Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

Localized waves in optical systems with periodic dispersion and nonlinearity management

We overview our recent developments in the theory of dispersion-managed (DM) solitons within the context of optical applications. First, we present a class of localized solutions with a period multiple to that of the standard DM soliton in the nonlinear Schrödinger equation with periodic variations of the dispersion. In the framework of a reduced ordinary differential equation-based model, we discuss the key features of these structures, such as a smaller energy compared to traditional DM solitons with the same temporal width. Next, we present new results on dissipative DM solitons, which occur in the context of mode-locked lasers. By means of numerical simulations and a reduced variational model of the complex Ginzburg-Landau equation, we analyze the influence of the different dissipative processes that take place in a laser.

On the theory of self-similar parabolic optical pulses

Self-similar optical pulses (or “similaritons”) of parabolic intensity profile can be found as asymptotic solutions of the nonlinear Schr¨odinger equation in a gain medium such as a fiber amplifier or laser resonator. These solutions represent a wide-ranging significance example of dissipative nonlinear structures in optics. Here, we address some issues related to the formation and evolution of parabolic pulses in a fiber gain medium by means of semi-analytic approaches. In particular, the effect of the third-order dispersion on the structure of the asymptotic solution is examined. Our analysis is based on the resolution of ordinary differential equations, which enable us to describe the main properties of the pulse propagation and structural characteristics observable through direct numerical simulations of the basic partial differential equation model with sufficient accuracy.

On the theory of self-similar parabolic optical pulses

Self-similar optical pulses (or “similaritons”) of parabolic intensity profile can be found as asymptotic solutions of the nonlinear Schr¨odinger equation in a gain medium such as a fiber amplifier or laser resonator. These solutions represent a wide-ranging significance example of dissipative nonlinear structures in optics. Here, we address some issues related to the formation and evolution of parabolic pulses in a fiber gain medium by means of semi-analytic approaches. In particular, the effect of the third-order dispersion on the structure of the asymptotic solution is examined. Our analysis is based on the resolution of ordinary differential equations, which enable us to describe the main properties of the pulse propagation and structural characteristics observable through direct numerical simulations of the basic partial differential equation model with sufficient accuracy.

Function interval arithmetic

We propose an arithmetic of function intervals as a basis for convenient rigorous numerical computation. Function intervals can be used as mathematical objects in their own right or as enclosures of functions over the reals. We present two areas of application of function interval arithmetic and associated software that implements the arithmetic: (1) Validated ordinary differential equation solving using the AERN library and within the Acumen hybrid system modeling tool. (2) Numerical theorem proving using the PolyPaver prover. © 2014 Springer-Verlag.

Localized waves in optical systems with periodic dispersion and nonlinearity management

We overview our recent developments in the theory of dispersion-managed (DM) solitons within the context of optical applications. First, we present a class of localized solutions with a period multiple to that of the standard DM soliton in the nonlinear Schrödinger equation with periodic variations of the dispersion. In the framework of a reduced ordinary differential equation-based model, we discuss the key features of these structures, such as a smaller energy compared to traditional DM solitons with the same temporal width. Next, we present new results on dissipative DM solitons, which occur in the context of mode-locked lasers. By means of numerical simulations and a reduced variational model of the complex Ginzburg-Landau equation, we analyze the influence of the different dissipative processes that take place in a laser.

Coherence and incoherence in an optical comb

We demonstrate a coexistence of coherent and incoherent modes in the optical comb generated by a passively mode-locked quantum dot laser. This is experimentally achieved by means of optical linewidth, radio frequency spectrum, and optical spectrum measurements and confirmed numerically by a delay-differential equation model showing excellent agreement with the experiment. We interpret the state as a chimera state. © 2014 American Physical Society.

