Entropy extension

We prove an “entropy extension-lifting theorem.” It consists of two inequalities for the covering numbers of two symmetric convex bodies. The first inequality, which can be called an “entropy extension theorem,” provides estimates in terms of entropy of sections and should be compared with the extension property of ?_∞. The second one, which can be called an “entropy lifting theorem,” provides estimates in terms of entropies of projections.     

An extension of the Brown-Robinson equivalence theorem

In this paper we present an extension of the Brown–Robinson equivalence theorem on the core and competitive allocations of a nonstandard exchange economy. This has, as its implication, a corresponding extension of their result on the cores of large but finite economies. The extension is based on a result which shows that the core allocations of a nonstandard exchange economy with “integrable” endowments are “integrable.”     

A Chevalley theorem for difference equations

By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on “extension of specializations” or “lifting of prime ideals”. We present a difference analog of this theorem. The approach is based on the philosophy that occasionally one needs to pass to higher powers of σ, where σ is the endomorphism defining the difference structure. In other words, we consider difference pseudo fields (which are finite direct products of fields) rather than difference fields. We also prove a result on compatibility of pseudo fields and present some applications of the main theorem, e.g. constrained extension and uniqueness of differential Picard–Vessiot rings with a difference parameter.     

A note on the core

This note studies an exchange economy in which there are n traders and n “kinds” of commodities. Each trader has n utility functions corresponding to n “kinds” of commodities, respectively. Thus, a multiple non-transferable utility game can be derived from this exchange economy. It is shown that a sufficient condition for non-emptiness of the core of a multiple non-transferable utility game. The result is an extension of Scarf-Billera theorem.     

Almost convexity on Abelian groups

The notion of almost convexity is studied and an extension of Kuczma’s theorem, originally proved in finite-dimensional spaces, is presented. The phrase “almost” is meant in the sense of abstract σ-ideals. The main result also generalizes the theorem proved in Jarczyk and Laczkovich (Math Inequal Appl 13:217–225, 2009).     

An edge-coloration theorem for bipartite graphs with applications

An edge-coloration theorem for bipartite graphs, announced in [4], is proved from which some well-known theorems due to König [5] and the author [2, 3] are deduced. The theorem is further applied to prove the “dual” of a theorem due to Lovász [6].     

Homogenization and Enhancement of the G‐Equation in Random Environments

We study the homogenization of a G‐equation that is advected by a divergence free “small mean” stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G‐equation, and we give necessary and sufficient conditions for enhancement. Since the problem is not assumed to be coercive, it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition that is necessary to apply the subadditive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence that has both a long‐time averaged limit (due to the subadditive ergodic theorem) and stays asymptotically close to the minimal time. © 2013 Wiley Periodicals, Inc.     

Combinatorial Generalizations of Jung’s Theorem

We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.     

On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares

A new proof of a recent theorem of Ellis, Gohberg, and Lay, which identifies the number of roots of a “continuous” matrix orthogonal polynomial in the open upper halfplane with the number of negative eigenvalues of a related integral operator is presented. A related extension problem is then formulated and solved in assorted classes of functions which are analytic in the open upper half plane, apart from a finite number of poles. A discrete analogue of this extension problem is also formulated and solved. © 1994 John Wiley & Sons, Inc.     

Vitushkin’s germ theorem for engel-type CR manifolds

We study real analytic CR manifolds of CR dimension 1 and codimension 2 in the three-dimensional complex space. We prove that the germ of a holomorphic mapping between “nonspherical” manifolds can be extended along any path (this is an analog of Vitushkin’s germ theorem). For a cubic model surface (“sphere”), we prove an analog of the Poincaré theorem on the mappings of spheres into ?². We construct an example of a compact “spherical” submanifold in a compact complex 3-space such that the germ of a mapping of the “sphere” into this submanifold cannot be extended to a certain point of the “sphere.”     

The affine representation theorem for abstract convex geometries

A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of “convexity” shared by some objects including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and show that every convex geometry can be represented as a generalized convex shelling. This is “the representation theorem for convex geometries” analogous to “the representation theorem for oriented matroids” by Folkman and Lawrence. An important feature is that our representation theorem is affine-geometric while that for oriented matroids is topological. Thus our representation theorem indicates the intrinsic simplicity of convex geometries, and opens a new research direction in the theory of convex geometries.     

The construction of all locally compact extensions

The paper presents one of the ways to construct all the locally compact extensions of a given Tychonoff space T. First, there proved the “local” variant of the Stone-C?ech theorem on “completely regular” Riesz spaces X(T) of continuous bounded functions on T with no unit function, in general, but with a collection of local units. In Theorem 1 it is proved that all the functions from X(T) can be “completely regularly” extended on the largest locally compact extension β_xT. Theorem 3 states, that β_xT are presenting, in fact, all the locally compact extensions of T.     

