On a System of Differential Equations Connected with the Gravitational Instability of a Multicomponent Medium in Newtonian Cosmology

The gravitational clustering of a multicomponent medium in an expanding universe has been considered by many authors in recent years. The system of differential equations associated with such a multicomponent medium is generalized in this article by incorporating arbitrary parameters. Explicit analytic solutions of this generalized system are given for various situations. Physical interpretations of the solutions are not considered, but the various solutions of physical problems obtained by various authors can be seen to be special cases of the general solutions given in this article.     

Twisted K-Theory and K-Theory of Bundle Gerbes

In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of K-brane charges in nontrivial backgrounds are briefly discussed. Received: 29 June 2001 / Accepted: 15 October 2001     

Bromination of cyclopropanes using potassium bromide and cerium(IV) ammonium nitrate (CAN): synthesis of 1,3-dibromides

The bromination of cyclopropanes using potassium bromide and cerium(IV) ammonium nitrate (CAN) in a two-phase system consisting of water and dichloromethane affording 1,3-dibromides in excellent yields is reported.     

Some properties of complex matrix-variate generalized Dirichlet integrals

Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive definite or hermitian positive definite, are available [4]. Real scalar variables case of the Dirichlet models are generalized in various directions. One such generalization of the type-2 or inverted Dirichlet is looked into in this article. Matrix-variate analogue, when the matrices are hermitian positive definite, are worked out along with some properties which are mathematically and statistically interesting.     

Some results on functions of matrix argument

A number of integrals of scalar variables are extended to the case of matrix variables. Functions where the argument is a positive definite symmetric matrix are considered in this article. With the help of a generalized matrix transform a number of results are derived which are extensions of the corresponding results in the univariate cases.     

T-Duality Simplifies Bulk-Boundary Correspondence

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice \({{{\mathbb{Z}}}^{4}}\), for the weakly coupled n-component \({|\varphi|^{4}}\) spin model for all \({n \ge 1}\), and for the continuous-time weakly self-avoiding walk. For the \({|\varphi|^{4}}\) model, we prove that the critical two-point function has |x|^?2 (Gaussian) decay asymptotically, for \({n \ge 1}\). We also determine the asymptotic decay of the critical correlations of the squares of components of \({\varphi}\), including the logarithmic corrections to Gaussian scaling, for \({n \ge 1}\). The above extends previously known results for n = 1 to all \({n \ge 1}\), and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the “watermelon” network consisting of p weakly mutually- and self-avoiding walks, for all \({p \ge 1}\), including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove the existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the \({|\varphi|^{4}}\) model, and provides a unified treatment of both models, and of all the above results.     

Simultaneous determination of bioactive compounds in Piper nigrum L. and a species comparison study using HPLC-PDA

Piper nigrum L. is a traditional medicine widely used in India for illnesses such as constipation, diarrhoea, earache, gangrene, heart disease, hernia, hoarseness, indigestion, insect bites, insomnia, joint pain, liver problems, lung disease, oral abscesses, sunburn, tooth decay and toothaches. In this study, six bioactive compounds, namely piperine (1), pellitorine (2), guineensine (3), pipnoohine (4), trichostachine (5) and piperonal (6) were quantified in different extracts of P. nigrum L. and compared with those of P. longum L. and P. chaba Hunter. To evaluate the quality of P. nigrum, a simple, accurate and precise HPLC-PDA method was developed for the simultaneous determination of the above-mentioned six compounds. The separation was achieved by Phenomenex Luna RP C(18) column (150?×?4.6?mm, 5?μm, Phenomenex Inc, CA, USA) with a binary gradient solvent system of water-acetonitrile, at a flow rate of 1.0?mL?min(-1) and detected at 210, 232, 262 and 343?nm. All six calibration curves showed good linearity (R (2)?>?0.9966). The method was reproducible with intra- and inter-day variations of less than 2% and 5%, respectively. The results demonstrated that this method is simple, reliable and suitable for the quality control of these plants.     

A pathway from Bayesian statistical analysis to superstatistics

Superstatistics and Tsallis statistics in statistical mechanics are given an interpretation in terms of Bayesian statistical analysis. Subsequently superstatistics is extended by replacing each component of the conditional and marginal densities by Mathai’s pathway model and further both components are replaced by Mathai’s pathway models. This produces a wide class of mathematically and statistically interesting functions for prospective applications in statistical physics. It is pointed out that the final integral is a particular case of a general class of integrals introduced by the authors earlier. Those integrals are also connected to Krätzel integrals in applied analysis, inverse Gaussian densities in stochastic processes, reaction rate integrals in the theory of nuclear astrophysics and Tsallis statistics in nonextensive statistical mechanics. The final results are obtained in terms of Fox’s H-function. Matrix variate analogue of one significant specific case is also pointed out.