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Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.

Given an uncountable, compact metric space X , we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on X .

The book description for \"Foundations of Algebraic Topology\" is currently unavailable.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.

Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a \"desingularization,\" improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

There are more than 50 matrix classes discussed in the literature of the Linear Complementarity Problem. This guide is offered as a compendium of notations, definitions, names, source information, and commentary on these many matrix classes. Also included are discussions of certain properties possessed by some (but not all) of the matrix classes considered in this guide. These properties—fullness, completeness, reflectiveness, and sign-change invariance—are the subject of another table featuring matrix classes that have one or more of them. Still another feature of this work is a matrix class inclusion map depicting relationships among the matrix classes listed herein.

This paper deals with the generalized vector quasivariational inclusion Problem (P 1 ) (resp. Problem (P 2 )) of finding a point ( z 0 , x 0 ) of a set E × K such that ( z 0 , x 0 )∈ B ( z 0 , x 0 )× A ( z 0 , x 0 ) and, for all η ∈ A ( z 0 , x 0 ), where A : E × K →2 K , B : E × K →2 E , C : E × K →2 Y , F , G : E × K × K →2 Y are some set-valued maps and Y is a topological vector space. The nonemptiness and compactness of the solution sets of Problems (P 1 ) and (P 2 ) are established under the verifiable assumption that the graph of the moving cone C is closed and that the set-valued maps F and G are C -semicontinuous in a new sense (weaker than the usual sense of semicontinuity).

This paper studies the existence of mild solutions and the compactness of a set of mild solutions to a nonlocal problem of fractional evolution inclusions of order α ∈ ( 1 , 2 ) . The main tools of our study include the concepts of fractional calculus, multivalued analysis, the cosine family, method of measure of noncompactness, and fixed-point theorem. As an application, we apply the obtained results to a control problem.

By providing a new iterative method our aim is finding a common element of the set of fixed points of two nonexpansive mappings, the set of solutions to a variational inclusion and the set of solutions of a generalized equilibrium problem in a real Hilbert space. We review the strong convergence of the new iterative method in the framework of Hilbert spaces. Finally, we show that our main result is a generalization for some known theorems in this field.

Microtubule-associated protein/microtubule affinity-regulating kinase 4 (MARK4) is a member of the family Ser/Thr kinase and involved in numerous biological functions including microtubule bundle formation, nervous system development, positive regulation of programmed cell death, cell cycle control, cell polarity determination, cell shape alterations, cell division etc. For various biophysical and structural studies, we need this protein in adequate quantity. In this paper, we report a novel cloning strategy for MARK4. We have cloned MARK4 catalytic domain including 59 N-terminal extra residues with unknown function and catalytic domain alone in PQE30 vector. The recombinant MARK4 was expressed in the inclusion bodies in M15 cells. The inclusion bodies were solubilized effectively with 1.5 % N -lauroylsarcosine in alkaline buffer and subsequently purified using Ni–NTA affinity chromatography in a single step with high purity and good concentration. Purity of protein was checked on sodium dodecyl sulphate–polyacrylamide gel electrophoresis and identified by using mass spectrometry immunoblotting. Refolding of the recombinant protein was validated by ATPase assay. Our purification procedure is quick, simple and produces adequate quantity of proteins with high purity in a limited step.