Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Asymptote of the eigenvalues of a completely continuous operator

It is proved that if(x) is the majorant of the s-numbers of a completely continuous operator A (i.e.,'(x)- 0, S_n(A) (n)) and if there are found numbers [0, 1] and r₀ > 0 such that r⁰ (r)/(r) will be monotonic in (r₀,), then for some > 0,((x) will be a majorant of the eigenvalues of A.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 487–492, October, 1973.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

Lévy homeomorphic parametrization and exponential families

Summary Let P={P : } be an exponential family of probability distributions with the canonical parameter and consider the one to one mapping : P . It is shown that, under mild regularity assumptions, and ^–1 are continuous with respect to the Lévy metric in P and Euclidean metric in .     

Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = × Z of the formU (, z)=(T,, z+f ()) ( , z Z), where is a complete separable metric space, T is a t.t. homeomorphism of onto itself, Z is a separable banach space, andf is a continuous map: z. For the special case W = S¹×R, T=+ ( is incommensurable with 2) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S¹R for which the corresponding homeomorphism (1) is t.t. for all incommensurable with 2.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.The author thanks D. V. Anosov for advice and interest in the work.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Sûto's method applied to a problem of Färe and Mitchell

Summary A real solution of the functional equation(x + (y – x)) = f(x) + g(y) + h(x)k(y) on a set ² is a 6-tuple (f, g, h, k, , ) of real valued functions such that the equation is identically fulfilled on. Except for cases known before—e.g. when is linear—we present all real solutions in an arbitrary region where the functions have derivatives of second order.     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

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.     

On Constructing Biharmonic Maps and Metrics

We construct biharmonic nonharmonic maps between Riemannian manifoldsM and N by first making the ansatz that M N be aharmonic map and then deforming the metric conformally on M to render biharmonic. The deformation will, in general, destroy theharmonicity of . We call a metric which renders the identity mapbiharmonic, a biharmonic metric. On an Einstein manifold, theonly conformally equivalent biharmonic metrics are defined byisoparametric functions.     

Wave operators and similarity of certain non-normal one-parameter groups

This paper starts from a self-adjoint Schrödinger operator H(0) for three particles. If the interaction is dilation-analytic, H(0) has an analytic continuation H() (>0). G(t,) (–(±,a,) defined as strong limits, when t±, of t-dependent operators. The wave operators establish transformations under which the subgroups are similar to unitary groups. The scattering matrix determined by G(t,) is diagonal with respect to a.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Special directions on contact metric three-manifolds

Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group (2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991.     

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.     

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = ₁ (x),..., _r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector ₀, exact estimates are obtained of the norms f(x)-s (f, ₀; x)_C[0,1] and f (x)-s (f, ₀; x)_{L[0, 1]} on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

Invariant measures ofC 2-diffeomorphisms of Markov type inR d

A class of uniformly expanding, piecewiseC ²-diffeomorphisms from domainsIR ^d (bounded or not) into themselves is considered. It is shown that the number of the extreme points of Fix (P )={gG:Pg=g} whereP is the Frobenius-Perron operator associated with andG={gL ¹: g0 g=1}, can be determined in an effective way. Moreover, it is shown that the sequence {P ^j g} is convergent inL ¹ for anygG, and in the topology of uniform convergence for anygG(1). The limit is a linear projectionR inL ¹ (defined by (3.1)) which mapsG onto Fix (P ) (see Th. 3.1).Dedicated to professor A. Lasota on the occasion of his 60th birthday