On optimality of spline interpolation for recovery of differentiable functions

W _p ^r (1p<) , . - r -1. =1 r-2.     

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.     

A constructive approach to the quantum (cosh ϕ)2 model. I. The method of the Gel'fand-Levitan-Marchenko equations

In a Hubert , with the aid of the postulated Gel'fand-Levitan-Marchenko quantum equations, one introduces the fields ₁(x) and ₂(x), which are the quantum analogues of the classical fields cosh (x) and sinh (x) in the sinh-Gordon model. It is shown that the fields _j(x) satisfy the Wightman axioms, including the invariance relative to reflections of space-time and mutual local commutativity. In addition, one proves the asymptotic completeness of the theory and one computes explicitly the scattering operator. In the developed approach, no cut-offs are used and, therefore, there are no renormalization effects.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 147–190, 1985.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by -blocking, for Euler-Lagrange DAEs of index 2. Thus we may use nonstiff multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed half-explicit Runge-Kutta methods.     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

Central limit theorems for stochastic processes under random entropy conditions

Summary Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array ( _nj ) of stochastic processes, indexed by a set S, to a sample-continuous Gaussian process, when the array satisfies a random entropy condition, analogous to one used by Giné and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-ervonenkis property and each _nj (f)fdnj is of the form njc for some reasonable random finite signed measure v _nj. As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik-ervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.Research supported by an NSF Postdoctoral Fellowship, grant no. MCS83-111686     

Convex Separable Minimization Subject to Bounded Variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.     

Phragmén-Lindelöf theorems for second-order quasilinear elliptic equations

Analogues are formulated of the well-known, in the theory of analytic functions, Phragmen-Lindelöf theorem for the gradients of solutions of a broad class of quasilinear equations of elliptic type. Examples are given illustrating the accuracy of the results obtained for the gradients of solutions of the equations of the form div(|U|^–2u)=f(x, u, u), where f(x, u, u) is a function locally bounded in ²ⁿ⁺¹. f(x, 0, u)=0, uf(x, u, u) c¦u¦1+q(1+ ¦u|), > 1, c > 0, q > 0, is an arbitrary real number, and n >- 2. The basic role in the technique employed in the paper is played by the apparatus of capacitary characteristics.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1376–1381, October, 1992.The author sincerely appreciates E. M. Landis's permanent attention and numerous useful discussions.     

Абсолютная сходимос ть рядов Фурье—Хаара от суперпозиций функци й

A necessary and sufficient condition is found for the function to ensure absolute convergence of the Haar—Fourier series of all functions(f) provided that the Haar—Fourier series off converges absolutely. Absolute convergence means absolute convergence of the series of coefficients, and the condition is that should be in Lip 1.

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Relaxation of a Quadratic Functional Defined by a Nonnegative Unbounded Matrix

We study the lower semicontinuous envelope in L^p(), F, of a functional F of the form F(u)=A uudx where A=A(x) is not strictly elliptic and not bounded. We prove that F; may also be written as F;(u)= Buudx with B=AP A for a matrix P which is the matrix of an orthogonal projection. In the one-dimensional case, we characterize the domain of F and we explicit the matrix P.     

Die Nuklearität der Ultradistributionsräume und der Satz vom Kern II

Continuing the research of part I conditions equivalent to ()- or ()-nuclearity of spaces of ultradifferential functions and their duals as well as some applications are given. To get these results it is shown that tensor products of smooth sequence spaces, power series spaces, and spaces S(M_q) introduced in part I are isomorphic to suitable sequence spaces of the same class, which are stable provided the factors are stable power series spaces. Hence it is possible to establish isomorphisms between different functions spaces, to calculate the nuclearity types of tensor products by the nuclearity types of the factors, and to prove that the class of ()- or ()-nuclear spaces is closed under forming tensor products iff is multiplicatively stable.     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

On Upper and Lower Almost β-continuous Multifunctions

In this paper, the authors define a multifunction F : X Y to be upper (lower) almost -continuous if F+(V) (F- (V)) is -open in X for every regular open set V of Y. They obtain some characterizations and several properties concerning upper (lower) almost -continuous multifunctions.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

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     

A note on sub-multiplicative norms

Summary IfX is a finite-dimensional linear space andL(X) the linear space of linear operators onX thenL(X) may be represented asXX ^*. IfE={e ₁, ...,e _n } is a basis forX and e _j y _j ^* is a typical element ofXX ^*, then norms can be introduced onL(X) in the form y _j ^* e _j . Given that the norm onX isE-absolute we derive a necessary and sufficient condition for the norm onL(X) to be submultiplicative.