On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Explicit expression for the transition density of the solution of a stochastic diffusion equation with piecewise-constant drift coefficient

The explicit form of the transition density is determined for the solution (t) of the stochastic diffusion equation d(t)=a((t))dt+dw(t), where a(z)= for x [a, b] and a(x)=0 for x [a, b], w(t) is a Wiener process.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 99–105, 1987.     

Inverse problem of the scattering of plane waves for a class of layered media

The direct and inverse problems of the scattering of plane waves in a layered, inhomogeneous medium are considered in the paper. In the appropriate variables the wave equation of the problem has the formu (z,)=Q(Z)u _ZZ(Z,), – < Z, <, Q(Z)|_Z<01. A special feature of the case considered, in contrast to those studied earlier, is that Q(Z)|_Z0 may change sign; because of this, the equation of the problem is, in general, an equation of mixed type. The correct formulation of the direct problem for such an equation and the study of the properties of its solution form a necessary step in the investigation. For a very broad class of media including cases of Q(z) of variable sign (Q(z) can change sign by a jump a finite number of times without vanishing anywhere) a procedure is developed for solving the corresponding inverse problem of determining Q(z) on the basis of the scattering datau(0,)|_(–,). This procedure makes it possible to recover Q(z) for all z[0,). The solution of the inverse problem is unique in this class.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 78, pp. 30–53, 1978.The author thanks his scientific supervisor A. S. Blagoveshchenskii for his constant attention and assistance in the work.     

Equations de convolution et formes quadratiques

Résumé On étudie, sans hypothèse de convexité, les équations f=g, f=g et f=g.

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Summary We study, without any convexity hypothesis, equations f=g, f=g and f=g where and respectively denote infimal convolution and deconvolution. We give an explicit formulation of these results in the quadratic hilbertian frame, and we interpret them in terms of parallel addition and subtraction of non necessarily semi-definite positive operators.

     

The problem of controlling a Wiener process when the number of switchings is bounded

We are considering the problem of controlling a one-dimensional Wiener process (t) (0)=0,E=0,D= ²t.Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 53–55.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

On a class of 3-dimensional contact metric manifolds

Let M³ be a 3-dimensional contact metric manifold with contact structure (, , , g), such that and =R(.,)) commute. Such a manifold is called 3--manifold. We prove that every 3--manifold with -parallel Weyl tensor is either flat or a Sasakian manifold with constant curvature 1.     

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

Comparison results for the lower tail of Gaussian seminorms

Let =( _n ) be i.i.d.N(0, 1) random variables andq(x), q(x):R [0, ) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q()<) andP(q()<) goes to a positive constant as 0⁺. We give satisfactory answers forl ₂-norms and also some results for sup-norms andl _p-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL ₂-norms.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

On the Lusternik-Schnirelman theory of a real cohomology class

Farber developed a Lusternik-Schnirelman theory for finite CW-complexes X and cohomology classes H ¹ (X;). This theory has similar properties as the classical Lusternik-Schnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X,) as a generalization of the Lusternik-Schnirelman category. If X is a closed smooth manifold this invariant relates to the number of zeros of a closed 1-form representing . Namely, a closed 1-form representing which admits a gradient-like vector field with no homoclinic cycles has at least cat(X,) zeros. In this paper we define an invariant F(X,) for closed smooth manifolds X which gives the least number of zeros a closed 1-form representing can have such that it admits a gradient-like vector field without homoclinic cycles and give estimations for this number. Mathematics Subject Classification (2000): Primary 37C29; Secondary 58E05     

Constructing Representations of the Nonstandardly Deformed Algebra sℓξ(2)

The generators of the algebra s(2), which result from the nonstandard (Jordanian) deformation of the algebra s(2), are realized in the form of finite-difference operators acting in a function space. This allows realizing arbitrary-dimensional representations of s(2) in the polynomial space that are in one-to-one correspondence with usual matrices of an appropriate dimension. We discuss using the suggested realization to construct and investigate the universal R-matrix invariant with respect to the action of the algebra s(2).     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

An explicit solution of the linear filtering problem for certain classes of nonrational spectral densities

We present an explicit solution of the problem of optimal linear filtering: the recovery of the useful signal(s) at the instantt+, (>0,<0, or=0) from known values of the received signal(s)=(s)+(s) in the past, i.e., at the instantt–s, s0. In doing so we assume the random processes(s) and /gr(s) are stationary and jointly stationary, while the stationary process of noise (s) with zero mean is assumed to be mutually correlated and jointly stationary with the process(s) under the assumption that there exists a common spectral densityf() for these processes.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 83–91, 1986.     

On a new class of contact metric 3-manifolds

We study 3---manifolds having Q parallel to characteristic field and the scalar curvature constant along the geodesic foliation generated by e or e. We find out a new class of contact metric 3-manifolds and we give non-flat examples of this class.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

Estimate of a random field based on the trajectory of a point moving in it

We consider the stochastic differential equationd _t =( _t )dt+ _t ( _t )dw _t in Euclidean space, where (x) is a Gaussian random field andw _t is a standard Wiener process. Let f _t ={ _s ,st}. Equations are obtained for the conditional meansm _t (x)=f _t } andB _t (x, y)=M{(x)(y)|f _t }.Translated fromTeariya Sluchaínykh Protsessov, Vol. 14, pp. 7–9, 1986.     

Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter

Let M be a Riemannian manifold, a differentiable form on M with values in a bundle Hom (TM, ), and let G be an open subset of M such that Ker forms a vector bundle of constant fibre dimension k>0 over G. We prove: If satisfies some analytical conditions, then is completely integrable, the integral manifolds of are spherically bent in M, and in some interesting cases they are complete.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .