The Solution of a Free Boundary Problem Related to Environmental Management Systems

Abstract

Explicit solutions of free boundary problems are notoriously difficult to find. In this article, we consider two log-normal diffusions. One represents the level of pollution, or degradation, in some environmental area. The second models the social, political, or financial cost of the pollution. A single control parameter is considered that reduces the rate of pollution. The optimal time to implement the change in the parameter is found by explicitly solving a free boundary problem. The novelty is that the smooth pasting conditions, which are difficult to justify, are not used in the derivation.     

Portfolio Optimization Model with Transaction Costs

Abstract The purpose of the article is to formulate,under the l_∞ risk measure,a model of portfolio selectionwith transaction costs and then investigate the optimal strategy within the proposed.The characterization of aoptimal strategy and the efficient algorithm for finding the optimal strategy are given.     

Image Restoration Using Modifications of Simulated Annealing

Abstract

This article uses a modified version of the simulated annealing algorithm to restore degraded spatial patterns. Standard simulated annealing is used to find an image that is a posterior mode when the number of images under consideration precludes sequential search for a maximum. I incorporate jumping probabilities of the annealing algorithm without randomization. The convergence of our algorithm is proven under a practical annealing schedule. The same idea is also implemented to improve the performance of other modifications of simulated annealing. These include forcing proportions of labels in an image, using posterior marginals, and incorporating an edge process. This article also studies nonlinear presmoothing of the observations.     

Minimal pairs in initial segments of the recursively enumerable degrees

We show that for every r.e. Turing degreea>0, there is an r.e. degreeb<a which is not half of a minimal pair in the initial segment [0, a]. This work was partially supported by National Science Foundation Grant DMS88-00030 to Stob, the New Zealand Marsden Fund for Basic Research in Science Grant VIC-509 to Downey, and New Zealand-United States Cooperative Science Program Grant INT90-20558 from the National Science Foundation to both authors. The authors thank the referee for several helpful comments.     

Errata for “The linear programming bound for codes over finite Frobenius rings”

The authors discovered some mistakes in the article that appeared on pp. 289–301 of the March volume of DCC 42 (2007). The validity of its results is not affected, nor is that of the examples since their computation did not involve the dual version of the LP bound. In any case, we feel the errors need to be rectified here. The online version of the original article can be found under doi:.     

A new way to evaluate the Probability and Fresnel integrals

In this article, we show how Laplace Transform may be used to evaluate variety of nontrivial improper integrals, including Probability and Fresnel integrals. The algorithm we have developed here to evaluate Probability, Fresnel and other similar integrals seems to be new. This method transforms the evaluation of certain improper integrals into evaluation of improper integrals of the corresponding Laplace transform, which in many cases are much easier.     

Cobordism,Relative Indices and Stein Fillings

In this article we build on the framework developed in Ann. Math. 166, 183–214 ([2007]), 166, 723–777 ([2007]), 167, 1–67 ([2008]) to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the Spin _ℂ-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the Spin _ℂ-Dirac operator, and gluing results for the indices of these boundary problems to Spin _ℂ-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds. This material is based upon work supported by the National Science Foundation under Grant No. 0603973, and the Francis J. Carey term chair. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.     

On Stochastic Stabilization of Discrete‐Time Markovian Jump Systems with Delay in State

Abstract

In this paper, we investigate the stochastic stabilization problem for a class of linear discrete time‐delay systems with Markovian jump parameters. The jump parameters considered here is modeled by a discrete‐time Markov chain. Our attention is focused on the design of linear state feedback memoryless controller such that stochastic stability of the resulting closed‐loop system is guaranteed when the system under consideration is either with or without parameter uncertainties. Sufficient conditions are proposed to solve the above problems, which are in terms of a set of solutions of coupled matrix inequalities.     

