ℋ-tensors and nonsingular ℋ-tensors

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ?-tensors. In this paper, we establish important properties of diagonally dominant tensors and ?-tensors. Distributions of eigenvalues of nonsingular symmetric ?-tensors are given. An ?₊-tensor is semi-positive, which enlarges the area of semi-positive tensor from ?-tensor to ?₊-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ?-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ?-tensor if and only if all of its principal sub-tensors are nonsingular ?-tensors. An irreducible tensor Ais an ?-tensor if and only if it is quasi-diagonally dominant.     

Two non-parameter iterative algorithms for identifying strong ℋ $\mathcal {H}$ -tensors

Numerical Algorithms - The strong $\mathcal {H}$ -tensors have important applications in many areas of science and engineering, e.g., the determination of positive definiteness for an even-order...     

ℋ-matrix techniques for approximating large covariance matrices and estimating its parameters

In this work the task is to use the available measurements to estimate unknown hyper-parameters (variance, smoothness parameter and covariance length) of the covariance function. We do it by maximizing the joint log-likelihood function. This is a non-convex and non-linear problem. To overcome cubic complexity in linear algebra, we approximate the discretised covariance function in the hierarchical (ℋ-) matrix format. The ℋ-matrix format has a log-linear computational cost and storage O(knlogn), where rank k is a small integer. On each iteration step of the optimization procedure the covariance matrix itself, its determinant and its Cholesky decomposition are recomputed within ℋ-matrix format. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonnegativity preserving algorithm for multilinear systems with nonsingular ℳ ${mathcal M}$ -tensors

Numerical Algorithms - This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the...     

On Weakly ℋ-embedded Subgroups of Finite Groups

Let G be a finite group and H a subgroup of G. We say that H is an ?-subgroup in G if N_G(H) ∩ H^g ≤ H for all g ∈ G; H is called weakly ?-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ?-subgroup in G. We say that H is weakly ? -embedded in G if G has a normal subgroup K such that H^G = HK and H ∩ K is an ?-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ?-embedded in G. Our results improve and generalize several recent results in the literature.     

On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

RANK t ℋ-PRIMES IN QUANTUM MATRICES

ABSTRACT

Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (2002 Goodearl , K. R. , Lenagan , T. H. ( 2002 ). Prime ideals invariant under winding automorphisms in quantum matrices . Internat. J. Math 13 : 497 – 532 . [CROSSREF]  [Google Scholar]) have shown that the number of ?-primes in R = O _q (? _n (𝕂)) that contain all (t + 1) × (t + 1) quantum minors but not all t × t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!) ² S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (1997 Kaneko , M. ( 1997 ). Poly-Bernoulli numbers . J. Théorie Nombres Bordeaux 9 : 221 – 228 . [Google Scholar]) and Arakawa and Kaneko (1999 Arakawa , T. , Kaneko , M. ( 1999 ). On poly-Bernoulli numbers . Comment Math. Univ. St. Paul 48 ( 2 ): 159 – 167 . [Google Scholar]).     

Generalized inverses modulo ℋ in semigroups and rings

The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

Can matching improve the performance of boosting for identifying important genes in observational studies?

When two groups of individuals are to be compared with respect to gene expression there will often be some potentially confounding variables that differ between the groups. Matching is an established approach for obtaining comparable groups and enabling subsequent univariate tests for each gene. Alternatively, the confounders might be incorporated directly into a multivariable regression model for adjustment. In contrast to univariate tests, such models can consider all genes simultaneously. Aiming to combine the advantages of both approaches, matching and multivariable modeling, we consider a matching-based boosting procedure for fitting risk prediction models in two-group settings. This possibly allows to identify and automatically remove problematic observations that might negatively affect the regression model. Therefore, we compare the ability to identify important covariates for this combination of matching and boosting with only boosting for different covariate correlation structures in a simulation study. Furthermore, we analyze the prediction performance of these approaches on two gene expression microarray studies. The first study comprises patients with B-cell and T-cell type acute lymphoblastic leukemia and the second patients with acute megakaryoblastic leukemia. While the matching component can in principle guard against problematic observations, the combined approach is seen to neither improve identification of important covariates nor to improve prediction performance. Therefore, a combination of the two approaches cannot be recommended. Adjustment for potential confounders is seen to provide the best performance, i.e. a pure multivariable regression modeling strategy seems to be promising even in presence of considerable heterogeneity.     

ℋ p stability of solutions of stochastic differential equations

Summary Solutions of systems of stochastic differential equations are shown to be stable in ^p under ^p perturbations of semimartingale differentials. Analogous results are obtained inp ^p when the solutions are not semimartingales but are only cadlag, adapted processes. Also, the solutions are shown to be stable under almost sure perturbations. These results are contrasted with the lack of stability under non- ^p perturbations, a result originally obtained by Wong and Zakai.This research was supported in part by NSF Grant No. MCS77-00095     

Finite Groups Whose Second Maximal Subgroups are ℋ-QC-Subgroups

A group G is called an “?-QC-group” if for any element x of order 2 or 4 of G, ?x? ?y? = ?y? ?x? for all y in G. In this article, we investigate the structure of the groups G of the title. Suppose that G is non-2-closed and non-2-nilpotent, then it turns out that such a group is either S ₄ or a uniquely determined group of order 48 or A ₅ or SL(2,5). The corresponding problem for G 2-closed or 2-nilpotent is open but very difficult.     

The structure of finite monoids satisfying the relation ℛ = ℋ

It is shown that any finite monoid S on which Green’s relations R and H coincide divides the monoid of all upper triangular row-monomial matrices over a finite group. The proof is constructive; given the monoid S, the corresponding group and the order of matrices can be effectively found. The obtained result is used to identify the pseudovariety generated by all finite monoids satisfying R = H with the semidirect product of the pseudovariety of all finite groups and the pseudovariety of all finite R-trivial monoids.     

Stabilization of Linear Switched Delay Systems: ℋ<Subscript>2</Subscript> and ℋ<Subscript>∞</Subscript> Methods

This paper presents new results pertaining to the delay-dependent stability and control synthesis of a class of linear switched continuous-time systems with time-varying delays. A new state transformation is introduced to exhibit the delay-dependent dynamics in the slow-time scale. For stability, we construct an appropriate selective Lyapunov functional to derive delay-dependent LMI-based sufficient conditions under arbitrary switching and without relying to overbounding. For the control synthesis, we design switched feedback schemes based on quadratic ℋ₂, ℋ_∞ and simultaneous ℋ₂/ℋ_∞ performance criteria. Under the developed transformation, it is established that both the instantaneous and delayed feedback control yield identical results. Numerical examples are presented to illustrate the analytical development.