A quantitative notion of redundancy for finite frames

The objective of this paper is to improve the customary definition of redundancy by providing quantitative measures in its place, which we coin upper and lower redundancies, that match better with an intuitive understanding of redundancy for finite frames in a Hilbert space. This motivates a carefully chosen list of desired properties for upper and lower redundancies. The means to achieve these properties is to consider the maximum and minimum of a redundancy function, which is interesting in itself. The redundancy function is defined on the sphere of the Hilbert space and measures the concentration of frame vectors around each point. A complete characterization of functions on the sphere which coincide with a redundancy function for some frame is given. The upper and lower redundancies obtained from this function are then shown to satisfy all of the intuitively desirable properties. In addition, the range of values they assume is characterized.     

p-adic valuations and k-regular sequences

A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x)∈Q_p[x], we consider the sequence , where v_p is the p-adic valuation. We show that this sequence is p-regular if and only if f(x) factors into a product of polynomials, one of which has no roots in Z_p, the other which factors into linear polynomials over Q. This answers a question of Allouche and Shallit.     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

Absolute optimal solution for a compact and convex game

This paper investigates the existence of absolute optimal solutions for a partition P in continuous and quasiconcave games. We show that the P-consistency property introduced in the paper, together with the quasiconcavity and continuity of payoffs, permits the existence of P-absolute optimal solutions in games with compact and convex strategy spaces. The P-consistency property is a general condition that cannot be dispensed with for the existence of P-absolute optimal solutions. We also characterize the existence of P-absolute optimal solutions by providing necessary and sufficient conditions. Moreover, we suggest an algorithm for efficiently computing P-absolute optimal solutions.     

Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP^(k)(I), MP^(k)(II), and MP^(k)(IV) families have closure property under finite sample minima. The MP^(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP^(k)(III) distribution is developed. Moreover, the MP^(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP^(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP^(k)(II) family is shown to have the truncation invariant property.     

Monotone optimal control for a class of Markov decision processes

This paper provides a unified framework to study monotone optimal control for a class of Markov decision processes through D-multimodularity. We demonstrate that each system in this class can be classified as either a substitution-type or a complement-type system according to the possible transition set, which can be used as a classification mechanism that integrates a variety of models in the literature. We develop a generic proof of the structural properties of both types of system. In particular, we show that D-multimodularity is a generally sufficient condition for monotone optimal control of different types of system in this class. With this unified theory, there is no need to pursue each problem ad hoc and the structural properties of this class of MDPs follow with ease.     

A new method for constructing infinite families of k-tight optimal double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. DLN has been widely used in the designing of local area networks and distributed systems. In this paper, a new method for constructing infinite families of k-tight optimal DLN is presented. For k = 0, 1, ..., 40, the infinite families of k-tight optimal DLN can be constructed by the new method, where the number n _k (t, a) of their nodes is a polynomial of degree 2 in t and contains a parameter a. And a conjecture is proposed.     

Algorithm for a general discrete k-out-of-n: G system subject to several types of failure with an indefinite number of repairpersons

A discrete k-out-of-n: G system with multi-state components is modelled by means of block-structured Markov chains. An indefinite number of repairpersons are assumed and PH distributions for the lifetime of the units and for the repair time are considered. The units can undergo two types of failures, repairable or non-repairable. The repairability of the failure can depend on the time elapsed up to failure. The system is modelled and the stationary distribution is built by using matrix analytic methods. Several performance measures of interest, such as the conditional probability of failure for the units and for the system, are built into the transient and stationary regimes. Rewards are included in the model. All results are shown in a matrix algorithmic form and are implemented computationally with Matlab. A numerical example of an optimization problem shows the versatility of the model.     

Optimal reconstruction systems for erasures and for the q-potential

In this paper we introduce the q-potential as an extension of the Benedetto-Fickus frame potential, defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call q-fundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko’s and Fulton’s theory on sums of hermitian operators.     

Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations

Commonly studied models of the consecutive-k-out-of-n: F repairable systems in the existing literatures were considering the systems which had one repairman without vacation or infinite repairmen without vacations. In addition to those models, multiple repairmen without vacations are studied occasionally. However, technical personnel are very short in some fields. Some failed components cannot be repaired in time. This paper deals with the phenomenon of waiting for repair by supposing R repairmen with multiple vacations in the system. Using the pairs (i, |j|), the factor that the R repairmen taking multiple vacations was embedded into the classical C(k, n: F) system. Reliability indexes are presented. Finally, the Runge–Kutta method was used to a special case, and the experimental results demonstrate the necessity and validity of the new model.     

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm , the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property.     

Ratio limit theorems and Tauberian theorems for vector-valued functions and sequences

We prove ratio limit theorems for (C,γ)-means (γ?0) and Abel means of functions and sequences in Banach spaces, and ratio Tauberian theorems for (C,γ)-means (γ?1) and Abel means of functions and sequences in Banach lattices.     

Variational formula and overdetermined problem for the principle eigenvalue of k-Hessian operator

We first deduce the first variational formula and some overdetermined problems for the principle eigenvalue of the k-Hessian operator, and then prove Serrin type symmetry result for our overdetermined problems.     

Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, Lp or ?p spaces, 1<p<∞), and K a nonempty closed convex (not necessarily bounded) subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that (I−T) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.     

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.     

Non-dominance and potential optimality for partial preference relations

In this paper we obtain new theoretical results relating the notions of potential optimality and non-dominance without an assumption that a value function exists. In particular, we investigate a decision problem involving the choice of single or multiple best objects. Our results show that the notions of potential optimality and non-dominance are equivalent in a special case of preferences of the decision maker expressed by partial quasi-orders.     

On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings

In this paper, we consider a normalized biholomorphic mapping f(x) defined on the unit ball in a complex Banach space, where the origin 0 is a zero of order k+1 of f(x)−x. The precise growth and covering theorem for f(x) is obtained when f(x) is a starlike mapping or a starlike mapping of order α. Especially, the precise growth and covering theorem for f(x) is also established when f(x) is a quasi-convex mapping. Moreover, the precise distortion theorem for f(x) is given when f(x) is a convex mapping. Our result includes many known results.     

Existence and uniqueness of positive periodic solutions for a neutral Logarithmic population model

In this paper, by using an abstract continuous theorem of k-set contractive operator, the criteria is established for the existence, global attractivity of positive periodic solution of a neutral delay Logarithmic population model with multiple delays. The result improve the known ones in [S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral Logarithmic population model with multiple delays, J. Comput. Appl. Math. 166 (2) (2004) 371-383].