Abstract:
Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee's novel use of Stein's method for $\{-1,+1\}$ valued variables to a general class of discrete distributions, and provide $\log n$ rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge

Abstract:
A grand unified field $\mathcal{M}^{\mu\nu}$ is constructed from Maxwell's Field tensor and appropriately modified flow field, both non-minimally coupled to gravity, to analyze the dynamics of hot charged fluids in curved background space-time. With a suitable $3+1$ decomposition, this new formalism of the hot fluid is then applied to investigate the vortical dynamics of the system. Finally, the equilibrium state for plasma with non-linear coupling through Ricci scalar $R$ to gravity is investigated to derive a double Beltrami equation in curved space-time.

Abstract:
Vorticity generation in accretion disks around Schwarzschild and Kerr black holes is investigated in the context of magnetofluid dynamics derived for both General Relativity (GR), and modified gravity formulations. In both cases, the Kerr geometry leads to a "stronger" generation of vorticity than its Schwarzschild counterpart. Of the two principal sources, the relativistic drive peaks near the innermost stable circular orbit (isco), whereas the baroclinic drive dominates at larger distances. Consequences of this new relativistic vorticity source are discussed in several astrophysical settings.

Abstract:
Using the magnetofluid unification framework, we show that the accretion disk plasma (embedded in the background geometry of a blackhole) can relax to a class of states known as the Beltrami-Bernoulli (BB) equilibria. Modeling the disk plasma as a Hall MHD system, we find that the space-time curvature can significantly alter the magnetic/velocity decay rate as we move away from the compact object; the velocity profiles in BB states, for example, deviate substantially from the predicted corresponding geodesic velocity profiles. These departures imply a rich interplay of plasma dynamics and general relativity revealed by examining the corresponding Bernoulli condition representing "homogeneity" of total energy. The relaxed states have their origin in the constraints provided by the two helicity invariants of Hall MHD. These helicities conspire to introduce a new oscillatory length scale into the system that is strongly influenced by relativistic and thermal effects.

Abstract:
3D particle-in-cell simulations demonstrate that the enhanced transparency of a relativistically hot plasma is sensitive to how the energy is partitioned between different degrees of freedom. For an anisotropic electron distribution, propagation characteristics, like the critical density, will depend on the polarization of the electromagnetic wave. Despite the onset of the Weibel instability in such plasmas, the anisotropy can persist long enough to affect laser propagation. This plasma can then function as a polarizer or a waveplate to dramatically alter the pulse polarization.

Abstract:
A simple, specific, precise, and accurate RP-HPLC method has been developed and validated for simultaneous estimation of Methylparaben (MP), Ketoconazole (KT), and Mometasone Furoate (MF) topical pharmaceutical dosage formulation. The separation was achieved by Waters X Terra C18 column using mobile phase consisting of buffer (triethyl amine in water, pH adjusted to 6.5 with glacial acetic acid)-acetonitrile (40？:？60,？v/v) at a flow rate of 1.5？mL/min and detection at 250？nm. The method showed linearity with correlation coefficient <0.9999 over the range of 0.12–15.2？μg/mL, 0.67–149.4？μg/mL, and 0.42–7.6？μg/mL for MP, KT, and MF, respectively. The mean recoveries were found to be in the range of 99.9–101.1% for all the components. The method was validated as per the ICH guidelines for linearity, limit of detection, limit of quantification, accuracy, precision, robustness and solution stability. Stability indicating capability of the developed method was established by analyzing forced degradation of samples in which spectral purity of MP, KT, and MF along with separation of degradation products from analytes peak was achieved. The method can be successfully applied for routine analysis of quantitative determination of MP, KT, and MF in pharmaceutical dosage form. 1. Introduction Mometasone Furoate (MF), (11β, 16α)-9, 21-dichloro-11-hydroxy-16-methyl-3, 20-dioxopregna-1, 4-dien-17-yl 2-furoate (Figure 1(a)) is a topical corticosteroid; it has anti-inflammatory, antipruritic, and vasoonstrictive properties. Mometasone inhibits the action of allergic reactions, eczema, and psoriasis that cause inflammation, redness, and swelling [1, 2]. Ketoconazole (KT), an imidazole derivative, chemically 1-[4-(4-{[2-(2,4-dichlorophenyl)-2-(1H-imidazol-1-ylmethyl)-1,3-dioxolan-4-yl]methoxy}phenyl)piperazin-1-yl]ethan-1-one (Figure 1(b)) is an antifungal agent with topical and systemic action and can be incorporated into several pharmaceutical forms. As for example a ketoconazole shampoo is effective against seborrhoeic dermatitis as well as Pityriasis versicolor [3–7]. KT and MF in combined dosage form are used in the treatment of mild to moderate inflamed cutaneous mycoses with antimycotic activity. Figure 1: Chemical structure of (a) Mometasone Furoate, (b) Ketoconazole, and (c) Methylparaben. The preservative system is an important part of semisolid formulations in preventing the deterioration of formulations from microbial contamination. Methylparaben (MP) (Figure 1(c)) and its salts are most commonly used as preservatives for many years. To establish their

