Ilya D. Shkredov
We obtain a series of lower bounds for the product set of combinatorial cubes, as well as some non--trivial upper estimates for the multiplicative energy of such sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We obtain upper bounds on the cardinality of Hilbert cubes in finite fields, which avoid large product sets and reciprocals of sum sets. In particular, our results replace recent estimates of N. Hegyvári and P. P. Pach (2020), which appear to be void for all admissible parameters. Our approach is different from that of N. Hegyvári and P. P. Pach and is based on some well-known bounds of double character and exponential sums over arbitrary sets, due to A. A. Karatsuba (1991) and N. G. Moshchevitin (2007), respectively.
Smooth Combinatorial Cubes are IDP
本文针对Tadao Oda提出的所有光滑多面体都具有整数分解性质的猜想，证明了任意维度的光滑组合立方体这一多面体子类满足该性质。
We show that, for every linear ordering of $[2]^n$, there is a large subcube on which the ordering is lexicographic. We use this to deduce that every long sequence contains a long monotone subsequence supported on an affine cube. More generally, we prove an analogous result for linear orderings of $[k]^n$. We show that, for every such ordering, there is a large subcube on which the ordering agrees with one of approximately $\frac{(k-1)!}{2(\ln 2)^k}$ orderings.
Consider a pyramid made out of unit cubes arranged in square horizontal layers, with a ledge of one cube's length around the perimeter of each layer. For any natural number $k$, we can count the number of ways of choosing $k$ unit cubes from the pyramid such that no two cubes are in the same horizontal layer; we can also count the number of ways of choosing $k$ unit cubes from the pyramid such that no two cubes come from the same vertical slice (taken parallel to a fixed edge of the pyramid) {\it or} from two adjacent slices. Djakov and Mityagin first established, using functional analysis, th…
Combinatorial cube packings in cube and torus
我们考虑立方体$z+[0,1]^n$与$z\in \frac{1}{N}\ZZ^n$的顺序随机包装成立方体$[0,2]^n$和环面$\QuotS{\RR^n}{2\ZZ^n}$作为$N\to\infty$。在立方体情况下，$[0,2]^n$作为$N\to\infty$，由此获得的随机立方体填充以概率$1-O(\frac{1}{N})$被简化为单个立方体。在环面的情况下，情况是不同的：对于$n\leq 2$，顺序随机立方体包装产生立方体平铺，但对于$n\geq 3$严格正概率，一个获得不可扩展的立方体包装。因此，我们引入了组合立方体填充的概念，它不是依赖于$N$，而是依赖于一些参数。我们用它们来导出堆积密度的$\frac{1}{N}$幂展开式.显式计算是在立方体的情况下进行的。在环面的情况下，情况更复杂，我们限制自己的情况下，严格正概率$N\to\infty$。证明了环面组合立方填充的下列结果：给出了一般的笛卡尔积结构。我们证明了参数的数量至少是$\frac{n(n+1)}{2}$，我们推测它是最多$2^n-1$。证明了至少含有$2^n-3$立方体的立方体包装是可扩展的。我们发现的最小数目的立方体在不可扩展的立方体包装$n$奇数和$n\leq 6$。
Reconstruction of slices of cubes
本文证明立方体切片可由其图唯一确定，并构造反例说明该性质不适用于一般简单多胞体。
Combinatorics of slices of cubes
本文通过结合组合、代数与数值方法，对维度至六的立方体超平面切片进行了完整计算分类，确定了各维度类型数，并扩展至七维过原点情形，同时分析了分布并提出了新理论结果与猜想。
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
The Cube Recurrence
我们构建了一个组合模型，描述了立方递归，一个非线性递归关系引入Propp，生成家庭的劳伦多项式索引的点在$\mathbb{Z}^3$。在这个过程中，我们证明了Propp和Fomin和Zelevinsky的几个命题，我们得到的Gale-Robinson序列的条款的组合解释。我们还指出，该模型可能会被用来获得一些有趣的结果，某些二部可平面图的完美匹配。
We suggest a combinatorial method of encoding continuous symbolic dynamical systems. A~continuous phase space, the infinite-dimensional cube, turns into the path space of a tree, and the shift is mapped to a transformation which was called a "transfer." The central problem is that of distinguishability: does the encoding separate almost all points of the space? The main result says that the partition of the cube into Weyl simplices satisfies this property.\footnote{{\it Keywords:} combinatorial encoding, transfer, Bernoulli scheme, graded graph.
