Riesz-Haviland criterion for incomplete data

The aim of this paper is to analyze a “support-free” version of the Riesz-Haviland theorem proved recently by the present authors, which characterizes truncations of the complex moment problem via positivity condition on appropriate families of polynomials in z and . The attention is focused on modifications of the positivity condition as well as the assumption on admissible truncations. The former results in truncations for which the corresponding “support-free” Riesz-Haviland condition locates a representing measure on the distinguished subset of the complex plane, while the latter effects a non-integral variant of the Riesz-Haviland theorem.     

On equivalencies between design-based and regression-based variance estimators for randomized experiments

This paper demonstrates that the randomization-based “Neyman” and constant-effects estimators for the variance of estimated average treatment effects are equivalent to a variant of the White “heteroskedasticity-robust” estimator and the homoskedastic ordinary least squares (OLS) estimator, respectively.     

The weighted Fermat-Torricelli problem for tetrahedra and an “inverse” problem

We study the weighted Fermat-Torricelli problem for tetrahedra in R³ and solve an “inverse” problem by introducing a method of differentiation. The solution of the inverse problem is the main result which states that: Given the Fermat-Torricelli point A₀ with the vertices lie on four prescribed rays, find the ratios between every pair of non-negative weights of two corresponding rays such that the sum of the four non-negative weights is a constant number. An application of the inverse weighted Fermat-Torricelli problem is the strong invariance principle of the weighted Fermat-Torricelli point which gives some classes of tetrahedra that could be named “evolutionary tetrahedra”.     

The Logic of Uncertain Justifications

In Artemov?s Justification Logic, one can make statements interpreted as “t is evidence for the truth of formula F.” We propose a variant of this logic in which one can say “I have degree r of confidence that t is evidence for the truth of formula F.” After defining both an axiomatic approach and a semantics for this Logic of Uncertain Justifications, we will prove the usual soundness and completeness theorems.     

On the three-dimensional Wigner-Poisson problem with inflow boundary conditions

We study the Wigner-Poisson problem in a bounded spatial domain, with non-homogeneous and time-dependent “inflow” boundary conditions. This system is a quantum model of charge transport in a semiconductor device coupled with reservoirs, in presence of a self-consistent potential and of an external one. We state a local-in-time well-posedness result for the problem. The main difficulty is proving in the three-dimensional case that the non-linear potential term is a Lipschitz perturbation of the “affine” streaming operator, in an appropriately weighted L²-space.     

The distance between two convex sets

In this paper we explore the duality relations that characterize least norm problems. The paper starts by presenting a new Minimum Norm Duality (MND) theorem, one that considers the distance between two convex sets. Roughly speaking the new theorem says that the shortest distance between the two sets is equal to the maximal “separation” between the sets, where the term “separation” refers to the distance between a pair of parallel hyperplanes that separates the two sets.The second part of the paper brings several examples of applications. The examples teach valuable lessons about the role of duality in least norm problems, and reveal new features of these problems. One lesson exposes the polar decomposition which characterizes the “solution” of an inconsistent system of linear inequalities. Another lesson reveals the close links between the MND theorem, theorems of the alternatives, steepest descent directions, and constructive optimality conditions.     

The efficient solution of the (quietly constrained) noisy, linear regulator problem

In a previous paper we gave a new, natural extension of the calculus of variations/optimal control theory to a (strong) stochastic setting. We now extend the theory of this most fundamental chapter of optimal control in several directions. Most importantly we present a new method of stochastic control, adding Brownian motion which makes the problem “noisy.” Secondly, we show how to obtain efficient solutions: direct stochastic integration for simpler problems and/or efficient and accurate numerical methods with a global a priori error of O(h3/2) for more complex problems. Finally, we include “quiet” constraints, i.e. deterministic relationships between the state and control variables. Our theory and results can be immediately restricted to the non “noisy” (deterministic) case yielding efficient, numerical solution techniques and an a priori error of O(h²). In this event we obtain the most efficient method of solving the (constrained) classical Linear Regulator Problem. Our methods are different from the standard theory of stochastic control. In some cases the solutions coincide or at least are closely related. However, our methods have many advantages including those mentioned above. In addition, our methods more directly follow the motivation and theory of classical (deterministic) optimization which is perhaps the most important area of physical and engineering science. Our results follow from related ideas in the deterministic theory. Thus, our approximation methods follow by guessing at an algorithm, but the proof of global convergence uses stochastic techniques because our trajectories are not differentiable. Along these lines, a general drift term in the trajectory equation is properly viewed as an added constraint and extends ideas given in the deterministic case by the first author.     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

Heteroclinic and Periodic Cycles in a Perturbed Convection Model

Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.     

Meta-data envelopment analysis: Finding a direction towards marginal profit maximization

This paper discusses a new meta-DEA approach to solve the problem of choosing direction vectors when estimating the directional distance function. The proposed model emphasizes finding the “direction” for productivity improvement rather than estimating the “score” of efficiency; focusing on “planning” over “evaluation”. In fact, the direction towards marginal profit maximization implies a step-by-step improvement and “wait-and-see” decision process, which is more consistent with the practical decision-making process. An empirical study of U.S. coal-fired power plants operating in 2011 validates the proposed model. The results show that the efficiency measure using the proposed direction is consistent with all other indices with the exception of the direction towards the profit-maximized benchmark. We conclude that the marginal profit maximization is a useful guide for determining direction in the directional distance function.     

