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.     

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     

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.     

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.     

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 Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x( ⁿ )+A ₁ x(^n–1 +...+ ⁿ A _n x, the matrices A₁,...,A_n are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ^m , F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in ^m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991.     

Some properties of the delay probability inM/M/s/s +c systems

We look at an extension of the steady state delay probability inM/M/s/s + c systems to nonintegral number of serverss and queue capacityc, which we call GED function. We show that this function is increasing and concave in the queue capacity. We find that if c 1, the reciprocal of the GED function is convex in the traffic intensity and the GED function is increasing in the traffic intensity if is below some ^* _s,c, and decreasing if is greater than ^* _s,c, Moreover, ^* _s,c is increasing in the number of servers and, fors 1, ^* _i,c=1 p^* _s,c < 2.Research supported by Grant BD/645/90-RM from Junta Nacional de Investigação Científica e Tecnológica.On leave from: Departamento de Matemätica, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal.     

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.     

Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

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.     

Commutativity of operators on the lattice of existence varieties

In a previous paper the authors introduced seven complete congruences on the lattice _ev(I of e-varieties of regular semigroups of the form _P :U _P VPU=PV, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form _P :U _P VPU=PV, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence on _ev(I) and eachU_ev(I), the -class ofU is an interval [U ,U ] so that there is associated with each such congruence an idempotent operatorUU on _ev(I). This paper establishes numerous results concerning the commutativity of operators of this form.This work was supported in part by NSERC Grant 4044.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Index of exponential growth for a class of stochastic semigroups

We consider finite-dimensional homogeneous stochastic semigroups X _s ^t , 0 s t < assuming values in the space of real square matrices. For stochastic semigroups assuming values in the class of upper triangular matrices we compute the index of exponential growth , where · is the operator norm of a matrix. The answer is given in terms of the characteristic Y_t of the generating process Y_t of the semigroup X_s ^t:x=–(1/2), where is the smallest eigenvalue of the matrix B which defines the characteristic Y_t=Bt.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 78–84, 1988.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

Finite clusters in high-density continuous percolation: Compression and sphericality

Summary A percolation process inR ^d is considered in which the sites are a Poisson process with intensity and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other. The distribution of the number of sites in the clusterC of the origin is examined, and related to the geometry ofC. It is shown that when andk are large, there is a characteristic radius such that conditionally on |C|=k, the convex hull ofC closely approximates a ball of radius , with high probability. When the normal volumek/ thatk points would occupy is small, the cluster is compressed, in that the number of points per unit volume in this -ball is much greater than the ambient density . For larger normal volumes there is less compression. This can be compared to Bernoulli bond percolation on the square lattice in two dimensions, where an analog of this compression is known not to occur.Research supported by NSF grant number DMS-9006395     

Preprojective components of wild tilted algebras

If A is a finite dimensional, connected, hereditary wild k-algebra, k algebraically closed and T a tilting module without preinjective direct summands, then the preprojective componentP of the tilted algebra B=End_A (T) is the preprojective component of a concealed wild factoralgebra C of B. Our first result is, that the growth number (C) of C is always bigger or equal to the growth number (A). Moreover the growth number (C) can be arbitrarily large; more precise: if A has at least 3 simple modules and N is any positive integer, then there exists a natural number n>N such that C is the Kronecker-algebraK _n, that is the path-algebra of the quiver (n arrows).     

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space H _p, p 1, in disks of radii , ₁, and ₂, 0 < ₁ < < ₂ < 1.     

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.     

Lower types of δ-subharmonic functions of fractional order

It is proved that the lower types of functions T(r, u) and N(r, u)=N(r, u₁)+N(z, u₂) relative to the proximate order (r) of a function u=U₁–u₂ of fractional order -subharmonic in ^m, m>- 2, coincide, that is, are simultaneously minimal or mean. In the case of an arbitrary proximate order (r), the assertion is, in general, false.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1280–1284, September, 1992.