A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group     

Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past:

G-convergence and the weak operator topology

We show that a bounded sequence (a_n)_n of symmetric d × d-matrix valued functions is G-convergent if and only if ((ι^∗︁a_nι)⁻¹ )_n converges in the weak operator topology. Here ι: R(grad₀) ↪ L²(Ω)^d denotes the (canonical) embedding from the range of the weak gradient grad₀ defined on H¹₀(Ω) into L²(Ω)_d, where Ω ⊆ ℝ^d is open and bounded. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

SOME EULER SPACES OF DIFFERENCE SEQUENCES OF ORDER m

Kizmaz [13] studied the difference sequence spaces e∞(△), c(△), and c0(△).Several article dealt with the sets of sequences of m-th order difference of which are bounded, convergent, or convergent to zero. Altay and Basar [5] and Altay, Basar, and Mursaleen [7] introduced the Euler sequence spaces eτ0, eτ0, andeτ∞, respectively. The main purpose of this article is to introduce the spaces eτ0(△(m)), eτc(△(m)), and eτ∞(△(m)) consisting of all sequences whose mth order differences are in the Euler spaces eτ0, eτc, and eτ∞, respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the α-, β-, and γ-duals of the spaces eτ0(△(m)), eτc(△(m)), and eτ∞(△(m)), and the Schauder basis of the spaces eτ0(△(m)), eτc(△(m)). The last section of the article is devoted to the characterization of some matrix mappings on the sequence space eτc(△(m)).     

Smoothk-gonal curves with another fixed pencil

Let X be a smooth curve of genus g with two pencils, sayg _k ¹ andg _m ¹ . Here we show that, in general, dim|rg _k ¹ +ug _m ¹ | has the expected dimension and construct examples in which dim |rg _k ¹ +ug _m ¹ | has a prescribed value.     

Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Van Geemen's families of lines on special quintic threefolds

We study the families of lines on the quintic threefolds of the pencilx ₀ ⁵ +x ₁ ⁵ +x ₂ ⁵ +x ₃ ⁵ +x ₄ ⁵ -5tx ₀ x ₁ x ₂ x ₃ x ₄=0. We show that on the generic threefold of the pencil there exists a 1-dimensional family of lines that is not a cone. Partially supported by funds M.P.I.     

Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A ^*)–B ^*(A+A ^*)B>0, whereB=–(P+L) ^–1 R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ converges if and only ifA+A ^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A _ij ] ₁ ² , whereA ₂₁=–A ₁₂ ^* ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ₁₁ ^* =A₂₂+A ₂₂ ^* ; and the proper values ofA ₁₂ ^–1 A ₁₁,A ₁₂ ^*–1 A ₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.     

关于环Fpm+uFpm+u2Fpm上循环码的注记

本文研究了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码分类.通过建立环F_p^m+uF_p^m+u²F_p^m到环F_p^m+uF_p^m的同态,给出了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码的新分类方法.应用这种方法,得到了环F_p^m+uF_p^m+u²F_p^m长度为p^s的循环码的码词数.     

Some Hierarchies of Primitive Recursive Functions on Term Algebras

We compare two different Grzegorczyk hierarchies {H_n^σ}n≥0 and {L_n^σ}n≥1 on term algebras, which grow according to the height and length of terms, respectively. The solution of almost all inclusion problems among the Grzegorczyk classes and the (simultaneous) recursion number classes R_n^σ and S_n^σ on term algebras shows {H_n^σ}n≥0 to generalize Weihrauch's Grzegorczyk hierarchy on words {E_n^k}n≥0 to arbitrary term algebras. However, by regarding terms as words, {L_n^σ}n≥1 turns out to be computationally equivalent to Weihrauch's hierarchy {E_n^σ}n≥0 on the whole. Especially, L₂^σ} is equivalent to polynomial time computability and contains several natural term algebra functions. This establishes a notion of feasible term algebra functions and predicates.