Cylindrical multidimensional surfaces in Lobachevsky space

We study complete, strongly parabolic metrics with constant relative index of nullity _c=k and complete, strongly parabolic surfaces with constant index of relative nullityv=k in a constant curvature space Rⁿ(c) under the assumption that there exists a surface orthogonal to the fibers of total geodesity, and if c<0, under the additional condition that the orthogonal surface be totally umbilical. For c>0 we have _c=l;v=l, for c=0 the Riemannian manifold is a metric product of the metric of R^l–k and the Euclidean space E^k while the surface is a cylinder in Euclidean space. For c<0 the metric has a special form and the surface is a cylindrical surface in Lobachevsky space.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 18–27, 1990.     

On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ such that the curvature operator of the metric of Mⁿ satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y _Q , X, Z #x2208; T_QMⁿ. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T₁Mⁿ be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mⁿ, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T₁(Mⁿ, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.     

Cohomology of Compact Locally Symmetric Spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S()=X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S _H()=Y of X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ⁿ and ^m , then low degree cohomology classes on the variety S() restrict non-trivially to the subvariety S _H (); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S().     

Cylindricity of tangent bundles of strongly parabolic metrics and strongly parabolic surfaces

It is proved that if the intrinsic zero-index of the Sasaki metric of a tangent bundleTM ⁿ isk, thenk is even andM ⁿ is the metric product of a Riemannian manifoldM ^n–k/2 by a Euclidean spaceE ^k/2, whileTM ⁿ is the metric product ofTM ^n–k/2 byE ^k . An expression is obtained for the second fundamental forms of the imbeddingTF ^l TM ⁿ in terms of the second fundamental forms of the imbeddingF ^l M ⁿ and the curvature tensor ofM ⁿ . It is proved thatTF ^l is totally geodesic inTM ⁿ if and only ifF ^l is totally geodesic inM ⁿ .Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 12–32.     

Boundary feedback stabilizability of parabolic equations

A parabolic equation defined on a bounded domain is considered, with input acting on theboundary expressed as a specifiedfeedback of the solution. Both Dirichlet and mixed (in particular, Neumann) boundary conditions are treated. Algebraic conditions based on the finitely many unstable eigenvalues are given, ensuring the existence ofboundary vectors, for which all the solutions to theboundary feedback parabolic equation decay exponentially to zero ast+ in (essentially) the strongest possible space norm. A semigroup approach is employed.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-77-3338.A preliminary version of this paper has appeared in the Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, held at Delft University, The Netherlands, July 3–6, 1979; pp. 428–433.     

Global controllability of conditionally periodic linear system

We use the notation: Rⁿ Is n-dimensional Euclidean space;S _a (x⁰)={x Rⁿ: ¦x-x ⁰ ¦ }; int Q is the interior of setQ Rⁿ. With any linear systemx=A (t)x +B (t) u, x Rⁿ,u R^m, (1)Translated from Matematicheskie Zametki, Vol. 32, No. 2, pp. 169–174, August, 1982.     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

An algorithm for constructing an optimal structural schedule

The article is devoted to the problem of finding an optimal schedule for a class of functionals ƒ which allows for the existence of a structural set of activities. The functionalƒ(R), where, is defined in the following way: where {_i(t)} is a structural set of functions, and the function F is defined on any finite set of arguments and satisfies the following conditions: 1)F(x)=(x); 2) F(x₁,x₂)=(x₁,x₂), F(x₁,x₂,...x₃)= (x₁, F(x₂,...,x_s)), S2; 3) and do not decrease in each of their arguments, and moreover, 3a) strictly increases with the increase of both arguments, 3b) if (x₁,x₂)>(x₁, x₂ (x₂, x₃)> (x₂,x₃), then F(x₁,x₂,x₃)>F(x₁,x₂,x₃).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 5–20, 1983.     