Chevalley–Weil theorem and subgroups of class groups

We prove, under some mild hypothesis, that an ´etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an “absolute” version of the Chevalley–Weil theorem. Using this result, we are able to generalise the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.     

New reals: Can live with them,can live without them

We give a self‐contained proof of the preservation theorem for proper countable support iterations known as “tools‐preservation”, “Case A” or “first preservation theorem” in the literature. We do not assume that the forcings add reals. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact, once one understands “irrational” merely as “not rational”, then the theorem becomes equivalent to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive reverse mathematics, of logical combinations of “rational” and “irrational” as predicates of real numbers.     

Théorèmes de Perron-Frobenius et Stein-Rosenberg booléens

The eigenelements of a Boolean matrix are defined. A “normal form” is given, which allows one to characterize those Boolean matrices the (Boolean) spectral radius of which is 0 or 1. Then the following results are proved: a Boolean Perron-Frobenius theorem, a “Truncated” Boolean Stein-Rosenberg theorem, and a Boolean Stein-Rosenberg theorem, which are the exact Boolean analoques of the usual corresponding theorems concerning real nonnegative matrices. Applications of these results are given elsewhere.     

Some topological structures of extensions of abstract reachability problems

We consider the problem of the reachability of states that are elements of a topological space under constraints of asymptotic nature on the choice of an argument of a given objective mapping. We study constructions that have the sense of extensions of the original space and are implemented with the use of methods that are natural for applied mathematics but employ elements of extensions used in general topology. The study is oriented towards the application in the problem on the construction and investigation of properties of reachability sets for control systems.Constructions involving an approximate observation of constraints in control problems, as well as various generalized regimes, were widely used by N.N. Krasovskii and his students. In particular, this approach was applied in the proof of N.N. Krasovskii and A.I. Subbotin’s fundamental theorem of the alternative, which made it possible to establish the existence of a saddle point in a nonlinear differential game. In the investigation of impulse control problems, Krasovskii used techniques from the theory of generalized functions, which formed the basis for many studies in this direction. A number of A.B. Kurzhanski’s papers are devoted to the solution of control problems related in one way or another to the construction of reachability sets. Control problems with incomplete information, duality issues for control and observation problems, and team control problems constitute a far from exhaustive list of research areas where Kurzhanskii obtained profound results. These studies are characterized by the use of a wide range of tools and methods from applied mathematics and various constructions as well as by the combination of theoretical investigations and procedures related to the possibility of computer modeling.The research direction developed in the present paper mainly concerns the problem of constraint observation (including “asymptotic” constraints) and involves other issues. Nevertheless, the idea of constructing generalized elements of various nature (in particular, generalized controls) seems to be useful for the purpose of asymptotic analysis of control problems that do not possess stability as well as problems on the comparison of different tendencies in the choice of control in the form of dependences on a complex of factors inherent in the original real-life problem. The use of such tools as the Stone–?ech compactification and Wallman’s extension is, of course, oriented toward the study of qualitative issues. In the authors’ opinion, the combined application of the approaches to the construction of extensions used in control theory and in general topology holds promise from the point of view of both pure and applied mathematics. Apparently, the present paper can be considered as a certain step in this direction.     

Forms and currents defining generalized <Emphasis Type="Italic">p</Emphasis>-Kähler structures

This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kähler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant role than “closed” forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.     

Extension spaces of oriented matroids

We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.     

A “planar” representation for generalized transition kernels

In [9], Mauldin, Preiss and von Weizsäcker have given a theorem representing transition kernels (atomless and between standard Borel spaces) by a planar model. Here, motivated by measure-theoretic as well as probabilistic considerations, we generalize by allowing the parametrizing spaceX to be arbitrary, with an arbitrary σ-field of “Borel” subsets, and allowing the corresponding measures to have atoms. (We also, for convenience rather than generality, allow arbitrary finite measures rather than probability ones.) The transition kernel is replaced by a substantially equivalent one fromX toX ×I that is “sectioned”, hence completely orthogonal. This is shown to be isomorphic to a model in which the image space consists of 3 specifically defined subsets ofX × ?: an ordinate set (in which vertical sections have Lebesgue measure), an “atomic” set contained inX × (??), and a “singular” set with null sections. The method incidentally produces and exploits a “reverse” transition kernel fromX toX ×I. Some further extensions are briefly discussed; in particular, allowing “uniformly σ-finite” measures (in the “standard” case) leads to a generalization that includes the planar representation theorem of Rokhlin [10] and the author [5]; cf. also [7, 2].