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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. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

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. InHigher 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. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

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).

Background Finding sensitive clinical outcome measures has become crucial in natural history studies and therapeutic trials of neuromuscular disorders. Here, we focus on 1‐year longitudinal data from quantitative magnetic resonance imaging (MRI) and phosphorus magnetic resonance spectroscopy (31P MRS) in a placebo‐controlled study of sirolimus for inclusion body myositis (IBM), also examining their links to functional, strength, and clinical parameters in lower limb muscles. Methods Quantitative MRI and 31P MRS data were collected at 3 T from a single site, involving 44 patients (22 on placebo, 22 on sirolimus) at baseline and year‐1, and 21 healthy controls. Assessments included fat fraction (FF), contractile cross‐sectional area (cCSA), and water T2 in global leg and thigh segments, muscle groups, individual muscles, as well as 31P MRS indices in quadriceps or triceps surae. Analyses covered patient‐control comparisons, annual change assessments via standard t‐tests and linear mixed models, calculation of standardized response means (SRM), and exploration of correlations between MRI, 31P MRS, functional, strength, and clinical parameters. Results The quadriceps and gastrocnemius medialis muscles had the highest FF values, displaying notable heterogeneity and asymmetry, particularly in the quadriceps. In the placebo group, the median 1‐year FF increase in the quadriceps was 3.2% (P < 0.001), whereas in the sirolimus group, it was 0.7% (P = 0.033). Both groups experienced a significant decrease in cCSA in the quadriceps after 1 year (P < 0.001), with median changes of 12.6% for the placebo group and 5.5% for the sirolimus group. Differences in FF and cCSA changes between the two groups were significant (P < 0.001). SRM values for FF and cCSA were 1.3 and 1.4 in the placebo group and 0.5 and 0.8 in the sirolimus group, respectively. Water T2 values were highest in the quadriceps muscles of both groups, significantly exceeding control values in both groups (P < 0.001) and were higher in the placebo group than in the sirolimus group. After treatment, water T2 increased significantly only in the sirolimus group's quadriceps (P < 0.01). Multiple 31P MRS indices were abnormal in patients compared to controls and remained unchanged after treatment. Significant correlations were identified between baseline water T2 and FF at baseline and the change in FF (P < 0.001). Additionally, significant correlations were observed between FF, cCSA, water T2, and functional and strength outcome measures. Conclusions This study has demonstrated that quantitative MRI/31P MRS can discern measurable differences between placebo and sirolimus‐treated IBM patients, offering promise for future therapeutic trials in idiopathic inflammatory myopathies such as IBM.

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.

Growing work highlights the importance of diversity, equity, and inclusion (DEI) initiatives in graduate medical education. Few studies have employed participatory research approaches to solicit trainee perspectives related to DEI. Our goal was to utilize group concept mapping (GCM), a mixed-methods participatory research approach, to describe resident perspectives on DEI within a large pediatric residency program. To organize and represent their perspectives on DEI, trainees completed brainstorming, sorting/rating, and interpretation activities in accordance with GCM methodology. Activities occurred via two synchronous discussion sessions (brainstorming and interpretation) and an electronic survey (sorting/rating). Items from the brainstorming session were sorted (i.e. grouped into categories by similarity) and rated (i.e. ranked on perceived importance and likelihood to change) by participants individually. Multidimensional scaling and hierarchical clustering were used to generate point maps, cluster maps, and go-zone illustrations for use in the interpretation session. We present data regarding participant characteristics and engagement, results from GCM activities, and action steps from this process. There were a total of 127 trainees in the residency program in 2021-2022. Participation varied across activities (brainstorming session: 21 participants; sorting/rating: 48 participants; interpretation session: 20 participants). A total of 64 unique items were generated from brainstorming. Five clusters emerged: 1) city factors, 2) institutional factors, 3) program representation, 4) program components, 5) non-DEI program perks. Program representation and program components clusters were rated as the most important and the most likely to be changed. Participants identified several action steps during the interpretation session which were shared with institutional leadership to direct programmatic reform. We demonstrate the utility of GCM, a structured and scalable participatory research method, to characterize trainee perceptions of DEI in graduate medical education.