On the congruence problem in infinite dimensional sesquilinear spaces

If two subspaces V and V of a sesquilinear space E are congruent (i.e., there is an isometry : E E with (V)=V) then their corresponding quadratic lattices V(V, E) and V(V, E) are isomorphic. It is shown that the converse holds for important types of sesquilinear spaces E, provided that dim(E) ₃. However, the converse generally fails if dim(E) ₃.     

Optimal Transient Response Shaping of the Servomechanism Problem

The problem of constructing a controller which gives a desirable transient response, e.g., a time response which is smooth, ripple-free, fast, and with negligible interaction effects, is considered in this paper for the servomechanism problem for continuous linear time invariant (LTI) systems. It is shown that this may be accomplished by constructing a controller which minimizes the following cheap control performance index: J=₀ {(+e)(+e)+(u)(u)}dwhere e is the error in the system and (u) is a polynomial function of u and its derivatives; here, >0 and is a diagonal matrix with nonnegative elements. A number of examples are included to illustrate the type of results which may be obtained.     

A characterization of semimartingales on nuclear spaces

Summary This work is devoted to prove the following fact: Suppose that is a nuclear space whose dual is nuclear under the strong topology. IfX is a weakly adapted mapping with values in such that for any,X'() has a modification which is a semimartingale then there exists a unique projective system of Hubert space-valued semimartingales indexed by the Hilbert-Schmidt neighbourhood base of the dual space whose projective limit isX.In the last part we study in detail a semimartingale defined as the convolution of a distribution by a random Dirac measure whose support is determined by the trajectories of a real-valued semimartingale.     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Biclosed Binary Relations and Galois Connections

Given two closure spaces (E,) and (E,), a relation RE×E is said biclosed if every row of its matrix representation corresponds to a closed subset of E, and every column to a closed subset of E. An isomorphism between, on the one hand, the set of all biclosed relations and, on the other hand, the set of all Galois connections between the two lattices of closed sets is established. Several computational applications are derived from this result.     

Zur Kennzeichnung der Kollineationen zwischen Quadriken

In projective spaces Pⁿ(K), Pⁿ(K) over commutative fields K, K of characteristic 2 we consider the quadrics Q^n–1Pⁿ(K), Q,^n–1Pⁿ(K) which have the same rank and index and each of which is different from its singularity space. A one-to-one correspondence of Q^n–1 onto Q'^n–1 can be extended uniquely to a collineation of Pⁿ(K) onto Pⁿ(K) if and only if satisfies the following condition: maps the (n–2)-dimensional quadrics of Q,^n–1 of maximal rankonto (n–2)-dimensional quadrics of Q^n–1 of maximal rank (Theorem 1).

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Herrn Prof. Dr. Helmut Karzel zum 60. Geburtstag gewidmet     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

Interval stability and interval covering property in finite posets

Given a finite ranked posetP, let (P) be the maximum size of a subset ofP such that no two elements of it belong simultaneously to some interval ofP and let (P) be the minimum number of intervals covering all elements ofP. We say thatP has the strong interval stability property (resp. the strong interval covering property) if for each subposetP induced by consecutive levels ofP, i.e.,P=P _(l)...P _(u), one has (P)=max{|P _(l)|, |P _(u)|} (resp. (P)=max{|P _(l)|, |P _(u)|}).We prove these properties for several classes of posets and discuss some general facts concerning the numbers (P) and (P), e.g., NP-completeness and min-max relations.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Brownian processes in infinite dimension

LetE be a locally convex space endowed with a centered gaussian measure . We construct a continuousE-valued brownian motionW _t with covariance . The main goal is to solve the SDE of Langevin type dX _t= dW _t–AX _t wherea andA are unbounded operators of the Cameron-Martin space of (E, ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv(t)= u–Av(t).     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

Kato's equality and spectral decomposition for positive C0-groups

Let A be the generator of a C₀-semigroup {T(t); t0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies <¦x¦, Ax>= (xD(A), x D(A)) (where q: EE is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

Über konvergente Zerlegungen von Matrizen

LetA, M, N ben × n real matrices, letA=M–N, letA andM be nonsingular. LetMy0 implyNy0 (where the prime denotes the transpose). ThenAy0 impliesNy0 if and only if the spectral radius (M ^–1 N) ofM ^–1 N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA, M, andN are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.

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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

On context patterns associated with concept lattices

In this paper, we consider the following reconstruction problem: Given two ordered sets (G, ) and (M, ) representing join- and meet-irreducible elements, respectively together with three relationsJ,, onG×M modelling comparability (gm) and maximal noncomparability with respect tog (gm, butgm*) and with respect tom (gm, butgm*). We determine necessary and sufficient conditions for the existence of a finite latticeL and injections :GJ(L) and :MM(L) such that the given order relations and the abstract relations coincide with the one induced by the latticeL.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

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 .     

Support functions of convex compacta

The properties of the space (X_x) of all sublinear functionals, defined on a space X' (topologically adjoint to a Hausdorff locally convex barrelled space X) and continuous in the Arens topology × (X, X), equipped with topology of uniform convergence on bounded subsets of X are studied. It is shown that completeness and separability of a space X are hereditary for (X_x). Criteria for the compactness of subsets of (X_x) and conditions for the metrizability of compacta in (X_x) are given. The topological isomorphism between (X_x) and the space of all nonempty convex compacta in X with the Vietoris topology is established. The results obtained here are applied for the study of the properties of multiple-valued integrals.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 203–213, August, 1977.The author thanks S. S. Kutateladze for useful discussions regarding this article.     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.