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.     

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.     

Poisson boundary of GLd(?)

We construct the Poisson boundary for a random walk supported by the general linear group on the rational numbers as the product of flag manifolds over the p-adic fields. For this purpose, we prove a law of large numbers using Oseledets’ multiplicative ergodic theorem. The only assumption we need is some moment condition on the measure governing the jumps of the random walk, but no irreducibility hypothesis is made.     

Gaussian Estimates of Order α and L p -Spectral Independence of Generators of C 0-Semigroups

We prove L^p-spectral independence for generators of C₀-semigroups estimated by the positive C₀-semigroup . In the preliminary process of the proof, we obtain the asymptotic expansion formula for the integral kernel of the C₀-semigroup .     

Equality of norm groups of subextensions of Sn(n?5) extensions of algebraic number fields

Let E/k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group S_n on n?5 letters. For any field extensions k⊆K, L⊆E a necessary and a sufficient condition is given for the equality to hold, where is the group of norms from K to k of the elements of the multiplicative group K∗ of K.     

Uniqueness of Solution of the Initial Value Problem for ut=ΔUm-up

The uniqueness of solution of the Cauchy problem u_t = Δu^m - u^p, \qquad S_T = R^n × (0, T) u(x, 0) = Φ(x), \qquad × ∈ R^n is obtained. Where n ≥ 1, m, p > 0, Φ(x) ∈ L^∞(R^n), Φ(x) ≥ 0.     

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.     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

On Γ-convergence of pairs of dual functionals

The paper considers a slightly modified notion of the Γ-convergence of convex functionals in uniformly convex Banach spaces and establishes that under standard coercitivity and growth conditions the Γ-convergence of a sequence of functionals {Fj} to implies that the corresponding sequence of dual functionals converges in an analogous sense to the dual to functional .     

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.      

Controlled algebraic K-theory of integral group ring of SL(3, ?)

We calculate the lower Controlled Algebraic K-theory of any finitely generated infinite subgroup of SL(3,Z), the group of 3×3 integral matrices of determinant 1.     

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.     

Embedding of Lipschitz classes into classes of functions of Λ-bounded variation

In this note, sufficient or necessary conditions for embedding of Lipschitz classes , 1?p<∞ into classes ΛBV of functions of Λ-bounded variation are considered. Based on the technique of decomposition of functions, we obtain sufficient and necessary conditions for the embedding , 0<α,β?1.     

On estimates of the density of Feynman-Kac semigroups of α-stable-like processes

Suppose that α∈(0,2) and that X is an α-stable-like process on Rd. Let F be a function on Rd belonging to the class Jd,α (see Introduction) and be _∑s?tF(Xs−,Xs), t>0, a discontinuous additive functional of X. With neither F nor X being symmetric, under certain conditions, we show that the Feynman-Kac semigroup defined by

     

A class of compact group extensions of β-transformations

We investigate the dynamical properties of a skew product transformation Tφ on [0,1)×G defined by Tφ(x,g)=(Tx,g⋅φ(x)) where T is the β-transformation for β?2 and φ(x) is a compact group G-valued step function with a finite number of discontinuities. We give several sufficient conditions for ergodicity and strong mixing of Tφ. As an application, we describe a class of step functions which satisfy the Central Limit Theorem for the β-transformations. As another application, we also consider a class of skew product transformations Tβ,a,w on [0,1)×[0,1) which maps where a,w∈R and give necessary and sufficient conditions for ergodicity and strong mixing.     

Uniform convexity of ψ-direct sums of Banach spaces

Let X and Y be Banach spaces and ψ a continuous convex function on the unit interval [0,1] satisfying certain conditions. Let X⊕_ψY be the direct sum of X and Y equipped with the associated norm with ψ. We show that X⊕_ψY is uniformly convex if and only if X,Y are uniformly convex and ψ is strictly convex. As a corollary we obtain that the ?_p,q-direct sum (not p=q=1 nor ∞), is uniformly convex if and only if X,Y are, where ?_p,q is the Lorentz sequence space. These results extend the well-known fact for the ?_p-sum . Some other examples are also presented.