Fast solving of weighted pairing least-squares systems

This paper presents a generalization of the “weighted least-squares” (WLS), named “weighted pairing least-squares” (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual “Cholesky factorization-backward substitution” method for solving linear systems.     

Public policy structuring incorporating reciprocal expectation analysis

This paper proposes a method of structuring public policy by incorporating reciprocal expectation analysis. The proposed method is characterized by three components: identification of the problem structure perceived by stakeholders using cognitive maps, policy structuring analysis with a value–driver matrix and a reciprocal expectation matrix, and feasibility analysis of agreements among the stakeholders. The three types of relationship among stakeholders are derived from the feasibility analysis, which are “Dosho-imu”, “Isho-imu”, and “Domu”. Three tests of feasibility to reach the agreement are then proposed: “information-sharing test”, “bargaining test”, and “reframing test”. A case study is presented, applying the method to strategic transportation planning in the Kanto region of Japan. Finally, the potential functions of the proposed method in practice are discussed.     

Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.     

An axiomatic analysis of concordance–discordance relations

Outranking methods propose an original way to build a preference relation between alternatives evaluated on several attributes that has a definite ordinal flavor. Indeed, most of them appeal the concordance/non-discordance principle that leads to declaring that an alternative is “superior” to another, if the coalition of attributes supporting this proposition is “sufficiently important” (concordance condition) and if there is no attribute that “strongly rejects” it (non-discordance condition). Such a way of comparing alternatives is rather natural. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. This explains why the axiomatic foundations of outranking methods have not been much investigated, which is often seen as one of their important weaknesses. This paper uses conjoint measurement techniques to obtain an axiomatic characterization of preference relations that can be obtained on the basis of the concordance/non-discordance principle. It emphasizes their main distinctive feature, i.e. their very crude way to distinguish various levels of preference differences on each attribute. We focus on outranking methods, such as ELECTRE I, that produce a reflexive relation, interpreted as an “at least as good as” preference relation. The results in this paper may be seen as an attempt to give such outranking methods a sound axiomatic foundation based on conjoint measurement.     

Competitive dynamics of e-commerce web sites

The characteristic of “rich gets richer” is much more conspicuous on the Internet, which is the important cause of “power-law” characteristic of the “winner-takes-all” markets. A competitive model of e-commerce web sites is presented in order to explore the effects of competition and the characteristic of “rich gets richer” on Internet economy, and also to investigate how the “winner-takes-all” phenomenon comes forth. Then these ordinary differential equations are qualitatively analyzed and numerically simulated to study the dynamics of competition. The effects of parameters on the dynamic behavior in the model are also discussed.     

A comparison of different quantity discount pricing policies in a two-echelon channel with stochastic and asymmetric demand information

In this paper, we study quantity discount pricing policies in a channel of one manufacturer and one retailer. The paper assumes that the channel faces a stochastic price-sensitive demand but the retailer can privately observe the realization of an uncertain demand parameter. The problem is analyzed as a Stackelberg game in which the manufacturer declares quantity discount pricing schemes to the retailer and then the retailer follows by selecting the retail price and associated quantity. Proposed in the paper are four quantity-discount pricing policies: “regular quantity discount”; “fixed percentage discount”; “incremental volume discount” and “fixed marginal-profit-rate discount”. Optimal solutions are derived, and numerical examples are presented to illustrate the efficiency of each discount policy.     

Fast partial differential equation de‐noising filter for mechanical vibration signal

A novel approach for mechanical vibration signal de‐noising filter using PDE and its numerical solution were presented. The proposed method is computationally fast compared with other conventional PDE‐based de‐noising methods. It enables: (i) by incorporating unconditional stable finite difference backward Euler scheme, the de‐noising process has no requirements of grid ratio; (ii) developing variational matrix‐based fast filter while the de‐noising process can be completed instantly, which will be accomplished by only one iteration; and (iii) effective de‐noising method for mechanical vibration signal interfered by Gauss white noise. The method is performed efficiently, and the de‐noising tests on different artificial Gauss white noise as well as natural mechanical noise are conducted. Experimental tests have been rigorously compared with different de‐noising methods to verify the efficacy of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Local and global solutions for some parabolic nonlocal problems

We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a “Newtonian” nonlocal term.     

A viscosity method with no spectral radius requirements for the split common fixed point problem

This paper is concerned with an algorithmic solution to the split common fixed point problem in Hilbert spaces. Our method can be regarded as a variant of the “viscosity approximation method”. Under very classical assumptions, we establish a strong convergence theorem with regard to involved operators belonging to the wide class of quasi-nonexpansive operators. In contrast with other related processes, our algorithm does not require any estimate of some spectral radius. The technique of analysis developed in this work is new and can be applied to many other fixed point iterations. Numerical experiments are also performed with regard to an inverse heat problem.     

Combining two pheromone structures for solving the car sequencing problem with Ant Colony Optimization

The car sequencing problem involves scheduling cars along an assembly line while satisfying capacity constraints. In this paper, we describe an Ant Colony Optimization (ACO) algorithm for solving this problem, and we introduce two different pheromone structures for this algorithm: the first pheromone structure aims at learning for “good” sequences of cars, whereas the second pheromone structure aims at learning for “critical” cars. We experimentally compare these two pheromone structures, that have complementary performances, and show that their combination allows ants to solve very quickly most instances.