Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

On the temporal analysis of special GERT networks using a modified Markov renewal process

In this paper we show how to represent a STEOR network (GERT network with only nodes of the stochastic exclusive-or Type) by a modified Markov renewal process. The computation of the renewal kernel of this process yields the temporal analysis of the network.

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Zusammenfassung Es wird gezeigt, wie ein STEOR-Netzplan (GERT-Netzplan, der nur Knoten vom Typ stochastisch exklusiv-oder enthält) durch einen modifizierten Markowschen Erneuerungsprozeß beschrieben werden kann. Die Bestimmung der Erneuerungsfunktionen dieses Prozesses liefert die Auswertung des STEOR-Netzplans (im Sinne der Zeitplanung).

     

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     

Linear operators generated by stochastic processes

Let X_t(0 t < ) be a homogeneous stochastically continuous stochastic process with independent increments; (, A, P) be the corresponding probability space; p, q1 be fixed numbers. Necessary and sufficient conditions are found for a stochastic integral defined on finite-valued functions to be extendable to a bounded operator from L_p (0, ) to L_q() (p q).Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 7–13, 1988.     

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.     

Adaptive boundary control of stochastic linear distributed parameter systems described by analytic semigroups

A stochastic adaptive control problem is formulated and solved for some unknown linear, stochastic distributed parameter systems that are described by analytic semigroups. The control occurs on the boundary. The highest-order operator is assumed to be known but the lower-order operators contain unknown parameters. Furthermore, the linear operators of the state and the control on the boundary contain unknown parameters. The noise in the system is a cylindrical white Gaussian noise. The performance measure is an ergodic, quadratic cost functional. For the identification of the unknown parameters a diminishing excitation is used that has no effect on the ergodic cost functional but ensures sufficient excitation for strong consistency. The adaptive control is the certainty equivalence control for the ergodic, quadratic cost functional with switchings to the zero control.This research was partially supported by NSF Grants ECS-9102714, ECS-9113029, and DMS-9305936.     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

Über die asymptotische Darstellung der Aufspaltung von Paaren benachbarter Eigenwerte der Differentialgleichung der Sphäroidfunktionen

Summary Let be the difference of a pair of adjacent eigen-values of the differential equation (1) for the spheroidal functions. Until now only the two first terms of the asymptotic expansion (2) of for large ^*2=–² had been known. In this note the next term, i. e. the coefficient of ^*–2, is given, and the way of calculating it is described.

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The preparation of this note containing partial results, was sponsored by the European Office Air Research and Development Command, U. S. Air Force, Project No. AF 61 (514)-443.     

Milnor numbers and the topology of polynomial hypersurfaces

Summary LetF: n + 1 be a polynomial. The problem of determining the homology groupsH _q (F ^–1 (c)), c , in terms of the critical points ofF is considered. In the best case it is shown, for a certain generic class of polynomials (tame polynomials), that for allc,F ^–1 (c) has the homotopy type of a bouquet of - ^c n-spheres. Here is the sum of all the Milnor numbers ofF at critical points ofF and ^c is the corresponding sum for critical points lying onF ^–1 (c). A second best case is also discussed and the homology groupsH _q (F ^–1 (c)) are calculated for genericc. This case gives an example in which the critical points at infinity ofF must be considered in order to determine the homology groupsH _q (F ^–1 (c)).     

Beitrag zur Reduktion des Entscheidungsproblems auf Klassen von Hornformeln mit kurzen Alternationen

