Weighted Hardy inequality with higher dimensional singularity on the boundary

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R ^N\) with \(N\ge 3\) and let \(\Sigma _k\) be a closed smooth submanifold of \(\partial \Omega \) of dimension \(1\le k\le N-2\) . In this paper we study the weighted Hardy inequality with weight function singular on \(\Sigma _k\) . In particular we provide necessary and sufficient conditions for existence of minimizers.     

Wave Kernels with Bi-Inverse Square Potentials on Euclidean Plane

The Cauchy problemfor the wave equation with bi-inverse square potential on Euclidean plane is solved in terms of the two variables Appell F₂ hypergeometric functions. Our principal tools are the Hankel transforms and the special functions of mathematical physics.     

Some local eigenvalue estimates involving curvatures

We first establish a local Faber–Krahn isoperimetric comparison in terms of scalar curvature pinching. Secondly we derive estimates of Cheeger constants related to the Dirichlet and Neumann problems via the (relative) isoperimetric profiles which allow us to obtain, in particular, lower bounds for first non-zero eigenvalues of the problem of Dirichlet and Neumann. These estimates involve scalar curvature and mean curvature respectively.     

Graph operations characterizing rank-width

Graph complexity measures such as tree-width, clique-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. These algorithms are based on hierarchical decompositions of the considered graphs, and on boundedness conditions on the graph operations that, more or less explicitly, recombine the components of decompositions into larger graphs. Rank-width is defined in a combinatorial way, based on a tree, and not in terms of graph operations. We define operations on graphs that characterize rank-width and help for the construction of Fixed Parameter Tractable algorithms, especially for problems specified in monadic second-order logic.     

Successive Linear Approximation Procedure for Generalized Fixed-Charge Transportation Problems

In this paper, we develop a new heuristic procedure for solving a generalization of the fixed-charge transportation problem in which there are resource losses in addition to the fixed charges. The losses may be evaporation losses when the commodity is a liquid, heat losses in an electrical distribution network, or deterioration losses in distribution networks involving perishable commodities such as, for example, food items. The proposed procedure consists of solving a sequence of pro-rated problems. It is different from heuristic procedures that have been developed for solving the standard fixed charge transportation problem, in that it is not based on extreme point enumeration. We experiment with problems involving up to 2100 arcs with fixed charges and resource losses. The results show that the proposed approach is viable for solving medium-to-large sized problems.     

Fast and simple online sample preparation coupled with capillary LC-MS/MS for determination of prostaglandins in cell culture supernatants

An online 2-D strong cation exchange (SCX)-RP capillary liquid chromatographic (cLC) method with IT mass spectrometric (IT-MS/MS) detection for the simultaneous determination of prostaglandin (PG) A(1), PGD(2), PGE(1), PGE(2), PGF(2a), 6-keto-(6k)PGF(1a), and 15-Delta(12,14)-deoxy-PGJ(2) (15dPGJ(2)) in cell culture supernatants was developed and validated. Pretreatment of the cell culture supernatants included dilution and filtration, and the analysis time including all sample preparation steps was less than 50 min per sample. Peptides/proteins contained in the matrix were removed by the SCX column. LODs in the range of 0.4-2.2 ng/mL cell culture supernatant, recoveries higher than 80% and within- and between-day precisions of less than 30% RSDs were obtained. Human mesenchymal stem cells (hMSCs) were stimulated with cytokine-containing supernatants derived from activated T lymphocytes, and PG production was analyzed using the developed method. PGE(2 )was found in cultures from both untreated and stimulated hMSCs, while PGE(1) was present above the detection limit only in stimulated cells.