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This paper deals with several ways to generalize the Flanders’ theorem to matrix triples. We consider six invertible matrices and try to write them as the possible products of three matrices. Initially, we describe a wide set of necessary conditions so that this system be solvable, showing that they are not sufficient. Next, we study the simultaneous solvability of two equations, selected appropriately among the matrix system. The rest of the paper is devoted to the study of a particular case, in which the six given matrices are simultaneously diagonalizable, with distinct nonzero eigenvalues. In this case, we obtain a necessary and sufficient condition for the solvability of the full matrix system. Moreover, an explicit solution to it is constructed. Certain technical results necessary for this work may be of independent interest.

In this paper we present some results on the question of existence of generalized or measure valued solutions for semilinear impulsive evolution equations on Banach spaces with the nonlinear parts being merely measurable and bounded on bounded sets. This is a far reaching generalization of the previous results of the author and others. It may be interesting to consider extension of the results of this paper to cover impulsive differential inclusions and their potential applications to control theory and uncertain dynamics.

A generalization of Mallat's classic multiresolution analysis (MRA), based on the theory of spectral pairs, was considered in two papers by Gabardo and Nashed. In this nonstandard setting, the translation set is no longer a subgroup or a translate of a subgroup of R, but is a spectrum associated with a one-dimensional spectral pair. In this paper, we continue the study based on this nonstandard setting and give the characterization for nonuniform wavelets associated with a nonuniform MRA. These characterizations are consistent with both the known necessary and sufficient conditions for the existence of nonuniform MRA wavelets and the known characterization for standard dyadic wavelets associated with an MRA.

In the first part of the paper we give a characterization for an associative algebra to be standardly stratified in the sense of Cline, Parshall and Scott, generalizing a theorem of V. Dlab. In the second part of the paper we construct characteristic tilting modules for standardly stratified algebras and use them to estimate the finitistic dimension of such algebras. These tilting modules give rise to the Ringel duality concept for stratified algebras. We also define and investigate a generalization of the notion of properly stratified algebras to the above setup.

A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system.On the other hand, the maximum even factor problem is a generalization of the maximum matching problem into digraphs. When the given digraph has a certain property called odd-cycle-symmetry, this problem is polynomially solvable.The main result of this paper is that the degree sequences of all even factors in a digraph form a jump system if and only if the digraph is odd-cycle-symmetric. Furthermore, as a generalization, we show that the weighted even factors induce an M-convex (M-concave) function on a constant-parity jump system. These results suggest that even factors are a natural generalization of matchings and the assumption of odd-cycle-symmetry of digraphs is essential.

The sequence space l ( p ) was introduced and studied by Maddox [I.J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2) 18 (1967) 345–355]. In the present paper, the sequence spaces l ( u , v ; p ) of non-absolute type which is derived by the generalized weighted mean is defined and proved that the spaces l ( u , v ; p ) and l ( p ) are linearly isomorphic. Besides this, the β- and γ-duals of the space l ( u , v ; p ) are computed and the basis of that space is constructed. Further, it is established that the sequence space l p ( u , v ) has AD property and given f-dual of the space l p ( u , v ) . Finally, the matrix mappings from the sequence spaces l ( u , v ; p ) to the sequence space μ and from the sequence space μ to the sequence space l ( u , v ; p ) are characterized.

Difference systems of sets (DSS) are combinatorial structures that are a generalization of cyclic difference sets and arise in connection with code synchronization. The paper surveys recent constructions and open problems concerning DSS obtained as partitions of cyclic difference sets.

In this paper analysis of the concept of associated homogeneous distributions (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist only associated homogeneous distributions of order k=1. We introduce a definition of quasi associated homogeneous distributions which is a natural generalization of the notion of associated homogeneous distributions and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transforms. It is proved that the class of quasi associated homogeneous distributions coincides with the class of distributions introduced in [I.M. Gel'fand, G.E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York, 1964, Chapter I, §4] as the class of associated homogeneous distributions. For the multidimensional case it is proved that f is a quasi associated homogeneous distribution if and only if it satisfies an Euler type system of differential equations. A new type of Γ-functions generated by quasi associated homogeneous distributions is defined.

Given a finite abelian group G (written additively), and a subset S of G, the size r(S) of the set {(a,b):a,b,[email protected]?S} may range between 0 and |S|^2, with the extremal values of r(S) corresponding to sum-free subsets and subgroups of G. In this paper, we consider the intermediate values which r(S) may take, particularly in the setting where G is Z/pZ under addition (p prime). We obtain various bounds and results. In the Z/pZ setting, this work may be viewed as a subset generalization of the Cauchy-Davenport Theorem.

The paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis (GMRA). We show the existence of a scaling vector of a GMRA such that its first component exhausts the spectrum of the core space near the origin. The corresponding low-pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector. We also prove a generalization of the Unitary Extension Principle for an infinite number of generators. This results in the construction scheme for tight framelets using low-pass and high-pass matrix masks generalizing the classical MRA constructions. We prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function.

The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian–Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K⊆L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived.

Using the Grothendieck–Lefschetz theory (see Grothendieck, 1968 [15] ) and a generalization (due to Cutkosky, 1997 [10] ) of a result from Grothendieck (1968) [15] concerning the simple connectedness, we prove that many closed subvarieties of P n of dimension ⩾2 need at least n − 1 equations to be defined in P n set-theoretically, i.e. their arithmetic rank is ⩾ n − 1 (Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the second part of the paper we prove that the arithmetic rank of a rational normal scroll of dimension d ⩾ 2 in P N is N − 2 , by producing an explicit set of N − 2 homogeneous equations which define these scrolls set-theoretically (see Theorem 2 of the Introduction).

Extensible (polynomial) lattice point sets have the property that the number N of points in the node set of a quasi-Monte Carlo algorithm may be increased while retaining the existing points. Explicit constructions for extensible (polynomial) lattice point sets have been presented recently by Niederreiter and Pillichshammer. It is the aim of this paper to establish extensibility for a powerful generalization of polynomial lattice point sets, the so-called hyperplane nets.

Recently Lagarias introduced the Wild semigroup, which is intimately connected to the 3x+1 conjecture. Applegate and Lagarias proved a weakened form of the 3x+1 conjecture while simultaneously characterizing the Wild semigroup through the Wild Number Theorem. In this paper, we consider a generalization of the Wild semigroup which leads to the statement of a Weak qx+1 Conjecture for q any prime. We prove our conjecture for q=5 together with a result analogous to the Wild Number Theorem. Next, we look at two other classes of variations of the Wild semigroup and prove a general statement of the same type as the Wild Number Theorem.

In this paper, we propose a generalization for the class of laura algebras, which we call almost laura. We show that this new class of algebras retains most of the essential features of laura algebras, especially concerning the important role played by the non-semiregular components in their Auslander-Reiten quivers. Also, we study more intensively the left supported almost laura algebras, showing that these are characterized by the presence of a generalized standard, convex and faithful component. Finally, we prove that almost laura algebras behave well with respect to full subcategories, split-by-nilpotent extensions and skew group algebras., 21 pages, very minor changes, to appear in J. Algebra

This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A : R n → R n is generalized totally positive (GTP), if its jth exterior power ∧ j A preserves a proper cone K j ⊂ ∧ j R n for every j = 1 , … , n . We also define generalized strictly totally positive (GSTP) operators. We prove that the spectrum of a GSTP operator is positive and simple, moreover, its eigenvectors are localized in special sets. The existence of invariant cones of finite ranks is shown under some additional conditions. Some new insights and alternative proofs of the well-known results of Gantmacher and Krein describing the properties of TP and STP matrices are presented.

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.

Let Φ be a root system of type G 2 , and let G be a group generated by non-trivial subgroups A α , α ∈ Φ , satisfying some generalized Steinberg relations, which are also satisfied by root subgroups corresponding to a Moufang hexagon. These relations can be considered as a generalization of the element-wise commutator relations in Chevalley groups. The Steinberg presentation specifies the groups satisfying the Chevalley commutator relations. In the present paper some sort of generalized Steinberg presentation for groups with root system of type G 2 is provided. As a main result we classify the possible structures for G.

In this paper, we establish a generalization of Hadamard's inequality to r-convex functions on Carnot groups.

If P ( z ) = ∑ ν = 0 n c ν z ν is a polynomial of degree n having no zeros in | z | 1 , then for | β | ⩽ 1 , it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45–51] that | z P ′ ( z ) + n β 2 P ( z ) | ⩽ n 2 { | 1 + β 2 | + | β 2 | } max | z | = 1 | P ( z ) | , | z | = 1 . In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next we improve upon the above inequality for the polynomials with restricted zeros. Our results refine and generalize certain well-known polynomial inequalities including some results of Bernstein, Lax, Malik and Vong, and Aziz and Dawood.

In this paper we study further the extended abelian rank one Stark conjecture contained in Emmons and Popescu (2009) [4] and Erickson (2009) [5] . We formulate a stronger question (Question 4.2) which seems easier to investigate both theoretically and computationally. Question 4.2 includes a generalization of the Brumer–Stark conjecture on annihilation of class groups (see Question 4.7). We link it with a conjecture of Gross (contained in Gross (1988) [6] ), and in the process find some new integrality properties of the Stickelberger element (Theorem 4.30). Finally, we provide some numerical examples with base field Q for which Question 4.2, and thus the extended abelian rank one Stark conjecture, have an affirmative answer.

