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.     

ON HEYDE’S THEOREM ON THE GROUP $$\mathbb{R}$$ × $$\mathbb{T}$$

Doklady Mathematics - According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of...     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

Character formulas for Feigin–Stoyanovsky’s type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules

Exact sequences of Feigin–Stoyanovsky’s type subspaces for affine Lie algebra \mathfraksl(l+1,\mathbbC)^[\tilde]\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}} lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for \mathfraksl(3,\mathbbC)^[\tilde]\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}, we obtain a new family of character formulas for all Feigin–Stoyanovsky’s type subspaces at the general level.     

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

Compactification de Chabauty de l’espace des sous-groupes de Cartan de {\text{ SL}}_n(\mathbb{R })

Let $G$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $G$ , which can be also seen as the space of maximal flats of the symmetric space of $G$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $G$ , endowed with the Chabauty topology. We show that when the real rank of $G$ is 1, or when $G={\text{ SL}}_3(\mathbb{R })$ or ${\text{ SL}}_4(\mathbb{R })$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $G$ , with real spectrum. And in the case of ${\text{ SL}}_3(\mathbb{R })$ , we study its topology more closely and we show that it is simply connected.     

The Quasiapproximate $(\mathcal{U} + \mathcal{K})$ –Invariants of Essentially Normal Operators

For a class of essentially normal operators, we characterize their norm closures of –orbits. Moreover, we introduce a notion of the quasiapproximate – equivalence of essentially normal operators and determine completely the quasiapproximate –invariants. Finally, we give the canonical forms of essentially normal operators under this quasiapproximate –equivalence.     

A $\mathbb{P}_N × \mathbb{P}_N$ Spectral Element Projection Method for the Unsteady Incompressible Navier-Stokes Equations

In this paper, we present a $\mathbb{P}_N × \mathbb{P}_N$ spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equations. The main purpose of this work consists of: (i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral element approaches in space; (ii) construction of a stable $\mathbb{P}_N × \mathbb{P}_N$ method together with a $\mathbb{P}_N → \mathbb{P}_{N-2}$post-filtering. The link of different methods will be clarified. The key feature of our method lies in that only one grid is needed for both velocity and pressure variables, which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis, the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.     

Parallel and semi-parallel hypersurfaces of \mathbb{S}^n × ℝ

We give a complete classification of totally umbilical, parallel and semi-parallel hypersurfaces of the Riemannian product space × ℝ. The author is a postdoctoral researcher supported by the Research Foundation — Flanders (F.W.O.).     

关于$\mathbb{C}P^3$ 中的伪脐Lagrange子流形

本文获得$\mathbb{C}P^3$中非极小的紧致伪脐Lagrange子流形常数数量曲率的一个估计. 作为其应用, 我们证明了$\mathbb{C}P^3$中紧致Einstein伪脐Lagrange子流形必是极小的.     

系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调

研究了系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调, 其中~$\mathbb{F}$ 是一个素特征的代数闭域且~$\frak{gl}(2,\mathbb{F})$ 是系数在~$\mathbb{F}$ 上的~$2\times 2$ 阶矩阵李代数. 计算出所有~$\frak{gl}(2,\mathbb{F})$ 到模李超代数~$W(m,3,\underline{1})$ 的子模的导子和内导子. 从而一维上同调~$\textrm{H}^{1}(\frak{gl}(2,\mathbb{F}),W(m,3,\underline{1}))$ 可以完全用矩阵的形式表示.     

一类复映射$z\leftarrow e^{i\phi}(\bar{z})^\alpha+c~ (\alpha<0)$的广义M集

本文分析了一类复映射$z \leftarrow e^{i\phi }(\bar {z})^\alpha +c\{\alpha < 0,\phi \in [0,2\pi)\}$的临界点的性质,给出了广义Mandelbrot集 (简称广义M集)的定义,并构造出一系列广义M集.利用复变函数理论和计算机制图相结合的实验数学的方法,本文对广义M集的结构和演化进行了研究,结果表明: 1). 广义M集的几何结构依赖于参数$\alpha$, $R$和$\phi$; 2). 整数阶广义M集具有对称性和分形特征; 3). 小     

\mathbb{R}�϶Գ�Cauchy�����񼯵�ȷ��Hausdorff���

??This paper establishes limsup type law of the iterated logarithm of the occupation measure, using the asymptotic equivalence relation between the occupation measure and the number of excursion process of a symmetric Cauchy process. Furthermore, by using the density theorem and the economic coverage method, it derives the exact Hausdorff measure for the range of a symmetric Cauchy process in \mathbb{R}.     

The identification problem for complex-valued Ornstein–Uhlenbeck operators in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$

In this paper we study perturbed Ornstein–Uhlenbeck operators

$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$

for simultaneously diagonalizable matrices \(A,B\in \mathbb {C}^{N,N}\). The unbounded drift term is defined by a skew-symmetric matrix \(S\in \mathbb {R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by

$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$

One key assumption is a new \(L^p\)-dissipativity condition

$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$

for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). Then we prove that the Schwartz space is a core of \(A_p\) and apply an \(L^p\)-solvability result of the resolvent equation for \(A_p\). In addition, we derive \(W^{1,p}\)-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

     

�˻�ͼ\mathbb{Z}^2\times\{0,1,\ldots,l-1\}�ij����Եij���֤��

It is well known that the two dimensional integer lattice \mathbb{Z}^2 is recurrent, while the three dimensional integer lattice is transient. In this paper we show that the product graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\} is recurrent. The proof approach only utilizes the elementary methods in probability theory (without the words of electric networks).     

Irreducible Modules Over Witt Algebras $\mathcal {W}_{n}$ and Over 𝖘𝖑n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$

In this paper, by using the “twisting technique” we obtain a class of new modules A _b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\). We give necessary and sufficient conditions for A _b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\)-modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\), all the above irreducible \(\mathcal {W}_{n}\)-modules A _b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules, denoted by Ω_1?a (λ ₁, λ ₂, ? ,λ _n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\), we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module Ω_1?a (λ ₁, λ ₂,? ,λ _n ) is reducible, we prove that it has a unique nontrivial submodule W _1?a (λ ₁, λ ₂,...λ _n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module with highest weight mΛ _n for some non-negative integer \(m\in \mathbb {Z}_{+}\). We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules of the form Ω_1?a (λ ₁, λ ₂,? ,λ _n ) or of the form W _1?a (λ ₁, λ ₂,...λ _n ) to be isomorphic.     

���ܶȹ��ƵĶԳƼ�����~$L_1(\mathbb{R}^d)$~�µ���ƫ��

Let f_n be a non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in \mathbb{R}^d. In this paper we prove two moderate deviation theorems in L_1(\mathbb{R}^d) for \{f_n(x)-f_n(-x),\,n\ge1\}.     

Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$

We prove that if a countable group Γ contains a copy of \mathbbF₂\mathbb{F}_{2}, then it admits uncountably many non orbit equivalent actions.     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m,&nbsp;n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...