Measurable vectors for von Neumann algebras

Let A be a semifinite von Neumann algebra, with countably decomposable center, on the Hilbert space H. A measurable vector is a linear functional on H whose domain contains a strongly dense domain and which satisfies certain continuity conditions. H can be embedded as a dense subspace of the topological vector space of measurable vectors. The measurable vectors are a module over the measurable operators, and the action of measurable operators on measurable vectors is jointly continuous with respect to suitable topologies. If A is standard, then the measurable operators and measurable vectors are isomorphic as topological vector spaces. If the center of A is not countably decomposable, the results hold with minor changes.     

The tame dimension vectors for trees

We consider representations of quivers over an algebraically closed field K. A dimension vector of a quiver is called hypercritical, if there is an m-parameter family of indecomposable representations for the dimension vector with m?2, but every family of representations for all smaller dimension vectors depends on a single parameter. We characterise the hypercritical dimension vectors for trees via their Tits forms and those of their decompositions and present the complete list of the hypercritical dimension vectors.Finally, this leads to a combinatorial classification of the tame dimension vectors for trees which is also given by the Tits forms.     

Constrained principal component analysis of standardized data for biplots with unit-length variable vectors

Principal component analysis (PCA) of an objects ×  variables data matrix is used for obtaining a low-dimensional biplot configuration, where data are approximated by the inner products of the vectors corresponding to objects and variables. Borg and Groenen (Modern multidimensional scaling. Springer, New York, 1997) have suggested another biplot procedure which uses a technique for approximating data by projections of object vectors on variable vectors. This technique is formulated as constraining the variable vectors in PCA to be of unit length and can be called unit-length vector analysis (UVA). However, an algorithm for UVA has not yet been developed. In this paper, we present such an algorithm, discuss the properties of UVA solutions, and demonstrate the advantage of UVA in biplots for standardized data with homogeneous variances among variables. The advantage of UVA-based biplots is that the projections of object vectors onto variable vectors express the approximation of data in an easy way, while in PCA-based biplots we must consider not only the projections, but also the lengths of variable vectors in order to visualize approximations.     

k-convexn-person games and their cores

For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to

1. all marginal worth vectors ofv belong to the core ofv; or
2. the core ofv is the convex hull of the set consisting of all marginal worth vectors ofv; or
3. the extreme points of the core ofv are exactly the marginal worth vectors ofv.

Examples ofk-convexn-person games are also treated.     

Standard orthogonal vectors in semilinear spaces and their applications

This paper investigates the standard orthogonal vectors in semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings. First, we discuss some characterizations of standard orthogonal vectors. Then as applications, we obtain some necessary and sufficient conditions that a set of vectors is a basis of a semilinear subspace which is generated by standard orthogonal vectors, prove that a set of linearly independent nonstandard orthogonal vectors cannot be orthogonalized if it has at least two nonzero vectors, and show that the analog of the Kronecker–Capelli theorem is valid for systems of equations.     

Common supercyclic vectors for a path of operators

We examine common supercyclic vectors for a path of operators. In particular, we show that the path consisting of convex combinations of two arbitrary unilateral weighted backward shifts has a dense Gδ set of common supercyclic vectors. Moreover, we show there exists a path with a dense Gδ set of common supercyclic vectors between a unilateral weighted backward shift which satisfies the Supercyclicity Criterion, and an operator which does not. Lastly, we provide an example of a path of unilateral weighted backward shifts that fails to have a common supercyclic vector.     

Simulation of geometric stable and other limiting multivariate distributions arising in random summation scheme

A method for simulation of ν_α-stable random vectors based on their representation as location and scale mixtures of α-stable random vectors is discussed. The method has applications in modeling multivariate financial portfolios and Monte Carlo estimation procedures.     

