On 𝔬𝔰𝔭(1 | 2)-Relative Cohomology on S 1|1

We compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the module of λ-densities 𝔉_λ on S ^1|1. This result allows us to compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the Poisson superalgebra 𝒮𝒫. We explicitly give 2-cocycles spanning these cohomology spaces.     

Subspace Inclusion Graph of a Vector Space

In this paper, the authors introduce a graph structure, called subspace inclusion graph ?n(𝕍) on a finite dimensional vector space 𝕍 where the vertex set is the collection of nontrivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. The diameter, girth, clique number, and chromatic number of ?n(𝕍) are studied. It is shown that two subspace inclusion graphs are isomorphic if and only if the base vector spaces are isomorphic. Finally, some properties of subspace inclusion graph are studied when the base field is finite.     

On small homotopies of loops

Two natural questions are answered in the negative:

• “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?”

• “Can adding arcs to a space cause an essential curve to become nulhomotopic?”

The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π₁-shape injective.

Asymptotic Behavior of Gaussian Samples in Fréchet Spaces

Abstract

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ?^∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ? ? 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.     

Elliptic equations,principal eigenvalue and dependence on the domain

We consider a general second order uniformly elliptic differential operator L and also the set θ of all open sets (not necessarily smooth) in the unit ball of ?ⁿ. We define a metric d in this space (up to an equivalence relation ～) that makes the space (θ/ ～,d) a complete metric space. We show that the principal eigenvalue and eigenfunction of L are continuous with the metric d. Similar results are obtained for the solutions of the equation Lv = ?.     

Symmetry in the geometry of metric contact pairs

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ? is a Calabi‐Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ?, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ?‐symmetric spaces or the product of a globally ?‐symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.     

Box ladders in a noninteger dimension

We construct a family of triangle-ladder diagrams that can be calculated using the Belokurov-Usyukina loop reduction technique in d=4?2? dimensions. The main idea of the approach we propose is to generalize this loop reduction technique existing in d=4 dimensions. We derive a recurrence relation between the result for an L-loop triangle-ladder diagram of this family and the result for an (L-1)-loop triangleladder diagram of the same family. Because the proposed method combines analytic and dimensional regularizations, we must remove the analytic regularization at the end of the calculation by taking the double uniform limit in which the parameters of the analytic regularization vanish. In the position space, we obtain a diagram in the left-hand side of the recurrence relations in which the rung indices are 1 and all other indices are 1 - ? in this limit. Fourier transforms of diagrams of this type give momentum space diagrams with rung indices 1 - ? and all other indices 1. By a conformal transformation of the dual space image of this momentum space representation, we relate such a family of triangle-ladder momentum diagrams to a family of box-ladder momentum diagrams with rung indices 1 - ? and all other indices 1. Because any diagram from this family is reducible to a one-loop diagram, the proposed generalization of the Belokurov-Usyukina loop reduction technique to a noninteger number of dimensions allows calculating this family of box-ladder diagrams in the momentum space explicitly in terms of Appell’s hypergeometric function F ₄ without expanding in powers of the parameter ? in an arbitrary kinematic region in the momentum space.     

Unification of a differential inclusion with parameter in the guidance problem

A differential inclusion (d.i.) with a parameter α ∈ ?, where ? is a compact set in a finite-dimensional Euclidean space is considered. The problem under study is one of guiding a d.i. to a given compact set M in the phase space of the d.i. at a fixed moment of time. Issues concerning the unification of the d.i. with a parameter are discussed. A specific feature of the present unification is that it is performed for a parameterdependent d.i. that cannot be initially structured by a vector function similar to those determining the dynamics of systems in control and conflict control problems.     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

An Erdős-Ko-Rado theorem for finite classical polar spaces

Consider a finite classical polar space of rank \(d\ge 2\) and an integer n with \(0<n<d\). In this paper, it is proved that the set consisting of all subspaces of rank n that contain a given point is a largest Erd?s-Ko-Rado set of subspaces of rank n of the polar space. We also show that there are no other Erd?s-Ko-Rado sets of subspaces of rank n of the same size.     

Categorical Abstract Algebraic Logic: Local Characterization Theorems for Classes of Systems

Let ? = ?F, R, ρ? be a system language. Given a class of ?-systems K and an ?-algebraic system A = ?SEN,?N,F??, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K _A of A-systems in K as the collection of all members of K of the form 𝔄 = ? SEN,?N,F?,R ^𝔄 ?, for some set of relation systems R ^𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K _A and global properties connecting K _A and K _A′, whenever there exists an ?-morphism ? F,α? : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K _A is an algebraic closure system, for every ?-algebraic system A, provided that K is closed under subsystems and reduced products.     

An Integral Jensen Inequality For Convex Multifunctions

We prove the following multivalued version of the Jensen integral inequality. Let X, Y be Banach spaces and D ? X an open and convex set. If F: D ? cl(Y) is a continuous convex function, then for each normalized measure space (Ω, S, μ), and for all μ-integrable functions ? : Ω ? D such that conv?(Ω) ? D, $$\int_{\Omega}(F\ o\ \phi)d\mu \subset F\Bigg(\int_{\Omega}\phi d\mu\Bigg).$$      

On Segal�CBargmann analysis for finite Coxeter groups and its heat kernel

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal?CBargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Sa?d and ?rsted and independently by Soltani) and the Dunkl heat kernel, due to R?sler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal?CBargmann analysis on Euclidean space. Hall??s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in???-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal?CBargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall??s Version C generalized Segal?CBargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal?CBargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal?CBargmann kernels for Versions A and C.     

Cantor’s intersection theorem for <Emphasis Type="Italic">K</Emphasis>-metric spaces with a solid cone and a contraction principle

We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenovi? and Shahzad (2011) obtained for a \({K}\)-metric space over a solid strongly minihedral cone. Next we show that our Cantor’s theorem yields a special case of a generalization of Banach’s contraction principle given very recently by Cvetkovi? and Rako?evi? (2014): we assume that a mapping \({T}\) satisfies the condition “\({d(Tx, Ty) \preceq \Lambda (d(x, y))}\)” for \({x, y \in X}\), where \({\preceq}\) is a partial order induced by \({K}\), and \({\Lambda : E \rightarrow E}\) is a linear positive operator with the spectral radius less than one. We also obtain new characterizations of convergence in the sense of Huang and Zhang in a \({K}\)-metric space.     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

Zero product determined Jordan algebras,I

We show that the Jordan algebra 𝒮 of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {.,?.} from 𝒮?×?𝒮 into a vector space X satisfies {x, y}?=?0 whenever x?○?y?=?0, then there exists a linear map T : 𝒮?→?X such that {x,?y}?=?T(x?○?y) for all x, y?∈?𝒮 (here, x?○?y?=?xy?+?yx).     

Notes on the geometry of the space of polynomials

We show that the symmetric injective tensor product space is not complex strictly convex if E is a complex Banach space of dim E ≥ 2 and if n ≥ 2 holds. It is also reproved that ?_∞ is finitely represented in if E is infinite‐dimensional and if n ≥ 2 holds, which was proved in the other way in [3]. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(X,p,μ)d,θ be a space of homogeneous type,(?) ∈(0,θ],|s|＜(?) andmax{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces (?)_∞q~s(X) and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~s(X)with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p＜∞ and max{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞ are also presented.Moreover,the au-thor introduces the new TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n ? 1, 𝕂) be the symplectic polar space defined in PG(2n ? 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n ? 1, 𝕂) ? Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n ? 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n ? 1, 𝕂). We show that, in spite of this fact, W(2n ? 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n ? 1, 𝕂) of W(2n ? 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).