Abstract:
A novel, simple, validated stability indicating HPLC method was developed for determination of Koptrizon and Tinosorb S. Stability indicating power of the method was established by forced degradation study. The chromatographic separation was achieved with Waters X Bridge column, by using mobile phase consisting of a mixture of acetonitrile？:？tetrahydrofuran？:？water (38？:？38？:？24, v/v/v). The method fulfilled validation criteria and was shown to be sensitive, with limits of detection (LOD) and quantitation (LOQ) of 0.024 and 0.08？μg？ for Koptrizon and 0.048 and 0.16？μg？ for Tinosorb S, respectively. The developed method is validated for parameters like precision, accuracy, linearity, solution stability, specificity, and ruggedness as per ICH norms. Design expert with ANOVA software with linear model was applied and a 23 full factorial design was employed to estimate the model coefficients and also to check the robustness of the method. Results of the two-level full factorial design, 23 with 10 runs including two-centre-point analysis based on the variance analysis (ANOVA), demonstrated that all three factors, as well as the interactions between retention time of Koptrizon, Tinosorb S, and USP plate count for Koptrizon, are statistically significant. 1. Introduction Frequent exposure to UV radiation causes pronounced harmful effects on human health. UV radiation-induced effects are manifested as acute responses, namely, sunburn, hyperplasia, and immunosuppressant, and as chronic responses, primarily photo carcinogenesis and photo ageing [1]. The molecules employed in cosmetic products to protect skin from the sun are classified as physical and chemical sunscreens. Physical sunscreens are represented mainly by zinc and titanium oxides which interrupt the path of UV light by scattering or reflection. Chemical sunscreens are generally aromatic compounds conjugated with an electron-donating group in “ortho” or “para” position and an electron acceptor group. This chemical structure favours electron delocalization and therefore helps excitation of molecules from ground state to an excited state. The energy required for this transition corresponds to the energies of ultraviolet A (UVA) and Ultraviolet B (UVB) radiations [2, 3]. Chronic exposure to UVB (280–320？nm) wavelengths induces damage to human skin, such as burns and erythema, but increases evidence of demonstrates that UVA radiation (320–400？nm) contributes to photo aging, which results in the accumulation of massive amounts of abnormal elastic material in the dermis of photo-aged skin and modification in

Abstract:
We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical results in graph theory: the theorem of K\H{o}v\'{a}ri, S\'{o}s and Tur\'{a}n on the minimum number of edges in a bipartite graph that has no large independent set, and the theorem of Hansel (also Katona and Szemer\'{e}di, Krichevskii) on the sum of the sizes of bipartite graphs that can be used to construct a graph (non-necessarily bipartite) that has no large independent set. As an application of our results, we show how they unify the underlying combinatorial principles developed in the proof of tight lower bounds for depth-two superconcentrators.

Abstract:
We present a technique of proving lower bounds for noisy computations. This is achieved by a theorem connecting computations on a kind of randomized decision trees and sampling based algorithms. This approach is surprisingly powerful, and applicable to several models of computation previously studied. As a first illustration we show how all the results of Evans and Pippenger (SIAM J. Computing, 1999) for noisy decision trees, some of which were derived using Fourier analysis, follow immediately if we consider the sampling-based algorithms that naturally arise from these decision trees. Next, we show a tight lower bound of $\Omega(N \log\log N)$ on the number of transmissions required to compute several functions (including the parity function and the majority function) in a network of $N$ randomly placed sensors, communicating using local transmissions, and operating with power near the connectivity threshold. This result considerably simplifies and strengthens an earlier result of Dutta, Kanoria Manjunath and Radhakrishnan (SODA 08) that such networks cannot compute the parity function reliably with significantly fewer than $N\log \log N$ transmissions. The lower bound for parity shown earlier made use of special properties of the parity function and is inapplicable, e.g., to the majority function. In this paper, we use our approach to develop an interesting connection between computation of boolean functions on noisy networks that make few transmissionss, and algorithms that work by sampling only a part of the input. It is straightforward to verify that such sampling-based algorithms cannot compute the majority function.

Abstract:
To address the issue we employed two approaches: gain of function by overexpressing FAK and loss of function by siRNA-mediated silencing of FAK. We observed that overexpression of FAK induces actin remodeling in skeletal muscle cells in presence of insulin. Concomitant to this Glut-4 molecules were also observed to be present in the vicinity of remodeled actin, as indicated by the colocalization studies. FAK-mediated actin remodeling resulted into subsequent glucose uptake via PI3K-dependent pathway. On the other hand FAK silencing reduced actin remodeling affecting Glut-4 translocation resulting into insulin resistance.The data confirms that FAK regulates glucose uptake through actin reorganization in skeletal muscle. FAK overexpression supports actin remodeling and subsequent glucose uptake in a PI3K dependent manner. Inhibition of FAK prevents insulin-stimulated remodeling of actin filaments resulting into decreased Glut-4 translocation and glucose uptake generating insulin resistance. To our knowledge this is the first study relating FAK, actin remodeling, Glut-4 translocation and glucose uptake and their interrelationship in generating insulin resistance.The uptake of glucose by skeletal muscle, a major insulin responsive organ, is mediated by the insulin-responsive glucose transporters, Glut-4 [1,2]. Insulin stimulation elicits Glut-4 recruitment to the plasma membrane of skeletal muscle thereby increasing glucose influx [3,4]. Under insulin resistant condition, loss of this response triggers an elevation of circulating insulin, resulting into hyperinsulinemia. This worsens insulin resistance resulting into type 2 diabetes [5-7]. Therefore, an in-depth study of insulin signaling at the cellular and molecular level is critical to understand the pathogenesis of type 2 diabetes.In order to better understand insulin resistance at a molecular level we have previously developed an insulin resistant skeletal muscle cell model by differentiating C2C12 skeletal muscle