Comment on path integration of a quadratic action with a generalized memory

Using the promeasure technique, we give an alternative evaluation of a path integral corresponding to a quadratic action with a generalized memory.

Comment on “path integration of a quadratic action with a generalized memory

Using the promeasure technique, we give an alternative evaluation of a path integral corresponding to a quadratic action with a generalized memory.

Non-linear systems with quadratic and cubic damping - an analytical approach

The possible equivalence of second-order non-linear systems having quadratic and cubic damping with third-order linear systems is studied in this paper. It is shown that this equivalence can be established through transformation techniques under certain constraints on the form of the non-linearity of the given system.

Quadratic Nonlinearity of One- and Two-Electron Oxidized Metalloporphyrins and Their Switching in Solution

We report the quadratic nonlinearity of one- and two-electron oxidation products of the first series of transition metal complexes of meso-tetraphenylporphyrin (TPP). Among many MTPP complexes, only CuTPP and ZnTPP show reversible oxidation/reduction cycles as seen from cyclic voltammetry experiments. While centrosymmetric neutral metalloporphyrins have zero first hyperpolarizability, β, as expected, the cation radicals and dications of CuTPP and ZnTPP have very high β values. The one- and two-electron oxidation of the MTPPs leads to symmetry-breaking of the metal−porphyrin core, resulting in a large β value that is perhaps aided in part by contributions from the two-photon resonance enhancement. The calculated static first hyperpolarizabilities, β0, which are evaluated in the framework of density functional theory by a coupled perturbed Hartree−Fock method, support the experimental trend. The switching of optical nonlinearity has been achieved between the neutral and the one-electron oxidation products but not between the one- and the two-electron oxidation products since dications that are electrochemically reversible are unstable due to the formation of stable isoporphyrins in the presence of nucleophiles such as halides.

The effect of halogen atom on the molecular quadratic nonlinearity of half sandwich complexes in the presence of an acceptor

Half sandwich complexes of the type [CpM(CO)(n)X] {X=Cl, Br, I; If, M=Fe, Ru; n=2 and if M=Mo; n=3} and [CpNiPPh3X] {X=Cl, Br, I} have been synthesized and their second order molecular nonlinearity (beta) measured at 1064 nm in CHCl3 by the hyper-Rayleigh scattering technique. Iron complexes consistently display larger beta values than ruthenium complexes while nickel complexes have marginally larger beta values than iron complexes. In the presence of an acceptor ligand such as CO or PPh3, the role of the halogen atom is that of a pi donor. The better overlap of Cl orbitals with Fe and Ni metal centres make Cl a better pi donor than Br or I in the respective complexes. Consequently, M-pi interaction is stronger in Fe/Ni-Cl complexes. The value of beta decreases as one goes down the halogen group. For the complexes of 4d metal ions where the metal-ligand distance is larger, the influence of pi orbital overlap appears to be less important, resulting in moderate changes in beta as a function of halogen substitution. (C) 2006 Elsevier B.V. All rights reserved.

On the design of quadratic weirs

This paper considers the problem of the design of the quadratic weir notch, which finds application in the proportionate method of flow measurement in a by-pass, such that the discharge through it is proportional to the square root of the head measured above a certain datum. The weir notch consists of a bottom in the form of a rectangular weir of width 2W and depth a over which a designed curve is fitted. A theorem concerning the flow through compound weirs called the “slope discharge continuity theorem” is discussed and proved. Using this, the problem is reduced to the determination of an exact solution to Volterra's integral equation in Abel's form. It is shown that in the case of a quadratic weir notch, the discharge is proportional to the square root of the head measured above a datum Image a above the crest of the weir. Further, it is observed that the function defining the shape of the weir is rapidly convergent and its value almost approximates to zero at distances of 3a and above from the crest of the weir. This interesting and significant behaviour of the function incidentally provides a very good approximate solution to a particular Fredholm integral equation of the first kind, transforming the notch into a device called a “proportional-orifice”. A new concept of a “notch-orifice” capable of passing a discharge proportional to the square root of the head (above a particular datum) while acting both as a notch, and as an orifice, is given. A typical experiment with one such notch-orifice, having A = 4 in., and W = 6 in., shows a remarkable agreement with the theory and is found to have a constant coefficient of discharge of 0.61 in the ranges of both notch and orifice.

Computationally efficient optical tomographic reconstructions through waveletizing the normalized quadratic perturbation equation - art. no.61640R

In this paper, we present a wavelet - based approach to solve the non-linear perturbation equation encountered in optical tomography. A particularly suitable data gathering geometry is used to gather a data set consisting of differential changes in intensity owing to the presence of the inhomogeneous regions. With this scheme, the unknown image, the data, as well as the weight matrix are all represented by wavelet expansions, thus yielding the representation of the original non - linear perturbation equation in the wavelet domain. The advantage in use of the non-linear perturbation equation is that there is no need to recompute the derivatives during the entire reconstruction process. Once the derivatives are computed, they are transformed into the wavelet domain. The purpose of going to the wavelet domain, is that, it has an inherent localization and de-noising property. The use of approximation coefficients, without the detail coefficients, is ideally suited for diffuse optical tomographic reconstructions, as the diffusion equation removes most of the high frequency information and the reconstruction appears low-pass filtered. We demonstrate through numerical simulations, that through solving merely the approximation coefficients one can reconstruct an image which has the same information content as the reconstruction from a non-waveletized procedure. In addition we demonstrate a better noise tolerance and much reduced computation time for reconstructions from this approach.

