On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

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.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

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.     

A study of the rate of rational approximation of a class of analytic functions

This paper is devoted to results belonging to the theory of rational approximation of analytic functions. We study the rate of decrease of the best approximations _n of a functionf holomorphic in a disk ¦z ¦z¦< , >1> by rational functions of order at most n in the uniform metric on the unit disk with center at z=0. We prove theorems that connect the rate of decrease of the quantities _n with the order 0 and type 0 of the functionf. The proofs of these results are based on the methods of the theory of Hankel operators.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 32–39.The author is grateful to Prof. V. A. Prokhorov for posing this problem and assistance with the preparation of this article.     

Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

Local completeness I

An infinite algebra is locally complete if its local closure is the set of all finitary operations. We give a local completeness criterion in terms of a system R of finitary relations on A such that the polymorph Pol of each R is locally incomplete and for every locally incomplete algebra A; F we have F Pol for some R. This system consists of (i) certain natural relations whose polymorphs are best possible in the sense that they are co-atoms in the lattice of locally closed incomplete algebras, (ii) five types of binary relations, (iii) one type of ternary relations and (iv) at least ternary totally reflexive and symmetric relations that are not locally central.Presented by K. A. Baker.     

Problem of the maximum of the n-th diameter in the hyperbolic plane

Let E be a compactum in the circle and let d_n(E) be the n-th Euclidean diameter of E: Let K^(h)() be the family of all continua in U of hyperbolic capacity , 0<<1. Let E ₁ ^* ()=[0, R₁()] and let for n=2,3,..., where R_n() is the solution of the equation K(k) being the elliptic integral of the first kind with module k and. In Sec. 1 of the paper one shows that for all even n=2m 4- and all 0<<1, the symmetric continuum E _n ^* () does not yield a maximum for d _n ^(h) (E) in the family K^(h)(). This complements the known result of a negative character in the problem of the maximum of the n-th Euclidean diameter in the family of all continua of a prescribed capacity. In Sec. 2 one shows that for any 0<<1, the maximum of d₃(E) in the family K^(h)() is attained only by the continua of the form, being a real number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 113–130, 1980.     

Further results on the covering radii of the Reed-Muller codes

LetR(r, m) by therth order Reed-Muller code of length2 ^m , and let (r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. (r+1,m+2) 2(r, m)+2 if 0rm–2. This improves the successive use of the known inequalities (r+1,m+2)2(r+1,m+1) and (r+1,m+1) (r, m).2.(2, 7)44. Previously best known upper bound for (2, 7) was 46. 3. The covering radius ofR(1,m) inR(m–1,m) is the same as the covering radius ofR(1,m) inR(m–2,m) form4.     

On liftings and the regularization of stochastic processes

Summary We show that not all liftings are suitable tools for the regularization of stochastic processes. Under Continuum hypothesis, we construct a Glivenko-Cantelli class Z on [0, 1] and a lifting ofL such that Z is not a Glivenko-Cantelli class. This strongly contrasts with the fact, shown earlier by the author, that some special liftings have exellent regularization properties.     

An Extension of the Pizzetti Formula for Polyharmonic Functions

We represent the integral over the unit ball B in R ⁿ of any poly-harmonic function u(x) of degree m as a linear combination with constant coefficients of the integrals of its Laplacians ^j u (j = 0,...,m - 1) over any fixed(n - 1)-dimensional hypersphere S() of radius (0 1). In case = 0 theformula reduces to the classical Pizzetti formula. In particular, the cubature formula derived here integrates exactly all algebraic polynomials of degree 2m - 1.     

Improvement of some bounds on the stability of fast Helmholtz solvers

Three fast Helmholtz solvers based on the Fast Fourier Transform are studied with respect to their stability. It is proved that they are strongly stable, and that the backward relative errors can grow at most like O(log₂ n)₀, where ₀ is the machine roundoff unit.     

Rational Approximation of Analytic Functions Having Generalized Orders of Rate of Growth

Let f be holomorphic on a domain G C¯ and _n be the error in best approximation of f in the supremum norm on a compact set E G by rational functions of order n. We obtain results characterizing the degree of decrease of the best approximation _n in terms connected with the condenser (E,F), F=C¯ \ G¯, and the rate of growth of the maximum modulus of f(z). In particular, if f has a generalized order (, , f) in the domain G, thenlim sup_n (n)/ (log (1/log⁺((_{0 1} ....... _n)^1/n(n+1) ))) (, , f),where = exp (1/C(E,F)), C(E,F) is the capacity of the condenser (E,F).     

Canonical correlation and reduction of multiple time series

It is shown that a degenerate rankd-variate stationary time series can be reduced to a full rank time series of lower dimension via an orthogonal transformationT provided that , the canonical correlation between past and future of the time series is strictly less than one. Procedures for estimation of rank of the multiple time series,T and testing =1 are outlined, the latter is related to testing the unit root hypothesis in ARMA models.     

Generalization of the theorem of Marcinkiewicz on entire characteristic functions of probability distributions

The relation is studied between the distribution of the zeros and the order of growth of entire analytic functions for which ¦p(z)¦ (i Imz) for Imz 0, in particular, of entire characteristic functions of probability distributions. The main result is the following: if ₁ is the exponent of convergence of the sequence of zeros of such a function of order which lie in a half plane Imz d > 0, then the inequality ₁ < implies the inequality p 3. This estimate is precise.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 94–103, 1979.The author is grateful to I. V. Ostrovskii for posing the problem and for his constant assistance with the work.     

Asymptotics of the fundamental system of solutions of a linear functional-differential equation with respect to a parameter

We study a functional-differential equation, whereF is a linear operator acting from the Hölder spaceH into the Sobolev space W _p ^s [0, 1] and is a complex parameter. For large absolute values of , we construct a one-to-one correspondence between the solutionsx(;t) andy(;t) of the equations andy ⁽ⁿ⁾+y ⁿ=0. We also establish conditions that should be imposed on the operatorF in order that specially selected fundamental systems of solutions of these equationsx _j (;t) andy _j (;t), j=1,...,n, satisfy the estimate with constantsc, >0 for the functional space=W _q ^l [0, 1] or.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 811–836, June, 1995.This work was financially supported by the International Science Foundation and the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let be a linear representation of a finite group over a field of characteristic 0. Further, let R be the corresponding algebra of invariants, and let P (t) be its Hilbert–Poincaré series. Then the series P (t) represents a rational function (t)/(t). If R is a complete intersection, then (t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where is an almost regular (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.     

Balanced ternary designs with block size three,any Λ and anyR

A balanced ternary design onV elements is a collection ofB blocks (which are multisets) of sizeK, such that each element occurs 0, 1 or 2 times per block andR times altogether, and such that each unordered pair of distinct elements occurs times. (For example, in the blockxxyyz, the pairxy is said to occur four times and the pairsxz, yz twice each.) It is straightforward to show that each element has to occur singly in a constant number of blocks, say ₁, and so each element also occurs twice in a constant number of blocks, say ₂, whereR= ₁+2 ₂. If ₂=0 the design is a balanced incomplete block design (binary design), so we assume ₂>0, andK<2V (corresponding to incompleteness in the binary case). Necessarily >1 if ₂>0 (andK>2).In 1980 and 1982 the author gave necessary and sufficient conditions for the existence of balanced ternary designs withK=3, =2 and ₂=1, 2 or 3. In this paper work on the existence of balanced ternary designs with block size three is concluded, in that necessary and sufficient conditions for the existence of a balanced ternary design withK=3, any >1 and any ₂ are given.     

Asymptotic Behavior of Loss Probability in GI/M/1/K Queue as K Tends to Infinity

We obtain an asymptotic behavior of the loss probability for the GI/M/1/K queue as K for cases of <1, >1 and =1.