Calculating the Fundamental Group of Galois Cover of the(2, 3)-embedding of CP~1× T

As known to all,it is quite difficult to compute the fundamental group of a surface of general type.In this paper,applying Moishezon-Teicher's algorithm,we investigate the fundamental group of a special surface of general type with zero topological index,namely,the Galois cover of the(2,3)-embedding of CP~1×T.Because the full presentation of the group is very complicated,we compute its special quotient and get an interesting result about its structure.     

The Structure of the Unit Group of the Group Algebra 𝔽3 k (C 3 × D 6)

The Structure of the Unit Group of the Group Algebra of the group C ₃ × D ₆ over a finite field of characteristic 3 is established in terms of split extensions of cyclic groups.     

Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P1 - {0,1, ∞}

Fix a prime number . We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro- completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).     

Global Well-Posedness of mKdV in L 2 (𝕋, ℝ)

ABSTRACT

We show that the Miura map L ²(𝕋) → H ^?1(𝕋), r?r _x  + r ² is a global fold and then apply our results on global well-posedness of KdV in H ^?1(𝕋) to show that mKdV is globally well-posed in L ²(𝕋).     

Canonical forms of 2 × 2 × 2 and 2 × 2 × 2 × 2 arrays over 𝔽2 and 𝔽3

We consider arrays of size 2?×?2?×?2 and 2?×?2?×?2?×?2 over the fields with two and three elements. We use computer algebra to determine the canonical forms of these arrays with respect to the action of the semidirect product of general linear groups with the symmetric group. For each canonical form, we determine the size of its orbit and the rank of the arrays in its orbit.     

The Euclidean Distortion of the Lamplighter Group

We show that the cyclic lamplighter group C ₂ ? C _n embeds into Hilbert space with distortion $\mathrm{O}(\sqrt{\log n})We show that the cyclic lamplighter group C ₂ ≀ C _n embeds into Hilbert space with distortion O(?{logn})\mathrm{O}(\sqrt{\log n}) . This matches the lower bound proved by Lee et al. (Geom. Funct. Anal., 2009), answering a question posed in that paper. Thus, the Euclidean distortion of C ₂ ≀ C _n is \varTheta(?{logn})\varTheta(\sqrt{\log n}) . Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni et al. (Isr. J. Math. 52(3):251–265, 1985) and by Gromov (see de Cornulier et. al. in Geom. Funct. Anal., 2009), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.     

Gelfand-Kirillov dimensions of the ℤ-graded oscillator representations of 𝔬(n,ℂ) and 𝔰𝔭(2n,ℂ)

In this paper, we give a method to compute the Gelfand–Kirillov dimensions of some polynomial–type weight modules. These modules are infinite-dimensional irreducible 𝔬(n,?)-modules and 𝔰𝔭(2n,?)-modules that appeared in the ?-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.     

Functoriality of the BGG Category 𝒪††

This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category 𝒪. We study functorial properties of 𝒪 across various setups. The first setup is over a skew group ring, involving a finite group Γ acting on a regular triangular algebra A. We develop Clifford theory for A?Γ, and obtain results on block decomposition, complete reducibility, and enough projectives. 𝒪 is shown to be a highest weight category when A satisfies one of the “Conditions (S);” the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module.

Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories 𝒪; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup, and which yield information about 𝒪.     

The Intrinsic Diameter of the Surface of a Parallelepiped

In the paper we obtain an explicit formula for the intrinsic diameter of the surface of a rectangular parallelepiped in 3-dimensional Euclidean space. As a consequence, we prove that an parallelepiped with relation for its edge lengths has maximal surface area among all rectangular parallelepipeds with given intrinsic diameter.     

The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real square matrices ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$, one of which has rank 1, where $2?d<+\infty$. Let $\rho (A)$ denote the spectral radius of a square matrix A. Then we prove that there always exists a finite-length word $(i_1^1\text{,}\dots \text{,}{i}_m^1)\in \{1\text{,}2{\}}^m$, for some finite $m?1$, such that$\sqrt[m]{\rho \left({\mathrm{S}}_{i_1^1}?{\mathrm{S}}_{i_m^1}\right)}={\sup }_{n?1}\left\{{\displaystyle \underset{(i_1\text{,}\dots \text{,}{i}_n)\in \{1\text{,}2{\}}^n}{\max }}\sqrt[n]{\rho ({\mathrm{S}}_{i_1}?{\mathrm{S}}_{i_n})}\right\}\text{.}$In other words, there holds the spectral finiteness property for $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$. Explicit formula for computation of the joint spectral radius is derived. This implies that the stability of the switched system induced by $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$ is algorithmically decidable in this case.     

The Hypoellipticity for a Class of Nonhomogeneous Convolution Operator on the Nilpotent Lie Group H~n×R~k

In this paper,we use the method of representation of Lie group to study a class of nonhomoge-neous convolution operator on the nilpotent Lie group H~M×R~k,and give a criteerion of their hypoellipticity.     

The heat kernel of the Laplacian defined on a uniform grid

We adapt a method originally developed by E.B. Davies for second order elliptic operators to obtain an upper heat kernel bound for the Laplacian defined on a uniform grid on the plane.     

Coherent Continuation on the Category of (𝔤, K)-Modules

Abstract

Let 𝔤 be a complex semisimple Lie algebra. Let K be an algebraic group acting on the flag variety of 𝔤 with finitely many orbits. We give a geometric interpretation of the coherent continuation on the category of finitely generated (𝔤, K )-modules in terms of the intertwining functors on the category of K-equivariant 𝒟-modules.     

Higher Secant Varieties of ℙ n × ℙ1 Embedded in Bi-Degree (a,b)

In this article, we compute the dimension of all the higher secant varieties to the Segre–Veronese embedding of ? ⁿ × ?¹ via the section of the sheaf 𝒪(a, b) for any n, a, b ∈ ?⁺. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann (1, s ? 1)-defective Veronese varieties.     

The Crossed Burnside Ring,the Drinfel’d Double,and the Dade Group of a <Emphasis Type="Italic">p</Emphasis>-Group

Let p be a prime number. This paper solves the question of the difference between the rank of the crossed Burnside ring B ^c (P) of a finite p-group P and of the rational representation ring R_\mathbbQ (D(P))R_{{\mathbb{Q}}} (\mathcal{D}(P)) of the Drinfel’d double D(P)\mathcal{D}(P) of the group algebra ℚ P. The difference is represented by using the Dade groups of certain subgroups of P.