We prove existence, regularity and nonexistence results for problems whose model is -Lambda u = f(x)/u gamma in Omega, with zero Dirichlet conditions on the boundary of an open, bounded subset Omega of R(N). Here gamma > 0 and f is a nonnegative function on Omega. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of gamma (which can be equal, larger or smaller than 1)

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In this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model iswith homogeneous Cauchy Dirichlet boundary conditions, where p>1 and + is a bounded measure on 0_(0, T ). We also study how the summability of the gradient improves if the measure + is a function in L m (0_(0, T )), with m``small.'' Moreover we give a new proof of the existence of a solution for problem (P). Academic Press

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We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with Singular lower order terms that have natural growth with respect to the gradient, whose model is {-Delta u + vertical bar del u vertical bar(2)/u(gamma) = f in Omega. u = 0 on partial derivative Omega. where Omega is an open bounded subset of R, gamma > 0 and f is a function which is strictly positive on every compactly contained subset of Omega. As a consequence of our main results, we prove that the condition gamma < 2 is necessary and sufficient for the existence of solutions in H(0)(1) (Omega) for every sufficiently regular f as above. (C) 2009 Elsevier Inc. All rights reserved

Abstract. We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q, t)-analogue of the Catalan number Cn. These (q, t)-Catalan numbers count, on the one hand, ad-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

Anatomical morphometric studies of the coracoid process and coraco-glenoid space were carried out on 204 dry scapulae. No statistically significant correlations were found between length, or thickness of the coracoid process, prominence of the coracoid tip, coracoid slope, coraco-glenoid distance, or position of the coracoid tip with respect to the uppermost point of the glenoid. These anatomical characteristics were independent of the dimensions of the scapulae. Three configurations of the coraco-glenoid space were identified. Type I configuration was found in 45% of scapulae and Type II and Type III, in 34% and 21% of specimens, respectively. The lowest value of the coraco-glenoid distance were seen in Type I scapulae. Morphometric characteristics which might predispose to subcoracoid impingement were found in 4% of Type I scapulae. A total of 27 scapulae, nine with each type of configuration were submitted to CT scanning. Scapulae with a Type I configuration were found to have low values for the coraco-glenoid angle and coracoid overlap, which are known to be associated with a short coraco-humeral distance. Subjects with a Type I configuration, and severe narrowing of the coraco-glenoid space, appear to be predisposed to coraco-humeral impingement. These morphometric characteristics may be easily evaluated on CT scans.