For a given parameterized problem, π, a kernelization algorithm is a polynomial-time pre-processing procedure that transforms an arbitrary instance of π into an equivalent one whose size depends only on the input parameter(s). The resulting instance is called a problem kernel. In this paper, a kernelization algorithm for the 3-Hitting Set problem is presented along with a general kernelization for d-Hitting Set. For 3-Hitting Set, an arbitrary instance is reduced into an equivalent one that contains at most 5k2+k elements. This kernelization is an improvement over previously known methods that guarantee cubic-order kernels. Our method is used also to obtain quadratic kernels for several other problems. For a constant d⩾3, a kernelization of d-Hitting Set is achieved by a non-trivial generalization of the 3-Hitting Set method, and guarantees a kernel whose order does not exceed (2d−1)kd−1+k.

A formula for the sum of any positive-integral power of the first N positive integers was published by Johann Faulhaber in the 1600s. In this paper, we generalize Faulhaber's formula to non-integral complex powers with real part greater than −1.

For the test of sphericity, Ledoit and Wolf [Ann. Statist. 30 (2002) 1081–1102] proposed a statistic which is robust against high dimensionality. In this paper, we consider a natural generalization of their statistic for the test that the smallest eigenvalues of a covariance matrix are equal. Some inequalities are obtained for sums of eigenvalues and sums of squared eigenvalues. These bounds permit us to obtain the asymptotic null distribution of our statistic, as the dimensionality and sample size go to infinity together, by using distributional results obtained by Ledoit and Wolf [Ann. Statist. 30 (2002) 1081–1102]. Some empirical results comparing our test with the likelihood ratio test are also given.

Suppose we are given an n × n matrix, M , and a set of values, { λ i } i = 1 m ( m ⩽ n ), and we wish to find the smallest perturbation in the 2-norm (i.e., spectral norm), Δ M , such that M - Δ M has the given eigenvalues λ i . Some interesting results have been obtained for variants of this problem for fixing two distinct eigenvalues, fixing one double eigenvalue, and fixing a triple eigenvalue. This paper provides a geometric motivation for these results and also motivates their generalization. We also present numerical examples (both “successes” and “failures”) of fixing more than two eigenvalues by these generalizations.

This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group A⋊G over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension.Generic polynomials for several small groups are constructed.

In the present paper we define the generalization of the Phillips operators. Direct and converse theorems are established for Phillips operators and its new generalization. The direct theorems are given for the second order Ditzian–Totik modulus of smoothness. The inverse results are derived in the terminology of [J. Anal. Math. 61 (1993) 61–111].

A remarkable fundamental theorem established by Mehta plays an important role in proving existence of fixed points, maximal elements, and equilibria in abstract economies. In this paper, we extend Himmelbergʼs measure of precompactness to the general setting of l . c . -spaces and obtain a generalization of Mehtaʼs theorem. As an application, we develop some new fixed point theorems involving a kind of condensing mappings.

In this paper, we propose a new preconditioner for solving linear systems by the preconditioned conjugate gradient (PCG) method. The preconditioner can be thought of as a generalization of the well-known T. Chan’s preconditioner. For Hermitian positive definite matrix, we observe that our preconditioner is also Hermitian positive definite. The operation cost and convergence of the PCG method are discussed. Numerical experiments have been performed on structured problems to show the competitiveness of this preconditioner.

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably $Q$ and $T$-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras [BZ], the Kontsevich evolution [DFK09b] and the $T$-systems themselves [DFK09a]. In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data., 46 pages, minor typos corrected

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.

Consider a renewal process, and let K ⩾ 0 denote the random duration of a typical renewal cycle. Assume that on any renewal cycle, a rare event called “success” can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence can be relatively slow, because each success corresponds to a time interval, not a point. If K is an arithmetic variable, a “finite-size correction” (FSC) is known to speed Poisson convergence by providing a second, subdominant term in the appropriate asymptotic expansion. This paper generalizes the FSC from arithmetic K to general K . Genomics applications require this generalization, because they have already heuristically applied the FSC to p -values involving absolutely continuous distributions. The FSC also sharpens certain results in queuing theory, insurance risk, traffic flow, and reliability theory.

We study recently meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (GCD) matrices. Analogously, in this paper we consider join matrices on lattices as an abstract generalization of least common multiple (LCM) matrices. A formula for the determinant of join matrices on join-closed sets, bounds for the determinant of join matrices on all sets and a formula for the inverse of join matrices on join-closed sets are given. The concept of a semi-multiplicative function gives us formulae for meet matrices on join-closed sets and join matrices on meet-closed sets. Finally, we show what new the study of meet and join matrices contributes to the usual GCD and LCM matrices.