Gale diagrams of convex polytopes and positive spanning sets of vectors

The Gale diagram construction assigns a finite set of vectors to a given convex polytope. The resulting correspondence between sets of vectors and convex polytopes is explored here from a new point of view, utilizing the concept of a positive k-spanning set of vectors and a new matrix construction for Gale diagrams.     

l k,s -Singular values and spectral radius of rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s -singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s -singular values /vectors, some properties of the related l ^k,s -spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.     

On the problem of characterizing the diversity vectors of balls

Under study are the diversity vectors of balls (the ith entry of the vector is equal to the number of different balls of radius i) for ordinary connected graphs. The problem is solved of characterizing the diversity vectors of balls in graphs of small diameter.     

Classification and generation of disturbance vectors for collision attacks against SHA-1

The main contribution of this paper is to provide a classification of disturbance vectors used in differential collision attacks against ${\tt{SHA}-1}$ . We show that all published disturbance vectors can be classified into two types of vectors, type-I and type-II. We present a deterministic algorithm which produce efficient disturbance vectors with respect to any given cost function. We define two simple cost functions to evaluate the efficiency of a candidate disturbance vector. Using our algorithm and those cost function we retrieved all previously known vectors and found that the most efficient disturbance vector is the one first reported as Codeword2 by Jutla and Patthak, A matching lower bound on the minimum weight of SHA-1 expansion code. Cryptology ePrint Archive, Report 2005/266, (2005). We also present a statistical evaluation of local collisions?? holding probabilities and show that the common assumption of local collision independence is flawed.     

On the eigenvalues of specially low-rank perturbed matrices

We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

The generating rank of the space of short vectors in the Leech lattice mod 2

Consider the partial linear space on the images in Λ/2Λ of the shortest nonzero vectors in the Leech lattice Λ, where the lines are the triples of vectors adding up to zero. We determine the universal embedding dimension and the generating rank of this space (both are 24) and classify its hyperplanes.     

Profile polytopes of some classes of families

The profile vector of a family F of subsets of an n-element set is (f ₀,f ₁,…,f _n ) where f _i denotes the number of the i-element members of F. The extreme points of the set of profile vectors for some class of families has long been studied. In this paper we introduce the notion of k-antichainpair families and determine the extreme points of the set of profile vectors of these families, extending results of Engel and P.L. Erd?s regarding extreme points of the set of profile vectors of intersecting, co-intersecting Sperner families. Using this result we determine the extreme points of the set of profile vectors for some other classes of families, including complement-free k-Sperner families and self-complementary k-Sperner families. We determine the maximum cardinality of intersecting k-Sperner families, generalizing a classical result of Milner from k = 1.     

Characterizations of adjacency on the branching polyhedron

Given two distinct branchings of a directed graph G, we present several conditions which are equivalent to the corresponding incidence vectors of the branchings being adjacent on the branching polyhedron of G. The proof of these equivalences uses a “shrinking algorithm” which will determine in O(n²) time and space whether or not the incidence vectors are adjacent.     

More on the generalized macaulay theorem — II

Let k₁ ? k₂? ? ? k_n be given positive integers and let S denote the set of vectors x = (x₁, x₂, … ,x_n) with integer components satisfying 0 ? x₁ ? k_ni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l.     

Isometric reflection vectors in Banach spaces

The aim of this paper is to study the set of isometric reflection vectors of a real Banach space X. We deal with geometry of isometric reflection vectors and parallelogram identity vectors, and we prove that a real Banach space is a Hilbert space if the set of parallelogram identity vectors has nonempty interior. It is also shown that every real Banach space can be decomposed as an Ir-sum of a Hilbert space and a Banach space with some points which are not isometric reflection vectors. Finally, we give a new proof of the Becerra-Rodríguez result: a real Banach space X is a Hilbert space if and only if is not rare.     

On supercyclicity of operators from a supercyclic semigroup

We show that for every supercyclic strongly continuous operator semigroup {Tt}_t?0 acting on a complex F-space, every Tt with t>0 is supercyclic. Moreover, the set of supercyclic vectors of each Tt with t>0 is exactly the set of supercyclic vectors of the entire semigroup.