数论/群论/组合学学术速递[1.10]
Update！H5支持摘要折叠，体验更佳！点击阅读原文访问arxivdaily.com，涵盖CS|物理|数学|经济|统计|金融|生物|电气领域，更有搜索、收藏等功能！
math.NT数论，共计3篇
math.GR群论，共计4篇
math.CO组合学，共计15篇
1.math.NT数论:
【1】 Summation formulae for quadrics
标题：二次曲面的求和公式
链接：https://arxiv.org/abs/2201.02583
摘要：We prove a Poisson summation formula for the zero locus of a quadratic form
in an even number of variables with no assumption on the support of the
functions involved. The key novelty in the formula is that all "boundary terms"
are given either by constants or sums over smaller quadrics related to the
original quadric. We also discuss the link with the classical problem of
estimating the number of solutions of a quadratic form in an even number of
variables. To prove the summation formula we compute (the Arthur truncated)
theta lift of the trivial representation of $\mathrm{SL}_2(\mathbb{A}_F)$. As
previously observed by Ginzburg, Rallis, and Soudry, this is an analogue for
orthogonal groups on vector spaces of even dimension of the global
Schr\"odinger representation of the metaplectic group.
【2】 Exponential multiple mixing for commuting automorphisms of a nilmanifold
标题：零流形交换自同构的指数多重混合
链接：https://arxiv.org/abs/2201.02556
备注：13 pages
摘要：Let $l\in \mathbb{N}_{\geq 1}$ and $\alpha : \mathbb{Z}^l\rightarrow
\text{Aut}(\mathscr{N})$ be an action of $\mathbb{Z}^l$ by automorphisms on a
compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha(z)$ is
ergodic for $z\in \mathbb{Z}^l\smallsetminus\{0\}$ and show that $\alpha$
satisfies exponential $n$-mixing for any integer $n\geq 2$. This extends
results of Gorodnik and Spatzier [Acta Math., 215 (2015)].
【3】 The regulator dominates the rank
标题：监管者占主导地位
链接：https://arxiv.org/abs/2201.02383
摘要：After noticing that the regulator of a number field dominates the rank of its
group of units, we bound from below the regulator of the Mordell-Weil group of
elliptic curves over global function fields of characteristic $p\geq5$. The
lower bound is an increasing function of the rank and of the height.
2.math.GR群论:
【1】 Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs
标题：无限图的纯映射类群的粗几何
链接：https://arxiv.org/abs/2201.02559
备注：41 pages, 10 figures
摘要：We discuss the large-scale geometry of pure mapping class groups of locally
finite, infinite graphs, motivated from recent work by Algom-Kfir--Bestvina and
the work of Mann--Rafi on the large-scale geometry of mapping class groups of
infinite-type surfaces. Using the framework of Rosendal for coarse geometry of
non-locally compact groups, we classify when the pure mapping class group of a
locally finite, infinite graph is globally coarsely bounded (an analog of
compact) and when it is locally coarsely bounded (an analog of locally
compact). Our techniques also give lower bounds on the first integral
cohomology of the pure mapping class group for some graphs and show that some
of these groups have continuous actions on simplicial trees.
【2】 The automorphism groups of Artin groups of edge-separated CLTTF graphs
标题：边分离CLTTF图的Artin群的自同构群
链接：https://arxiv.org/abs/2201.02502
备注：30 pages, 6 figures
摘要：We provide an explicit presentation of the automorphism group of an
edge-separated CLTTF Artin group.
【3】 Not all nilpotent monoids are finitely related
标题：并非所有的幂零幺半群都是有限相关的
链接：https://arxiv.org/abs/2201.02375
摘要：A finite semigroup is finitely related (has finite degree) if its term
functions are determined by a finite set of finitary relations. For example, it
is known that all nilpotent semigroups are finitely related. A nilpotent monoid
is a nilpotent semigroup with adjoined identity. We show that every
$4$-nilpotent monoid is finitely related. We also give an example of a
$5$-nilpotent monoid that is not finitely related. This is the first known
example where adjoining an identity to a finitely related semigroup yields a
semigroup which is not finitely related. We also provide examples of finitely
related semigroups which have subsemigroups, homomorphic images, and in
particular Rees quotients, that are not finitely related.