We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments.

The paper is devoted to a matrix generalization of a problem studied by Grenander and Rosenblatt (Trans. Amer. Math. Soc. 76 (1954) 112–126) and deals with the computation of the infimum Δ of ∫TQ∗(z)M(dz)Q(z), where M is a non-negative Hermitian matrix-valued Borel measure on the unit circle T and Q runs through the set of matrix-valued polynomials with prescribed values of some of their derivatives at a finite set J of complex numbers. Under some additional assumptions on M and J, the value of Δ is computed and the results are applied to linear prediction problems of multivariate weakly stationary random sequences. A related truncated problem is studied and further extremal problems are briefly discussed.

We study the distribution properties of sequences which are a generalization of the well-known van der Corput–Halton sequence on the one hand, and digital (T,s)-sequences in the sense of Niederreiter on the other. In this paper we completely answer the question under which conditions such a sequence is uniformly distributed in the s-dimensional unit cube, by using methods based on the q-additive property of the weighted q-ary sum-of-digits function.

Ray nonsingular (RNS) matrices are a generalization of sign nonsingular (SNS) matrices from the real field to complex field. The problem of how to recognize ray nonsingular matrices is still open. In this paper, we give an algorithm to recognize the ray nonsingularity for the so called “cycle chain matrices”, whose associated digraphs are of certain simple structure. Furthermore, the cycle chain matrices that are not ray nonsingular are classified into two classes according to whether or not they can be majorized by some RNS cycle chain matrices, and the case when they can is characterized.

In this paper, we consider the nonlinear elliptic systems under controllable growth condition. We use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. We extend previous partial regularity results under the natural growth condition to the case of the controllable growth condition, and directly establishing the optimal Hölder exponent for the derivative of a weak solution.

This paper contains two results concerning linear embeddings of subsets of Euclidean space in low-dimensional normed spaces. The first is an improvement of the known dependence on ɛ in Dvoretzky's theorem from order of ɛ 2 to order of ɛ (except for log factors). The second is a joint generalization of (Milman's version of) Dvoretzky's theorem and (a recent generalization by Klartag and Mendelson of) the Johnson–Lindenstrauss Lemma.

As a generalization of attenuated spaces, the concept of singular linear spaces was introduced in [K. Wang, J. Guo, F. Li, Association schemes based on attenuated spaces, European J. Combin. 31 (2010) 297–305]. This paper first gives two anzahl theorems in singular linear spaces, and then discusses their applications to the constructions of Deza digraphs, quasi-strongly regular graphs, lattices and authentication codes.

This paper introduces a concept of diagonalization that uses not a basis of eigenvectors, but a frame. A frame is a generalization of a basis which is used in a number of signal and image processing applications. We first investigate the properties of frame diagonalization, drawing parallels with those of basis diagonalization. We then describe several methods of constructing frames for frame diagonalization. In particular, we prove the existence of a universal diagonalizer for each n ∈ N that simultaneously diagonalizes all matrices in M n ( C ) , and create a method of frame diagonalization that works for any matrix in M n ( C ) , uses at most ⌊ 3 n / 2 ⌋ frame vectors and retains information about the eigenvalues of the matrix.

We present a short proof of a generalization of a result of Cheriyan and Thurimella: a simple graph of minimum degree k can be augmented to a k-edge connected simple graph by adding ⩽knk+1 edges, where n is the number of nodes. One application (from the previous paper) is an approximation algorithm with a guarantee of 1+2k+1 for the following NP-hard problem: given a simple undirected graph, find a minimum-size k-edge connected spanning subgraph. For the special cases of k=4,5,6, this is the best approximation guarantee known.

In this paper, we consider a class of nonlinear autoregressive (AR) processes with state-dependent switching, which are two-component Markov processes. The state-dependent switching model is a nontrivial generalization of Markovian switching formulation and it includes the Markovian switching as a special case. We prove the Feller and strong Feller continuity by means of introducing auxiliary processes and making use of the Radon–Nikodym derivatives. Then, we investigate the geometric ergodicity by the Foster–Lyapunov inequality. Moreover, we establish the V-uniform ergodicity by means of introducing additional auxiliary processes and by virtue of constructing certain order-preserving couplings of the original as well as the auxiliary processes. In addition, illustrative examples are provided for demonstration.

In this paper we look for a generalization of a property of matrix pairs, to quadruples of matrices. It is known that it is possible to analyze the feedback equivalence of matrix pairs in terms of the similarity of the square matrices belonging to sets related to the pairs. We will determine when it is possible to analyze the equivalence of matrix quadruples in terms of the feedback equivalence of matrix pairs.

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg–Treves condition ( P ) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann–Hilbert problem is considered.