Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σ_k∈ℤ ^Pk Ф(nx−k), where n≥2 is an integer, the coefficients {P_k} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.     

Norms of powers of absolutely convergent fourier series

Letf(t) = ∑a _k e ^ikt be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖ⁿ‖ converges to a finite limitl asn → ∞ (l ² = sec(arga),a = (f′(0))² -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn ^-1/2) for the Fourier coefficientsa _nk off ⁿ , which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn ^-1 are present!) for ‖f ⁿ ‖; (iii) the fact that ifi ^j f ^(j)(0) is real forj = 1,2,..., 2h + 2 then ‖f ⁿ ‖ = l + o(n ^-h ),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times.     

On an infinite series of Abel occurring in the theory of interpolation

The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σ_{n = 1}^∞ f⁽ⁿ⁾(nβ) z(z − nβ)^{n − 1}/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed.     

On the theory of mappings with finite area distortion

It is shown that mappings in ℝⁿ with finite distortion of area in all dimensions 1 ≤ k ≤ n − 1 satisfy certain modulus inequalities in terms of inner and outer dilatations of the mappings; in particular, generalizations of the well-known Poletskii inequality for quasiregular mappings are proved. The theory developed is applicable, for example, to the class of finitely bi-Lipschitz mappings, which is a natural generalization of the bi-Lipschitz mappings, as well as isometries and quasi-isometries in ℝⁿ.     

New perturbation theory representation of the conformal symmetry breaking effects in gauge quantum field theory models

We propose a hypothesis on the detailed structure for the representation of the conformal symmetry breaking term in the basic Crewther relation generalized in the perturbation theory framework in QCD renormalized in the [`(MS)]\overline {MS} scheme. We establish the validity of this representation in the O(α _s ⁴ ) approximation. Using the variant of the generalized Crewther relation formulated here allows finding relations between specific contributions to the QCD perturbation series coefficients for the flavor nonsinglet part of the Adler function D _A ^ns for the electron-positron annihilation in hadrons and to the perturbation series coefficients for the Bjorken sum rule S _Bjp for the polarized deep-inelastic lepton-nucleon scattering. We find new relations between the α _s ⁴ coefficients of D _A ^ns and S _Bjp . Satisfaction of one of them serves as an additional theoretical verification of the recent computer analytic calculations of the terms of order α _s ⁴ in the expressions for these two quantities.     

On the structure theory of the Iwasawa algebra of a p-adic Lie group

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a p-adic analytic group G. For G without any p-torsion element we prove that Λ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-nullΛ-module. This is classical when G=ℤ ^k _p for some integer k≥1, but was previously unknown in the non-commutative case. Then the category of Λ-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the ℤ _p -torsion part of a finitely generated Λ-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere. Received May 12, 2001 / final version received July 5, 2001?Published online September 3, 2001     

Finitely-presented groups with long lower central series

We construct examples of finitely presented groupsG with the property that, ifG _ω is the intersection of the lower central series ofG, then [G,G _ω ] ≠G _ω .     

On one extremal problem for positive series

The approximation properties of the spaces S _ϕ ^p introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of n-term approximations of q-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set. Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1677–1683, December, 2005.     

Convergence of double Fourier series

Summary Let a2π-periodic function f(x, y) be continuous in some neighbourhood of the point (x, y) except possibly along finitely many lines l ₁ , l ₂ , ..., l_k terminating at (x, y). The problem of convergence of the Fourier series of f(x, y) at the point (x, y) is examined in some detail. It is established that under certain restrictions on the variation of f(x, y), and also on the lines l ₁ , l ₂ , ..., l_k, the fourier series converges to a value bounded above by the limit superior, and below by the limit inferior of f(x+u, y+v), u, v →0, this value depending on the manner in which the series is summed. The preparation of this paper was financed, in part, by a Canadian Mathematical Congress Summer Research Grant (1968). Entrata in Redazione il 3 febbraio 1969.     

