Stability inL 1 of some integral operators

A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL ¹, i.e. lim _n P ⁿ f=0 forfL ¹ and f(x) dx=0.     

Problems with one quarter

In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation (r(t)u(t)) + p(t)u(t) = 0 are derived. One of them deals with the case dt/r(t) = , and the other with the case dt/r(t) < .This work was supported by the grant VGA of Slovak Republic No. 1/7466/20.     

Non removable edges in 3-connected cubic graphs

LetG be a cyclicallyk-edge-connected cubic graph withk 3. Lete be an edge ofG. LetG be the cubic graph obtained fromG by deletinge and its end vertices. The edgee is said to bek-removable ifG is also cyclicallyk-edge-connected. Let us denote by S _k (G) the graph induced by thek-removable edges and by N _k (G) the graph induced by the non 3-removable edges ofG. In a previous paper [7], we have proved that N ₃(G) is empty if and only ifG is cyclically 4-edge connected and that if N ₃(G) is not empty then it is a forest containing at least three trees. Andersen, Fleischner and Jackson [1] and, independently, McCuaig [11] studied N ₄(G). Here, we study the structure of N _k (G) fork 5 and we give some constructions of graphs such thatN _k (G) = E(G). We note that the main result of this paper (Theorem 5) has been announced independently by McCuaig [11].

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Résumé SoitG un graphe cubique cyliquementk-arête-connexe, aveck 3. Soite une arête deG et soitG le graphe cubique obtenu à partir deG en supprimante et ses extrémités. L'arêtee est ditek-suppressible siG est aussi cycliquementk-arête-connexe. Désignons par S _k (G) le graphe induit par les arêtesk-suppressibles et par N _k (G) celui induit par les arêtes nonk-suppressibles. Dans un précédent article [7], nous avons montré que N ₃(G) est vide si et seulement siG est cycliquement 4-arête-connexe et que si N ₃(G) n'est pas vide alors c'est une forêt possédant au moins trois arbres. Andersen, Fleischner and Jackson [1] et, indépendemment, McCuaig [11] ont étudié N ₄(G). Ici, nous étudions la structure de N _k (G) pourk 5 et nous donnons des constructions de graphes pour lesquelsN _k (G) = E(G). Nous signalons que le résultat principal de cet article (Théorème 5) a été annoncé indépendamment par McCuaig [11].

     

Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

The Flooding Time in Random Graphs

Based on our analysis of the hopcount of the shortest path between two arbitrary nodes in the class G _p (N) of random graphs, the corresponding flooding time is investigated. The flooding time T _N (p) is the minimum time needed to reach all other nodes from one node. We show that, after scaling, the flooding time T _N (p) converges in distribution to the two-fold convolution ^(2*) of the Gumbel distribution function (z)=exp (–e ^–z ), when the link density p _N satisfies Np _N /(log N)³ if N .     

Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Bijectivity of -endomorphisms of ℬ(H) and the unilateral shift

LetH be a separable infinite-dimensional complex Hilbert space. We prove that if : (H)(H) is a^*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then is an isometric linear or conjugate-linear algebra automorphism of (H). In particular, if the unilateral shift is contained in the range of a^*-endomorphism of (H), then is bijective.Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

Factorization and reflexivity on Fock spaces

The framework of the paper is that of the full Fock space and the Banach algebraF which can be viewed as non-commutative analogues of the Hardy spacesH ² andH respectively.An inner-outer factorization for any element in as well as characterization of invertible elements inF are obtained. We also give a complete characterization of invariant subspaces for the left creation operatorsS ₁ ,..., S _n of . This enables us to show that every weakly (strongly) closed unital subalgebra of {(S ₁ ,..., S _n ) F } is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.The first author was supported in part by NSF DMS 93-21369 1991Mathematics Subject Classification. Primary 47D25, Secondary 32A35, 47A67.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Tangential chow forms — rank 1 hypersurfaces in a Grassmannian

Let X ⁿ P ^N be an n-dimensional projective variety, and N–n–1kN–1. The closure in the Grassmannian G(k+1, N+1) of the set of k-planes meeting the smooth locus of X nontransversally is a tangential Chow form (TCF) of X.TCF's are generally hypersurfaces. We show that a hypersurface is a TCF iff its conormal form has rank 1, and that a TCF is a hypersurface iff some quadric in the second fundamental form of X has rank n+k+1–N.     

Optimal quadratures for analytic functions

For integrals _–1 ¹ w(x)f(x)dx with and with analytic integrands, we consider the determination of optimal abscissasx _i ^o and weightsA _i ^o , for a fixedn, which minimize the errorE _n (f)= _–1 ¹ w(x)f(x)dx – _i ⁼¹ⁿ A _i f(x _i ) over an appropriate Hilbert spaceH ²(E ; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA _i *, corresponding to (preassigned) Gaussian abscissasx _i ^G , which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA _i * are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA _i ^G . In each case,A _i ^G =A _i *+O( ^–4n ), . For , a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x _i ^G ,A _i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x _i ^G ,A _i *;i=1,...,n} to the optimal solution {x _i ^o ,A _i ^o ;i=1,...,n} in terms of _n (), the maximum absolute remainder in the second set ofn normal equations. In each case, _n () is, at least, of the order of ^–4n for large.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Mixed-norm estimates for a class of integral operators

(, ) — R ^m ×R ⁿ . f R ^m ×R ⁿ f_p,q, f L _p (R ^m) x y, L^q(Rⁿ). ׃ _q,r cƒ _p,r , ׃ R ^m ×R ⁿ , , , q r . , ( ¦¦) K ₀ (y); p, g r , K ₀.     

A note on the optimal quadrature inH p

Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH ^p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA