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.     

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.     

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).     

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.     

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.     

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.     

Regular Hadamard Matrices Generating Infinite Families of Symmetric Designs

If H is a regular Hadamard matrix with row sum 2h, m is a positive integer, and q = (2h ? 1)², then (4h ²(q ^{m + 1} ? 1)/(q ?1),(2h ² ? h)q ^m ,(h ²-h)q ^m ) are feasible parameters of a symmetric designs. If q is a prime power, then a balanced generalized weighing matrix BGW((q ^{m +1} ? 1)/(q?1),q ^m ,q ^m ?q ^{m ?1}) can be applied to construct such a design if H satisfies certain structural conditions. We describe such conditions and show that if H satisfies these conditions and B is a regular Hadamard matrix of Bush type, then B×H satisfies these structural conditions. This allows us to construct parametrically new infinite families of symmetric designs.     

Completely inverse AG ∗∗-groupoids

A completely inverse AG ^??-groupoid is a groupoid satisfying the identities (xy)z=(zy)x, x(yz)=y(xz) and xx ^?1=x ^?1 x, where x ^?1 is a unique inverse of x, that is, x=(xx ^?1)x and x ^?1=(x ^?1 x)x ^?1. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse AG ^??-groupoid; namely: the maximum idempotent-separating congruence, the least AG-group congruence and the least E-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse AG ^??-groupoids. In particular, we describe congruences on completely inverse AG ^??-groupoids by their kernel and trace.     

Dirichlet Heat Kernel Estimates for Stable Processes with Singular Drift In Unbounded C 1,1 Open Sets

Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C ^1,1 open set in ? ^d and μ = (μ ¹, . . . , μ ^d ) where each μ ^j is a signed measure on ? ^d belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let X ^μ be an α-stable process with drift μ in ? ^d and let X ^μ,D be the subprocess of X ^μ in D. In this paper, we derive sharp two-sided estimates for the transition density of X ^μ,D .     

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.     

Toward a stochastic calculus for several Markov processes

In this paper we investigate a class of harmonic functions associated with a pair xt = (xt11, xt22) of strong Markov processes. In the case where both processes are Brownian motions, a smooth function f is harmonic if Δx1Δx2f(x1,x2) = 0. For these harmonic functions we investigate a certain boundary value problem which is analogous to the Dirichlet problem associated with a single process. One basic tool for this study is a generalization of Dynkin's formula, which can be thought of as a kind of stochastic Green's formula. Another important tool is the use of Markov processes xti?i obtained from xtii by certain random time changes. We call such a process a stochastic wave since it propogates deterministically through a certain family of sets; however its position on a given set is random.     

On pseudoinverses of operator products

The analytical structure of the Moore-Penrose pseudoinverse of the product ab of any two operators over finite-dimensional unitary spaces is studied. The existence of the unique representation of the form (ab)⁺=b⁺(h+g)a⁺ is proved. Here h:= (a⁺abb⁺)⁺ is an (oblique) projector and g is an operator with a number of special properties. In particular, h+g is a projector, g is orthogonal to h in some metric, and g³=0. A necessary and sufficient condition for the case (ab)⁺=b⁺ha⁺ is established. This case contains the classical one (ab)⁺=b⁺a⁺ (the reverse-order law). For the latter a new necessary and sufficient condition is given.     

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).     

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.     

Conditional expectations and operator decompositions

For aC ^*-algebraA with a conditional expectation Φ:A → A onto a subalgebraB we have the linear decompositionA=B⊕H whereH=ker(Φ). Since Φ preserves adjoints, it is also clear that a similar decomposition holds for the selfadjoint parts:A ^s =B ^s ⊕H ^s (we useV ^s ={aεV;a ^*=a} for any subspaceV of A). Apply now the exponential function to each of the three termsA ^s ,B ^s , andH ^s . The results are: the setG ⁺ of positive invertible elements ofA, the setB ⁺ of positive invertible elements ofB, and the setC={e^h;h ^*=h, Φ(h)=0}, respectively. We consider here the question of lifting the decompositionA ^s =B ^s ⊕H ^s to the exponential sets. Concretely, is every element ofG ⁺ the product of elements ofB ⁺ andC, respectively, just as any selfadjoint element ofA is the sum of selfadjoint elements ofB andH? The answer is yes in the following sense: Eacha ε G ⁺ is the positive part of a productbe of elementsb ε B ⁺ and c εC, and bothb andc are uniquely determined and depend analytically ona. This can be rephrased as follows: The map (6, c) →(bc) ⁺ is an analytic diffeomorphism fromB ⁺ x C ontoG ⁺, where for any invertiblex ε A we denote with x⁺ the positive square root ofxx ^*. This result can be expressed equivalently as: The map (b, c) →bcb is a diffeomorphism between the same spaces. Notice that combining the polar decomposition with these results we can write every invertibleg ε A asg=bcu, whereb ε B ⁺,c ε C, andu is unitary. This decomposition is unique and the factorsb, c, u depend analytically ofg. In the case of matrix algebras with Φ=trace/dimension, the factorization corresponds tog=| det(g)|cu withc > 0,det(c)=1, andu unitary. This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the ”general context” for regularity indicated in the introduction of the last mentioned paper.     

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.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

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).     

The relations among the three kinds of conditional risk measures

Let(Ω,E,P)be a probability space,F a sub-σ-algebra of E,Lp(E)(1 p+∞)the classical function space and Lp F(E)the L0(F)-module generated by Lp(E),which can be made into a random normed module in a natural way.Up to the present time,there are three kinds of conditional risk measures,whose model spaces are L∞(E),Lp(E)(1 p+∞)and Lp F(E)(1 p+∞)respectively,and a conditional convex dual representation theorem has been established for each kind.The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems.We first establish the relation between Lp(E)and Lp F(E),namely Lp F(E)=Hcc(Lp(E)),which shows that Lp F(E)is exactly the countable concatenation hull of Lp(E).Based on the precise relation,we then prove that every L0(F)-convex Lp(E)-conditional risk measure(1 p+∞)can be uniquely extended to an L0(F)-convex Lp F(E)-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter,which shows that the study of Lp-conditional risk measures can be incorporated into that of Lp F(E)-conditional risk measures.In particular,in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L0-convex conditional risk measures.∞     

On sesquilinear forms over fields with involution

Let k be a field of characteristic ≠2 with an involution σ. A matrix A is split if there is a change of variables Q such that (Q^σ)^TAQ consists of two complementary diagonal blocks. We classify all matrices that do not split. As a consequence we obtain a new proof for the following result. Given a square matrix A there is a matrix S such that (S^σ)^TAS=A^T and S^σS=I.