Spectral functions of zeros in the Besselq-functions

Zeta functions _v(z; q)= _n=1 [j_vn(q)]^–z and partition functions Z_v(t; q)=_n exp[–tj _vn ² (q)] related to the zeros j_vn(q) of the Bessel q-functions J_v(x; q) and J _v ⁽²⁾ (x; q) are studied and explicit formulas for _v(2n; q) at n=±1, ±2, ... are obtained. The poles of _v(z; q) in the complex plane and the corresponding residues are found. Asymptotics of the partition functions Z_v(t; q) at t 0 are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 397–414, June, 1996.     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation : GF(q^s) GF(q^s) can be used to treat a linear cyclic code of wordlength v over GF(q^s) as a linear code () of Wordlength sv over GF(q). This paper determines those linear cyclic codes and transformations for which the resulting linear code () is also cyclic.     

The properties of matricesαI+βJ and some related problems

LetF be a field,v(2) an integer,I _v the identity matrix of orderv, andJ _v the all-one matrix of orderv. LetD={I _v+J _vGL _v(F)|,F}, and ={CGL _v(F)| the row-sums and the column-sums ofC are all equal}. In this paper, we study (1) the center and the centralizer of , (2) the centralizer and the normalizer ofD whenv=2, and (3) the orders ofD, and the normalizer ofD.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Summierbarkeit der geometrischen Reihe auf vorgeschriebenen Mengen

In this paper we are concerned with the summability of the geometric series by matrix methods. We prove the following theorem: Suppose M_o:={z:|z|<1}, M₁, M₂, is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2, let f_k be a prescribed function, analytic on . Then there exists a triangular matrix , such that the V-transform {_n(z)} of the geometric series has the following properties: {_n(z)} converges compactly to on M_o; for k=1,2, there are sets B_k, such that has Lebesgue-measure zero and _n(z)f_k(z) for zB_k; if there is a set B^*, such that B^*M^* has Lebesgue-measure zero and {_n(z)} diverges for zB^*.     

Spectral properties of certain operators on Hardy spaces of planar regions

If G is any region in the complex plane and H^P (G), 1p, denotes the Hardy space of analytic functions on G, then the principal aim of this paper is to present the spectral properties of the multiplication operators M defined on H^P(G) by Mf=f, where is any bounded analytic function on G. In order to do this, the speccial case in which G is bounded and (z)=z is studied in detail. In addition an operator called the truncation operator that is not a multilication operator and acts on the Hardy spaces of regions of the form /K for a compact set K is studied.Dedicated to the memory of my brotherThis research was partially supported by National Science Foundation grant MCS 83-204-26.     

Sur un problème de Rényi

Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev _q (x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv _q (x); this improves on previous results.

     

On the Infinitesimal Rigidity of Homogeneous Varieties

Let XP be a variety (respectively an open subset of an analytic submanifold) and let xX be a point where all integer valued differential invariants are locally constant. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre P× P, n,m2, a Grassmaniann G(2,n+2), n4, or the Cayley plane OP², then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P²×P² had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v₂(P) and the Fubini cubic form of X at x is zero, then X=v₂ (P) (resp. an open subset of v₂(P)). All these results are valid in the real or complex analytic categories and locally in the C category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I₂(P P) S²C, I₂(P¹× P) S²C and I₂(S₅) S²C¹⁶ are stable in the sense that if A S^* is an analytic family such that for t0,AA, then A₀A. We also make some observations related to the Fulton–:Hansen connectedness theorem.     

Modified critical behavior in the ϕ4(O n ) model

In the standard ⁴(O _n ) model, a critical regime in which the coupling constantg of the ⁴ decreases as a certain given power as T-T _c0 is considered. From the point of view of physics, such a formulation of the problem corresponds to a certain class of trajectories of approach to the triple point in the two-dimensional plane of the physical parameters of the system. It is shown that in such a modified critical regime all the critical dimensions can be expressed in terms of the specified value of the exponent and the ordinary critical dimensions of the ⁴ model known in the form of 4- expansions.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 168–172, April, 1992.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

Nielsensche Realisierungssätze für Seifertfaserungen

Let M be a closed aspherical manifold and A a finite subgroup of the outer automorphism group Out ₁M of ₁M. A necessary (and in many cases also sufficient) condition for realising A by the induced action of an isomorphic group of homeomorphisms of M is the existence of an extension 1₁MEA1 to the abstract kernel (A,₁M, AOut ₁M). If the center of ₁M is nontrivial, this condition need not be fulfilled ([14]). We showed in [25] however that one can always find a surjection BA of a finite group B with abelian kernel such that there exists an extension to the abstract kernel (B,₁M,BAOut₁M), and one can try to realize B instead of A. The main result of the present paper is a characterisation of all such groups B (for a given A) which can be realized by a group of homeomorphisms. The class of manifolds considered here consists of certain Seifert fiber spaces in arbitrary dimensions but the main result is purely algebraic and can be applied to other classes of manifolds, for example to flat Riemannian manifolds.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