Extended Real-Valued Functions

This paper is concerned with the properties of the sets of strategic measures induced by admissible team policies in decentralized stochastic control and the convexity properties in dynamic team problems. To facilitate a convex analytical approach, strategic measures for team problems are introduced. Properties such as convexity, compactness, and Borel measurability under weak convergence topology are studied, and sufficient conditions for each of these properties are presented. These lead to existence of and structural results for optimal policies. It will be shown that the set of strategic measures for teams which are not classical is in general nonconvex, but the extreme points of a relaxed set consist of deterministic team policies, which lead to their optimality for a given team problem under an expected cost criterion. Externally provided independent common randomness for static teams or private randomness for dynamic teams do not improve the team performance. The problem of when a sequential team p...

Convexity properties are vital in the classical calculus of variations, and many notions of convex analysis, such as the Legendre-Fenchel transform, have their origin there. Conceptual developments in contemporary convex analysis have in turn enriched that venerable subject by making it possible to treat a vastly larger class of problems effectively in a “neoclassical” framework of extended-real-valued functions and their subgradients.

New sequential exact Euclidean distance transform algorithms based on convex analysis

The purpose of this section is to remind the reader essential notions about convex sets and functions which will be useful in the sequel. For a more detailed exposition on convexity, the reader is referred to specialized literature (Rockafellar, Convex analysis, Princeton University Press, Princeton, 1970; Rockafellar and Wets, Variational analysis, Springer, New York, 1998; Boyd and Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, 2004).

Fixed-Point Algorithms for Inverse Problems in Science and Engineering presents some ofthe most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems. This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry. Topics presented include: Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory. Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods. Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas. Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

Following an idea suggested by Rockafellar in his classical book on Convex Analysis, the first author defined in his doctoral thesis the enlarged space, by adjoining to the ordinary n-dimensional space “improper points” defined by directions, and developed a theory of convexity in such a space. Later the second author introduced a new model for the enlarged space, suitable for applying tools from the theory of cones, and closely related to the model that Rockafellar & Wets called “ray model” of the “cosmic closure” of the Euclidean space in their recent book on Variational Analysis. In this paper we introduce that model and use it to get some new theorems and proofs.2000 Mathematics Subject ClassificationPrimary 52A01Secondary 52A20Keywords and phrasesGeneralized ConvexityUnbounded Convex SetsCosmic ClosureEnlarged Space

Potato proteins for technical applications: Nutrition, isolation, modification and functional properties - A review

We analyze a class of norms defined via an optimal interpolation problem involving the composition of norms and a linear operator. This construction, known as infimal postcomposition in convex analysis, is shown to encompass various norms which have been used as regularizers in machine learning, signal processing, and statistics. In particular, these include the latent group lasso, the overlapping group lasso, and certain norms used for learning tensors. We establish basic properties of this class of norms and we provide dual norms. The extension to more general classes of convex functions is also discussed. A stochastic block-coordinate version of the Douglas-Rachford algorithm is devised to solve minimization problems involving these regularizers. A prominent feature of the algorithm is that it yields iterates that converge to a solution in the case of nonsmooth losses and random block updates. Finally, we present numerical experiments with problems employing the latent group lasso penalty.

This article presents new results that extend the scope of the Young integral inequality. They are formulated by different functional composition schemes of the integrals involved. These schemes depend on a unique auxiliary function. In total, six theorems are proved. Four of them are derived from a simple composition scheme based on sub-additive, super-additive, convex and concave properties of the auxiliary function, while the remaining two use more sophisticated composition schemes based on convex and concave properties. Our framework also unifies several established variants of the Young integral inequality. Three secondary propositions complete the theorems. The theory is supported by a number of tractable examples involving power, exponential, logarithmic, ratio and hyperbolic functions.

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes.

Convexity properties axe often useful in characterising and finding bounds for special function and their zeros, as well as in questions concerning the existence and uniqueness of zeros in certain intervals. In this survey paper, we describe some work related to the gamma function, the q-gamma function, Bessel and cylinder functions and the Hermite function.

Although there is general agreement that the human middle temporal (MT)/V5+ complex corresponds to monkey area MT/V5 proper plus a number of neighboring motion-sensitive areas, the identification of human MT/V5 within the complex has proven difficult. Here, we have used functional magnetic resonance imaging and the retinotopic mapping technique, which has very recently disclosed the organization of the visual field maps within the monkey MT/V5 cluster. We observed a retinotopic organization in humans very similar to that documented in monkeys: an MT/V5 cluster that includes areas MT/V5, pMSTv (putative ventral part of the medial superior temporal area), pFST (putative fundus of the superior temporal area), and pV4t (putative V4 transitional zone), and neighbors a more ventral putative human posterior inferior temporal area (phPIT) cluster. The four areas in the MT/V5 cluster and the two areas in the phPIT cluster each represent the complete contralateral hemifield. The complete MT/V5 cluster comprises 70% of the motion localizer activation. Human MT/V5 is located in the region bound by lateral, anterior, and inferior occipital sulci and occupies only one-fifth of the motion complex. It shares the basic functional properties of its monkey homolog: receptive field size relative to other areas, response to moving and static stimuli, as well as sensitivity to three-dimensional structure from motion. Functional properties sharply distinguish the MT/V5 cluster from its immediate neighbors in the phPIT cluster and the LO (lateral occipital) regions. Together with similarities in retinotopic organization and topological neighborhood, the functional properties suggest that MT/V5 in human and macaque cortex are homologous.

In the study of property testing, a particularly important role has been played by linear invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed‐Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear‐invariant properties, drawing on techniques from additive combinatorics and from graph theory.Our main contributions here are the following: We introduce a simple combinatorial condition, which we call subspace‐heredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with one‐sided error, thus addressing a challenge posed by Sudan recently. We introduce a new technique for proving the testability of Boolean functions. The main tool we develop is an extension of Green's arithmetic regularity lemma. Using it, we verify a special case of the conjecture. Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.© 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 232–260, 2015

Effect of high pressure homogenization combined with juice ratio on water-soluble pectin characteristics, functional properties and bioactive compounds in mixed juices