A proof for the positive definiteness of the Jaccard index matrix

In this paper we provide a proof for the positive definiteness of the Jaccard index matrix used as a weighting matrix in the Euclidean distance between belief functions defined in Jousselme et al. [13]. The idea of this proof relies on the decomposition of the matrix into an infinite sum of positive semidefinite matrices. The proof is valid for any size of the frame of discernment but we provide an illustration for a frame of three elements. The Jaccard index matrix being positive definite guaranties that the associated Euclidean distance is a full metric and thus that a null distance between two belief functions implies that these belief functions are strictly identical.     

A detailed study of the generation of optically detectable watermarks using the logistic map

A digital watermark is a visible, or preferably invisible, identification code that is permanently embedded in digital media, to prove owner authentication and provide protection for documents. Given the interest in watermark generation using chaotic functions a detailed study of one chaotic function for this purpose is performed. In this paper, we present an approach for the generation of watermarks using the logistic map. Using this function, in conjunction with seed management, it is possible to generate chaotic sequences that may be used to create highpass or lowpass digital watermarks. In this paper we provide a detailed study on the generation of optically detectable watermarks and we provide some guidelines on successful chaotic watermark generation using the logistic map, and show using a recently published scheme, how care must be taken in the selection of the function seed.     

A detailed study of the generation of optically detectable watermarks using the logistic map

A digital watermark is a visible, or preferably invisible, identification code that is permanently embedded in digital media, to prove owner authentication and provide protection for documents. Given the interest in watermark generation using chaotic functions a detailed study of one chaotic function for this purpose is performed. In this paper, we present an approach for the generation of watermarks using the logistic map. Using this function, in conjunction with seed management, it is possible to generate chaotic sequences that may be used to create highpass or lowpass digital watermarks. In this paper we provide a detailed study on the generation of optically detectable watermarks and we provide some guidelines on successful chaotic watermark generation using the logistic map, and show using a recently published scheme, how care must be taken in the selection of the function seed.     

A correct proof of the McMorris-Powers’ theorem on the consensus of phylogenies

McMorris and Powers proved an Arrow-type theorem on phylogenies given as collections of quartets. There is an error in one of the main lemmas used to prove this theorem. However, this lemma (and thereby the theorem) is still true, and we provide a corrected proof.     

A primer of substitution tilings of the Euclidean plane

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.     

The Ground State Energy of the Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.     

Approximation of the joint spectral radius using sum of squares

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.     

On the complexity of robust geometric programming with polyhedral uncertainty

Geometric Programming is a useful tool with a wide range of applications in engineering. As in real-world problems input data is likely to be affected by uncertainty, robust geometric programming has been introduced. It is an open question whether there exists a tractable reformulation of the robust geometric programming model. We provide a negative answer by showing that robust geometric programming with polyhedral uncertainty is co-NP hard, and provide positive results for the case of interval uncertainty.     

On the pseudo-inverse of the Laplacian of a bipartite graph

We provide an efficient method to calculate the pseudo-inverse of the Laplacian of a bipartite graph, which is based on the pseudo-inverse of the normalized Laplacian.     

On the complexity of the car sequencing problem

In this note we give an easier proof of the known result that the car sequencing problem is NP-hard, and point out that it is NP-hard in the strong sense. We show that a previous claim of NP-completeness is incorrect, and instead we give a sufficient condition of membership of NP. We also provide a pseudo-polynomial algorithm for a special case.     

Inversion of Rational Matrices by the Reinforcement Method

The reinforcement method is extended to provide an algorithmfor obtaining the inverse of a matrix whose terms are ratiosof polynomials.     

Toward an Elementary Axiomatic Theory of the Category of LP-Matroids

The purpose of this paper is to provide the beginnings of an elementary theory for the category of loopless pointed matroids and strong maps. We propose a finite set of elementary axioms that is the beginning of an elementary axiomatic theory for this category.     

The Existence of a Waver Point on the Omitted Arc of a Support Point of the Class S

The purpose of this article is to present new information about the geometry of the omitted arc of a support point of the class S. We provide an example of an omitted arc whose tangent vector reverses direction from clockwise to counterclockwise at an interior point of the arc.     

On Measuring the Contribution of OR

Cost-benefit analysis offers a common-sense approach to measuring the contribution of OR but encounters difficulties which have been recognized by Ackoff. Ackoff's concept of adaptivity appears to offer a neat solution to those difficulties but can be seen to provide only a methodology and to lack a background theory. The authors argue that: in general the development of background theory should precede the development of a methodology; for OR the background theory required is one of decision-making; and that the social sciences provide the resource from which an adequate theory of decision-making can be developed.     

On the differentiability of the support function

The support function is strongly related to the cost function in economics. Because the differentiability of the cost function is an important property, being related to the celebrated Shephard’s lemma, our propose is to study the differentiability of the support function of a nonempty closed and convex subset of a finite dimensional normed space. So we provide characterizations of the differentiability of the support function on three subsets of its domain.     

The emergence of groups in the evolution of friendship networks

Friendship networks usually show a certain degree of segmentation: subgroups of friends. The explanation of the emergence of such groups from initially dyadic pair friendships is a difficult but important problem. In this paper we attempt to provide a first contribution to the explanation of subgroup formation in friendship networks by using the LS set as a definition for a friendship group. We construct a dynamic individual oriented model of friendship formation and provide preliminary simulation results that give an idea of how to continue the process of explaining group formation.     

On the failure of the co-hopf property for subgroups of word-hyperbolic groups

We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic groupG such thatH is one-ended, andH does not split over a cyclic group, andH is isomorphic to one of its proper subgroups. Supported as an NSF Postdoctoral Fellow under grant no. DMS-9627506.     

On the Existence of B -Root Subgroups on Affine Spherical Varieties

Doklady Mathematics - Let X be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group G. In this paper, we provide sufficient conditions,...     

A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements

We consider a supply chain setting where multiple uncapacitated facilities serve a set of customers with a single product. The majority of literature on such problems requires assigning all of any given customer??s demand to a single facility. While this single-sourcing strategy is optimal under linear (or concave) cost structures, it will often be suboptimal under the nonlinear costs that arise in the presence of safety stock costs. Our primary goal is to characterize the incremental costs that result from a single-sourcing strategy. We propose a general model that uses a cardinality constraint on the number of supply facilities that may serve a customer. The result is a complex mixed-integer nonlinear programming problem. We provide a generalized Benders decomposition algorithm for the case in which a customer??s demand may be split among an arbitrary number of supply facilities. The Benders subproblem takes the form of an uncapacitated, nonlinear transportation problem, a relevant and interesting problem in its own right. We provide analysis and insight on this subproblem, which allows us to devise a hybrid algorithm based on an outer approximation of this subproblem to accelerate the generalized Benders decomposition algorithm. We also provide computational results for the general model that permit characterizing the costs that arise from a single-sourcing strategy.