Transcendence of solutions of q-Painlevé equation of type {A_7^{(1)}}

In this paper we prove that solutions of q-P(A ₇) are all transcendental over C(t). We also investigate transcendence of solutions of q-P(A ₆) and prove transcendence of hypergeometric solutions of q-P(A ₆).     

Transcendence of solutions of q-Painlev�� equation of type {A^{(1)}_{6}}

In this paper we prove the transcendence of the solutions of the q-Painlevé equation of type ${A_6^{(1)}}$ . The q-Painlevé equation of type ${A_6^{(1)}}$ is also called d-P _II or q-P _II and has a continuous limit to the Painlevé equation of type II. Its symmetry is (A ₁?+?A ₁)⁽¹⁾.     

Boundaries of Holomorphic 1-Chains Within Holomorphic Line Bundles over $mathbb{CP}^{1}$

We show that boundaries of holomorphic 1-chains within holomorphic line bundles of mathbbCP¹mathbb{CP}^{1} can be characterized using a single generating function of Wermer moments. In the case of negative line bundles, a rationality condition on the generating function plus the vanishing moment condition together form an equivalent condition for bounding. We provide some examples which reveal that the vanishing moment condition is not sufficient by itself. These examples also can be used to demonstrate one point of caution about the use of birational maps in this topic.     

Recursion operators and hierarchies of $$text{mKdV}$$ equations related to the Kac–Moody algebras $$D_4^{(1)}$$ , $$D_4^{(2)}$$ , and $$D_4^{(3)}$$

Theoretical and Mathematical Physics - We construct three nonequivalent gradings in the algebra $$D_4simeq so(8)$$ . The first is the standard grading obtained with the Coxeter automorphism...     

On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$

In this paper, based on the basis composed of two sets of splines with distinct local supports, cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation. The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points, which can reproduce any polynomial of nearly best degrees. And by means of the modulus of continuity, the estimation of the operator approximating a real sufficiently smooth function is reviewed as well. Moreover, the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation. And then the convergence results are worked out.     

$ \mathfrak{G} $-separability of the lattice of τ-closed totally saturated formations

It is proved that the lattice of all τ -closed totally saturated formations of finite groups is \mathfrakG \mathfrak{G} -separable.     

Two-dimensional Meixner Random Vectors of Class ${mathcal{M}}_{L}$

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.     

The $$text{m}$$ KdV-type equations related to $$A_5^{(1)}$$ and $$A_5^{(2)}$$ Kac–Moody algebras

Theoretical and Mathematical Physics - We explain the details of the algebraic constructions on nontrivial examples of the mKdV equations related to the $$A_5^{(1)}$$ and $$A_5^{(2)}$$...     

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.     

Zigzag Strip Bundle Realization of B(Λ0) over $U_{q}(C_{n}^{(1)})$

Zigzag strip bundles are new combinatorial models realizing the crystals B(∞) for the quantum affine algebras \(U_{q}(\mathfrak {g})\), where \(\mathfrak {g}=B_{n}^{(1)},D_{n}^{(1)}\), \(D_{n+1}^{(2)}\), \(C_{n}^{(1)}\), \(A_{2n-1}^{(2)}\), \(A_{2n}^{(2)}\). Recently, these models were used to the realization of highest weight crystals except for the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\). In this paper, we construct the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\) using zigzag strip bundles, which completes the realizations of all highest weight crystals over \(U_{q}(\mathfrak {g})\).     

Convergence of simultaneous Hermite-Padé approximants to then-tuple ofq-hypergeometric series\{ {}_1\Phi _1 (_{(c,\gamma _j )}^{(1,1)} ;z)\} _{j = 1}^n

We investigate the convergence of simultaneous Hermite-Padé approximants to then-tuple of power series $$f_i (z) = \sum\limits_{k = 0}^\infty {C_k^{(i)} z^k ,} i = 1,2,...,n,$$ where $$C_0^{(i)} = 1;C_k^{(i)} = \prod\limits_{p = 0}^{k - 1} {\frac{1}{{(C - q^{\gamma i + p} )}},} k \ge 1.$$ HereC, q∈?, γ _i ∈?,i=1, 2,...,n. For |C|≠1, ifq=e^iθ, θ∈(0, 2π) and θ/2π is irrational, eachf _i (z),i=1,...,n, has a natural boundary on its circle of convergence. We show that “close-to-diagonal” and other sequences of Hermite-Padé approximants converge in capacity to (f ₁(z),..., f_n (z)) inside the common circle of convergence of eachf _i (z),i=1,...,n.     

