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.     

The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x)

In this paper the regularity of the Lagrangiansf(x, )=||^(x)(1< ₁(x)₂< +) is studied. Our main result: If(x) is Holder continuous, then the Lagrangianf(x, )=f(x, )=||^(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.Supported by the National Natural Science Foundation of China.     

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.     

Multilevel large deviations and interacting diffusions

Summary Let ( ^N ) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ^{M, N} denote the empirical measure associated withM independent copies of ^N . As a main result, we show that ( ^{M, N} ) also satisfies the large deviation principle asM,N. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ^{M, N} (t) =M ^–1 _i=1 ^N _i ^N (t) 0tT, where ( ₁ ^N ,..., _M ^N (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant     

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.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

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.     

Necessary convergence conditions for series ΣαnSn in the case of identically distributed independent random quantities

Necessary (in some cases also sufficient) conditions are obtained for convergence of the series a _n S _n whereS _n = ₁ ⁿ _k k are independent random quantities. The cases in which k are symmetrical or identically distributed quantities are investigated in more detail.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 529–536, October, 1976.     

Balayages on Excessive Measures,their Representation and the Quasi-Lindelöf Property

If Exc is the set of all excessive measures associated with a submarkovian resolvent on a Lusin measurable space and B is a balayage on Exc then we show that for any mExc there exists a basic set A (determined up to a m-polar set) such that B=(B^A)* for any Exc, m. The m-quasi-Lindelöf property (for the fine topology) holds iff for any B there exists the smallest basic set A as above. We characterize the case when any B is representable i.e. there exists a basic set such that B=(B^A)* on Exc.     

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.     

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     

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 .     

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.

     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

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     

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.     

Stability and uniqueness of the solution of the inverse kinematic problem of seismology in higher dimensions

The following inverse kinematic problem of seismology is considered. In the compact domain M of dimension ,2 with the metric, we consider the problem of constructing a new metricdu=nds according to the known formula where ,M and K_, is the geodesic in the metric du, connecting the points , . One proves uniqueness and one obtains a stability estimate, where the refraction indices n₁, n₂ are the solutions of the inverse kinematic problem, constructed relative to the functions ₁, ₂, respectively, is the differential form on M×Mwhere =₂–₁,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 3–6, 1979.     

On the Skitovich–Darmois Theorem for Compact Abelian Groups

Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f _n: XX a homomorphism f _n(x)=nx, and X ⁽ⁿ⁾=Im f _n. Denote by I(X) the set of idempotent distributions on X and by (X) the set of Gaussian distributions on X. Consider linear statistics L ₁= ₁( ₁)+ ₂( ₂) and L ₂= ₁( ₁)+ ₂( ₂), where _j are independent random variables taking on values in X and with distributions _j, and _j, _jAut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L ₁ and L ₂ implies that ₁, ₂I(X) if and only if X possesses the property: for each prime p the factor-group X/X ^(p) is finite. If X is connected, then there exist independent random variables _j taking on values in X and with distributions _j, and _j, _jAut(X) such that L ₁ and L ₂ are independent, whereas ₁, ₂(X) * I(X).