【4】 Systems of equations over the group ring of Thompson's group $F$
标题：Thompson群的群环上的方程组$F$
链接：https://arxiv.org/abs/2201.02308
备注：arXiv admin note: text overlap with arXiv:2101.01848
摘要：Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. It is known
that if $G$ is amenable then $R$ satisfies the Ore condition: for any $a,b\in
R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It is
also true for amenable groups that a non-zero solution exists for any finite
system of linear equations over $R$, where the number of unknowns exceeds the
number of equations. Recently Bartholdi proved the converse. As a consequence
of this theorem, Kielak proved that R.\,Thompson's group $F$ is amenable if and
only if it satisfies the Ore condition. The amenability problem for $F$ is a
long-standing open question.
In this paper we prove that some equations or their systems have non-zero
solutions in the group rings of $F$. We improve some results by Donnelly
showing that there exist finite sets $Y\subset F$ with the property $|AY| <
\frac43|Y|$, where $A=\{x_0,x_1,x_2\}$. This implies some result on the systems
of equations. We show that for any element $b$ in the group ring of $F$, the
equation $(1-x_0)u=bv$ has a non-zero solution. The corresponding fact for
$1-x_1$ instead of $1-x_0$ remains open. We deduce that for any $m\ge1$ the
system $(1-x_0)u_0=(1-x_1)u_1=\cdots=(1-x_m)u_m$ has nonzero solutions in the
group ring of $F$. We also analyze the equation $(1-x_0)u=(1-x_1)v$ giving a
precise explicit description of all its solutions in $K[F]$. This is important
since to any group relation between $x_0$, $x_1$ in $F$ one can naturally
assign such a solution. So this can help to estimate the number of relations of
a given length between generators.
3.math.CO组合学:
【1】 The Inverse of the Incidence Matrix of a Unicyclic Graph
标题：单圈图的关联矩阵的逆
链接：https://arxiv.org/abs/2201.02580
摘要：The vertex-edge incidence matrix of a (connected) unicyclic graph G is a
square matrix which is invertible if and only if the cycle of G is an odd
cycle. A combinatorial formula of the inverse of the incidence matrix of an odd
unicyclic graph was known. A combinatorial formula of the Moore-Penrose inverse
of the incidence matrix of an even unicyclic graph is presented solving an open
problem.
【2】 Incidence and Laplacian matrices of wheel graphs and their inverses
标题：轮图及其逆图的关联矩阵和拉普拉斯矩阵
链接：https://arxiv.org/abs/2201.02579
摘要：It has been an open problem to find the Moore-Penrose inverses of the
incidence, Laplacian, and signless Laplacian matrices of families of graphs
except trees and unicyclic graphs. Since the inverse formulas for an odd
unicyclic graph and an even unicyclic graph are quite different, we consider
wheel graphs as they are formed from odd or even cycles. In this article solve
the open problem for wheel graphs. This work has an interesting connection to
inverses of circulant matrices.
【3】 Standard Young tableaux and lattice paths
标题：标准Young画面和晶格路径
链接：https://arxiv.org/abs/2201.02540
备注：9 pages, 3 figures
摘要：Using lattice path counting arguments, we reproduce a well known formula for
the number of standard Young tableaux. We also produce an interesting new
formula for tableaux of height $\leq 3$ using the Fourier methods of Ault and
Kicey.
【4】 Four Blocks Cycles C(k,1,1,1) in Digraphs
标题：有向图中的四块圈C(k，1，1，1)
链接：https://arxiv.org/abs/2201.02520
摘要：A four blocks cycle C(k1,k2,k3,k4) is an oriented cycle formed by the union
of four internally disjoint directed paths of lengths k1,k2,k3 and k4
respectively. El Mniny proved that if D is a digraph having a spanning out-tree
T with no subdivisions of C(k, 1, 1, 1), then the chromatic number of D is at
most 8^{3}k. In this paper, we will improve this bound to 18k.
【5】 Skew Dyck paths with catastrophes
标题：具有灾变的偏向Dyck路径
链接：https://arxiv.org/abs/2201.02518
摘要：Skew Dyck paths are like Dyck paths, but an additional south-west step
$(-1,-1)$ is allowed, provided that the path does not intersect itself. Lattice
paths with catastrophes can drop from any level to the origin in just one step.
We combine these two ideas. The analysis is strictly based on generating
functions, and the kernel method is used.