Concerning a sufficient condition of sidon for the integrability of trigonometric series

We consider problems concerning integrability and convergence in the metric L of trigonometric series, the coefficientsa _k of which satisfy the conditions:a _k 0, there exist numbers A_k such that A_k 0, < , and ¦¦ A_k. The integrability of a series in cosines under these conditions is equivalent to a theorem of Sidon.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 317–328, September, 1973.     

Lax Representation for a Triplet of Scalar Fields

We construct a 3×3 matrix zero-curvature representation for the system of three two-dimensional relativistically invariant scalar fields. This system belongs to the class described by the Lagrangian L = [g _ij(u)u _x ⁱ u _t ^j]/2+f(u), where g _ij is the metric tensor of a three-dimensional reducible Riemannian space. We previously found all systems of this class that have higher polynomial symmetries of the orders 2, 3, 4, or 5. In this paper, we find a zero-curvature representation for one of these systems. The calculation is based on the analysis of an evolutionary system u _t = S(u), where S is one of the higher symmetries. This approach can also be applied to other hyperbolic systems. We also find recursion relations for a sequence of conserved currents of the triplet of scalar fields under consideration.     

Convergence of integrated processes of arbitrary Hermite rank

Summary Let {X(s), –<s<} be a normalized stationary Gaussian process with a long-range correlation. The weak limit in C[0,1] of the integrated process , is investigated. Here d(x) = x ^H L(x) with <H<1 and L(x) is a slowly varying function at infinity. The function G satisfies EG(X(s))=0, EG ² (X(s))< and has arbitrary Hermite rank m1. (The Hermite rank of G is the index of the first non-zero coefficient in the expansion of G in Hermite polynomials.) It is shown thatZ _x (t) converges for all m1 to some process ¯Z _m (t) that depends essentially on m. The limiting process ¯Z _m (t) is characterized through various representations involving multiple Itô integrals. These representations are all equivalent in the finite-dimensional distributions sense. The processes ¯Z _m (t) are non-Gaussian when m2. They are self-similar, that is,¯Z _m (at) and a ^H ¯Z _m (t) have the same finite-dimensional distributions for all a>0.Research supported by the National Science Foundation grants MCS 77-03543 and ENG 78-11454.This paper contains results closely connected to those of the paper by Dobrushin and Major, Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 27–52 (1979). The investigations were done independently and at about the same time. Different methods were usedDedicated to Professor Leopold Schmetterer on occasion of his 60th Birthday     

Extremal solutions for nonlinear parabolic problems with discontinuities

This paper examines nonlinear parabolic initial-boundary value problems with a discontinuous forcing term, which is locally of bounded variation. Assuming that there exist an upper solution and a lower solution , we prove the existence of a maximal and of a minimal solution within the order interval [,] L ^P (P xZ). Our approach is based on a Jordan-type decomposition for the discontinuous forcing term and on a fixed point theorem for nondecreasing maps in ordered Banach spaces.     

State constrained control problems for parabolic systems: Regularity of optimal solutions

Quadratic control problems for parabolic equations withstate constraints are considered. Regularity (smoothness) of the optimal solution is investigated. It is shown that the optimal control is continuous in time with the values inL ₂() and its time derivative belongs toL ₂[OT×].Research partially supported by National Aeronautics and Space Administration under Grant No. NSG 4015.     

A degenerate and strongly coupled quasilinear parabolic system not in divergence form

This paper deals with positive solutions of degenerate and strongly coupled quasi-linear parabolic system not in divergence form: u_t=v^p(u+au), v_t=u^q (v+bv) with null Dirichlet boundary condition and positive initial condition, where p, q, a and b are all positive constants, and p, q 1. The local existence of positive classical solution is proved. Moreover, it will be proved that: (i) When min {a, b} ₁ then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm (we can not prove the uniqueness result in general); (ii) When min {a, b} > ₁, there is no global positive classical solution (we can not still prove the uniqueness result), if in addition the initial datum (u₀v₀) satisfies u₀ + au₀ 0, v₀+bv₀ 0 in , then the positive classical solution is unique and blows up in finite time, where ₁ is the first eigenvalue of – in with homogeneous Dirichlet boundary condition.     

