Order Perturbation Analysis Research Articles

A PERTURBATIVE ANALYSIS OF SYNCHROTRON SPECTRAL INDEX VARIATION OVER THE MICROWAVE SKY

ABSTRACT In this paper, we implement a perturbative approach, first proposed by Bouchet & Gispert, to estimate the variation of the spectral index of galactic polarized synchrotron emission, using a linear combination of simulated Stokes Q polarization maps of selected frequency bands from WMAP and Planck observations on a region of sky dominated by the synchrotron Stokes Q signal. We find that a first order perturbative analysis recovers the input spectral index map well. Along with the spectral index variation map, our method provides a fixed reference index, , over the sky portion being analyzed. Using Monte-Carlo simulations, we find that , which matches very closely with the position of a peak at of the empirical probability density function of input synchrotron indices obtained from the same sky region. For thermal dust, the mean recovered spectral index from simulations, matches very well with the spatially fixed input thermal dust spectral index . As accompanying results of the method, we reconstruct cosmic microwave background, thermal dust, and a synchrotron template components with fixed spectral indices over the entire sky region. We use, in our analysis, full pixel–pixel noise covariance matrices of all frequency bands, estimated from the sky region being analyzed. The perturbative technique of this work (1) can build a model with an arbitrary but sufficient degree of accuracy (and precession) as allowed by the data and (2) can produce maximum likelihood estimators for reference indices and templates asymptotically.

Computational study on fused five membered heterocyclic compounds containing tertiary oxygen

The structure, stability and electronic properties of two fused five membered heterocyclic compounds containing tertiary oxygen have been investigated with quantum chemistry computation. The results show that 3H-4λ3-furo[1,2-a]furan (M1) and 2a1λ4-oxacyclopenta[cd]pentalene (M2) both own three CO bonds and especially in M2 the three CO bonds and angles are approximately identical. The two compounds are comparatively stable and M2 with three pentacycles is more stable than compound M1 according to the CO bond lengths, bond orders, Egap, electronegativity (χ), hardness (η), electrophilicity (ω), and aromaticity indexes. The aromaticity indexes such as NICS(0), NICS(1) and NICS(1)zz show that they both have strong aromaticity. The natural bonding orbital (NBO) analysis shows that oxygen atom takes hybrid orbitals composed of s and p orbital components to form σ(CO) bonds in compounds M1 and M2. The second order perturbation analysis shows that the occurrence of intramolecular charge transfer within the molecule exists in compound M1.

The Referendum Blues: shocking the system

When faced with a complex systems of inter-related equations, economists are used to undertaking perturbation analysis in order to understand key relationships, in which we shock the economic system and trace out the effects on certain variables. We may be interested in whether the system is stable, in plotting its return to equilibrium, or in the dynamics of how variables respond or jump from one resting point to some other. The decision first to hold a Referendum and then the decision to Leave the European Union have certainly shocked our economic and political settlement. The question is whether the Referendum shock has simply revealed something about the system or may go further and act to undermine its fundamentals. In the Economic Prospects section of this Review we present the National Institute’s estimates of the consequences of the Referendum, and as indicated in our May Review,1 we continue to expect a slowdown in overall economic activity following the Leave vote and a period of heightened uncertainty that will act to bear down further on activity in the UK and, to a lesser extent, abroad.

Mechanical structure optimization in minimum-time motion control of flexible bodies

This paper considers the problem of minimum-time motion control of a flexible body for which the dynamic properties can be optimized within a structural design. This allows a matching of modal characteristics to a control task such that reductions in time-of-motion can be achieved compared with unoptimized designs. The problem formulation is based on an elastic structure undergoing a motion task with specified boundary conditions and subject to limits on actuation forces. A numerical method for calculating minimum-time control input solutions based on an iterative construction of the reachable set is first considered. A structural optimization approach is further developed based on continuity properties of the solution set, these admitting a first order perturbation analysis from which an optimization of system design parameters can be achieved. A selection of numerical case studies are presented involving single and multi-mode structures. Possibilities for realization of the approach using variable stiffness/deformable structures are discussed and results from a linear motion stage with tunable-stiffness flexible armature presented that demonstrate the potential benefits.

