Simulation from Wishart Distributions with Eigenvalue Constraints

Abstract

This article provides an efficient algorithm for generating a random matrix according to a Wishart distribution, but with eigenvalues constrained to be less than a given vector of positive values. The procedure of Odell and Feiveson provides a guide, but the modifications here ensure that the diagonal elements of a candidate matrix are less than the corresponding elements of the constraint vector, thus greatly improving the chances that the matrix will be acceptable. The Normal hierarchical model with vector outcomes and the multivariate random effects model provide motivating applications.     

Efficient algorithms for the longest common subsequence problem with sequential substring constraints

In this paper, we generalize the inclusion constrained longest common subsequence (CLCS) problem to the hybrid CLCS problem which is the combination of the sequence inclusion CLCS and the string inclusion CLCS, called the sequential substring constrained longest common subsequence   (SSCLCS) problem. In the SSCLCS problem, we are given two strings A$A$ and B$B$ of lengths m$m$ and n$n$, respectively, formed by alphabet Σ$\Sigma$ and a constraint sequence C$C$ formed by ordered strings (C¹,C²,C³,…,C^l)$(C^1,C^2,C^3,\dots ,C^l)$ with total length r$r$. The problem is that of finding the longest common subsequence D$D$ of A$A$ and B$B$ containing C¹,C²,C³,…,C^l$C^1,C^2,C^3,\dots ,C^l$ as substrings and with the order of the C$C$’s retained. This problem has two variants, depending on whether the strings in C$C$ cannot overlap or may overlap. We propose algorithms with O(mnl+(m+n)(|Σ|+r))$O(mnl+(m+n)(\left|\Sigma \right|+r))$ and O(mnr+(m+n)|Σ|)$O(mnr+(m+n)\left|\Sigma \right|)$ time for the two variants. For the special case with one or two constraints, our algorithm runs in O(mn+(m+n)(|Σ|+r))$O(mn+(m+n)(\left|\Sigma \right|+r))$ or O(mnr+(m+n)|Σ|)$O(mnr+(m+n)\left|\Sigma \right|)$ time, respectively—an order faster than the algorithm proposed by Chen and Chao.     

Structures,stabilities and electronic properties of boron-doped silicon clusters B3Sin (n=1–17) and their anions

The structures, stabilities and electronic properties of neutral and anionic B₃Si_n (n?=?1–17) clusters have been systemically investigated on the basis of density functional theory at the B3LYP/6-311?+?G(d) level and CALYPSO structure prediction method. The structural searches show that three boron atoms tend to form B₃ triangle encapsulated into Si_n cages with the increasing number of silicon atoms. Most of the lowest energy structures can be derived by using the squashed pentagonal bipyramid structure of B₃Si₄ and B₃Si₄^? as the major building unit. The relative stabilities are studied based on the calculated binding energies, second-order difference of energies and HOMO–LUMO gaps of the lowest energy structures. In addition, Hirshfeld, natural population analysis, Bader approaches and natural electronic configuration are performed to explore the charge transfer. At last, molecular orbital, magnetic properties, IR, Raman and UV–vis spectra are also, respectively, analysed for providing strong support for essential theoretical and experimental research.     

On a Nonlinear,Nonlocal Parabolic Problem with Conservation of Mass,Mean and Variance

In this paper we prove that the steepest descent of certain porous-medium type functionals with respect to the quadratic Wasserstein distance over a constrained (but not weakly closed) manifold gives rise to a nonlinear, nonlocal parabolic partial differential equation connected to the study of the asymptotic behavior of solutions for filtration problems. The result by Carlen and Gangbo on constrained optimization for steepest descent of the negative Boltzmann entropy in the Wasserstein space is generalized to porous-medium type functionals. An interesting feature of the resulting Fokker-Planck equation is the nonlocality of its drift term occurring at the same time as its nonlinearity.     

Optimisation of gravity-fed water distribution network design: A critical review

This paper surveys the literature on the optimisation of water distribution network design. The water distribution network design (WDND) optimisation problem entails finding the material and diameter of each pipe in the network so that the total cost of the network is minimised without violating any hydraulic constraints. This is a difficult combinatorial optimisation problem, in which decision variables are discrete and both cost function and constraints are non-linear. Over the past 30 years, a large number of methods, especially in the field of (meta) heuristics, have been developed to solve this problem, most of which obtain good results on the available benchmark networks. In addition to outlining the basic features of each method, a detailed computational comparison is presented. Based on this comparison, some issues with the current state of the art in this domain are discussed, and some future research directions are suggested. Additionally, the need for an adequate set of benchmark instances is motivated, and the minimal requirements for an instance set generator are discussed.     

Minimising the number of gap-zeros in binary matrices

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.     

A hybrid metaheuristic method for the Maximum Diversity Problem

The Maximum Diversity Problem (MDP) consists in selecting a subset of m$m$ elements from a given set of n$n$ elements (n>m$n>m$) in such a way that the sum of the pairwise distances between the m$m$ chosen elements is maximized. We present a hybrid metaheuristic algorithm (denoted by MAMDP) for MDP. The algorithm uses a dedicated crossover operator to generate new solutions and a constrained neighborhood tabu search procedure for local optimization. MAMDP applies also a distance-and-quality based replacement strategy to maintain population diversity. Extensive evaluations on a large set of 120 benchmark instances show that the proposed approach competes very favorably with the current state-of-art methods for MDP. In particular, it consistently and easily attains all the best known lower bounds and yields improved lower bounds for 6 large MDP instances. The key components of MAMDP are analyzed to shed light on their influence on the performance of the algorithm.     

Macro-Hybrid Variational Formulations of Constrained Boundary Value Problems

In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping domain decompositions, and corresponding three-field versions are derived. Then, for regularization purposes, augmented formulations are established via preconditioned exact penalizations and expressed in terms of proximation operators. Optimization interpretations are given for potential problems, recovering the classic two- and three-field augmented Lagrangian formulations. Furthermore, associated parallel two- and three-field proximal-point algorithms are discussed for numerical resolution of finite element discretizations. Applications to dual mixed variational formulations of problems from mechanics illustrate the theory.     

Application of deuterated THENA for assigning the absolute configuration of chiral secondary alcohols

The structure of a constrained bicyclic chiral derivatizing agent (CDA), 1,2,3,4-tetrahydro-1,4-epoxynaphthalene-1-carboxylic acid, THENA 1, was modified by replacing both exo-methylene protons with deuterium atoms. The modified CDA, THENA-d₂2, could be used to assign the absolute configuration of chiral secondary alcohols with good reliability. Compared with THENA, the multiplicity of the methylene proton signals in the ¹H NMR spectra of THENA-d₂ derivatives is less complicated and the new CDA thus offers simpler NMR spectra for data interpretation.