Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

Constructive methods of approximation by ridge functions and radial functions

Let be a univariant function, and letg(x) be the average of (x,u) asu runs over the unit sphere in ⁿ . We give a necessary and sufficient condition forg to be a kernel function, i.e., thatg be inL ¹ ( ⁿ ) and have integral 1. The result is used to give a constructive proof of the density of the ridge functions based upon the function .     

Dilation-analytic wave operators for three particles

Let H(0) be a dilation-analytic three-particle Schrödinger operator with analytic continuation H() (>0). Let a be zero or the energy of a two-particle bound state. Let- (a) be the Laplace operator representing the kinetic energy of the relative motion of fragments scattered in channel a. By recent results, wave operators W (±, a, ) with conjugates W (±, a, ) exist such that W (±, a, ) W (±, a, ) is a projection P (a, ) commuting with H () while [H ()-a]W (±, a, ) equals-W(±, a, ) (a) e²ⁱ. This paper shows that the wave operators transform dilation-analytic functions of particle coordinates into dilation-analytic functions. Specifically, if the left shoulder of the spectrum of P (a,) H () does not sweep across eigenvalues of H() when , then W(-, a, ) and W (+, a, ) are dilation analytic in [, ]. If the right shoulder does not sweep across eigenvalues, W(+, a, ) and W(-, a, ) are dilation analytic in [,]. A semisimple eigenvalue of H () embedded in the spectrum of P (a, ) H () does not prevent the wave operators from being dilation analytic in an interval [, ] with as an interior point.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

The integrability of conjugate functions

For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968.     

Weakly isomorphic transformations that are not isomorphic

Summary A new method for construction of transformations T _i: (X _i, B _i, _i) , i=1,2, that are factors of each other but that are not measuretheoretically isomorphic is provided. This method uses ergodic product cocycles of the form S ⁱ ₁xS ⁱ ₂x...,, where : XZ ₂ is a cocycle, S belongs to the centralizer of T and T is an ergodic translation on a compact, monothetic group X.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Cauchy-type integrals in domains with smooth boundary

Let be a conformal mapping of the unit circle onto the domain bounded by a curve of classC ¹. There exist curves such that the function (f·) is not necessarily a Cauchy type integral in the unit circle when f is a Cauchy type integral in Int .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 173–174, 1987.In conclusion, I express my gratitude to S. A. Vinogradov and V. P. Khavin for advice and for their interest in this note.     

Joint torsion of Toeplitz operators with H∞ symbols

We give a new² index theorem for the basic example of Toeplitz operators on the circle. The joint torsion, a non zero complex valued analytic index, of a pair of Fredholm Toeplitz operatorsT andT withH symbols is computed by residues in the disk, and is determined by a monodromy integral which specifies the isomorphism class of a flat line bundle on the circle. When the symbols and are rational a product of joint torsions identifies the isomorphism class of the bundle inH ¹ (S ¹,C ^*), and the identification extends by rational approximation to the case of smooth symbols defined on the circle.Partially supported by National Science Foundation grants to both authors.     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.     

A characterization of multivariate quasi-interpolation formulas and its applications

Summary Let be a compactly supported function on ^s andS () the linear space withgenerator ; that is,S () is the linear span of the multiinteger translates of . It is well known that corresponding to a generator there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called -interpolation and a notion of higher order quasi-interpolation called -approximation. A characterization of -approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator at the multi-integers ^s facilitates the above study. An algorithm to yield this information for box splines is discussed.Supported by the National Science Foundation and the U.S. Army Research Office     

Compact and weakly compact weighted composition operators on weighted spaces of continuous functions

In this note we characterize the compact and weakly compact weighted composition operatorsW _, on certain weighted locally convex spacesCV _o(X, E) of vector-valued continuous functions induced by self maps ofX and the operator-valued mappings XB(E).The work of this author was supported in part by CSIR Grant 9/100/92-EMR-IThe work of this author was supported in part by UGC Grant F.8-7/91 (RBB-II)     

Maximal order for quadratures usingn evaluations

The problem is to find approximationsI (f; h) to the integralI(f; h)= ₀ ^h f. Such an approximation has local orderp ifI(f; h)–I (f; h)=O(h ^p ) ash0. Let(n) denote the maximal local order possible for a method usingn evaluations of a function or its derivatives. We show that (n)=2n+1 if the information used is Hermitian. This is conjectured to be true in general. The conjecture is established for all methods using three or fewer evaluations.This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422. Numerical results reported in this paper were obtained through the computing facilities of the University of Maryland.     

A bound for the error in the numerical integration of a first order quasilinear equation

We obtain a bound for the error in the numerical integration of the quasilinear equation u_t+((u))_x=0 by a finite difference method in the case when (u)0.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 207–215, February, 1973.     

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.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

The question of A*-summability of double trigonometric Fourier series

It is proved that for anyf(x, y) L(R), where R=[-,,-, ], a function (x, y), exists such that ¦(x, y) ¦=¦f(x, y) ¦ for almost all (x, y) R. The Fourier series of the function (x, y) and all conjugate trigonometric series are A^*-summable almost everywhere.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 145–150, February, 1972.     

Posets isomorphic to their extensions

A standard extension for a poset P is a system Q of lower ends (descending subsets) of P containing all principal ideals of P. An isomorphism between P and Q is called recycling if [Y]Q for all YQ. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.