Compound Poisson Processes with Two-Sided Reflection

We consider a compound oscillating Poisson process with two-sided reflection. This process is defined by an upper-semicontinuous compound Poisson process (t) and its functionals, namely the first-exit time of (t) from an interval and the first-exit time of (t) across the upper and lower levels. We study the main characteristics of this oscillating process in terms of the potential and resolvent of the process (t) introduced by Korolyuk. For this purpose, we refine the Pecherskii identities and some other results for upper-semicontinuous Poisson processes.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

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     

On pseudo-differential operators and smoothness of special Lie-group representations

Two algebras of global pseudo-differential operators over ⁿ are investigated, with corresponding classes of symbols A⁰=CB (all (x, )-derivatives bounded over ²ⁿ), and A₁ (all finite applications of _xj, _j, and _pq=_pq–_px_p on the symbol are in A₀). The class A₁ consists of classical symbols, i.e., ^x a= 0((1+||)–^||) for x Kc ;ⁿ, K, compact. It is shown that a bounded operator A of 210C=L²(Rⁿ) is a pseudo-differential operator with symbol aA_j if and only if the map AG^–1AG, G g_j is infinitely differentiable, from a certain Lie-group g_j c GL(210C) to (210C) with operator norm. g₀ is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

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.     

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.     

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.     

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 .     

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.     

Limit distribution function of inhomogeneities in regions with random boundary. I

A study is made of the interaction of systems of charged particles with a membrane consisting of inhomogeneities randomly distributed in accordance with the same law in the neighborhoods of corresponding sites of a planar crystal lattice. A system of equations for the self-consistent potentialU ₁(x, ⁰,..., ^N ,...) and density of surface changes (x, ⁰,..., ^N ,...) is derived and solved.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 98–112, July, 1992.     

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 ().     

A characterization of Gaussian measures on locally compact Abelian groups

Let and be independent random variables having equal variance. In order that + and – be independent, it is necessary and sufficient that and have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when and take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of + and –, functions of and are considered which satisfy the associativity equation.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 759–762, November, 1977.     

Local theorems for some statistics of integer-valued sequences

Let Js={j₁, ..., j_s} be a collection of nonnegative numbers, j₁+...+j_s=n, j_s1, R(Js) be the set of sequencesf=(f(1), ...,f(n)) in each of which the integer m occurs j_s times. Randomly and equiprobably one chooses a sequence f from R(Js). Let _n,r be the number of r-drops in f; _n be the r-principal index off. Local limit theorems are established in this paper for the random variables _n,r and _n as n .Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 61–66, 1988.     

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.     

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.

     

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.     

Minimax filtration of linear transformations of stationary sequences

We will consider the problem of determining a linear, mean-square optimal estimate of the transformation of a stationary random sequence (k) with density f() from observations of the sequence (k) + n(k) withk0, where (k) is a stationary sequence not correlated with (k) with density g(). The least favorable spectral densities and minimax (robust) spectral characteristics of an optimal estimate A for different classes of densities are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 92–99, January, 1991.     

Newton—Goldstein convergence rates for convex constrained minimization problems with singular solutions

Convergence rates of Newton-Goldstein sequences are estimated for convex constrained minimization problems with singular solutions, i.e., solutions at which the local quadratic approximationQ(, x) to the objective functionF grows more slowly than x – ² for admissible vectorsx near. For a large class of iterative minimization methods with quadratic subproblems, it is shown that the valuesr _n =F(x _n )–inf F are of orderO(n ^–1/3) at least. For the Newton—Goldstein method this estimate is sharpened slightly tor _n =O(n ^–1/2) when the second Fréchet differentialF is Lipschitz continuous and the admissible set is bounded. Still sharper estimates are derived when certain growth conditions are satisfied byF or its local linear approximation at. The most surprising conclusion is that Newton—Goldstein sequences can convergesuperlinearly to a singular extremal whenF(), x – Ax – ^v for someA > 0, somev (2,2.5) and allx in near, and that this growth condition onF() is entirely natural for a nontrivial class of constrained minimization problems on feasible sets = ¹{[0,1],U} withU a uniformly convex set in ^d . Feasible sets of this kind are commonly encountered in the optimal control of continuous-time dynamical systems governed by differential equations, and may be viewed as infinite-dimensional limits of Cartesian product setsU ^k in ^kd . Superlinear convergence of Newton—Goldstein sequences for the problem (,F) suggests that analogous sequences for increasingly refined finite-dimensional approximation (U ^kd ,F _k ) to (,F) will exhibit convergence properties that are in some sense uniformly good ink ask .Investigation partially supported by the U.S. Air Force through the Air Force Institute of Technology, and by NSF Grant ECS-8005958.     

A Functional Interpretation of Cycloid Processes: Application to the Study of Asymptotic Behaviour

We show that the classic Chapman–Kolmogorov equations of certain Markovian transition semigroups on finite state spaces have a formal analogy, of a homologic nature, in terms of cycloids ₁, ..., _B, and positive numbers w₁, ..., w_B. The collection _k ,w _k completely determines a Markov process {_n}, called a cycloid process, admitting an invariant probability distribution, and decomposes its distribution Prob(_n = , _{n + 1} = ) into a linear expression. The latter is further used in the study of the asymptotic behaviour of the cycloid process.