An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

Roth's removal rule revisited

The theory of companion matrices is used to give explicit representations for the matrices needed in Roth's removal rule. These are then used to give simple proofs for the cyclic decomposition theorem, as well as for Roth's similarity theorem for matrices over a field.     

Inequalities in completely convex stochastic programming

A two-stage stochastic programming problem in which the random variable enters in a convex manner is called completely convex. For such problems we give a sequence of inequalities and equalities showing the equivalence of optimality over plans and optimality of a two-stage procedure related to dynamic programming and giving upper bounds on the expected value of perfect information. Our assumptions are the weakest possible to guarantee the results in the completely convex case and supersede previous related results which have received erroneous proofs or have been established under highly restrictive conditions. In the course of our argument we exhibit a new measurable selection theorem and a rather general form of Jensen's inequality. We also present a multistage generalization of our central theorem.     

A new proof of Kiselman's minimum principle for plurisubharmonic functions

We give a new proof of Kiselman's minimum principle for plurisubharmonic functions, inspired by Demailly's regularization of plurisubharmonic functions by using Ohsawa–Takegoshi's extension theorem.     

Some bounds for the Ramsey-Paris-Harrington numbers

It has recently been discovered that a certain variant of Ramsey's theorem cannot be proved in first-order Peano arithmetic although it is in fact a true theorem. In this paper we give some bounds for the “Ramsey-Paris-Harrington numbers” associated with this variant of Ramsey's theorem, involving coloring of pairs. In the course of the investigation we also study certain weaker and stronger partition relations.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

The fuzzy integral

In this paper we define the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure. We show that the monotone convergence theorem and Fatou's lemma are still true in this new setting. We study some of the properties of this integral, and show that it coincides with another fuzzy integral defined in the literature. Our main result is a convergence theorem, that is in a way stronger than the Lebesgue-dominated convergence theorem. This holds when the fuzzy measure is also assumed to be subadditive.     

A simple proof that every matroid is an intersection of fundamental transversal matroids

We give a new proof of a theorem of Bondy and Welsh. Our proof is simpler than previous ones in that it makes no use of Hall's theorem on the existence of a transversal of a family of sets.     

Hurwitz' theorem implies Rouché's theorem

It is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem.     

A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger's equation

A Banach algebra A of functionals on C[a, b] is introduced and it is proved that the operator-valued Feynman integral recently defined by Cameron and Storvick exists for functionals in A. Two existence theorems of Cameron and Storvick are seen to be special cases of this result; in fact, even in these cases, the present theorem gives improved results.Cameron and Storvick have used their function space integral to give a solution to an integral equation formally equivalent to Schroedinger's equation; using our existence theorem, we give a relatively brief and transparent proof of this result.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.     

On pairs of multidimensional matrices

Recently Bapat applied a topological theorem of Kronecker and generalized a theorem of Sinkhorn on positive matrices. Here we give an alternative proof of a slightly stronger version of his generalization. This proof combines Kakutani's fixed point theorem and the duality theorem of linear programming and gives yet another proof of a theorem of Bacharach and Menon on pairs of nonnegative matrices.     

Helly-type theorems for pseudoline arrangements in P2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.     

Ramsey's theorem with sums or unions

Using Hindman's theorem as a strong pigeonhole principle, we prove strengthened versions of Ramsey's theorem and of various generalizations of Ramsey's theorem due to Nash-Williams, Galvin and Prikry, and Silver.     

An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space

J. B. Baillon [C. R. Acad. Sci. Paris Ser. A.280 (1975), 1511–1514] proved an ergodic theorem for a single nonexpansive mapping in a Hilbert space, which is a nonlinear version of von Neumann's mean ergodic theorem. In this paper, we study the ergodic behavior of a semigroup of nonexpansive mappings. We try to find a sequence of means on the semigroup, generalizing the Cesàro means on $\text{N}$, such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed-points. Our method of proof is an appropriate modification of A. Pazy's proof [Israel J. Math.26 (1977), 197–204] of Baillon's theorem.     

An improved version of Lloyd's theorem

The generalisation of Lloyd's theorem to distance-transitive graphs can be improved in the case of antipodal graphs by looking at the derived graph. In the case of binary perfect codes the roots of the Lloyd polynomial are even integers. This can be applied to give a short proof of the binary perfect code theorem.     

New generating functions for Jacobi and related polynomials

In this paper the authors prove a generalization of certain generating functions for Jacobi and related polynomials, given recently by H. M. Srivastava. The method used is due to Pólya and Szegö, and it is based on Rodrigues' formula for the Jacobi polynomials and Lagrange's expansion theorem. A number of special and limiting cases of the main result will give rise to a class of generating functions for ultraspherical, Laguerre and Bessel polynomials.     

An operator not satisfying Lomonosov's hypothesis

An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosov's invariant subspace theorem does not apply to every operator.     

Generalized controllability and inertia theory

Recently the idea of controllability has been used to generalize Lyapunov's theorem and the main inertia theorem. Corresponding results are established in this paper for a large class of linear transformations on the space of n×n Hermitian matrices.