Under which conditions is the Jacobi space L_{w^{(a,b)} }^p [ - 1,1] subset of L_{w^{(\alpha ,\beta )} }^1 [ - 1,1] ?

Exact conditions for α, β, a, b > ?1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $L_{w^{(a,b)} }^p [ - 1,1]$ ? $L_{w^{(\alpha ,\beta )} }^1 [ - 1,1]$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.     

双圆盘加权Hardy空间上$T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$的约化子空间

本文中, 我们主要刻画了Toeplitz算子$T=M_{z^k}+M^*_{z^l}$的约化子空间, 其中 $k_i, l_i$ ($i=1,2$) 均是正整数, $k=(k_1,k_2), l=(l_1,l_2)$ 且 $k\neq l$, $M_{z^k}$, $M_{z^l}$ 是双圆盘加权Hardy空间$\mathcal{H}_\omega^2(\mathbb{D}^2)$上的乘法算子. 对权系数 $\omega$ 适当限制, 我们证明了由 $z^m$ 生成的 $T$ 的约化子空间均是极小的. 特别地, Bergman 空间和加权 Dirichlet 空间 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 均是满足该限制条件的加权Hardy空间. 作为应用, 我们刻画了 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 上 Toeplitz 算子 $T_{z^k+\bar{z}^l}$ 的约化子空间, 该结论是对双圆盘Bergman 空间上相关结论的推广.     

Knaster’s Problem for (Z 2) k -Symmetric Subsets of the Sphere S^{2^{k}-1}

We prove a result about partitioning an absolutely continuous measure in ℝ ^d into 2d equal parts by a system of cones with common vertex, where d is an odd prime power. The proof is topological and based on the calculation of the equivariant Euler class of a certain vector bundle.     

Bases of Feigin–Stoyanovsky’s type subspaces for $$C_\ell ^{(1)}$$

In this paper, we construct combinatorial bases of Feigin–Stoyanovsky’s type subspaces of standard modules for level k affine Lie algebra \(C_\ell ^{(1)}\). We prove spanning by using annihilating field \(x_\theta (z)^{k+1}\) of standard modules. In the proof of linear independence, we use simple currents and intertwining operators whose existence is given by fusion rules.     

An efficient improvement of gift wrapping algorithm for computing the convex hull of a finite set of points in ℝ n $\mathbb {R}^{n}$

Numerical Algorithms - In this paper, we present an efficient improvement of gift wrapping algorithm for determining the convex hull of a finite set of points in $\mathbb {R}^{n}$ space, applying...     

On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

We are concerned with the derivation of Poincaré-Friedrichs type inequalities in the broken Sobolev space $W^{2,1}$($Ω$; $\mathcal{T}_h$) with respect to a geometrically conforming, simplicial triagulation $\mathcal{T}_h$ of a bounded Lipschitz domain $Ω$ in $\mathbb{R}^d$ , $d$ $∈$ $\mathbb{N}$. Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as ${\rm C}^0$ Discontinuous Galerkin (${\rm C}^0$${\rm DG}$) approximations of minimization problems in the Sobolev space $W^{2,1}$($Ω$), or more generally, in the Banach space $BV^2$($Ω$) of functions of bounded second order total variation. As an application, we consider a ${\rm C}^0$${\rm DG}$ approximation of a minimization problem in$BV^2$($Ω$) which is useful for texture analysis and management in image restoration.     

Formulas for Calculating the $ 3j $ -Symbols of the Representations of the Lie Algebra $ {mathfrak{gl}}_{3} $ for the Gelfand–Tsetlin Bases

Siberian Mathematical Journal - We give a simple explicit formula for an arbitrary $ 3j $ -symbol for the Lie algebra  $ {mathfrak{gl}}_{3} $ . The symbol is...     

Regular orthoscalar representations of the extended Dynkin graph $ {\tilde{E}_8} $ AND ∗-algebra associatedwith it

We obtain a classification of regular orthoscalar representations of the extended Dynkin graph [(E)\tilde]₈ {\tilde{E}_8} with special character. Using this classification, we describe triples of self-adjoint operators A, B, and C such that their spectra are contained in the sets {0, 1, 2, 3, 4, 5}, {0, 2, 4}, and {0, 3}, respectively, and the equality A + B + C = 6I is true.