of provability to escape and domination properties of computable functions.
- Published
- 2015
19. The commuting graph of the symmetric inverse semigroup
- Author
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Wolfram Bentz, Konieczny Janusz, and João Araújo
- Subjects
Combinatorics ,Discrete mathematics ,Nilpotent ,Cancellative semigroup ,Symmetric group ,Semigroup ,General Mathematics ,Bicyclic semigroup ,Inverse element ,Special classes of semigroups ,Mathematics ,Symmetric inverse semigroup - Abstract
The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dolžan and Oblak claimed that this upper bound is in fact the exact value. The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.
- Published
- 2015
20. Uncountably many permutation stable groups
- Author
-
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
21. Derived categories of skew quadric hypersurfaces
- Author
-
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
22. 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
23. 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
24. 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
25. 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
26. 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
27. 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
28. 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
29. 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
30. 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
31. 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
32. 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
33. 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
34. 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
35. Singular vectors on manifolds and fractals
- Author
-
Barak Weiss, Nikolay G. Moshchevitin, and Dmitry Kleinbock
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Fractal ,Dimension (vector space) ,Mathematics::Number Theory ,General Mathematics ,Diophantine equation ,Irrational number ,Affine space ,Algebra over a field ,Submanifold ,Mathematics - Abstract
We generalize Khintchine’s method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of ℝn, in particular on any analytic submanifold of ℝn of dimension ≥2 which is not contained in a proper rational affine subspace.
- Published
- 2021
36. Codimension growth of simple Jordan superalgebras
- Author
-
Mikhail Zaicev and Ivan P. Shestakov
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,Zero (complex analysis) ,Field (mathematics) ,ÁLGEBRAS DE JORDAN ,Codimension ,Superalgebra ,Integer ,Simple (abstract algebra) ,Lie algebra ,Algebraically closed field ,Mathematics - Abstract
We study asymptotic behaviour of graded and non-graded codimensions of simple Jordan superalgebras over a field of characteristic zero. It is known that the PI-exponent of any finite-dimensional associative or Jordan or Lie algebra A is a non-negative integer less than or equal to the dimension of algebra A. Moreover, the PI-exponent is equal to the dimension if and only if A is simple provided that the base field is algebraically closed. In the present paper we prove that for a Jordan superalgebra P(t) = H(Mt∣t, trp) its non-graded and ℤ2-graded exponents are strictly less than dim P(t). In particular, exp P(2) is fractional.
- Published
- 2021
37. Two-player stochastic games I: A reduction
- Author
-
Nicolas Vieille, Laboratoire d'économétrie de l'École polytechnique (CECO), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Groupe de Recherche en Economie Théorique et Appliquée (GREThA), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)
- Subjects
Computer Science::Computer Science and Game Theory ,Class (set theory) ,Reduction (recursion theory) ,General Mathematics ,05 social sciences ,Stochastic game ,ComputingMilieux_PERSONALCOMPUTING ,TheoryofComputation_GENERAL ,Action (physics) ,[SHS.ECO.ECO]Humanities and Social Sciences/Economics and Finance/domain_shs.eco.eco ,0502 economics and business ,stochastic games ,050206 economic theory ,Finite state ,050207 economics ,Algebra over a field ,Mathematical economics ,Mathematics - Abstract
This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper.