Constrained Nonparametric Maximum Likelihood via Mixtures

Abstract

This article discusses a new technique for calculating maximum likelihood estimators (MLEs) of probability measures when it is assumed the measures are constrained to a compact, convex set. Measures in such sets can be represented as mixtures of simple, known extreme measures, and so the problem of maximizing the likelihood in the constrained measures becomes one of maximizing in an unconstrained mixing measure. Such convex constraints arise in many modeling situations, such as empirical likelihood and estimation under stochastic ordering constraints. This article describes the mixture representation technique for these two situations and presents a data analysis of an experiment in cancer genetics, where a partial stochastic ordering is assumed but the data are incomplete.     

Factoring wavelet transforms into lifting steps

This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z⁻¹])=E(n;R[z, z⁻¹])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. Research Tutorial Acknowledgements and Notes. Page 264.     

Some basic properties of packing and covering constants

This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collectionK of convex bodies. There are two possibilities to construct such packings and coverings: One may permit that the convex bodies fromK are used repeatedly, or one may require that these bodies should be used at most once. In each case one can define the packing and covering constants ofK as, respectively, the least upper bound and the greatest lower bound of the densities of all such packings and coverings. Three theorems are proved. First it is shown that there exist always packings and coverings whose densities are equal to the corresponding packing and covering constants. Then, a quantitative continuity theorem is proved which shows in particular that the packing and covering constants depend, in a certain sense, continuously onK. Finally, a kind of a transference theorem is proved, which enables one to evaluate the packing and covering constants when no repetitions are allowed from the case when repetitions are permitted. Furthermore, various consequences of these theorems are discussed.Supported by National Science Foundation Research Grant DMS 8300825.     

Markov Decision Processes with Variance Minimization: A New Condition and Approach

Abstract

This article deals with the limiting average variance criterion for discrete-time Markov decision processes in Borel spaces. The costs may have neither upper nor lower bounds. We propose another set of conditions under which we prove the existence of a variance minimal policy in the class of average expected cost optimal stationary policies. Our conditions are weaker than those in the previous literature. Moreover, some sufficient conditions for the existence of a variance minimal policy are imposed on the primitive data of the model. In particular, the stochastic monotonicity condition in this paper has been first used to study the limiting average variance criterion. Also, the optimality inequality approach provided here is different from the “optimality equation approach” widely used in the previous literature. Finally, we use a controlled queueing system to illustrate our results.     

ON THE SEPARATION PROPERTY OF SYMMETRIC ORDINARY SECOND-ORDER DIFFERENTIAL EXPRESSIONS

A 1-variable calculus type argument is used to show that, for a function f : R ² → R, if for all (a, b) ? R ² we have that f o c is smooth for every smooth curve c : R → R ² nonsingular except at 0 and with c(0) = (a, b), then f is smooth. This strengthens Boman's theorem. In fact, we use an even more special collection of smooth curves to prove Boman's theorem. It is shown using a related special collection of smooth curves how the upper half cone can be viewed largely as a model for polar coordinates. Our proof here shows how the use of Frölicher spaces can reduce questions in several dimensions to those of one real variable.     

The Graphical Horseshoe Estimator for Inverse Covariance Matrices

We develop a new estimator of the inverse covariance matrix for high-dimensional multivariate normal data using the horseshoe prior. The proposed graphical horseshoe estimator has attractive properties compared to other popular estimators, such as the graphical lasso and the graphical smoothly clipped absolute deviation. The most prominent benefit is that when the true inverse covariance matrix is sparse, the graphical horseshoe provides estimates with small information divergence from the sampling model. The posterior mean under the graphical horseshoe prior can also be almost unbiased under certain conditions. In addition to these theoretical results, we also provide a full Gibbs sampler for implementing our estimator. MATLAB code is available for download from github at http://github.com/liyf1988/GHS. The graphical horseshoe estimator compares favorably to existing techniques in simulations and in a human gene network data analysis. Supplementary materials for this article are available online.     

Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces

We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.      

Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

For longitudinal data, the modeling of a correlation matrix ?R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (?1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of ?R or ?R^{? 1} for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online.     

Exploiting structure in piecewise-linear homotopy algorithms for solving equations

We consider the recent algorithms for computing fixed points or zeros of continuous functions fromR ⁿ to itself that are based on tracing piecewise-linear paths in triangulations. We investigate the possible savings that arise when these fixed-point algorithms with their usual triangulations are applied to computing zeros of functionsf with special structure:f is either piecewise-linear in certain variables, separable, or has Jacobian with small bandwidth. Each of these structures leads to a property we call modularity; the algorithmic path within a simplex can be continued into an adjacent simplex without a function evaluation or linear programming pivot. Modularity also arises without any special structure onf from the linearity of the function that is deformed tof. In the case thatf is separable we show that the path generated by Kojima's algorithm with the homotopyH ² coincides with the path generated by the standard restart algorithm of Merrill when the usual triangulations are employed. The extra function evaluations and linear programming steps required by the standard algorithm can be avoided by exploiting modularity.This research was performed while the author was visiting the Mathematics Research Center, University of Wisconsin-Madison, and was sponsored by the United States Army under Contract No. DAAG-29-75-C-0024 and by the National Science Foundation under Grant No. ENG76-08749.     

Smooth local sinusoidal bases on two-dimensional L-shaped regions

In this article, we construct two-dimensional continuous/smooth local sinusoidal bases (also called Malvar wavelets) defined onL-shaped regions. With this construction, one is able to construct local sinusoidal bases and lapped orthogonal transforms (LOT) on arbitrarily shaped regions. This work is motivated from and useful in object-based video coding, where a segmented moving object may have arbitrary shape and block transform coding of this object is needed. Acknowledgements and Notes. His work was partially supported by an initiative grant from the Department of Electrical and Computer Engineering, University of Delaware, the Air Force Office of Scientific Research (AFOSR) under Grant No. F49620-97-1-0253, and the National Science Foundation CAREER Program under Grant MIP-9703377.     

Solvability of Linear and Quasilinear Elliptic Boundary Value Problems via the A-Proper Mapping Theory

ABSTRACT

In this paper we use the theory of A-proper mappings and their uniform limits to obtain in a simple way general variational approximation-solvability and/or existence theorems for quasilinear elliptic BVProblems in divergence form of order 2m. In Section 1 we first show how our simple approach is used to obtain constructive solvability of linear elliptic equations under a somewhat different definition of strong ellipticity condition which allows us to dispense with the usage of the Garding inequality. In view of this, the same approach is also used in Section 2 to establish the constructive solvability of nonlinear BVProblems when the leading part satisfies a nonlinear version of the strong ellipticity condition used here. This result is then used to establish a new existence result (see Theorem 2.3) for a not necessarily coercive quasilinear elliptic BVProblem which properly includes earlier results of Vi?ik, Browder, Poho?ayev, Leray and Lions and others. In Section 3 some special cases of our abstract results are stated. These are applicable to ODE's and PDE's which are not in divergence form.     

Spline Estimation of Discontinuous Regression Functions

Abstract

This article deals with regression function estimation when the regression function is smooth at all but a finite number of points. An important question is: How can one produce discontinuous output without knowledge of the location of discontinuity points? Unlike most commonly used smoothers that tend to blur discontinuity in the data, we need to find a smoother that can detect such discontinuity. In this article, linear splines are used to estimate discontinuous regression functions. A procedure of knot-merging is introduced for the estimation of regression functions near discontinuous points. The basic idea is to use multiple knots for spline estimates. We use an automatic procedure involving the least squares method, stepwise knot addition, stepwise basis deletion, knot-merging, and the Bayes information criterion to select the final model. The proposed method can produce discontinuous outputs. Numerical examples using both simulated and real data are given to illustrate the performance of the proposed method.