METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Patroids

A matroid M over a set E of elements is semiseparated by a partition {S¹, S²} of E iff rank E = rank S¹ + rank S² + 1. Such a semiseparation defines in each Sⁱ a pair of matroids or patroid Pⁱ = (Mⁱ, mⁱ); the two patroids P¹, P² weld to form M. The operations of removing and contracting a non-degenerate element of a matroid produce a patroid. The properties of patroids, their bases, and circuits are discussed.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Quasirecognition by prime graph of 2 D n (3α) where n = 4m + 1 ≥ 21 and α is odd

Let G be a finite group. The prime graph of G is denoted by Γ(G). In this paper, as the main result, we show that if G is a finite group such that Γ(G) = Γ(² D _n (3^α)), where n = 4m+ 1 and α is odd, then G has a unique non-Abelian composition factor isomorphic to ² D _n (3^α). We also show that if G is a finite group satisfying |G| = |² D _n (3^α)|, and Γ(G) = Γ(² D _n (3^α)), then G ? ² D _n (3^α). As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for ² D _n (3^α). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that ² D _n (3^α) is quasirecognizable by the spectrum.     

Relative (pa,pb, pa,pa−b-Difference Sets: A Unified Exponent Bound and a Local Ring Construction

We show that for an odd prime p the exponent of an abelian group of order p^a+b containing a relative (p^a, p^b, p^a, p^a−b)-difference set cannot exceed p^⌊a/2⌋+1. Furthermore, we give a new local ring construction of relative (q^2u, q, q^2u, q^2u−1)-difference sets for prime powers q. Finally, we discuss an important open case concerning the existence of abelian relative (p^a, p, p^a, p^a−1)-difference sets.     

On dimensional rigidity of bar-and-joint frameworks

Let V={1,2,…,n}. A mapping p:V→R^r, where p¹,…,pⁿ are not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in R^r, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in R^r is said to be dimensionally rigid iff there does not exist a framework G(q) in R^s for s?r+1, such that ∥qⁱ-q^j∥²=∥pⁱ-p^j∥² for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p¹,…,pⁿ of the given r-configuration are in general position, is also investigated.     

Two-point inequalities,the Hermite semigroup and the Gauss-Weierstrass semigroup

Let e^?zH, Re z ? 0, be the Hermite semigroup on R with Gauss measure μ. Necessary and sufficient conditions for e^?zH to be a bounded map from L^p(μ) into L^q(μ), 1 ? p, q ? ∞, are found and in many cases it is proved that e^?zH: L^p(μ) → L^q(μ) is in fact a contraction. Furthermore, these results and a formula relating the Hermite semigroup with the Gauss-Weierstrass semigroup e^zΔ enable one to calculate the precise norm of e^zΔ:L^p(dx) → L^q(dx) in a large number of cases.     

On finite x-decomposable groups for X = {1, 2, 4}

A normal subgroup N of a finite group G is called an n-decomposable subgroup if N is a union of n distinct conjugacy classes of G. Each finite nonabelian nonperfect group is proved to be isomorphic to Q ₁₂, or Z ₂ × A ₄, or G = ??a, b, c | a ¹¹ = b ⁵ = c ² = 1, b ^?1 ab = a ⁴, c ^?1 ac = a ^?1, c ^?1 bc = b ^?1?? if every nontrivial normal subgroup is 2- or 4-decomposable.     

Further results on the reverse-order law

An explicit expression is obtained for a pair of generalized inverses (B^?,A^?) such that B^?A^?=(AB)⁺_MN, and a class of pairs (B^?,A^? of this property is shown. A necessary and sufficient condition for (AB)^? to have the expression B^?A^? is also given.     

Approximation of the semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann space

The Reggeon field theory is governed by a non-self adjoint operator constructed as a polynomial in A, A^*, the standard Bose annihilation and creation operators. In zero transverse dimension, this Hamiltonian acting in Bargmann space is defined by

Hλ^′,μ=λ^′A*2A²+μA^*A+iλA^*(A^*+A)A,     

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps f:S ² → S ² that are post-critically finite. In addition, we assume that f is expanding in a suitable sense. It is shown that each sufficiently high iterate F = f ⁿ of f is semi-conjugate to z ^d :S ¹ → S ¹, where d = deg F. More precisely, for such an F we construct a Peano curve γ:S ¹ → S ² (onto), such that F°γ(z) = γ(z ^d ) (for all z∈S ¹).     