Nonlinear tuning of microresonators for dynamic range enhancement.

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

Hugoniot experiments with unsteady waves

Recent development of transparent shock wave standard materials, such as quartz, enables continuous tracking of shock waves using optical velocimetry, providing information on shock wave steadiness and pressure perturbations in the target. From a first order perturbation analysis, we develop a set of analytical formulas that connect the pressure perturbations at the drive surface to the shock velocity perturbations observed in measurements. With targets that incorporate a calibrated transparent witness material, such as quartz, and with the analytical formulas describing the perturbation response, it is possible to determine the sound speed and Grüneisen coefficient of an unknown sample by using evolution of the non-steady perturbations as a probe. These formulas can also be used to improve the accuracy of traditional shock wave impedance match Hugoniot experiments of opaque samples driven with non-steady waves.

MODULATING THE ELECTRONIC STRUCTURE OF AMINO ACIDS: INTERACTION OF MODEL LEWIS ACIDS WITH ANTHRANILIC ACID

On the basis of theoretical B3LYP calculations, Yanez and co-workers (J. Chem. Theory Comput. 2012, 8, 2293) illustrated that beryllium ions are capable of significantly modulating (changing) the electronic structures of imidazole. In this computational organic chemistry study, the interaction of this β-amino acid and five model Lewis acids (BeF1+, Be2+, AlF21+, AlF2+, and Al3+) were investigated. Several aspects were addressed: natural bond orbitals, including second order perturbation analysis of intra-molecular charge delocalization and the natural population analysis atomic charges; molecular geometries; selected infrared stretching frequencies (C-N, C-O, and N-H), and selected 1H-NMR chemical shifts. The data illustrate that this interaction can weaken the H-O bond and goes beyond strengthening the intra-molecular hydrogen bond (N...H-O) to cause a spontaneous transfer of the proton to the nitrogen atom in five cases generating zwitterion structures. Many new features are observed. Most importantly, the zwitterion structures include a stabilizing hydrogen bond (N-H...O) that varies in relative strength according to the Lewis acid. These findings explain the experimental observations of α-amino acids (for example: J. Am. Chem. Soc. 2001, 123, 3577) and are the first reported fundamental electronic structure characterization of β-amino acids in zwitterion form.

Gravitational lensing with f (χ) = χ3/2 gravity in accordance with astrophysical observations

In this article we perform a second order perturbation analysis of the gravitational metric theory of gravity $ f(\chi) = \chi^{3/2} $ developed by Bernal et al. (2011). We show that the theory accounts in detail for two observational facts: (1) the phenomenology of flattened rotation curves associated to the Tully-Fisher relation observed in spiral galaxies, and (2) the details of observations of gravitational lensing in galaxies and groups of galaxies, without the need of any dark matter. We show how all dynamical observations on flat rotation curves and gravitational lensing can be synthesised in terms of the empirically required metric coefficients of any metric theory of gravity. We construct the corresponding metric components for the theory presented at second order in perturbation, which are shown to be perfectly compatible with the empirically derived ones. It is also shown that under the theory being presented, in order to obtain a complete full agreement with the observational results, a specific signature of Riemann's tensor has to be chosen. This signature corresponds to the one most widely used nowadays in relativity theory. Also, a computational program, the MEXICAS (Metric EXtended-gravity Incorporated through a Computer Algebraic System) code, developed for its usage in the Computer Algebraic System (CAS) Maxima for working out perturbations on a metric theory of gravity, is presented and made publicly available.

Perturbation analysis of the van den Berg Kesten inequality for determinantal probability measures

This paper describes a second order perturbation analysis of the BK property in the space of Hermitean determinantal probability measures around the subspace of product measures, showing that the second order Taylor approximation of the BK inequality holds for increasing events.

Study of a low Mach nuclear core model for single-phase flows

This paper deals with the modelling of the coolant (water) in a nuclear reactor core. This study is based on a monophasic low Mach number model (LMNC model) coupled to the stiffened gas law for a single-phase flow. Some analytical steady and unsteady solutions are presented for the 1D case. We then introduce a numerical scheme to simulate the 1D model in order to assess its relevance. Finally, we carry out a normal mode perturbation analysis in order to approximate 2D solutions around the 1D steady solutions.

