Concretely Locally Presentable and Locally Generated Categories

We give concrete versions of the characterizations of locally -presentable (resp. -generated) categories as categories of models of (resp. weakly) -ary limit-theories and of (resp. weakly) -ary essentially algebraic theories. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms. -presentable and -generated objects are also investigated from this viewpoint.     

Universally meager sets

We study category counterparts of the notion of a universal measure zero set of reals.

We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.

In particular, two problems emerging from an earlier work of Grzegorek are solved.

     

Some Generic Results on Nonattaining Functionals

We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the nonsupport functionals are a dense G subset of the polar set, and (b) any nonsemicoercive proper convex lsc [weak^*-lsc] function in a [dual] Banach space has a generic [dense G ] set of L -perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining L -perturbations. This results also in a criterion for reflexivity.     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Derived Lengths of Solvable Groups Having Five Irreducible Character Degrees I

Let G be a solvable group with five character degrees. We show that the derived length of G is at most 5. This verifies that the Taketa inequality, dl(G)|cd(G)|, is valid for solvable groups with at most five character degrees.     

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik-Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.     

Track Extensions of Categories and Cohomology

We introduce the notion of n-fold track extensions of a category C by a natural system D and prove that such extensions represent classes in the cohomology of C with coefficients in D introduced by Baues–Wirsching. This generalizes a result of Huebschmann on the cohomology of groups.     

Analysis of ordered categorical data from repeated measurements assuming a quantitative latent variable

Summary The purpose of the present paper is to propose an analytical method for ordered categorical responses obtained from a repeated measurement/longitudinal experiment. The ordered categorical scale is assumed to be a manifestation of a latent quantitative variable. A linear model is assumed for location parameters of the underlying distributions. Weighted least square method is applied to parameter estimation and subsequent analysis. Two data sets are analyzed to show several aspects of analysis by the proposed model and to discuss comparative characteristics of analysis compared with earlier analysis. A mention is made for a computer software program for the proposed model.     

Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.     

A(∞)TH上的辫子Monoidal范畴

Let A and H be Hopf algebra,T-smash product A (∞)T H generalizes twisted smash product A*H.This paper shows a necessary and sufficient condition for T-smash product module category A(∞)T H M to be braided monoidal category.