Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences

The Cubic Sieve Method for solving the Discrete Logarithm Problem in prime fields requires a nontrivial solution to the Cubic Sieve Congruence (CSC) x(3) equivalent to y(2)z (mod p), where p is a given prime number. A nontrivial solution must also satisfy x(3) not equal y(2)z and 1 <= x, y, z < p(alpha), where alpha is a given real number such that 1/3 < alpha <= 1/2. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. CSC can be parametrized as x equivalent to v(2)z (mod p) and y equivalent to v(3)z (mod p). In this paper, we give a deterministic polynomial-time (O(ln(3) p) bit-operations) algorithm to determine, for a given v, a nontrivial solution to CSC, if one exists. Previously it took (O) over tilde (p(alpha)) time in the worst case to determine this. We relate the CSC problem to the gap problem of fractional part sequences, where we need to determine the non-negative integers N satisfying the fractional part inequality {theta N} < phi (theta and phi are given real numbers). The correspondence between the CSC problem and the gap problem is that determining the parameter z in the former problem corresponds to determining N in the latter problem. We also show in the alpha = 1/2 case of CSC that for a certain class of primes the CSC problem can be solved deterministically in <(O)over tilde>(p(1/3)) time compared to the previous best of (O) over tilde (p(1/2)). It is empirically observed that about one out of three primes is covered by the above class. (C) 2013 Elsevier B.V. All rights reserved.

UNIQUENESS OF SOLUTIONS TO SCHRODINGER EQUATIONS ON H-TYPE GROUPS

This paper deals with the Schrodinger equation i partial derivative(s)u(z, t; s) - Lu(z, t; s) = 0; where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies vertical bar f(z, t)vertical bar less than or similar to q(alpha)(z, t), where q(s) is the heat kernel associated to L. If in addition vertical bar u(z, t; s(0))vertical bar less than or similar to q(beta)(z, t), for some s(0) is an element of R \textbackslash {0}, then we prove that u(z, t; s) = 0 for all s is an element of R whenever alpha beta < s(0)(2). This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1.

Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions

The paper discusses the frequency domain based solution for a certain class of wave equations such as: a second order partial differential equation in one variable with constant and varying coefficients (Cantilever beam) and a coupled second order partial differential equation in two variables with constant and varying coefficients (Timoshenko beam). The exact solution of the Cantilever beam with uniform and varying cross-section and the Timoshenko beam with uniform cross-section is available. However, the exact solution for Timoshenko beam with varying cross-section is not available. Laplace spectral methods are used to solve these problems exactly in frequency domain. The numerical solution in frequency domain is done by discretisation in space by approximating the unknown function using spectral functions like Chebyshev polynomials, Legendre polynomials and also Normal polynomials. Different numerical methods such as Galerkin Method, Petrov- Galerkin method, Method of moments and Collocation method or the Pseudo-spectral method in frequency domain are studied and compared with the available exact solution. An approximate solution is also obtained for the Timoshenko beam with varying cross-section using Laplace Spectral Element Method (LSEM). The group speeds are computed exactly for the Cantilever beam and Timoshenko beam with uniform cross-section and is compared with the group speeds obtained numerically. The shear mode and the bending modes of the Timoshenko beam with uniform cross-section are separated numerically by applying a modulated pulse as the shear force and the corresponding group speeds for varying taper parameter in are obtained numerically by varying the frequency of the input pulse. An approximate expression for calculating group speeds corresponding to the shear mode and the bending mode, and also the cut-off frequency is obtained. Finally, we show that the cut-off frequency disappears for large in, for epsilon > 0 and increases for large in, for epsilon < 0.

