Teichmüller curves in genus three and just likely intersections in $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).These constraints are akin to those that appear in Zilber and Pink’s conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\) (rather than the more standard \(\mathbf{G}_{m}^{n}\)). The ambient algebraic group lies outside the scope of Zilber’s Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum \(\Omega\mathcal{M}_{3}(1,1,1,1)\), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of \(\mathbf{G}_{m}^{9}\). These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.     

Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets D=(r_1D)∪(r_2D (1 r_1-r_2-r_3)/2)∪(r_3D 1 r_3) and E=(r_1E)∪(r_2E 1-r_2- r_3)∪(r_3E 1-r_3),and proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r_1~m=r_3~n.     

${R}_{0}$代数中的滤子格

Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.     

Covers ℘ for Abstract Regular Polytopes \mathcal {Q} such that \mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}

One key problem in the theory of abstract polytopes is the so-called amalgamation problem. In its most general form, this is the problem of characterising the polytopes with given facets  $\mathcal {K}$ and vertex figures ?. The first step in solving it for particular  $\mathcal{K}$ and ? is to find the universal such polytope, which covers all the others. This article explains a construction that may be attempted on an arbitrary polytope ?, which often yields an infinite family of finite polytopes covering ? and sharing its facets and vertex figures. The existence of such an infinite family proves that the universal polytope is infinite; alternatively, the construction can produce an explicit example of an infinite polytope of the desired type. An algorithm for attempting the construction is explained, along with sufficient conditions for it to work. The construction is applied to a few  $\mathcal{K}$ and ? for which it was previously not known whether or not the universal polytope was infinite, or for which only a finite number of finite polytopes was previously known. It is conjectured that the construction is quite broadly applicable.     

广义{1,3,5}-{1,4,5}自相似集的Lipschitz 等价

讨论广义{1,3,5}-{1,4,5}自相似集D=(r₁D)∪(r₂D+ (1+r₁-r₂-r₃)/2)∪(r₃ D+1-r₃)以及E=(r₁E)∪(r₂E+1-r₂-r₃)∪(r₃E+1-r₃)的Lipschitz等价性, 并证明了D与E为Lipschitz等价当且仅当存在正整数m和n, 使得 r₁^m=r₃ⁿ.     

r模同态的{1}-逆与{2}-逆

本文探讨模同态广义逆在环模理论中的应用．利用模同态的{1}-逆与{2}-逆，分别给出了一类环及一类重要模的特征刻画．     

����ϡ��ͼģ�͵�\mbox{PM}_{2.5}�ֲ�������ṹѧϰ

??We consider the structure learning problem of the \mbox{PM}_{2.5} pollution data over 31 provincial capitals in China. Specifically, we make use of the graphical model tools to study the hubs and the community structures of the \mbox{PM}_{2.5} pollution networks. The results show that the hubs in the \mbox{PM}_{2.5}pollution networks are always seriously polluted cities, and the \mbox{PM}_{2.5} pollution networks have significant community structures which consist of cities which in some sense can be regarded as blocks with similar cause of pollution. In view of the results, we suggest that the government should strengthen the effort to treat the seriously polluted areas and western China areas. Moreover, the management of the \mbox{PM}_{2.5} pollution should be region-dependent.     

面积法推导数列{n}、{n^2}、{n^3}的前n项和公式

数列与函数、不等式等有着密切的联系,又是今后学习高等数学的基础内容之一.其中等差数列作为一种特殊的数列,是高中生探究特殊数列的开始,它对后续数列的学习无论是在内容上还是在方法上都具有积极的意义.     

关于数列{2~n}

易知等比数列{2~n}前n项的和为S_n=a_(n+1)-2。对于这个关系式,我们有三点联想。 (一)简便求和。若a_5=32,则S_4=30。 (二)判定由关系式a_(n+1)=rS_n+S给出的数列是否为等比数列。事实上a_(n+1)=rS_n+S (1) a~n=rS_(n-1)+S (2) (1)-(2)得a_(n+1)-a_n=r(s_n-S_(n-1))=ra_n a_(n+1)/a_n=r+ 因此,r≠-1,a_1=S,{a_n}为等比数列。 (三)等比数列前n项和公式的新法推导。     

集合{sinn}的探讨

本文分析了集合{Sinn}的性质,得出{Sinn}是超越数集。     

斐数列{Fn}浅探

意大利数学家斐波那契(Fibonacci)在观察，研究兔子的繁殖这一问题中，发现了一个十分重要的数列{Fn}：     

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

数列{sinn~k}发散的证明

研究了当k是正整数时,数列{sinnk}的敛散性,文献[1]给出了{sinn}和{sinn2}发散的证明,但是证明方法复杂,不便阅读.为此,本文进行了两个方面的研究:一方面是运用简单的方法证明了{sinn}和{sinn2}的发散;另一方面证明了一个重要结论,即{sinnk}发散.     

$\mathbb{Z}^{k}$-作用一维子系统的Lipschitz跟踪性

本文主要研究了$\mathbb{Z}^{k}$-作用一维子系统的跟踪性质. 文中运用两种等价的方式引入了$\mathbb{Z}^{k}$-作用一维子系统的伪轨以及跟踪性的概念. 对于一个闭黎曼流形上的光滑$\mathbb{Z}^{k}$-作用$T$, 我们通过诱导的非自治动力系统提出了Anosov方向的概念. 借助Bowen几何的方法, 我们证明了$T$沿着任意Anosov方向具有Lipschitz跟踪性.     

PA���е�H\'{a}jek-R\'{e}nyi�Ͳ���ʽ��ǿ��������

In this paper, a new H\'{a}jek-R\'{e}nyi-type inequality for mean zero associated random variables is obtained, which generalizes and improves the result of Theorem 2.2 of \ncite{9}. In addition, a Brunk-Prokhorov-type strong law of large numbers is also given.     

( m,n)-Equidista nt Sets i n $\ mathbb{R}^{k},\ mathbb{S}^{k}$ , a nd ℙk

We call a metric space X (m,n)-equidistant if, when A⊆X has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝ ^k contains an (m,1)-equidistant set if and only if k≥m. Although the sphere is (3,2)-equidistant, and ℝ⁴ contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about analogous to the Double Midset Conjecture.     

二部图形式的Erd\H{O}s-S\'{o}s猜想

二部图形式的Erd\H{O}s-S\''{o}s猜想     

图的{P_4}—分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上.记Pl长度为l-1的路,如果G能够分解成若干个Pl,则称G存在{Pl}—分解.关于图的给定长路分解问题主要结果有:(i)连通图G存在{P3}—分解当且仅当G有偶数条边(见[1]);(ii)连通图G存在{P3,P4}—分解当且仅当G不是C3和奇树,这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树(见[3]).本文讨论了3正则图的{P4}—分解情况,并构造证明了边数为3k(k热∈Z且k≥2)的完全图Kn和完全二部图Kr,s存在{P4}—分解.