Flame wrinkle destruction processes in harmonically forced, laminar premixed flames

This paper describes numerical and theoretical analyses of the nonlinear dynamics of harmonically forced, stretch-sensitive premixed flames. A key objective of this work is to analyze the relative contributions of kinematic restoration and flame stretch upon the rate at which flame wrinkles, excited by harmonic forcing, are smoothed out. Kinematic restoration is an intrinsically nonlinear process with a two spatial-zone structure, whose amplitude dependence is fundamentally different near and far from the wrinkle excitation source. Flame stretch processes appear even in the small perturbation limit, and smooth out flame wrinkles in thermodiffusively stable mixtures. Which process dominates is a function of the perturbation amplitude, frequency, stretch sensitivity of the mixture, and spatial location. This paper presents computed results illustrating the solution characteristics, as well as key dimensionless parameters controlling the solution based upon a third order perturbation analysis.

Utility Maximization for Layered Transmission Using the Broadcast Approach

We investigate the power allocation problem in layered transmission with successive enhancement using the broadcast approach. This problem is particularly important for multimedia streaming applications and multicasting services over slowly fading channels with partial channel information at the transmitter. We consider the practical scenario of limited and pre-specified source layers, and we propose a generic and efficient solution to optimize the transmission power allocated to each layer in order to maximize the average user satisfaction defined by a utility function of the decoded layers. We assume a block-fading Rayleigh distributed channel model. However, our solution can be extended to a number of other channel distributions. Although the direct formulation of the problem is not convex, we propose change of optimization variable, and we use the dual problem and the perturbation analysis in order to obtain the optimal solution. The solution is based on a one-dimensional bisection search irrespective of the total number of layers. Our optimality analysis indicates that in some cases it may not be optimal to transmit all the layers.

Random transverse field Ising model in dimensionsd = 2, 3: infinite disorder scaling via a non-linear transfer approach

The ‘cavity-mean-field’ approximation developed for the random transverse field Isingmodel on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) hasbeen found to reproduce the known exact result for the surface magnetization ind = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the presentpaper, we propose to extend these ideas in finite dimensionsd > 1 via a non-linear transfer approach for the surface magnetization. In the disorderedphase, the linearization of the transfer equations corresponds to the transfermatrix for a directed polymer in a random medium of transverse dimensionD = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011arXiv:1110.3145). We present numerical results of the non-linear transfer approach in dimensionsd = 2 and 3. In both cases, we find that the critical point is governed by infinite disorderscaling. In particular, exactly at criticality, the one-point surface magnetizationscales as , where ωc(d) coincides withthe droplet exponent ωDP(D = d − 1) of the corresponding directed polymer model, withωc(d = 2) = 1/3 and . The distribution P(v) of thepositive random variable v of order O(1) presents a power-law singularity near the origin with a(d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-lawdecay with xs = ωc(1 + a) independentlyof the order k.

Group velocity of elastic waves as eigensensitivity of a parameter dependent eigenvalue problem

We consider the problem of calculating group velocity of elastic waves along different branches of the dispersion curves without carrying out direct numerical differentiation along these curves. We obtain an expression for the rate of change of eigenvalues with respect to a parameter for a parameter-dependent ‘non-linear eigenvalue problem’. Eigenvalues have the meaning of frequency (or its square) and the role of the parameter is played by the wave number. We then define a quotient which shows Rayleigh-quotient-like behaviour in that it differs from unity by a quantity that scales as ∼ϵ2, where ϵ is the magnitude of a parameter used to perturb an eigenvector. By carrying out first order perturbation analysis of a generalised eigenvalue problem involving two Hermitian matrices, at least one of which is positive definite, we obtain an exact expression for group velocity as a rate of change of an eigenvalue. This form is particularly suited for elastic waves. The connection of the eigensensitivity relationship with the stationarity of the Rayleigh's quotient is explored. Two specific examples of mechanical waveguides are presented: (i) the Timoshenko waveguide and (ii) the bending-torsion coupled elastic waveguide.

