Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

On the subspace L((x ∧ y)m) of Sm(∧2?4)

We take the exterior power ℝ⁴ ∧ ℝ⁴ of the space ℝ⁴, its mth symmetric power V = S ^m (∧²ℝ⁴) = (ℝ⁴ ∧ ℝ⁴) ∨ (ℝ⁴ ∧ ℝ⁴) ∨ ... ∨(ℝ⁴ ∧ ℝ⁴), and put V ₀ = L((x ∧ y)∨ ... ∨(x ∧ y): x, y ∈ ℝ⁴). We find the dimension of V ₀ and an algorithm for distinguishing a basis for V ₀ efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field. Original Russian Text Copyright ? 2009 Gubarev V. Yu. The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-344.2008.1). __________ Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 503–514, May–June, 2009.     

Embedding of Hpω in the class eL

We obtain the necessary conditions for the embedding H _p ^ω ⊂ e ^L (1≤p∞) with convex modulus of continuity ω in terms of this modulus. In the case p=1 these conditions are also sufficient.     

Vector subdivision schemes in (Lp(?s))r(1?p?∞) spaces

The purpose of this paper is to investigate the refinement equations of the form

KO?i groups of G3(?n), n odd

We use the work of Karoubi and Mudrinski on the real Grothendieck's groups of certain complex projective bundles to show that the torsion of the KO ^–i groups of G ₃( ⁿ ), n odd, are related to the known torsion of the KO ^–i groups of G₂( ⁿ ).     

On qualitative analysis of delay systems and xΔ = f(t, x, xσ) on time scales

Here we solve two problems presented in paper [9] (C C Tisdell and A Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008) 3504–3524). We study existence and uniqueness of solutions for delay systems and first-order dynamic equations of the form x ^Δ = f (t,x,x ^σ ) on time scales by using the Banach’s fixed-point theorem. Some examples are presented to illustrate the efficiency of the proposed results.     

Asymptotics of the Best Polynomial Approximation of?|x|p and?of?the Best Laurent Polynomial Approximation of sgn(x) on Two Symmetric Intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| ^p on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn(x) on [−1,−a]∪[a,1] by Laurent polynomials.      

Two-side estimates for sums of absolute values of Fourier coefficients of functions from Hω(Tm)

The class of functions H ^ω is considered, where ω(t) is a continuity modulus monotone in the sense of Hardy and satisfying some condition C. The behavior of the value is obtained, where is the sum of absolute values of Fourier coefficients of a function f ∈ L(T ^m ) in pth power. Original Russian Text ? D.M. D’yachenko, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 19–26.     

A new technique of integral representations in ?n

A new technique of integral representations in ℂ ⁿ , which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the ∂-equations on strictly pseudoconvex domains in ℂ ⁿ are obtained. These new formulas are simpler than the classical ones, especially the solutions of the ∂-equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in ℂ ⁿ so that all corresponding formulas are simplified.     

Lie Symmetries of the Equation ut(x, t)?+?g(u)ux(x, t) = 0

In this paper the invariance criterion is applied for the nonlinear equation

Projective line over the finite quotient ring GF(2)[x]/〈x3 ? x〉 and quantum entanglement: Theoretical background

We consider the projective line over the finite quotient ring R_⋄ ≡ GF(2)[x]/〈x ³ − x〉. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because R_⋄ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence stems from a nontrivial character of the Jacobson radical of the ring. The quotient ring is isomorphic to GF(2) ⊗ GF(2). The projective line over features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we will discuss in detail in Part II of this paper. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 44–53, April, 2007.     

Two-side estimates for sums of absolute values of Fourier coefficients of functions from Hω(Tm) with the use of partial moduli of smoothness

Two-side estimates of sums of absolute values of Fourier coefficients for functions of many variables from the classes H ^ω (T ^m ) are established in the paper with the help of partial moduli of smoothness.     

On solutions of functional-differential equations f ′( x ) = a ( x ) f ( g ( x )) + b ( x ) f ( x ) + c ( x ) in the large

We consider solutions of functional-differential equations

关于f+a(f(k))n值分布的一个注记

本文研究亚纯函数的值分布问题.利用值分布理论,获得了一个带精简密指量的模分布的不等式,改进了Xu和Yang等人的结果.     

On irreducible characters of the group Sn that are semiproportional on An or Sn?An. II

In the author’s previous paper, the hypothesis that the alternating groups A _n have no pairs of semiproportional irreducible characters is reduced to a hypothesis concerning the problem of describing the pairs of irreducible characters of the symmetric group S _n that are semiproportional on one of the sets A _n or S _n ∖A _n . In this hypothesis, properties of such a pair of characters are expressed in terms of Young’s diagrams corresponding to these characters. The theorem proved in this paper allows one to exclude from consideration some stages of the verification of this hypothesis.     

环Fp+vFp上互补对偶常循环码

本文研究了环F_p+vF_p上互补对偶（1-2v）-常循环码.利用环F_p+vF_p上（1-2v）-常循环码的分解式C=vC_1-v ⊕（1-v）C_v,得到了环F_p+vF_p上互补对偶（1-2v）-常循环码的生成多项式.然后借助从F_p+vF_p到F_p²的Gray映射,证明了环F_p+vF_p上互补对偶（1-2v）-常循环码的Gray像是F_p的互补对偶循环码.