Sufficient conditions for bang-bang control in Hilbert space

Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is,Ax=u,J(u)=(r,x)+(x,x), >0. For example, ifA is a bounded operator with a bounded inverse from a Hilbert spaceH into itself and the control setU is the unit ball inH, then an optimal control is bang-bang (has norm l) if 0<1/2;A ^–1*r·A ^–1–2, but is singular (an interior point ofU) if >1/2A ^–1*r·A².This work was supported by NRC Grant No. A-4047 and NSF Grant No. GP-7445.     

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 summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Asymptotic stability at infinity of planar vector fields

Let >0 andX be aC ¹ vector field on the plane such that: (i) for allq², Det(DX(q))>0; and (ii) for allp², with p, Trace(D(X(p))<0. IfX has a singularity and ² Trace(DX)dxdy is less than 0 (resp. greater or equal than 0), then the point at infinity of the Riemann sphere ²{} is a repellor (resp. an attractor) ofX.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Spectral synthesis on a system of unbounded domains starlike in a common direction

, . . - ₁, ..., ₄, — ; =(₁,)×...×H(₄), — H(₁, ..., H(₄), r H(₁) — , ₁ ; D: HH- . , D. , ₁..., ₄ , (.. z₁ z+te^ia ₁ t>0), W H .     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

Singularities of the Green function of a random walk on a discrete group

LetX be a countable discrete group and let be an irreducible probability onX. The radius of convergence of the Green function is finite, and independent ofx. Let 0} \right\}$$ " align="middle" border="0"> be the period of . We show that for eachxX the singularities of the analytic functionzG(x; z) on the circle {z:|z|=} are precisely the points e ^2ik/d k=0, ...,d–1. In particular, is the only singularity on the circle in the aperiodic cased=1 (which occurs, for example, when (e)>0). This affirms a conjecture ofLalley [5]. When is symmetric, i.e., (x ^–1)=(x) for allxX, d is either 1 or 2. As another particular case of our result, we see that- is then a singularity ofzG (x; z) if and only ifd=2, in which caseX is bicolored. This answers a question ofde la Harpe, Robertson andValette [2].     

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

ApproximatingL 2-invariants by their finite-dimensional analogues

LetX be a finite connectedCW-complex. Suppose that its fundamental group is residually finite, i.e. there is a nested sequence ... _m + 1 _m ... of in normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL ²-Betti number ofX is the limit of the sequenceb _p(X_m)/[: _m ] whereb _p(X_m) is the (ordinary)p-th Betti number of the finite covering ofX associated with _m .     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

On σ-discrete borel mappings via quasi-metrics

Let X and Y be metrizable spaces. We show that, for a mapping f : X Y, there exists a quasi-metric X inducing the topology of X such that f regarded as a mapping from (X, max{, ^–1}) to Y is continuous if and only if f in the original topology of X is a -discrete map of Borel class 1. Further, we prove that, for every -discrete mapping f: X Y of Borel class + 1, there exists a compatible quasi-metric on X such that f : (X, max{, ^–1}) Y is of Borel class . We also investigate a more general situation when the range of the mapping under consideration is not necessarily metrizable. In passing, we obtain some results related to the behaviour of absolutely Borel sets and absolutely analytic spaces with respect to compatible quasi-metrics.     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.     

Donsker's Delta Function of Lévy Process

Let X={X(t):tR} be a Lévy process and a non-decreasing, right continuous, bounded function with (–)=0 (((1+u ²)/u ²)d(u) is the Lévy measure). In this paper we define the Donsker delta function (X(t)–a), t>0 and aR, as a generalized Lévy functional under the condition that (0)–(0–)>0. This leads us to define F(X(t)) for any tempered distribution F, and as an application, we derive an Itô formula for F(X(t)) when has jumps at 0 and 1.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

Une approche transformationnelle a l'algebre universelle

In this paper we propose an axiomatization of the notion of system of terms of a theory by means of which we obtain a representation of equational classes (or varieties) of algebras. We define analgebraic transformational system (S.T.A.) as a quadruple (T,v,S,+) satisfying the axioms, where T is a set containing the variables v(n), n, and having operators S(): TT, . In addition there are operations Q⁺ on T commuting with the operators. A notion of morphism between S.T.A. 's is defined to obtain the category which is shown to be equivalent to the dual of the category of equational classes. In the last section we establish the equivalence between and Lawvere's category of algebraic theories in which every definable constant is present.

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Extrait de la Thèse de doctorat de l'auteur, Université de Montréal, 1971.