Observation of Combined Effect of Temperature and Pressure on Cubic to Hexagonal Phase Transformation in ZnS at the Nanoscale

The temperature of allotropic phase transformation in ZnS (cubic to wurtzite) changes with pressure and particle size. In this paper we have explored the interrelation among these through a detailed study of ZnS powders obtained by a temperature-controlled high energy milling process. By employing the combined effect of temperature and pressure in an indigenously built cryomill, we have demonstrated a large-scale, low-temperature synthesis of wurtzite ZnS nanoparticles. The synthesized products have been characterized for their phase and microstructure by the use of X-ray diffraction and transmission electron microscopic techniques. Further, it has been demonstrated that the synthesized materials exhibit photoluminescence emissions in the UV-visible region with an unusual doublet pattern due to the presence of both cubic and hexagonal wurtzite domains in the same particles. By further fine-tuning the processing conditions, it may be possible to achieve controlled defect related photoluminescence emissions from the ZnS nanoparticles.

Modified Adomian Decomposition Method for Van der Pol equations

In the paper, the well known Adomian Decomposition Method (ADM) is modified to solve the parabolic equations. The present method is quite different than the numerical method. The results are compared with the existing exact or analytical method. The already known existing Adomian Decomposition Method is modified to improve the accuracy and convergence. Thus, the modified method is named as Modified Adomian Decomposition Method (MADM). The Modified Adomian Decomposition Method results are found to converge very quickly and are more accurate compared to ADM and numerical methods. MADM is quite efficient and is practically well suited for use in these problems. Several examples are given to check the reliability of the present method. Modified Adomian Decomposition Method is a non-numerical method which can be adapted for solving parabolic equations. In the current paper, the principle of the decomposition method is described, and its advantages are shown in the form of parabolic equations. (C) 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L-2m-norms of vorticity D-m (1 <= m <= infinity). The first in this hierarchy, D-1, is the global enstrophy. Three regimes naturally occur in the D-1-D-m plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime 1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

REPRESENTING A CUBIC GRAPH AS THE INTERSECTION GRAPH OF AXIS-PARALLEL BOXES IN THREE DIMENSIONS

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes touch just at their boundaries.

Transition from dissipative to conservative dynamics in equations of hydrodynamics

We show, by using direct numerical simulations and theory, how, by increasing the order of dissipativity (alpha) in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case alpha -> infinity U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of alpha greater than a crossover value alpha(crossover). We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows.

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.

Simple three parameter equations for correlating liquid phase compositions in subcritical and supercritical systems

Two Chrastil type expressions have been developed to model the solubility of supercritical fluids/gases in liquids. The three parameter expressions proposed correlates the solubility as a function of temperature, pressure and density. The equation can also be used to check the self-consistency of the experimental data of liquid phase compositions for supercritical fluid-liquid equilibria. Fifty three different binary systems (carbon-dioxide + liquid) with around 2700 data points encompassing a wide range of compounds like esters, alcohols, carboxylic acids and ionic liquids were successfully modeled for a wide range of temperatures and pressures. Besides the test for self-consistency, based on the data at one temperature, the model can be used to predict the solubility of supercritical fluids in liquids at different temperatures. (C) 2014 Elsevier B.V. All rights reserved.

Facile green fabrication of iron-doped cubic ZrO2 nanoparticles by Phyllanthus acidus: Structural, photocatalytic and photoluminescent properties

Cubic ZrO2: Fe3+ (0.5-4 mol%) nanoparticles (NPs) were synthesized via bin-inspired, inexpensive and simple route using Phyllanthus acidus as fuel. PXRD, SEM, TEM, FTIR, UV absorption and PL studies were performed to ascertain the formation of NPs. Rietveld analysis confirmed the formation of cubic structure. The influence of Fe3+ on the structure, morphology, UV absorption, PL emission and photocatalytic activity of NPs were investigated. The CIE chromaticity coordinates (0.36, 0.41) show that NPs could be a promising candidate for white LEDs. The influence of Fe3+ on ZrO2 matrix for photocatalytic degradation of AO7 was evaluated under UVA and Sunlight irradiation. The enhanced photocatalytic activity of spherical shaped ZrO2: Fe3+ (2 mol%) under UVA light was attributed to dopant concentration, crystallite size, narrow band gap, textural properties and capability for reducing the electron-hole pair recombination. The trend of inhibitory effect in the presence of different radical scavengers were followed the order SO42- > Cl- > C2H5OH > HCO3- > CO32-. The recycling catalytic ability of the ZrO2: Fe3+ (2 mol%) was also evaluated with a negligible decrease in the degradation efficiency even after the sixth successive run. (C) 2014 Elsevier B.V. All rights reserved.