Multiplicative $q$-hypergeometric series arising from real quadratic fields
安德鲁斯，戴森和希克森表明，2 $q$ -超几何级数，回到拉马努金，有关的实二次域，这解释了有趣的性质，他们的傅里叶系数。这类级数与自守形式也有一个有趣的关系。在这里，我们构建更多这样的例子产生有趣的组合统计。
The present paper is a continuation of author's paper arXiv:1909.00940 [math.AT] devoted to the lemmas of Alexander and Sperner, but is independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-…
We show that there is a constant $c$ such that any colouring of the cube $[3]^n$ in $c \log \log n$ colours contains a monochromatic combinatorial line.
A combinatorial approach to exponential patterns in multiplicative $IP^{\star}$ sets in $\mathbb{N}$
本研究针对乘法$IP^{\star}$集的指数模式相关结论，提出一种组合式证明方法，替代此前基于Stone-Čech紧化代数结构的证明路径，为相关数论组合问题提供新的研究视角。
The Distance Approach to Approximate Combinatorial Counting
通过求解组合定义集合上的随机生成的最优化问题，得到了组合定义集合的基数的快速（多项式时间）估计的一般方法.在几何上，我们通过从立方体中的一个点到子集的平均距离来估计布尔立方体的子集的基数。作为应用，我们提出了一种新的随机多项式时间算法，它通过求解少量的分配问题来逼近0-1矩阵的积和式。
Limit shapes of multiplicative measures associated with coagulation-fragmentation processes and random combinatorial structures
我们发现极限形状的一个家庭的乘法措施的集合上的分区，引起的指数母函数的扩展参数，$a_k\sim Ck^{p-1}, k\to\infty, p>0$，其中$C$是一个正的常数。考虑的措施与可逆的凝聚-碎裂过程和某些组合结构，称为组件。从尺度随机分拆的极限形状出发，证明了其涨落的泛函中心极限定理。我们证明，当组件的大小超过门限值，组件的数量的独立性转换成它们的条件独立性。除其他事项外，本文还讨论了，在一般情况下，极限形状，阈值和凝胶化之间的相互作用。
The Interplay between Additive and Multiplicative Central Sets Theorems
本文建立了一个统一的中心集定理版本，基于拓扑动力学与Stone--Čech紧化代数框架，同时刻画了加性与乘性结构。
AlphaMapleSAT: An MCTS-based Cube-and-Conquer SAT Solver for Hard Combinatorial Problems
AlphaMapleSAT通过结合MCTS与演绎反馈，改进了立方与征服SAT求解器，显著提升了解决难题组合问题的效率。
A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in this paper. Several enumerative polynomials, e.g.\ rank generating function, cube polynomials and degree sequence polynomials are obtained. Some of these results relate to Fibonacci sequence and Padovan sequence.
With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph, which is the Alexander dual of the more well-known independence complex. In particular, the set of satisfiable formulae in k-conjunctive normal form with less than or equal to n variables has the homotopy type of Theta(Cube(n,n-k)), where Cube(n,n-k) is a hypergraph associated to the (n-k)-skeleton of an n-cube. We make partial progress in calculating the homotopy type of theta for these cubical hypergraphs, and we also give cal…
Let $N_1(m)=\max\{n \colon ϕ(n) \leq m\}$ and $N_1 = \{N_1(m) \colon m \in ϕ(\mathbb{N})\}$ where $ϕ(n)$ denotes the Euler's totient function. Masser and Shiu \cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient numbers an…
We present a positive, combinatorial, good formula for the complexity (= number of spanning trees) of the $q$-analog of the $n$-cube. Our method also yields the explicit block diagonalization of the commutant of the $GL(n,F_q)$ action on the $q$-analog of the Boolean algebra.