Extension of Darken's Quadratic Formalism to Dilute Multicomponent Solutions

Darken's quadratic formalism is extended to multicomponent solutions. Equations are developed for the representation of the integral and partial excess free energies, entropies and enthalpies in dilute multicomponent solutions. Quadratic formalism applied to multicomponent solutions is thermodynamically consistent. The formalism is compared with the conventional second order Maclaurin series or interaction parameter representation and the relations between them are derived. Advantages of the quadratic formalism are discussed.

A systolic evaluator for linear, quadratic, and cubic expressions

High-speed evaluation of a large number of linear, quadratic, and cubic expressions is very important for the modeling and real-time display of objects in computer graphics. Using VLSI techniques, chips called pixel planes have actually been built by H. Fuchs and his group to evaluate linear expressions. In this paper, we describe a topological variant of Fuchs' pixel planes which can evaluate linear, quadratic, cubic, and higher-order polynomials. In our design, we make use of local interconnections only, i.e., interconnections between neighboring processing cells. This leads to the concept of tiling the processing cells for VLSI implementation.

Geometry and quadratic nonlinearity of charge transfer complexes in solution using depolarized hyper-Rayleigh scattering

We report large quadratic nonlinearity in a series of 1:1 molecular complexes between methyl substituted benzene donors and quinone acceptors in solution. The first hyperpolarizability, beta(HRS), which is very small for the individual components, becomes large by intermolecular charge transfer (CT) interaction between the donor and the acceptor in the complex. In addition, we have investigated the geometry of these CT complexes in solution using polarization resolved hyper-Rayleigh scattering (HRS). Using linearly (electric field vector along X direction) and circularly polarized incident light, respectively, we have measured two macroscopic depolarization ratios D = I-2 omega,I-X,I-X/I-2 omega,I-Z,I-X and D' = I-2 omega,I-X,I-C/I-2 omega,I-Z,I-C in the laboratory fixed XYZ frame by detecting the second harmonic scattered light in a polarization resolved fashion. The experimentally obtained first hyperpolarizability, beta(HRS), and the value of macroscopic depolarization ratios, D and D', are then matched with the theoretically deduced values from single and double configuration interaction calculations performed using the Zerner's intermediate neglect of differential overlap self-consistent reaction field technique. In solution, since several geometries are possible, we have carried out calculations by rotating the acceptor moiety around three different axes keeping the donor molecule fixed at an optimized geometry. These rotations give us the theoretical beta(HRS), D and D' values as a function of the geometry of the complex. The calculated beta(HRS), D, and D' values that closely match with the experimental values, give the dominant equilibrium geometry in solution. All the CT complexes between methyl benzenes and chloranil or 1,2-dichloro-4,5-dicyano-p-benzoquinone investigated here are found to have a slipped parallel stacking of the donors and the acceptors. Furthermore, the geometries are staggered and in some pairs, a twist angle as high as 30 degrees is observed. Thus, we have demonstrated in this paper that the polarization resolved HRS technique along with theoretical calculations can unravel the geometry of CT complexes in solution. (C) 2011 American Institute of Physics. doi:10.1063/1.3514922]

Geometry and quadratic nonlinearity of charge transfer complexes in solution: A theoretical study

In this paper, we have computed the quadratic nonlinear optical (NLO) properties of a class of weak charge transfer (CT) complexes. These weak complexes are formed when the methyl substituted benzenes (donors) are added to strong acceptors like chloranil (CHL) or di-chloro-di-cyano benzoquinone (DDQ) in chloroform or in dichloromethane. The formation of such complexes is manifested by the presence of a broad absorption maximum in the visible range of the spectrum where neither the donor nor the acceptor absorbs. The appearance of this visible band is due to CT interactions, which result in strong NLO responses. We have employed the semiempirical intermediate neglect of differential overlap (INDO/S) Hamiltonian to calculate the energy levels of these CT complexes using single and double configuration interaction (SDCI). The solvent effects are taken into account by using the self-consistent reaction field (SCRF) scheme. The geometry of the complex is obtained by exploring different relative molecular geometries by rotating the acceptor with respect to the fixed donor about three different axes. The theoretical geometry that best fits the experimental energy gaps, beta(HRS) and macroscopic depolarization ratios is taken to be the most probable geometry of the complex. Our studies show that the most probable geometry of these complexes in solution is the parallel displaced structure with a significant twist in some cases. (C) 2011 American Institute of Physics. doi:10.1063/1.3526748]

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25: 137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6): 859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright (C) 2011 John Wiley & Sons, Ltd.

Solutions of cubic equations in quadratic fields

Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

Quadratic Lyapunov functions for linear systems Journal

The paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation S0305004100044777_inline1 The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable sy