Categorifying the concept of topological group, one obtains the notion of a ‘topological 2-group’. This in turn allows a theory of ‘principal 2-bundles’ generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M , principal G-bundles over M are classified by either the Čech cohomology Ȟ(M, G) or the set of homotopy clas...

In this paper, we have dened and studied a generalized form of topological vector spaces called s-topological vector spaces. s-topological vector spaces are dened by using semi-open sets and semi-continuity in the sense of Levine. Along with other results, it is proved that every s-topological vector space is generalized homogeneous space. Every open subspace of an s-topological vector space is...

the notion of generalized locally bounded $i$-topological vectorspaces is introduced. some of their important properties arestudied. the relationship between this kind of spaces and thelocally bounded $i$-topological vector spaces introduced by wu andfang [boundedness and locally bounded fuzzy topological vectorspaces, fuzzy math. 5 (4) (1985) 87$-$94] is discussed. moreover, wealso use the fam...

Introduction 2 1. Symmetric spectra 5 1.1. Simplicial sets 5 1.2. Symmetric spectra 6 1.3. Simplicial structure on Sp 8 1.4. Symmetric Ω-spectra 10 2. The smash product of symmetric spectra 11 2.1. Symmetric sequences 11 2.2. Symmetric spectra 14 2.3. The ordinary category of spectra 18 3. Stable homotopy theory of symmetric spectra 19 3.1. Stable equivalence 20 3.2. Model categories 26 3.3. Le...

The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A striking amount of work on pseudoisotopy theory [Hatcher, Waldhausen, Cohen-Carlsson-Goodwillie-Hsiang-Madsen . . . ] can be formulated as a T...

Studying protein interaction networks using functional and topological information is important for understanding cellular organization and functionality. This study deals with identifying important proteins in protein interaction networks using SWEMODE (Lubovac, et al, 2006) and analyzing topological and functional properties of these proteins with the help of information derived from modular ...

Building on the topological foundations constructed in Part I, we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field. In the present work, we develop two algebraic frameworks corresponding to two interpretations of Kubota’s n-Hilbert reciprocity formalism, presented in a...

This paper proposes a generation method of a subject-specific Facial Expression Map (FEMap) using the Self-Organizing Maps (SOM) of unsupervised learning and Counter Propagation Networks (CPN) of supervised learning together. The proposed method consists of two steps. In the first step, the topological change of a face pattern in the expressional process of facial expression is learned hierarch...

iv Contents Preface vii Chapter 1. Model categories 1 1.1. The definition of a model category 2 1.2. The homotopy category 7 1.3. Quillen functors and derived functors 13 1.3.1. Quillen functors 13 1.3.2. Derived functors and naturality 16 1.3.3. Quillen equivalences 19 1.4. 2-categories and pseudo-2-functors 22 Chapter 2. Examples 27 2.1. Cofibrantly generated model categories 28 2.1.1. Ordina...

The influence of cation-π interactions on the strength and nature of intramolecular O...H hydrogen bond has been investigated by quantum chemical calculations in orthohydroxy benzaldehyde (HBA) compound. Ab initio calculations have been performed at MP2/6-311++G** level of theory. Vibrational frequencies and physical properties such as chemical potential and chemical hardness of these compounds...