Let $ E=\{ (1, b_1, \ldots , b_n)\in R^{n+1} \mid \; b_i= \pm 1 ,\; i=1, \ldots, n \}$, $E^{\times n}_{\ne 0} := \{ W=(w_{i_1}, \ldots , w_{i_n}) \mid w_{i_k}\in E, \, k=1, \ldots, n, \, dim \, span(w_{i_1}, \ldots , w_{i_n}) = n \},$ and $q^W_l := |span(w_{i_{n-l+1}}, \ldots , w_{i_n}) \cap E|.$ Then for any weights $p=(p_1, \ldots, p_{2^n})$, $p_i\in R$, $\sum_{i=1}^{2^n}{p_i} =1$ we have for the number of threshold functions $P(2,n)$ the following lower bound $$P(2, n) \geq 2\sum_{W\in E^{\times n}_{\ne 0}}{\frac{1- p_{i_1} -p_{i_2} - \cdots - p_{i_{q_n^W}}}{q_n^W\cdot q_{n-1}^W\cdots q_1^W…
Boundary rigidity of 3D CAT(0) cube complexes
研究离散版边界刚性问题，证明有限3D CAT(0)立方复形及任意有限2D CAT(0)立方复形的组合类型均可由其边界距离重构，推广并加强了Haslegrave的结论。
We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$ denote the minimum number of points inside the $d$-dimensional unit cube $[0,1]^d$ such that they intersect every axis-aligned box inside $[0,1]^d$ of volume greater than $r$. We prove an upper bound on $N(r,d)$, matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on $r$. This fully determines the rate of growth of $N(r,d)$ if $r\in(0,1)$ is fixed.
We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an "edge slide" operation on spanning trees. This gives an answer in the case n=3 to a question implicitly raised by Stanley. Our argument also gives a bijective proof of the n=3 case of a weighted count of the spanning trees of the n-cube due to Martin and Reiner.
A new proof for the multiplicative property of the boolean cumulants with applications to operator-valued case
本文给出了布尔累积量的几个组合性质。一个推论是一个新的证明的乘法性质的布尔累积量系列，可以很容易地适应的情况下，布尔独立合并代数。
On the arithmetic product of combinatorial species
本文介绍了两种新的组合类二元运算：算术积和修正算术积。算术产品给予组合意义的产品的狄利克雷级数和兰伯特系列的背景下，物种。它允许我们引入乘法物种的概念，提升到乘法算术函数的经典概念的组合水平。有趣的组合结构介绍;克隆组装的结构，超克隆树，丰富的矩形等最近的研究卡梅隆，Gewurz和Merola，关于产品行动的上下文中的寡形群，动机引进修改算术产品。利用修正的算术积得到了新的计数结果。我们还推广和简化了Canfield和Pittel最近的一些结果，这些结果与具有限制满足的划分元组的计数有关。
On the interplay between notions of additive and multiplicative largeness and its combinatorial applications
本文研究自然数及一般环结构中加法与乘法“大”概念的关系，证明乘法大可三种方式推出加法大，并据此得到有限域、Diophantine 逼近、等差数列及经典定理强化等应用。
Lower bounds for cube-ideal set-systems
本文通过组合学、凸几何和多面体理论，为立方理想集系统的大小给出指数下界，并对其VC维给出线性下界，应用于图论和组合优化，包括强定向、完美匹配、diojins和理想clutters，涉及Lovász-Plummer猜想。
Cop number of partial cubes
本文研究部分立方体（超立方体的等距子图）的警察数，建立其下界，对子类给出上界，并针对斐波那契立方体改进这些界，该图类与化学图论中的共振图模型相关。
The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\mathbb R^d$ is $\lfloor(3d+1)/2\rfloor$.
