On a family of symmetric hypergeometric functions of several variables and their Euler type integral representation

This paper is devoted to the family {_Gn}$\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients (_?1+?+_?n)!/(_?1!…_?n!)$({?}_1+?+{?}_n)!/({?}_1!\dots {?}_n!)$, where n∈_Z?1$n\in {\mathbb{Z}}_{?1}$. All these series belong to the family of the general Appell–Lauricella?s series. It is shown that each function _Gn$G_n$ can be expressed by an integral involving the previous one, _Gn−1$G_{n-1}$. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand–Kapranov–Zelevinsky?s theory of A  -hypergeometric systems or with the Aomoto?s theory of hypergeometric functions. The quasi-invariance of each function _Gn$G_n$ with regard to the action of a finite number of involutions of C^?n${\mathbb{C}}^{?n}$ is also established. Finally, a particular attention is reserved to the study of the functions _G2$G_2$ and _G3$G_3$, each of which is proved to be algebraic or to be expressed by the Legendre?s elliptic function of the first kind.     

A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type method to compute selected eigenpairs of the product eigenvalue problem _Am?_A1x=λx,$A_m?A_{1}x=\lambda x,$ where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.     

Functions of perturbed n-tuples of commuting self-adjoint operators

Let (_A1,…,_An)$(A_1,\dots ,A_n)$ and (_B1,…,_Bn)$(B_1,\dots ,B_n)$ be n-tuples of commuting self-adjoint operators on Hilbert space. For functions f   on Rⁿ${\mathbb{R}}^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of f(_A1,…,_An)−f(_B1,…,_Bn)$f(A_1,\dots ,A_n)-f(B_1,\dots ,B_n)$ in terms of the corresponding norms of _Aj−_Bj$A_j-B_j$, 1?j?n$1?j?n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for n?3$n?3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary n  . We also get sharp estimates for quasicommutators f(_A1,…,_An)R−Rf(_B1,…,_Bn)$f(A_1,\dots ,A_n)R-Rf(B_1,\dots ,B_n)$ in terms of norms of _AjR−R_Bj$A_{j}R-R{B}_j$, 1?j?n$1?j?n$, for a bounded linear operator R.     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G   and 1<p<∞$1<p<\infty$ let _Ap(G)$A_p(G)$ be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)$A(G)$ (p=2$p=2$). It is shown that for any amenable group H  , a proper affine map α:Y⊂H→G$\alpha :Y\subset H\to G$ induces a p  -completely contractive algebra homomorphism _?α:_Ap(G)→_Ap(H)${?}_{\alpha }:A_p(G)\to A_p(H)$ by setting _?α(u)=u°α${?}_{\alpha }(u)=u°\alpha$ on Y   and _?α(u)=0${?}_{\alpha }(u)=0$ off of Y. Moreover, we show that if both G and H are amenable then any p  -completely contractive algebra homomorphism ?:_Ap(G)→_Ap(H)$?:A_p(G)\to A_p(H)$ is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2$p=2$) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.     

Polynomial numerical hulls of some normal matrices

In this note, polynomial numerical hulls of matrices of the form _A1⊕i_A2$A_1\oplus i{A}_2$, where _A1$A_1$ and _A2$A_2$ are Hermitian, are characterized.     

Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible _L2$L_2$-norm of the discrepancy function. We consider the discrepancy function of the Chen–Skriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a b$b$-adic generalization of the Haar system and corresponding characterizations of the Besov space norm. Results for further function spaces and integration errors are concluded.     

Applications of general monotone sequences to strong approximation by Fourier series

We consider the strong means of Fourier series generated by infinite nonnegative triangular matrices and prove some estimates of such means in the case of a matrix with rows stating the sequences from the class GM(₅β)$GM\left({}_5\beta \right)$. Our theorems correspond to the results of L. Leindler [L. Leindler, A note on strong approximation of Fourier series, Anal. Math. 29 (2003) 195–199] and essentially extend the result of S.M. Mazhar and V. Totik [S.M. Mazhar, V. Totik, Approximation of continuous functions by T -means of Fourier series, J. Approx. Theory, 60 (1990) 174–182].     