【6】 On the Distance Spectra of Extended Double Stars
标题：关于扩展双星的距离谱
链接：https://arxiv.org/abs/2201.02499
备注：13 pages, 13 figures, arXiv:1307.5931, arXiv:1606.07551, To be presented at JMM 2022
摘要：The distance matrix of a connected graph is defined as the matrix in which
the entries are the pairwise distances between vertices. The distance spectrum
of a graph is the set of eigenvalues of its distance matrix. A graph is said to
be determined by its distance spectrum if there does not exist a non-isomorphic
graph with the same spectrum. The question of which graphs are determined by
their spectrum has been raised in the past, but it remains largely unresolved.
In this paper, we prove that extended double stars are determined by their
distance spectra.
【7】 Neighborly partitions and the numerators of Rogers-Ramanujan identities
标题：邻域划分与Rogers-Ramanujan恒等式的分子
链接：https://arxiv.org/abs/2201.02481
摘要：We prove two partition identities which are dual to the Rogers-Ramanujan
identities. These identities are inspired by (and proved using) a
correspondence between three kinds of objects: a new type of partitions
(neighborly partitions), monomial ideals and some infinite graphs.
【8】 Triangles in intersecting families
标题：相交族中的三角形
链接：https://arxiv.org/abs/2201.02452
备注：7 pages
摘要：We prove the following the generalized Tur\'an type result. A collection
$\mathcal{T}$ of $r$ sets is an $r$-triangle if for every
$T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have
$\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in \mathcal{T}}T$ is empty. A
family $\mathcal{F}$ of sets is $r$-wise intersecting if for any
$F_1,F_2,\dots,F_r\in \mathcal{F}$ we have $\cap_{i=1}^rF_i\neq \emptyset$ or
equivalently if $\mathcal{F}$ does not contain any $m$-triangle for
$m=2,3,\dots,r$. We prove that if $n\ge n_0(r,k)$, then the $r$-wise
intersecting family $\mathcal{F}\subseteq \binom{[n]}{k}$ containing the most
number of $(r+1)$-triangles is isomorphic to $\{F\in \binom{[n]}{k}:|F\cap
[r+1]|\ge r\}$.
【9】 On some problems regarding distance-balanced graphs
标题：关于距离平衡图的几个问题
链接：https://arxiv.org/abs/2201.02430
备注：16 pages
摘要：A graph $\Gamma$ is said to be distance-balanced if for any edge $uv$ of
$\Gamma$, the number of vertices closer to $u$ than to $v$ is equal to the
number of vertices closer to $v$ than to $u$, and it is called nicely
distance-balanced if in addition this number is independent of the chosen edge
$uv$. A graph $\Gamma$ is said to be strongly distance-balanced if for any edge
$uv$ of $\Gamma$ and any integer $k$, the number of vertices at distance $k$
from $u$ and at distance $k+1$ from $v$ is equal to the number of vertices at
distance $k+1$ from $u$ and at distance $k$ from $v$.
In this paper we answer an open problem posed by Kutnar and Miklavi\v{c}
[European J. Combin. 39 (2014), 57-67] by constructing several infinite
families of nonbipartite nicely distance-balanced graphs which are not strongly
distance-balanced. We disprove a conjecture regarding characterization of
strongly distance-balanced graphs posed by Balakrishnan et al. [European J.
Combin. 30 (2009), 1048-1053] by providing infinitely many counterexamples, and
answer an open question posed by Kutnar et al. in [Discrete Math. 306 (2006),
1881-1894] regarding existence of semisymmetric distance-balanced graphs which
are not strongly distance-balanced by providing an infinite family of such
examples. We also show that for a graph $\Gamma$ with $n$ vertices and $m$
edges it can be checked in $O(mn)$ time if $\Gamma$ is strongly-distance
balanced and in $O(m^2)$ time if $\Gamma$ is nicely distance-balanced.
【10】 Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials
标题：用切比雪夫多项式证明某些苯类的Kekulé数猜想
链接：https://arxiv.org/abs/2201.02376
备注：28 pages,3 figures, 1 table
摘要：In chemistry, Cyvin-Gutman enumerates Kekul\'{e} numbers for certain
benzenoids and record it as $A050446$ on OEIS. This number is exactly the two
variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) +
\sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where
$T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this
number also appeared in the context of weighted graphs, graph polytopes, magic
labellings, and unit primitive matrices, studied by different authors. Several
interesting conjectures were made on the OEIS. These conjectures are related to
both the row and column generating function of $T(n,m)$. In this paper, give
explicit formula of the column generating function, which is also the
generating function $F(n,x)$ studied by B\'{o}na, Ju, and Yoshida. We also get
trig function representations by using Chebyshev polynomials of the second
kind. This allows us to prove all these conjectures.