Convergence of mock Fourier series

For certain Cantor measures μ on ℝⁿ, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse ^2πiγ·x for λεΛ. a discrete subset of ℝⁿ called aspectrum for μ. For anyL ¹ functionf, we define coefficientsc _γ(f)=∝f(y)e ^−2πiγiy dμ(y) and form the Mock Fourier series ∑_λ∈Λ^cλ(f)e ^2πiλ·x . There is a natural sequence of finite subsets Λ_n increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by We prove, under mild technical assumptions on μ and Λ, thats _n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL ¹. Research supported in part by the National Science Foundation, Grant DMS-0140194.     

On the derived series of some groups

We solve Problems 17.82 and 17.86(b) posed by Mikhailov in the Kourovka Notebook [1]. Namely, we construct: (1) an example of a finitely presented group H in which the intersection H ^(ω) of all terms of the derived series is distinct from its commutant; (2) an example of a balanced presentation 〈x ₁, x ₂, x ₃|r ₁, r ₂, r ₃〉 of the trivial group for which F(x ₁, x ₂, x ₃)/[R ₁,R ₂] is not a residually soluble group (here R _i (i = 1, 2) denotes the normal closure of r _i in F(x ₁, x ₂, x ₃)). The construction of the second example is related to some approach to the Whitehead asphericity conjecture.     

A generalization of Cohen- Eisenstein series and Shimura liftings and some congruences between Cusp forms and Eisenstein series

In this paper we introduce some modular forms of half-integral weight on congruence group Г_o(4N) withN an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we can prove that the restriction of Shimura lifting on Eisenstein spaceE _k+1/2 ⁺ (4N,χl) gives an isomorphism fromE _k+1/2 ⁺ (4N,χl) toE _2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.     

The summation of power series having positive coefficients

The paper develops a method for the summation of a series of positive terms given by Gismalla et al. [2]. Here we are concerned with the summation of power series ∑^∞_n = ₁ μ_nxⁿ where μ_n > 0 and 0 < x < 1, and x may be very close to 1 (but not equal to 1). Using ideas of Longman [4] we show how a bound can be obtained for the error when the process is curtailed at a particular stage. The method supposes that the coefficients in the summation are moments but, in practice, seems to work even when the coefficients are not identifiable as moments.     

Power series approximations for two-class generalized processor sharing systems

We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1−β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z ₁,z ₂) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z ₁,z ₂) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.     

Functional equations for Hecke-Maaass series

The Dirichlet (Hecke-Maass) series associated with the eigenfuctionsf andg of the invariant differential operator Δ_k=−y²(∂²/∂x²)+iky∂/∂x of weightk are investigated. It is proved that any relation of the form (f/kM)=g for thek-action of the groupSL ₂ SL ₂(ℝ) is equivalent to a pair of functional equations relating the Hecke-Maass series forf andg and involving only traditional gamma factors. This work was supported by the Russian Foundation for Basic Research (grant No. 96-01-10439). Institute of Applied Mathematics, Far East Division of Russian Academy of Sciences. Translated from Funktional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 23–32, April–June, 2000. Translated by V. M. Volosov     

The functional equation for Poincaré series of trace rings of generic 2×2 matrices

In this note we give a rational expression for the Poincaré series of Π_m,2, the trace ring ofm generic 2×2 matrices. This result extends the computations of E. Formanek form⩽4. As a consequence, we prove that the Poincaré series satisfies the functional equation(II_m,2;1/t)=-t^4m.P(II_m,2,t) (m>2) supporting the conjecture that Π_m,2 is a Gorenstein ring. Work supported by an NFWO/FNRS grant.     

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ _f ⁺ isomorphic to (I, f) generalizing the \-shift and another shift Σ _M over an infinite alphabet, which is of finite type given by an infinite transition matrixM. Σ _M has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].     

On the theory of the Beltrami equation

We study ring homeomorphisms and, on this basis, obtain a series of theorems on the existence of the so-called ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations that extends earlier results is formulated. In particular, we give new existence criteria for homeomorphic solutions f of the class W _loc ^1,1 with f ⁻¹ ∈ W _loc ^1,2 in terms of tangential dilatations and functions of finite mean oscillation. The ring solutions also satisfy additional capacity inequalities. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1571–1583, November, 2006.