Ohne Zusammenfassung Zusatz bei der Korrektur: Ein vollständiger und korrekter Beweis für die Entscheidbarkeit der eingangs angeführten Aanderaaschen Klasse ((0, ), (, , ...)) erscheint demnächst im JSL (S.O. Aanderaa/H.R.Lewis: Prefix classes of Krom formulas). Ebendort wird auch die Reduktionstypeneigenschaft für ((0, ), (0, 0, )) und ((0, )), (0, 0, )) nachgewiesen, während ((0, ), (, )) sich als entscheidbar herausgestellt hat (s. E. Börger: Eine entscheidbare Klasse von Kromformeln. ZMLG 19 (1973), 117–120.) Der Kromsche Reduktionstyp konnte mittlerweile einerseits zu ((0, ), (0, 4)) verschärft werden (s. D. Rödding, E. Börger: The undecidability of (0, 4)-formulae with binary disjunctions, vorgetragen auf dem Logic Coll. Bristol 1973, ein abstract erscheint im JSL), andererseits kündigt H.R.Lewis die Reduktionstypeneigenschaft für ((0, ), (0, 1)) an (s. H.R.Lewis: Krom formulas with one dyadic predicate letter. Notices AMS 20, 5 (1973) A-500, abstr. no. 73T-E78.)Dieser Aufsatz geht aus der Dissertation [2] hervor, die dem Fachbereich Mathematik der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Münster im Sommersemester 1971 vorgelegt worden ist. Die Ergebnisse stammen aus dem Wintersemester 1970/71. Eine Ankündigung der hauptsächlichen Resultate ist in den Notices of the American Mathematical Society 19, 2 (1972) A-333 unter der abstract no. * 72T-E24 erschienen.     

On the solvability of a complete second-order differential equation in Banach space

The paper deals with the Cauchy problem for a complete second-order differential equation with unbounded operator coefficientsu+A(t)u+B(t)u=f, u(0)=u0, u(0)=u ₁ . By using the commutant method, we construct a coercive solution of this problem in Holder space in the case where the operatorB is as strong as the operator A².Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1449–1454, October, 1993.     

A two‐level queueing network model with blocking and non‐blocking messages

We analyze a novel twolevel queueing network with blocking, consisting of N level1 parallel queues linked to M level2 parallel queues. The processing of a customer by a level1 server requires additional services that are exclusively offered by level2 servers. These level2 servers are accessed through blocking and nonblocking messages issued by level1 servers. If a blocking message is issued, the level1 server gets blocked until the message is fully processed at the level2 server. The queueing network is analyzed approximately using a decomposition method, which can be viewed as a generalization of the wellknown twonode decomposition algorithm used to analyze tandem queueing networks with blocking. Numerical tests show that the algorithm has a good accuracy.     

Cauchy problem for a semilinear wave equation. II

The existence and the uniqueness of a weak solution of the Cauchy problem, for a second-order semilinear pseudodifferential hyperbolic equation on a smooth Riemannian manifold (of dimension n3) without boundary, is proved. In particular, for the equation ü–u+u^p–1=0 one has unique solvability for the Cauchy problem when 1(n+2)/(n–2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 38–85, 1990.     

Waterman classes and spherical partial sums of double Fourier series

f(x,y) ₀BV(T²), ={1/n} _n=1 .

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Dedicated to Professor Károly Tandori, the outstanding mathematician and academician on his seventieth birthday

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This work was done under the financial support of the Russian Foundation for Fundamental Scientific Research, Grant 93-01-00240.     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Integrability of multiple Walsh series and Parseval's formula

f(x,y) _jk . , {c _jk} , f(x, )(, ) [0,1)&#x0445;[0,1) , - (0,0). , , f, - f. , , , [1] . . - [5] [6].

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This research is supported by National Science Council, Taipei, R.O.C. under Grant #NSC 84-2121-M-007-026.     

Differentiation of fractional order onp-groups,approximation properties

G- p- . [5] - (G) L _r(G) (1r<), . . , - . , , , . . , X. , . (. [1], [2] [4]).     

Stabilization theorem for the Milnor K2-functor

Let be an associative ring. For every natural number n there is a canonical homomorphism _n: K_2,n()K₂(), where K₂ is the Milnor functor and K_2,n() the associated unstable K-group. Dennis and Vasershtein have proved that if n is larger than the stable rank of , _n is an epimorphism. It is proved in the article that if n – 1 is greater than the stable rank of , the homomorphism _n is an isomorphism.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 131–152, 1976.     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.