【11】 On the intersection ideal graph of semigroups
标题：关于半群的交理想图
链接：https://arxiv.org/abs/2201.02346
备注：2 figures. arXiv admin note: text overlap with arXiv:2110.14194
摘要：The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple
undirected graph whose vertices are all nontrivial left ideals of $S$ and two
distinct left ideals $I, J$ are adjacent if and only if their intersection is
nontrivial. In this paper, we investigate the connectedness of $\Gamma(S)$. We
show that if $\Gamma(S)$ is connected then $diam(\Gamma(S)) \leq 2$. Further we
classify the semigroups such that the diameter of their intersection graph is
two. Other graph invariants, namely perfectness, planarity, girth, dominance
number, clique number, independence number etc. are also discussed. Finally, if
$S$ is union of $n$ minimal left ideals then we obtain the automorphism group
of $\Gamma(S)$.
【12】 Automorphisms of left Ideal relation graph over full matrix ring
标题：全矩阵环上左理想关系图的自同构
链接：https://arxiv.org/abs/2201.02345
摘要：The left-ideal relation graph on a ring $R$, denoted by
$\overrightarrow{\Gamma_{l-i}}(R)$, is a directed graph whose vertex set is all
the elements of $R$ and there is a directed edge from $x$ to a distinct $y$ if
and only if the left ideal generated by $x$, written as $[x]$, is properly
contained in the left ideal generated by $y$. In this paper, the automorphisms
of $\overrightarrow{\Gamma_{l-i}}(R)$ are characterized, where $R$ is the ring
of all $n \times n$ matrices over a finite field $F_q$. The undirected left
relation graph, denoted by $\Gamma_{l-i}(M_n(F_q))$, is the simple graph whose
vertices are all the elements of $R$ and two distinct vertices $x, y$ are
adjacent if and only if either $[x] \subset [y]$ or $[y] \subset [x]$ is
considered. Various graph theoretic properties of $\Gamma_{l-i}(M_n(F_q))$
including connectedness, girth, clique number, etc. are studied.
【13】 On f-Derangements and Decomposing Bipartite Graphs into Paths
标题：关于f-乱列与二部图的路分解
链接：https://arxiv.org/abs/2201.02332
摘要：Let $f: \{1, ..., n\} \rightarrow \{1, ..., n\}$ be a function (not
necessarily one-to-one). An $f-derangement$ is a permutation $ g:\{1,...,n\}
\rightarrow \{1,...,n\}$ such that $g(i) \neq f(i)$ for each $ i = 1, ..., n$.
When $f$ is itself a permutation, this is a standard derangement. We examine
properties of f-derangements, and show that when we fix the maximum number of
preimages for any item under $f$, the fraction of permutations that are
f-derangements tends to $ 1/e$ for large $n$, regardless of the choice of $f$.
We then use this result to analyze a heuristic method to decompose bipartite
graphs into paths of length 5
【14】 Identities involving the tribonacci numbers squared via tiling with combs
标题：用梳子平铺的包含Tribonacci数平方的恒等式
链接：https://arxiv.org/abs/2201.02285
备注：7 pages, 1 figure
摘要：The number of ways to tile an $n$-board (an $n\times1$ rectangular board)
with $(\frac12,\frac12;1)$-, $(\frac12,\frac12;2)$-, and
$(\frac12,\frac12;3)$-combs is $T_{n+2}^2$ where $T_n$ is the $n$th tribonacci
number. A $(\frac12,\frac12;m)$-comb is a tile composed of $m$ sub-tiles of
dimensions $\frac12\times1$ (with the shorter sides always horizontal)
separated by gaps of dimensions $\frac12\times1$. We use such tilings to obtain
quick combinatorial proofs of identities relating the tribonacci numbers
squared to one another, to other combinations of tribonacci numbers, and to the
Fibonacci, Narayana's cows, and Padovan numbers. Most of these identities
appear to be new.
【15】 A proof of the Graham Sloane conjecture
标题：格雷厄姆·斯隆猜想的一个证明
链接：https://arxiv.org/abs/2201.02240
备注：arXiv admin note: text overlap with arXiv:2111.12812
摘要：We settle in the affirmative the Graham-Sloane conjecture.
机器翻译，仅供参考
点击“阅读原文”获取带摘要的学术速递