A Note on n-Perinormal Operators

The study of operators satisfying

σ_ja(T)=σa(T)${\sigma }_{ja}\left(T\right)={\sigma }_a\left(T\right)$

is of significant interest. Does

σ_ja(T)=σa(T)${\sigma }_{ja}\left(T\right)={\sigma }_a\left(T\right)$

for n-perinormal operator

T∈B(H)?$T\in B\left(H\right)?$

This question was raised by Mecheri and Braha [Oper. Matrices 6 (2012), 725–734]. In the note we construct a counterexample to this question and obtain the following result: if T is a n-perinormal operator in B(H), then

σ_ja(T)\{0}=σa(T)\{0}.${\sigma }_{ja}\left(T\right)\backslash \left\{0\right\}={\sigma }_a\left(T\right)\backslash \left\{0\right\}.$

We also consider tensor product of n-perinormal operators.     

Double absolute summability factor theorems and applications

In an earlier paper the first author [Ekrem Savas, Factors for _|A|k${\left|A\right|}_k$ summability of infinite series, Comput. Math. Appl. 53 (7) (2007) 1045–1049. [3]] obtained a summability factor theorem for absolute summability of the order k≥1$k\ge 1$. In this paper we extend that result to doubly infinite matrices.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.     

Uniform hypergraphs containing no grids

A hypergraph is called an r×r$r\times r$grid   if it is isomorphic to a pattern of r$r$ horizontal and r$r$ vertical lines, i.e., a family of sets {_A1,…,_Ar,_B1,…,_Br}$\{A_1,\dots ,A_r,B_1,\dots ,B_r\}$ such that _Ai∩_Aj=_Bi∩_Bj=0?$A_i\cap A_j=B_i\cap B_j=0?$ for 1≤i<j≤r$1\le i<j\le r$ and |_Ai∩_Bj|=1$|A_i\cap B_j|=1$ for 1≤i,j≤r$1\le i,j\le r$. Three sets _C1,_C2,_C3$C_1,C_2,C_3$ form a triangle   if they pairwise intersect in three distinct singletons, |_C1∩_C2|=|_C2∩_C3|=|_C3∩_C1|=1$|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1$, _C1∩_C2≠_C1∩_C3$C_1\cap C_2\ne C_1\cap C_3$. A hypergraph is linear  , if |E∩F|≤1$|E\cap F|\le 1$ holds for every pair of edges E≠F$E\ne F$.     

Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times

In this paper, we use the formula for the Itô–Wiener expansion of the solution of the stochastic differential equation proven by Krylov and Veretennikov to obtain several results concerning some properties of this expansion. Our main goal is to study the Itô–Wiener expansion of the local time at the fixed point for the solution of the stochastic differential equation in the multidimensional case (when standard local time does not exist even for Brownian motion). We show that under some conditions the renormalized local time exists in the functional space defined by the _L2$L_2$-norm of the action of some smoothing operator.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

Projective planes over 2-dimensional quadratic algebras

The split version of the Freudenthal–Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras ,  and . The geometries appearing in the second row are Severi varieties [24]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry _A2×_A2$A_2\times A_2$ in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of “projective planes over 2-dimensional quadratic algebras”, in cases of the split and non-split Galois extensions, the inseparable extensions of degree 2 in characteristic 2 and the dual numbers.     

Dualizability of automatic algebras

We make a start on one of George McNulty's Dozen Easy Problems  : “Which finite automatic algebras are dualizable?” We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain _A1?_A2?_A3??${\mathbf{A}}_1?{\mathbf{A}}_2?{\mathbf{A}}_3??$ of finite automatic algebras that are alternately dualizable and non-dualizable.     

The inverse problem on subset sums

For a set A$A$, let P(A)$P\left(A\right)$ be the set of all finite subset sums of A$A$. We prove that if a sequence B={_b1<_b2<?}$B=\{b_1<b_2<?\}$ of integers satisfies _b1≥11,_b2≥3_b1+5,_b3≥3_b2+3$b_1\ge 11,b_2\ge 3b_1+5,b_3\ge 3b_2+3$ and _bn+1>3_bn−_bn−2$b_{n+1}>3b_n-b_{n-2}$  (n≥3)$(n\ge 3)$, then there exists a sequence of positive integers A={_a1<_a2<?}$A=\{a_1<a_2<?\}$ such that P(A)=N?B$P\left(A\right)=\mathbb{N}?B$. These lower bounds are optimal in a sense. We pose a problem for further research.