Shifts in Cayley graphs

An automorphism of an undirected simple graph is called a shift if it maps every vertex to an adjacent one. For all finite groups G, we determine the minimum nonzero valency of a Cayley graph on G that does not admit a shift. We also get a classification of groups with all involutions central and such that for every pair a,b of elements of the group, one of ab=ba, aba⁻¹=b⁻¹, bab⁻¹=a⁻¹ or a²=b^±2 holds.     

Greedy bases in L p spaces

We consider a weighted L ^p space L ^p (w) with a weight function w. It is known that the Haar system H _p normalized in L ^p is a greedy basis of L ^p , 1 < p < ∞. We study a question of when the Haar system H _p ^w normalized in L ^p (w) is a greedy basis of L ^p (w), 1 < p < ∞. We prove that if w is such that H _p ^w is a Schauder basis of L ^p (w), then H _p ^w is also a greedy basis of L ^p (w), 1 < p < ∞. Moreover, we prove that a subsystem of the Haar system obtained by discarding finitely many elements from it is a Schauder basis in a weighted norm space L ^p (w); then it is a greedy basis.     

Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire

Call a locally compact group G, C¹-unique, if L¹(G) has exactly one (separating) C¹-norm. It is easy to see that a ¹-regular group G is C¹-unique and that a C¹-unique group is amenable. For connected groups G it is proved that G is C¹-unique, if the interior R(G)⁰ of a certain part R(G) of Prim(G), called the regular part of Prim(G), is dense in Prim(G), and that C¹-uniqueness of G implies the density of R(G) in Prim(G). From this it is derived that a connected group of type I is C¹-unique if and only if R(G)⁰ is dense in Prim(G). For exponential G, a quite explicit version of this result in terms of the Lie algebra of G is given. As an easy consequence, examples of amenable groups, which are not C¹-unique, and C¹-unique groups, which are not ¹-regular are obtained. Furthermore it is shown that a connected locally compact group G is amenable if and only if L¹(G) has exactly one C¹-norm, which is invariant under the isometric ¹-automorphisms of L¹(G).     

Submanifolds Of Maximal Nullity In Symmetric Spaces

A non-totally-geodesic submanifold of relative nullity n — 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F ² of nullity 1 in a totally geodesic submanifold N³ ? M locally isometric to S ²(c) × ? or H ²(c) × ?; a submanifold F ^k+1 spanned by a totally geodesic submanifold F ^k(c) of constant curvature moving by a special curve in the isometry group of M; a submanifold F ^k+l of nullity k in a flat totally geodesic submanifold of M; a curve.     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

A geometric Gordan-Stiemke theorem

Let C and K be closed cones in Rⁿ. Denote by φ (K∩C) the face of C generated by K ∩ C, by φ(K ∩ D)^D the dual face of φ(K ∩ C) in C¹, and by φ(-K¹ ∩ C¹) the face of C¹ generated by -K¹ ∩ C¹. It is proved that φ(K ∩ C¹) if and only if -C¹ ∩ [span(K ∩ C)] ⊥ ? C¹ + K¹. In particular, the closedness of C¹ + K¹ is a sufficient condition. Our result contains a generalization of the Gordon-Stiemke theorem which appeared in a recent paper of Saunders and Schneider.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Nonnegative sectional curvature hypersurfaces in a real space form

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form M ⁿ⁺¹(c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M ⁿ⁺¹(c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S ^k (a) × S ^n-k (√1 ? a ²), 1 ≤ k ≤ n ? 1, in S ⁿ⁺¹(1) and the Riemannian product H ^k (tanh² r ? 1) × S ^n-k (coth² r ? 1), 1 ≤ k ≤ n ? 1, in H ⁿ⁺¹(?1).     

Franchissement et temps local pour l'oscillateur harmonique

We consider the second order Stochastic Differential Equation dP_t^β = V_t^β dt with P₀^β = p₀, dV_t^β = βV_t^βdt − βω²P_t^β + βdW_t with V₀^β = v₀, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that P^β converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of P^β at level u during the time interval [0, t] · (N_t^Pβ (u)) to the local time of P(L_t^P (u)).