Sure things — Dominance and independence rules for choice under uncertainty

The primary purpose of this paper is to show that differences between models of choice under uncertainty may be derived primarily from different assumptions about the appropriate ways in which states of the world may be compared and combined. It considers different concepts of stochastic dominance arising from different permitted transformations on the ordering of prizes during a comparison of two lotteries. These concepts imply various forms of the Independence axiom and correspond to various non-expected utility theories.     

On SIP algorithms for minimizing the mean-risk function in the multi-period single-source problem under uncertainty

We present an algorithmic framework for solving the strategic problem of assigning retailers to facilities in a multi-period single-sourcing product environment under uncertainty in the demand from the retailers and the costs of production, inventory holding, backlogging and distribution of the product. The functional to minimize is included by the expected objective function and the excess probability functional. By considering a splitting variable mathematical representation of the Deterministic Equivalent Model, we introduce several so-called Fix-and-Relax procedures that exploit the excess probability functional structure in addition to the structure of the special ordered sets related to the non-anticipativity constraints for the assignment variables. Some computational experience is reported. This research has been partially supported by the Grant TIC2003-05982-C05-05 from MCYT.     

“Value” of additional information in multicriterion analysis under uncertainty

The expected value of information in classical (monocriterion) decision analysis has been well covered in the literature. One cannot say the same thing about the multicriterion analysis, particularly when one is in the presence of multicriterion aggregation procedures based on outranking relations for a ranking problematic. The objective of this paper is to try to extend the Bayesian approach to a multicriterion analysis in the context of uncertainty. After illustrating the a posteriori analysis, we shall mention some difficulties associated with the pre a posteriori analysis and the concepts of the “expected value” of perfect or imperfect information.     

A derivative algorithm for inexact quadratic program – application to environmental decision-making under uncertainty

Inexact quadratic programming (IQP) is an extension of conventional quadratic programming for handling both nonlinearities in cost objectives and uncertainties with modeling parameters. It has been a useful tool for environmental systems analysis. However, inefficiency in its solution method has existed, leading to difficulties in its practical application. In this study, a derivative algorithm (DAM) is proposed for solving the IQP. It improves upon the existing method through provision of a quantitative expression for uncertain relationships between the quadratic objective function and the decision variables. The DAM requires much lower computational efforts than the existing algorithm, which is especially meaningful for the IQP's application to large-scale problems. The developed DAM is applied to a hypothetical problem of municipal solid waste management and planning. Detailed solution steps are provided to clearly demonstrate the method's advantages.     

On the determination of strength of belief for decision support under uncertainty—Part II: fusing strengths of belief

Valuation functions are used in decision making under uncertainty to enable comparisons of alternatives. They are based on a weighted averaging of the n possible payoffs available under an alternative. The weighting vectors used are a reflection of the decision-making agent's strength of belief a given outcome will occur. Our concern is with developing methods to fuse multiple sources of these weighting vectors. We first suggest a method based on a normalized product. Some methods are suggested for handling completely conflicting beliefs. We abstract the basic features of this product fusion method. Particularly notably among these properties is the fact that a source with all weights equal, 1/n, acts as an identity in the fusion process. We consider next a fusion method using a uninorm aggregation operator with identity 1/n. We carefully look at this new type of method for multi-source fusion and suggest some generations and modifications. Finally we consider the situation when the contributing sources have differing credibilities.     

A topological characterization for closed sets under polar duality in ℚ n

A topological characterization is given for closed sets in ⁿ under the restriction of (cone) polar duality to ⁿ .     

The correct proofs for the optimal ordering policy with?trade credit under two different payment methods in?a?supply chain system

Cheng et al. (Top, 2010. doi:10.1007/s11750-08-0062-3) consider the optimal ordering policy with trade credit under two different payment methods. Under Assumption (5) by Cheng et al., the annual total relevant cost TRC(T) is only defined on a finite interval. However, Cheng et al. treat the domain of TRC(T) to be the set of all positive numbers such that the formulation and optimal solution of TRC(T) cause some errors. So, the main purpose of this paper not only removes those shortcomings by Cheng et al. but also presents the correct proofs for Theorems?1 and?2 of Cheng et?al.     

Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We investigate boundedness of the evolutione ^itH in the sense ofL ²(ℝ³→L ²(ℝ³) as well asL ¹(ℝ³→L ^∞(ℝ³) for the non-selfadjoint operator where μ>0 andV ₁, V₂ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)–A4) below, but without imposing any restrictions on the edges±μ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen. This work was initiated in June of 2004, while the first author visited Caltech, and he wishes to thank that institution for its hospitality and support. The first author was partially supported by the NSF grant DMS-0303413. The second author was partially supported by a Sloan fellowship and the NSF grant DMS-0300081. The authors thank Avy Soffer for his interest in this work.     

Variance or spectral density in sampled data filtering?

Most physical systems operate in continuous time. However, to interact with such systems one needs to take samples. This raises the question of the relationship between the sampled response and the response of the underlying continuous-time system. In this paper we review several aspects of the sampling process. In particular, we examine the role played by variance and spectral density in describing discrete random processes. We argue that spectral density has several advantages over variance. We illustrate the ideas by reference to the problem of state estimation using the discrete-time Kalman filter.     

Super efficiencies or super inefficiencies? Insights from a joint computation model for slacks-based measures in DEA

The slacks-based measure (SBM) can incorporate input and output slacks that would otherwise be neglected in the classical DEA model. In parallel, the super-efficiency model for SBM (S-SBM) has been developed for the purpose of ranking SBM efficient decision-making units (DMUs). When implementing SBM in conjunction with S-SBM, however, several issues can arise. First, unlike the standard super-efficiency model, S-SBM can only solve for super-efficiency scores but not SBM scores. Second, the S-SBM model may result in weakly efficient reference points. Third, the S-SBM and SBM scores for certain DMUs may be discontinuous with a perturbation to their inputs and outputs, making it hard to interpret and justify the scores in applications and the efficiency scores may be sensitive to small changes/errors in data. Due to this discontinuity, the S-SBM model may overestimate the super-efficiency score. This paper extends the existing SBM approaches and develops a joint model (J-SBM) that addresses the above issues; namely, the J-SBM model can (1) simultaneously compute SBM scores for inefficient DMUs and super-efficiency for efficient DMUs, (2) guarantee the reference points generated by the joint model are Pareto-efficient, and (3) the J-SBM scores of a firm are continuous in the input and output space. Interestingly, the radial DEA efficiency and super-efficiency scores for a DMU are continuous in the input–output space. The J-SBM model combines the merits of the radial and SBM models (i.e., continuity and Pareto-efficiency).     

The method of p-harmonic approximation and optimal interior partial regularity for energy minimizing p-harmonic maps under the controllable growth condition

In this paper, we are concerned with the partial regularity for the weak solutions of energy minimizing p-harmonic maps under the controllable growth condition. We get the interior partial regularity by the p-harmonic approximation method together with the technique used to get the decay estimation on some Degenerate elliptic equations and the obstacle problem by Tan and Yan. In particular, we directly get the optimal regularity.     

Laws of Large Numbers for Cesàro alpha-integrable Random Variables under Dependence Condition AANA or AQSI

Both residual Cesàro alpha-integrability (RCI( α)) and strongly residual Cesàro alpha-integrability (SRCI(α)) are two special kinds of extensions to uniform integrability, and both asymp-totically almost negative association (AANA) and asymptotically quadrant sub-independence (AQSI) are two special kinds of dependence structures. By relating the RCI(α) property as well as the SRCI(α) property with dependence condition AANA or AQSI, we formulate some tail-integrability conditions under which for appropriate α the RCI(α) property yields L1-convergence results and the SRCI(α) property yields strong laws of large numbers, which is the continuation of the corresponding literature.     

Global well-posedness in the energy space for the Benjamin–Ono equation on the circle

We prove that the Benjamin-Ono equation is locally well-posed in . This leads to a global well-posedeness result in thanks to the energy conservation.     

Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains

We study the generalized Hénon equation in reflectionally symmetric or point symmetric domains and prove that a least energy solution is neither reflectionally symmetric nor even. Moreover, we prove the existence of a positive solution with prescribed exact symmetry.