We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $n\to +\infty$. We propose to use the Prim order on the vertices instead of the…
We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variet…
Edge-isoperimetric inequalities and influences
我们给出了一个组合证明的结果Kahn，Kalai，和Linial，其中指出，每一个平衡的布尔函数的$n$维布尔立方体有一个变量的影响，至少欧米茄（\frac{log n}{n}）。证明的方法，然后用于恢复额外的等周结果的立方体，改进的常数。本文还对最佳常数作了一些说明，并讨论了它们可能的含义
The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass of symmetric sets (subsets that are invariant under permutations of coordinates) of the Boolean cube, in positive characteristic. Our results subsume multiple statements on finite-degree Z-closures that have found applications in extremal combinatorial problems, for instance, pertaining to set systems (Hegedűs, Stud. Sci. Math. Hung. 2010; Hegedűs, arXiv 2021…
In this paper we use a formula for the $n$-th power of a $2\times2$ matrix $A$ (in terms of the entries in $A$) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if $m$ and $n$ are positive integers and $s \in \{0,1,2,\dots,$ $\lfloor (mn-1)/2 \rfloor \}$, then \begin{multline*} \sum_{i,j,k,t}2^{1+2t-mn+n} \frac{(-1)^{nk+i(n+1)}}{1+δ_{(m-1)/2,\,i+k}} \binom{m-1-i}{i} \binom{m-1-2i}{k}\times\\ \binom{n(m-1-2(i+k))}{2j}\binom{j}{t-n(i+k)} \binom{n-1-s+t}{s-t}\\ =\binom{mn-1-s}{s}. \end{multline*} 2) The generalized Fibonacci polynomial $f_{m}(x,s)$…
We investigate properties of a multivariate function $E(m_1,m_2,...,m_r)$, called {\it orbicyclic}, that arises in enumerative combinatorics in counting non-isomorphic maps on orientable surfaces. $E(m_1,m_2,...,m_r)$ proves to be multiplicative, and a simple formula for its calculation is provided. It is shown that the necessary and sufficient conditions for this function to vanish is equivalent to familiar Harvey's conditions that characterize possible branching data of finite cyclic automorphism groups of Riemann surfaces.
A combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there at most $2^{d+1}-2$ neighbourly simplices in $\mathbb R^d$, is presented.
Isoperimetric-Combinatorial Bounds for Range-Controlled Matchings and Quasi-Interpolation from Scattered Data
该研究开发介观框架分析有限点集扰动，通过施加HT组合约束证明$Y$与$Q$存在范围为$O(r)$的完美匹配，可转移积分逼近估计并锚定$Q$，还用平移不变拟插值方法获高阶估计，关键限制是HT条件及$r\leq Ch$。
A theorem of W. Derrick ensures that the volume of any Riemannian cube $([0,1]^n,g)$ is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick's inequality for weighted open covers of the cube $[0,1]^n$, which is motivated by a question about lower volume bounds in metric spaces. Our main theorem generalizes a previous result of the author, which gave a combinatorial version of Derrick's inequality and was used in the analysis of boundaries of hyperbolic groups. As an application, we answer a question of Y. B…
New combinatorial identity for the set of partitions and limit theorems in finite free probability theory
本文通过建立集合划分的精细化组合恒等式，研究了有限自由概率论中的极限定理，给出了 Sakuma–Yoshida 定理的有限自由类比、有限自由乘法卷积的中心极限定理，并为 Kabluchko 的相关极限定理提供了新证明。
The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.
We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular we obtain an alternative proof for a result ob…
Non-crossing linked partitions and multiplication of free random variables
材料给出了一个新的组合证明的乘法性质的S-变换。特别是，它的逆的系数的几个属性连接到非交叉链接分区和平面树。
Jack superpolynomials: physical and combinatorial definitions
杰克超多项式是量子三角函数Calogero-Moser-Sutherland的超对称扩张的本征函数。它们相对于标量积是正交的，被称为物理的，这是由这个量子力学问题自然引起的。但杰克超多项式也可以更组合地定义，从对称超多项式的乘法基开始，相对于组合标量积的单参数变形强制正交性。这两种结构是等价的。这为各种基本构造的正确性和杰克超多项式在对称超多项式理论中的关键作用提供了强有力的支持。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。