Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Epsilon efficiency

This paper considers the extension of -optimality for scalar problems to vector maximization problems, or efficiency problems, which havem objective functions defined on a set .It is shown that the natural extension of the scalar -optimality concepts [viz, given >0, given a solution setS, ifxS there exists an efficient solutiony with f(x)–f(y), and given an efficient solutiony, there exists anxS with f(x)–f(y)] do not hold for some methods used. Six concepts of -efficient sets are introduced and examined, to a very limited extent, in the context of five methods used for generating efficient points or near efficient points.In doing so, a distinction is drawn between methods in which the surrogate optimizations are carried out exactly, and those where terminal -optimal solutions are obtained.The author would like to thank the referees whose thoroughness was extremely helpful for the revised paper.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

Isomorphisms of Some Complex Poisson Brackets

Given a complex manifold M_i, structures of Poisson algebras on (M_i)=C(M_i,C) which are associated with a nondegenerate -closed (2,0)-form _i on M_i are considered. It is shown that every isomorphism of Poisson structures (M₁) (M₂) is generated by a biholomorphic map :M₂ M₁ such that ₂ = *₁     

Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb

The one-dimensional Helmholtz equation, ² u _xx –u=f(x), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for 1 because the homogeneous solutions are exp(±x/), which have boundary layers of thickness O(1/). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([x–1]/).) Two strategies for small are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f(x) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when is very, very tiny.     

On the equivalence of renormalizations in standard and dimensional regularizations of 2D four-fermion interactions

We discuss the problem of equivalence between the standard (integer-dimensional d=2) and the d=2+ dimensional renormalization schemes for the complete U_N-symmetrical four-fermion interaction model. To ensure the multiplicative renormalizability of the theory, we need three charges in the first case; in the second, we need an infinite series of independent charges g {g_n, n=0, 1, ...}. After the usual MS-renormalization. there exists a UV-finite renormalization of fields. Charges gg'(g) exist such that the renormalized Green's functions in the limit 0 depend only on the three lower charges g'_n(g) with n=0, 1, 2. rather than on the whole set. This ensures the possibility of establishing the equivalence of the two renormalization schemes. The results of calculations in the MS scheme up to two loops for the -functions, and up to three loops for the anomalous field dimension are presented. These are presented together with the derivation of the projection technique relations, which allows one to express the higher renormalized composite operators of the 4F-interaction via the lower ones in the limit 0.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 27–46, April, 1996.Translated by L. O. Chekhov.     

Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

Exchangeable random variables and the subsequence principle

Summary Call a sequence {X _n } of r.v.'s -exchangeable if on the same probability space there exists an exchangeable sequence {Y _n } such thatP(|X _n –Y _n |) for alln. We prove that any tight sequence {X _n } defined on a rich enough probability space contains -exchangeable subsequences for every >0. The distribution of the approximating exchangeable sequences is also described in terms of {X _n }. Our results give a convenient way to prove limit theorems for subsequences of general r.v. sequences. In particular, they provide a simplified way to prove the subsequence theorems of Aldous [1] and lead also to various extensions.     

Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction

Out of a right, circular cylinder of height H and cross-section a disc of radius R+ one removes a stack of nH/ parallel, equi-spaced cylinders C_j,j=1,2,...,n, each of radius R and height . Here , are fixed positive numbers and is a positive parameter to be allowed to go to zero. The union of the C_j almost fills in the sense that any two contiguous cylinders C_j are at a mutual distance of the order of and that the outer shell, i.e., the gap S=-_o has thickness of the order of (_o is obtained from by formally setting =0). The cylinder from which the C_j are removed, is an almost disconnected structure, it is denoted by , and it arises in the mathematical theory of phototransduction.For each >0 we consider the heat equation in the almost disconnected structure , for the unknown function u, with variational boundary data on the faces of the removed cylinders C_j. The limit of this family of problems as 0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u within the limiting cylinder _o.It is shown that the limiting problem consists of an interior diffusion in _o and a boundary diffusion on the lateral boundary S of _o. The interior diffusion is governed by the 2-dimensional heat equation in _o, for an interior limiting function u. The boundary diffusion is governed by the Laplace–Beltrami heat equation on S, for a boundary limiting function u_S. Moreover the exterior flux of the interior limit u provides the source term for the boundary diffusion on S. Finally the interior limit u, computed on S in the sense of the traces, coincides with the boundary limit u_S. As a consequence of the geometry of , local arguments do not suffice to prove convergence in _o, and also we have to take into account the behavior of the solution in S. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in -dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole ^N, by means of the Kirzbraun–Pucci extension technique.The biological origin of this problem is traced, and its application to signal transduction in the retina rod cells of vertebrates is discussed. Mathematics Subject Classification (2000) 35B27, 35K50, 92C37     

A Second Descent Problem for Quadratic Forms

Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied by Kahn and Laghribi. More precisely, we conjecture that for any quadratic form q over F and any Im(W(F) W(F(q))), there exists a quadratic form W(F) such that dim 2 dim and _F(q), where F(q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim 3 and any q, and get partial results for dim {4, 5,6}. We also give other related results.     

A Distortion Theorem for Univalent Gap Series

We prove a distortion theorem for conformal mappings of the unit disk for which log f is representable as the Hadamard gap series. This theorem implies in particular that such conformal mapping is almost bounded, i.e., for every >0, there is a positive constant C such that |f,(z)|C (1–|z|) ^–.     

Continuity of martingales in the Brownian excursion filtration

Summary Given a Brownian motionB, we consider the filtration ( _{x xR} ), where _x is defined as the -field generated by the excursions ofB belowx. In this paper we prove a conjecture of Walsh which says that all -martingales are continuous.     

Positivity Preserving Forms have the Fatou Property

We prove that if (,D) is a positivity preserving form on L ² (E;m), and if (u _n)_n is a sequence in D() converging m-almost everywhere to u L ² (E;m), then (u,u) lim inf_n (u _n ,u _n ).     

Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the O(n)-Symmetric φ4-Theory in the (4−ε)-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the (4–)-expansion for the scaling function of the pair correlator of the O(n)-symmetric ⁴-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the ³ order of the expansion of the scaling function in the small- domain.     

Generation of alternating groups by pairs of conjugates

Considering the conjugacy classes of the alternating group of degreen, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is 1 –(n), and(n) 0 asn .