Stress concentration and surface instability of anisotropic solids with slightly wavy boundary

This paper presents a first order perturbation analysis of stress concentration and surface morphology instability of elastically anisotropic solids. The boundary of the solids under consideration is periodic along two orthogonal directions. The magnitude of the undulation is sufficiently small so that a half-space model can be used for simplification. We derive expressions for the stress concentration factors and the critical wavelength of the perturbation in terms of the remote stresses, surface energy anisotropy and the elastic anisotropy of the solid. Numerical applications to cubic materials using Barnett–Lothe integrals are also given.

Hydrodynamic and thermal interaction of a periodically oscillating fluid with a porous medium lying over a thick solid plate

In this paper, the hydrodynamic and thermal interaction of a periodically oscillating compressible fluid with a layer of porous medium is analytically investigated. The velocity amplitude of the oscillating fluid is considered to be small when compared to the thermal penetration depth (a transverse length scale) of the thermal energy transport inside the porous layer. The solid wall adjacent to the bottom of the porous layer is considered to be thick and the conduction heat transfer inside that wall is included in this model. The entire problem is treated as a conjugate heat transfer thermoacoustic like problem. The governing momentum and energy equations are simplified and linearized by using a first order perturbation analysis. Such an analysis is usually used to solve the linear thermoacoustic problem in the low Mach number limit and available in the thermoacoustic literature. Analytical expressions for oscillating velocity, temperatures in the porous layer and in the solid wall, heat transfer from the solid wall to the porous medium, and the entropy generation rate are calculated and graphically presented.

Effect of the Darcy number on the energy flow and operating conditions of a thermoacoustic porous-medium system

In this paper, a simplified porous medium thermoacoustic system is modeled to observe its energy interaction characteristics and identify its operating conditions mainly as a function of porous medium Darcy number. The governing Darcy–Brinkman momentum equation and energy equation are simplified and linearized by using a first order perturbation analysis. Similar perturbation analysis is usually used to solve the linear thermoacoustic problem in the low Mach number limit. Simplified momentum and energy equations are solved, in the frequency domain, in order to obtain the expressions of the fluctuating velocity ( u 1) and temperature ( T 1). Time averaged and space averaged heat fluxes and work fluxes are calculated using the expressions of fluctuating velocity and temperature. The effects of the drive ratio (DR), Darcy number ( Da δ ), temperature gradient (∇ T m ), and frequency ( f) on the heat flux, work flux, and operating conditions are discussed and graphically presented.

DETERMINATION OF THE STRONG COUPLING CONSTANT FROM NLO QCD ANALYSIS OF PROTON STRUCTURE FUNCTION

In this article we present a determination of the strong coupling constant and parton distribution functions (PDFs) based on a next-to-leading order (NLO) perturbative QCD analysis of proton structure function. More precisely, we extract [Formula: see text] and PDFs by fitting perturbative QCD predictions to the data from the measurements of the proton structure function [Formula: see text] in deep inelastic scattering, which are based on perturbative QCD calculations up to NLO. We obtain at NLO [Formula: see text] in the variable-flavor number scheme.

Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor

The problem of computing the best rank-$(p,q,r)$ approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.

Effect of dielectric constant tuning on a photonic cavity frequency and Q-factor

For practical applications in quantum electrodynamics, it has been proposed to produce frequency tuning or Q-switching by dynamically changing the dielectric constant around a nano-cavity. Local changes in the dielectric constant of a photonic cavity with finite-lifetime, may affect not only the frequency of electromagnetic cavity modes but also their quality-factor (Q). Thus, it is important to have prediction capability regarding the combined effect of these changes. Here perturbation theory, usually applied to eigenmodes with real eigenvalues, is formulated in the complex domain, in which the eigen-frequency imaginary part is related to the Q-factor. Normalizable leaky modes, and bi-orthogonality in a finite volume are the basis for such a formulation. We introduce such capabilities by presenting semi-analytical expressions of first order perturbation analysis for a 3D cavity with radiation losses. The obtained results are in good agreement with numerical calculations.