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.     

On trees and tree dimension of ordered sets

We call an ordered set (X, ) a tree if no pair of incomparable elements ofX has an upper bound. It is shown that there is a natural way to associate a tree (T, ) with any ordered set (X, ), and (T, ) can be characterized by a universal property. We define the tree dimensiontd(X, ) of an ordered set as the minimal number of extensions of (X, ) which are trees such that the given order is the intersection of those tree orders. We give characterizations of the tree dimension, relations between dimension and tree dimension, and removal theorems.     

Approximations in thermal explosion theory and the nature of the degenerate critical point

Summary In studies of thermal explosion the Frank-Kamenetskii approximation sets exp(–E/RT)=exp(–E/RT ₀)exp (/(1+))exp(–E/RT ₀)exp, where=RT ₀/E i.e. it assumes0. When this approximation is not made, it is known that criticality vanishes for greater than a certain value _*, say. This may occur whether the Arrhenius form is used or some suitable approximation to it; many authors have proposed approximations involving the maximum dimensionless temperature in the reactant. The nature of the degeneracy near the value _* is examined for such approximations in general, some approximations are considered and the results compared.

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Zusammenfassung In Studien von thermischen Explosionen setzt man in der Näherung von Frank-Kamenetskii exp(–E/RT)=exp(–E/RT ₀)exp(/(1+))exp(–E/RT ₀) exp), wobei=RT ₀/E ist, d.h. man nimmt0 an. Wenn diese Näherung nicht benützt wird, so weiß man, daß die Kritikalität verschwindet wenn einen gewissen Wert _* überschreitet. Dies findet man mit Benützung der Formel von Arrhenius oder mit einer Näherung dazu; viele Autoren haben Näherungen vorgeschlagen mit Verwendung der maximalen dimensionslosen Temperatur im Reaktionsgemisch. Es wird für solche Näherungen die Natur der Entartung der Lösung in der Umgebung von _* untersucht; die Resultate für verschiedene Näherungen werden verglichen.

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Résumé Dans les études de la théorie de l'explosion thermale l'approximation de Frank-Kamenetskii pose exp(–E/RT)=exp(–E/RT ₀)exp(/(1+)]exp(–E/RT ₀) exp, avec=RT ₀/E, c'est à dire on admet0.On sait que, en dehors de cette approximation, la limite critique disparaît lorsque dépasse une certaine valeur dénommée _*. Ceci peut se produire soit en utilisant la forme d'Arrhenius ou une approximation adéquate.Plusieurs auteurs ont proposé des approximations utilisant la température maximum non-dimensionelle du réactif. Dans la présente étude on examine d'une façon générale le caractère de la dégénérescence aux alentours de la valeur _* pour ce genre d'approximations. Ensuite on considère quelques approximations particulières et les résultats sont comparés.

     

New integer pairs for Hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair(t, r); t is the order of the Hjelmslev plane andr is the order of the underlying projective plane. The aim of this paper is to give some new constructions of Hjelmslev planes with an invariant pair (t, 2). First we construct a PH-plane with the invariant pair (20, 2). Using this, 16 more invariant pairs (t, 2) witht 1000 are obtained. In all, we thus obtain 17 new PH-planes with invariant pairs (t, 2),t 1000.     

Diameters of a class of smooth functions in the space L2

The class V, consisting of the smooth functions f(t), ot1, satisfying the condition ₀ ¹ [f ^(r) (t)]dt1, where the function (t) is nonnegative and r is a natural number, is studied. Under certain restrictions on the function (t) ensuring the compactness of the class V, the order of decrease of the Kolmogorov diameters d_n(V) is computed. The analogous problem for the case r=1 is solved also for functions of several variables.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 671–678, November, 1977.     

Bang-bang optimal control for the dam problem

The dam problem with general geometry is considered. Fluid is drawn from the bottomS ₁ at a ratek where 0 k N, _{S 1} k M; the objective is to minimize the total pressure of the fluid in the dam. A bang-bang principle is established for any optimal controlk ₀, that is,k ₀ = 0 on a setA andk ₀ =N on the complement setS ₁ A. In the case of a rectangular dam the structure ofA is determined and the uniqueness of the minimizerk ₀ is established.This work is partially supported by National Science Foundation Grants DMS-8501397 and DMS-8420896.     

A note on merging

For a finite poset P and x, yP let pr(x>y) be the fraction of linear extensions which put x above y. N. Linial has shown that for posets of width 2 there is always a pair x, y with 1/3 pr(x>y)2/3. The disjoint union C ₁C ₂ of a 1-element chain with a 2-element chain shows that the bound 1/3 cannot be further increased. In this paper the extreme case is characterized: If P is a poset of width 2 then the bound 1/3 is exact iff P is an ordinal sum of C ₁C ₂'s and C ₁'s.     

