A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Cyphostemmins A-B, two new antifungal oligostilbenes from Cyphostemma crotalarioides (Vitaceae)

Two new antifungal resveratrol dimers, cyphostemmins A-B (1–2), have isolated from the roots of Cyphostemma crotalarioides planch (Vitaceae) together with resveratrol 3 and previously known reseveratrol dimers (4–7). Structures of these new compounds have been established on the basis of their MS and ¹H and ¹³C NMR spectroscopic data.     

1,3,4-Oxadiazole, 1,3,4-thiadiazole and 1,2,4-triazole derivatives as potential antibacterial agents

Since the introduction of the first antibiotic (penicillin, 1942) into medical practice, to date, there has been an ongoing “race” between scientists creating new drugs and pathogenic bacteria. Antibiotic-bacteria are becoming progressively common, and to make matters worse, more and more bacteria are becoming resistant to all known antibiotics. The traditional method for this problem is to introduce new antibiotics that kill the resistant mutants. This specific “arms race” resulted into thousands of potentially active chemicals are synthesized in laboratories around the world every day.1,3,4-Oxadiazole; 1,3,4-thiadiazole; 1,2,4-triazole and some of their derivatives are involved in modifications at the following axes: First, attaching a thio-group into heterocyclic rings. Second, introducing different substitutions at position 5 which often are the residuals of the synthetic starting materials such as simple aliphatic, substituted aliphatic chains, aromatic carbocyclic and heterocyclic residues.     

A commutativity theorem

LetR be an associative ring with identity which satisfies the identities(xy) ^k =(yx) ^k and(xy) ^l =(yx) ^l , wherek andl are relatively prime positive integers, depending onx andy. ThenR is commutative. Moreover, examples are given which show thatR need not be commutative if either of the above identities is dropped. This theorem is also true for groups. Presented by N. S. Mendelsohn.     

Stress in aerogel during depressurization of autoclave: II. Silica gels

Although supercritical drying avoids the capillary stresses that tend to warp and crack xerogels, there are other sources of stress that interfere with the preparation of monolithic aerogels. In this paper, we present experimental results showing that there is a limit to the rate at which the pressure can be released from the autoclave without causing cracking, and that the maximum rate decreases as the gel size increases. Using an analysis developed in a companion paper, the stresses generated during depressurization are compared to the modulus of rupture of our aerogels. The calculations require knowledge of the pressure-dependence of the density of the vapor (ethanol, in our experiments), as well as the permeability and modulus of the gel network. Measurements of those properties were performed on a series of silica gels made under basic and neutral conditions. We find that the calculated stresses are large enough to account for the cracking of our gels at high rates of depressurization; moreover, the predicted dependence of stress on gel diameter is in agreement with experiment.     

Converses of the Jensen inequality derived from the Green functions with applications in information theory

Jensen integral inequality has got much importance regarding their applications in different fields of mathematics. In this paper, two converses of Jensen integral inequality for convex function are obtained. The results are applied to establish converses of Hölder and Hermite-Hadamard inequalities as well. At the end, some useful applications in information theory of the obtained results are given. The idea used in this paper may inculcate further research.     

Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.     

On Hexagonal Structures in Higher Dimensional Theories

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2), SU(3), G ₂, Spin(7), O(8) as well as E ₈ and O(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.