Bayesian parameter identification in dynamic state space models using modified measurement equations

When Markov chain Monte Carlo (MCMC) samplers are used in problems of system parameter identification, one would face computational difficulties in dealing with large amount of measurement data and (or) low levels of measurement noise. Such exigencies are likely to occur in problems of parameter identification in dynamical systems when amount of vibratory measurement data and number of parameters to be identified could be large. In such cases, the posterior probability density function of the system parameters tends to have regions of narrow supports and a finite length MCMC chain is unlikely to cover pertinent regions. The present study proposes strategies based on modification of measurement equations and subsequent corrections, to alleviate this difficulty. This involves artificial enhancement of measurement noise, assimilation of transformed packets of measurements, and a global iteration strategy to improve the choice of prior models. Illustrative examples cover laboratory studies on a time variant dynamical system and a bending-torsion coupled, geometrically non-linear building frame under earthquake support motions. (C) 2015 Elsevier Ltd. All rights reserved.

Effect of Manganese (II) Oxide on microstructure and ionic transport properties of nanostructured cubic zirconia

Effect of MnO addition on microstructure and ionic transport properties of nanocrystalline cubic(c)-ZrO2 is reported. Monoclinic (m) ZrO2 powders with 10-30 mol% MnO powder are mechanically alloyed in a planetary ball mill at room temperature for 10 h and annealed at 550 degrees C for 6 h. In all compositions m-ZrO2 transforms completely to nanocrystalline c-ZrO2 phase and MnO is fully incorporated into c-ZrO2 lattice. Rietveld's refinement technique is employed for detailed microstructure analysis by analyzing XRD patterns. High resolution transmission electron microscopy (HRTEM) analysis confirms the complete formation of c-ZrO2 phase. Presence of stoichiometric Mn in c-ZrO2 powder is confirmed by Electron Probe Microscopy analysis. XPS analysis reveals that Mn is mostly in Mn2+ oxidation state. A correlation between lattice parameter and oxygen vacancy is established. A detailed ionic conductivity measurement in the 250 degrees-575 degrees C temperature range describes the effect of MnO on conductivity of c-ZrO2. The ionic conductivity (s) of 30 mol% MnO alloyed ZrO2 at 550 degrees C is 0.04 s cm(-1). Electrical relaxation studies are carried out by impedance and modulus spectroscopy. Relaxation frequency is found to increase with temperature and MnO mol fraction. Electrical characterization predicts that these compounds have potentials for use as solid oxide fuel cell electrolyte material. (C) 2015 Elsevier Ltd. All rights reserved.

Stabilization of the high-temperature and high-pressure cubic phase of ZnO by temperature-controlled milling

The reversible transition of wurtzite to rock salt phase under pressure is well reported in literature. The cubic phase is unstable under ambient conditions both in the bulk and in nanoparticles. This paper reports defect-induced stabilization of cubic ZnO phase in sub 20 nm ZnO particles and explores their optical properties. The size reduction was achieved by ball milling in a specially designed mill which allows a control of the milling temperature. The process of synthesis involved both variation of milling temperature (including low temperature similar to 150 K) and impact pressure. We show that these have profound influence in the introduction of defects and stabilization of the cubic phase. A molecular dynamics simulation is presented to explain the observed results. The measured optical properties have further supported the observations of defect-induced stabilization of cubic ZnO and reduction in particle size.

A modified approach to numerical solution of Fredholm integral equations of the second kind

A modified approach to obtain approximate numerical solutions of Fredholin integral equations of the second kind is presented. The error bound is explained by the aid of several illustrative examples. In each example, the approximate solution is compared with the exact solution, wherever possible, and an excellent agreement is observed. In addition, the error bound in each example is compared with the one obtained by the Nystrom method. It is found that the error bound of the present method is smaller than the ones obtained by the Nystrom method. Further, the present method is successfully applied to derive the solution of an integral equation arising in a special Dirichlet problem. (C) 2015 Elsevier Inc. All rights reserved.