Beitrag zur Reduktion des Entscheidungsproblems auf Klassen von Hornformeln mit kurzen Alternationen

Ohne Zusammenfassung Zusatz bei der Korrektur: Ein vollständiger und korrekter Beweis für die Entscheidbarkeit der eingangs angeführten Aanderaaschen Klasse ((0, ), (, , ...)) erscheint demnächst im JSL (S.O. Aanderaa/H.R.Lewis: Prefix classes of Krom formulas). Ebendort wird auch die Reduktionstypeneigenschaft für ((0, ), (0, 0, )) und ((0, )), (0, 0, )) nachgewiesen, während ((0, ), (, )) sich als entscheidbar herausgestellt hat (s. E. Börger: Eine entscheidbare Klasse von Kromformeln. ZMLG 19 (1973), 117–120.) Der Kromsche Reduktionstyp konnte mittlerweile einerseits zu ((0, ), (0, 4)) verschärft werden (s. D. Rödding, E. Börger: The undecidability of (0, 4)-formulae with binary disjunctions, vorgetragen auf dem Logic Coll. Bristol 1973, ein abstract erscheint im JSL), andererseits kündigt H.R.Lewis die Reduktionstypeneigenschaft für ((0, ), (0, 1)) an (s. H.R.Lewis: Krom formulas with one dyadic predicate letter. Notices AMS 20, 5 (1973) A-500, abstr. no. 73T-E78.)Dieser Aufsatz geht aus der Dissertation [2] hervor, die dem Fachbereich Mathematik der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Münster im Sommersemester 1971 vorgelegt worden ist. Die Ergebnisse stammen aus dem Wintersemester 1970/71. Eine Ankündigung der hauptsächlichen Resultate ist in den Notices of the American Mathematical Society 19, 2 (1972) A-333 unter der abstract no. * 72T-E24 erschienen.     

On coercive solvability of parabolic equations

In this paper we study the coercive solvability of abstract differential equations of parabolic type in the spaces of L. N. Slobodetskii W _p . It is established that the solution of equations with a constant operator A which generates an analytic semigroup belongs to the trace space E(, p, A). The results obtained are applied to the study of equations with a variable operator.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 409–419, April, 1972.     

Comparison of Green Functions and Harmonic Measures for Parabolic Operators

In this paper, we prove two-sided pointwise estimates for the Green function of a parabolic operator with singular first order term on a C^1,1-cylindrical domain . Basing on these estimates, we establish the equivalence of the parabolic measure, the adjoint parabolic measure and the surface measure on the lateral boundary of . These results are first studied by some authors for certain elliptic and less general parabolic operators. Mathematics Subject Classifications (2000) 31B25, 35B05, 35K10, 58J35.     

Boundary control of parabolic systems: Finite-element approximation

Finite-element approximation of a Dirichlet type boundary control problem for parabolic systems is considered. An approach based on the direct approximation of an input-output semigroup formula is applied. Error estimates inL ₂[OT; L ₂()] andL ₂[OT; L ₂()] norms are derived for optimal state and optimal control, respectively. It turns out that these estimates areoptimal with respect to the approximation theoretic properties.Research supported in part under Grant no. NSG 4015, National Aeronautics and Space Administration.     

Stability of stationary and periodic solutions of nonlinear parabolic systems

One investigates the problem of the stability of the solutions of nonlinear parabolic initial-boundary-value problems relative to small perturbations of the class C with any0 under suitable restrictions (depending on ) on the structure of the nonlinear terms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 286–304, 1979.The author expresses his thanks to his scientific advisor, V. A. Solonnikov, for his help in the completion of this paper.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25