Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

Flag enumerations of matroid base polytopes

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.     

The complexity of the weight problem for permutation and matrix groups

Given a metric d on a permutation group G, the corresponding weight problem is to decide whether there exists an element π∈G such that d(π,e)=k, for some given value k. Here we show that this problem is NP-complete for many well-known metrics. An analogous problem in matrix groups, eigenvalue-free problem, and two related problems in permutation groups, the maximum and minimum weight problems, are also investigated in this paper.     

A geometric realization of quantum groups of type D

We geometrize quantum groups of type D in the spirit of Beilinson et al. (1990) [1].     

Squarefree P-modules and the cd-index

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen–Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen–Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley–Reisner ring of the barycentric subdivision of an odd dimensional Cohen–Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function.     

Two local operators and the BLUE

For a given matrix A, a matrix P such that PA = A is said to be a local identity, and such that P²A = PA is said to be a local idempotent. In the paper, some simple properties of such operators are presented. Their relation to the best linear unbiased estimation in the general Gauss-Markov model is also demonstrated.     

Parseval Frame Wavelet Multipliers in L2（Rd）

Let A be a d × d real expansive matrix. An A-dilation Parseval frame wavelet is a function ?? ?? L ²(? ^d ), such that the set $ \left\{ {\left| {\det A} \right|^{\frac{n} {2}} \psi \left( {A^n t - \ell } \right):n \in \mathbb{Z},\ell \in \mathbb{Z}^d } \right\} $ forms a Parseval frame for L ²(? ^d ). A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of d??? is an A-dilation Parseval frame wavelet whenever ?? is an A-dilation Parseval frame wavelet, where ??? denotes the Fourier transform of ??. In this paper, the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)| = 2. As an application, the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L ²(? ^d ) is discussed.     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation—the one with the fewest non-zero entries, measured by ∥x∥₀. Such solutions find applications in signal and image processing, where the topic is typically referred to as “sparse representation”. Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.Armed with these recent results, the problem can be reversed, and formed as an implied matrix factorization problem: Given a set of vectors {b_i}, known to emerge from such sparse constructions, Ax_i = b_i, with sufficiently sparse representations x_i, we seek the matrix A. In this paper we present both theoretical and algorithmic studies of this problem. We establish the uniqueness of the dictionary A, depending on the quantity and nature of the set {b_i}, and the sparsity of {x_i}. We also describe a recently developed algorithm, the K-SVD, that practically find the matrix A, in a manner similar to the K-Means algorithm. Finally, we demonstrate this algorithm on several stylized applications in image processing.     

Improved T − ψ nodal finite element schemes for eddy current problems

The aim of this paper is to propose improved T − ψ finite element schemes for eddy current problems in the three-dimensional bounded domain with a simply-connected conductor. In order to utilize nodal finite elements in space discretization, we decompose the magnetic field into summation of a vector potential and the gradient of a scalar potential in the conductor; while in the nonconducting domain, we only deal with the gradient of the scalar potential. As distinguished from the traditional coupled scheme with both vector and scalar potentials solved in a discretizing equation system, the proposed decoupled scheme is presented to solve them in two separate equation systems, which avoids solving a saddle-point equation system like the traditional coupled scheme and leads to an important saving in computational effort. The simulation results and the data comparison of TEAM Workshop Benchmark Problem 7 between the coupled and decoupled schemes show the validity and efficiency of the decoupled one.     

On the projectors FF and FF

A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF^† and F^†F, determined by a complex square matrix F and its Moore-Penrose inverse F^†. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rako?evi? [J. Benítez, V. Rako?evi?, Matrices A such that AA^† − A^†A are nonsingular, Appl. Math. Comput. 217 (2010) 3493-3503]. Further characteristics of FF^† and F^†F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF^† and F^†F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.     

Kirchberger-type theorems for separation by convex domains

We say that a convex set K in ? ^d strictly separates the set A from the set B if A ? int(K) and B ? cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ? ^d with the property that for every T ? A?B of cardinality at most d + 2, there is a half space strictly separating T ? A and T ? B, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ? ^d is d + 2.In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

Gradient based and least squares based iterative algorithms for matrix equations AXB + CXD = F

This paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation AXB + CX^TD = F. The basic idea is to decompose the matrix equation (system) under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving Ax = b and AXB = F. The analysis shows that when the matrix equation has a unique solution (under the sense of least squares), the iterative solution converges to the exact solution for any initial values. A numerical example verifies the proposed theorems.     

The Rank of Abelian Varieties over Infinite Galois Extensions

G. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112-128) asked if every Abelian variety A defined over a number field k with dim A>0 has infinite rank over the maximal Abelian extension k^ab of k. We verify this for the Jacobians of cyclic covers of P¹, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).     

Splitting in 2-computably enumerable degrees with avoiding cones

In this paper we show that for any pair of properly 2-c. e. degrees 0 < d < a such that there are no c. e. degrees between d and a, the degree a is splittable in the class of 2-c. e. degrees avoiding the upper cone of d. We also study the possibility to characterize such an isolation in terms of splitting.     

Automata all of whose congruences are inner

A congruence on an automaton A is called inner if it is the kernel of a certain endomorphism on A. We propose a characterization of automata, all of whose congruences are inner.     

The Equivalence Between Seven Classes of Wavelet Multipliers and Arcwise Connectivity They Induce

Let A be a d×d expansive matrix with |detA|=2. An A-wavelet is a function $\psi\in L^{2}(\mathbb{R}^{d})$ such that $\{2^{\frac{j}{2}}\psi(A\cdot-k):\,j\in \mathbb{Z},\,k\in \mathbb{Z}^{d}\}$ is an orthonormal basis for $L^{2}(\mathbb{R}^{d})$ . A measurable function f is called an A-wavelet multiplier if the inverse Fourier transform of $f\hat{\psi}$ is an A-wavelet whenever ψ is an A-wavelet, where $\hat{\psi}$ denotes the Fourier transform of ψ. A-scaling function multiplier, A-PFW multiplier, semi-orthogonal A-PFW multiplier, MRA A-wavelet multiplier, MRA A-PFW multiplier and semi-orthogonal MRA A-PFW multiplier are defined similarly. In this paper, we prove that the above seven classes of multipliers are equivalent, and obtain a characterization of them. We then prove that if the set of all A-wavelet multipliers acts on some A-scaling function (A-wavelet, A-PFW, semi-orthogonal A-PFW, MRA A-wavelet, MRA A-PFW, semi-orthogonal MRA A-PFW), the orbit is arcwise connected in $L^{2}(\mathbb{R}^{d})$ , and that if the generator of an orbit is an MRA A-PFW, the orbit is equal to the set of all MRA A-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRA A-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.