- Published
- 2000
38. 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
39. 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
40. Combinatorial generation via permutation languages. II. Lattice congruences
- Author
-
Torsten Mütze and Hung P. Hoang
- Subjects
General Mathematics ,Polytope ,Congruence relation ,Hamiltonian path ,QA76 ,Combinatorics ,symbols.namesake ,Lattice (module) ,Permutation ,Cover (topology) ,Symmetric group ,symbols ,Hypercube ,QA ,Mathematics - Abstract
This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.\ud \ud
- Published
- 2021
41. 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
42. Subset selection for matrices with fixed blocks
- Author
-
Jiaxin Xie and Zhiqiang Xu
- Subjects
FOS: Computer and information sciences ,Selection (relational algebra) ,General Mathematics ,Matrix norm ,Block (permutation group theory) ,Interlacing ,Combinatorics ,Matrix (mathematics) ,Asymptotically optimal algorithm ,Computer Science - Data Structures and Algorithms ,Data Structures and Algorithms (cs.DS) ,Moore–Penrose pseudoinverse ,Column (data store) ,Mathematics - Abstract
Subset selection for matrices is the task of extracting a column sub-matrix from a given matrix $B\in\mathbb{R}^{n\times m}$ with $m>n$ such that the pseudoinverse of the sampled matrix has as small Frobenius or spectral norm as possible. In this paper, we consider a more general problem of subset selection for matrices that allows a block to be fixed at the beginning. Under this setting, we provide a deterministic method for selecting a column sub-matrix from $B$. We also present a bound for both the Frobenius and spectral norms of the pseudoinverse of the sampled matrix, showing that the bound is asymptotically optimal. The main technology for proving this result is the interlacing families of polynomials developed by Marcus, Spielman, and Srivastava. This idea also results in a deterministic greedy selection algorithm that produces the sub-matrix promised by our result.
- Published
- 2021
43. 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
44. 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
45. 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
46. Minimal varieties of PI-superalgebras with graded involution
- Author
-
Viviane Ribeiro Tomaz da Silva, Ernesto Spinelli, and Onofrio Mario Di Vincenzo
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,Rank (linear algebra) ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Subalgebra ,Zero (complex analysis) ,Triangular matrix ,$ast$-graded polynomial identities ,Field (mathematics) ,0102 computer and information sciences ,Graded algebras ,involutions ,exponent ,minimal varieties ,01 natural sciences ,010201 computation theory & mathematics ,Exponent ,Involution (philosophy) ,0101 mathematics ,Variety (universal algebra) ,Mathematics - Abstract
In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed *-graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary ℤ2-grading and graded involution.
- Published
- 2021
47. A Tverberg type theorem for collectively unavoidable complexes
- Author
-
Duško Jojić, Gaiane Panina, and Rade T. Živaljević
- Subjects
Combinatorics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,Discrete Morse theory ,0102 computer and information sciences ,Join (topology) ,Extension (predicate logic) ,0101 mathematics ,Algebra over a field ,Type (model theory) ,01 natural sciences ,Mathematics - Abstract
We prove that the symmetrized deleted join SymmDelJoin( $$\mathcal{K}$$ ) of a “balanced family” $$\mathcal{K}$$ = 〈Ki〉 =1 of collectively r-unavoidable subcomplexes of 2[m] is (m−r−1)-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result which is more conceptual and more general than previously known results. Already the case r = 2 of this result seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman’s discrete Morse theory.
- Published
- 2021
48. Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces
- Author
-
Lien-Yung Kao
- Subjects
Pure mathematics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,Entropy (information theory) ,0102 computer and information sciences ,0101 mathematics ,Mathematics::Geometric Topology ,01 natural sciences ,Convexity ,Mathematics - Abstract
In this paper, we extend the construction of pressure metrics to Teichmuller spaces of surfaces with punctures. This construction recovers Thurston’s Riemannian metric on Teichmuller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.
- Published
- 2020
49. 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
50. Codimensions of star-algebras and low exponential growth
- Author
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Daniela La Mattina, Antonio Giambruno, Giambruno A., and La Mattina D.
- Subjects
Involution (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Settore MAT/02 - Algebra ,Exponential growth ,010201 computation theory & mathematics ,Bounded function ,Exponent ,0101 mathematics ,polynomial identity, involution, growth ,Mathematics - Abstract
In this paper we prove that if A is any algebra with involution * satisfying a non-trivial polynomial identity, then its sequence of *-codimensions is eventually non-decreasing. Furthermore, by making use of the *-exponent we reconstruct the only two *-algebras, up to T*-equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.
- Published
- 2020
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