The crossing number of posets

The purpose of this paper is to investigate some properties of the crossing number (P) of a posetP. We first study the crossing numbers of the product and the lexicographical sum of posets. The results are similar to the dimensions of these posets. Then we consider the problem of what happens to the crossing number when a point is taken away from a poset. We show that ifP is a poset such that P and (P–)1, then 1/2 (P)(P–)(P). We don't know yet how to improve the lower bound. We also determine the crossing numbers of some subposets of the Boolean latticeB _n which consist of some specified ranks. Finally we show that _n is crossing critical where _n is the subposet ofB _n which is restricted to rank 1, rankn–1 and middle rank(s). Some open problems are raised at the end of this paper.     

Homogeneity in finite ordered sets

A tower in an ordered set (X, ) is defined to be a subsetS ofX which has the property that for everysS there is a maximal chainC in {xX|xs} which is wholly contained inS. An ordered set (X, ) is called tower-homogeneous if every order isomorphism between towers in (X, ) can be extended to an automorphism of (X, ). It is shown that a finite ordered set is tower-homogeneous if and only if it can be built up from singletons stepwise by constructions of three different types.     

One-third-integrality in the max-cut problem

Given a graphG = (V, E), the metric polytopeS (G) is defined by the inequalitiesx(F) – x(CF) |F| – 1 for , |F| odd,C cycle ofG, and 0 x _e 1 fore E. Optimization overS (G) provides an approximation for the max-cut problem. The graphG is called 1/d-integral if all the vertices ofS(G) have their coordinates in{i/d 0 i d}. We prove that the class of 1/d-integral graphs is closed under minors, and we present several minimal forbidden minors for 1/3-integrality. In particular, we characterize the 1/3-integral graphs on seven nodes. We study several operations preserving 1/d-integrality, in particular, thek-sum operation for 0 k 3. We prove that series parallel graphs are characterized by the following stronger property. All vertices of the polytopeS (G) {x x u} are 1/3-integral for every choice of 1/3-integral bounds, u on the edges ofG. Research by this author was partially done at CWI in Amsterdam.Research by this author was done at the Institut für Diskrete Mathematik of Bonn, supported by the A. von Humboldt Foundation.Deceased on April 2nd, 1995.     

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x( ⁿ )+A ₁ x(^n–1 +...+ ⁿ A _n x, the matrices A₁,...,A_n are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ^m , F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in ^m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991.     

Infinite ordered sets spanned by finitely many chains

We say that an ordered set P is spanned by a family C of chains if P=(P, ) is the transitive closure of {(C, | C) C C. It is shown that there is a function h: such that if P is spanned by k< chains, then P has a finite cutset-number h(k) (i.e. for any xP, there is a finite set F of size |F|h(k)–1, such that the elements of F are incomparable with x and {x}F meets every maximal chain of P). The function h is exponentially bounded but eventually dominates any polynomial function, even if it is only required that there are at most h(k) pairwise disjoint maximal chains in P, whenever P is spanned by k< chains.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.     

Bounds of eigenvalues of a graph

LetG be a simple graph withn vertices. We denote by _i(G) thei-th largest eigenvalue ofG. In this paper, several results are presented concerning bounds on the eigenvalues ofG. In particular, it is shown that –1₂(G)(n–2)/2, and the left hand equality holds if and only ifG is a complete graph with at least two vertices; the right hand equality holds if and only ifn is even andG2K _n/2.     

On well-ordered mono-unary algebras

Given a linearly ordered set (A, _R ) and an R-monotone function f: AA, we give a necessary and sufficient condition on A, f, _R , involving generating sets and forbidden subalgebras, for _R to be a well-ordering.Partially supported by Hungarian National Foundation for Scientific Research Grant nr. 1813.     

Linear operators generated by stochastic processes

Let X_t(0 t < ) be a homogeneous stochastically continuous stochastic process with independent increments; (, A, P) be the corresponding probability space; p, q1 be fixed numbers. Necessary and sufficient conditions are found for a stochastic integral defined on finite-valued functions to be extendable to a bounded operator from L_p (0, ) to L_q() (p q).Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 7–13, 1988.     

Inequalities involving the inradii of simplexes

An inequality involving the inradii ofk-simplexes andl-simplexes (3 n) whose vertices lie in a finite set of points inE ⁿ has been proved in this paper.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

On the postulation of a general projection of a curve inP N,N⩾4

Summary Fix a curve X of genus g and L Pic ^d (X). Let _L(X) be the image of X through the complete linear system H⁰(X, L). Here we prove that a general projection of _L(X) intoP ^N has maximal rank if either (a) N4, 0gN–1, dg+N, or (b) dd (g, N) for suitable d(g, N).