Weak form equation–based finite-element modeling of viscoelastic asphalt mixtures

The objective of this study is to demonstrate using weak form partial differential equation (PDE) method for a finite-element (FE) modeling of a new constitutive relation without the need of user subroutine programming. The viscoelastic asphalt mixtures were modeled by the weak form PDE-based FE method as the examples in the paper. A solid-like generalized Maxwell model was used to represent the deforming mechanism of a viscoelastic material, the constitutive relations of which were derived and implemented in the weak form PDE module of Comsol Multiphysics, a commercial FE program. The weak form PDE modeling of viscoelasticity was verified by comparing Comsol and Abaqus simulations, which employed the same loading configurations and material property inputs in virtual laboratory test simulations. Both produced identical results in terms of axial and radial strain responses. The weak form PDE modeling of viscoelasticity was further validated by comparing the weak form PDE predictions with real laboratory test results of six types of asphalt mixtures with two air void contents and three aging periods. The viscoelastic material properties such as the coefficients of a Prony series model for the relaxation modulus were obtained by converting from the master curves of dynamic modulus and phase angle. Strain responses of compressive creep tests at three temperatures and cyclic load tests were predicted using the weak form PDE modeling and found to be comparable with the measurements of the real laboratory tests. It was demonstrated that the weak form PDE-based FE modeling can serve as an efficient method to implement new constitutive models and can free engineers from user subroutine programming.

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

Integration of the Manakov-PMD Equation with Precomputed M(w) Matrices for PMD Simulation

A novel direct integration technique of the Manakov-PMD equation for the simulation of polarisation mode dispersion (PMD) in optical communication systems is demonstrated and shown to be numerically as efficient as the commonly used coarse-step method. The main advantage of using a direct integration of the Manakov-PMD equation over the coarse-step method is a higher accuracy of the PMD model. The new algorithm uses precomputed M(w) matrices to increase the computational speed compared to a full integration without loss of accuracy. The simulation results for the probability distribution function (PDF) of the differential group delay (DGD) and the autocorrelation function (ACF) of the polarisation dispersion vector for varying numbers of precomputed M(w) matrices are compared to analytical models and results from the coarse-step method. It is shown that the coarse-step method achieves a significantly inferior reproduction of the statistical properties of PMD in optical fibres compared to a direct integration of the Manakov-PMD equation.

Ethnic groups as market segments: implications of the demand structure

Ethnic market potential in Britain has not yet been thoroughly researched. Important recent trends have focused mainly on the affective and emotional aspects of ethnicity, and included deliberations on the emergence of a revitalised neo-ethnic consciousness; its identification; politicisation, and the impact on it; of a rising third-world consciousness. This investigation attempts to take cognizance of the consuner demand of the ethnic Asian and West Indian groups, as specific market segments. It discusses the rationale for ethnic segmentation on the underlying premise, that the starting point for all product marketing is a response to perceived market opportunities. On the basis of this approach, the UK laundry detergent and automobile markets were investigated; as being representative of product categories constitutirg extremes along the purchase-search-time continuun in consuner decision-making. Ethnic groups were further analysed for their retail patronage patterns; media usage, and the differential effectiveness of alternative advertisirg strategies. The basic technique of marketing research namely the sample survey, was used with the aim of applying scientific techniques in obtaining information on ethnic groups. The integrated marketirg framework utilised allowed, moreover, for the collection of market research data on the specific issues of ethnic product penetration dealing with retailing, advertising and product promotion. The evidence highlights the fact that the cultural orientations of ethnic groups are instrunental in providing for differential demand structures. It points to the answer that ethnicity is an anchor not only for a deeper sense of identity; but also serves as a focus for the economic interests of ethnic groups. On this basis it is argued here, that since cultural levelling would eventually produce stagnation; current marketing strategies should utilise ethnic diversity as an econanic artifact; which; per se is necessary for profitability and growth; especially in innovative product design and development.

Differential perception of innovation with particular reference to the organisational context

Differential perception of innovation is a research area which has been advocated as a suitable topic for study in recent years. It developed from the problems encountered within earlier perception of innovation studies which sought to establish what characteristics of an innovation affected the ease of its adoption. While some success was achieved In relating perception of innovation to adoption behaviour, variability encountered Within groups expected - to fercelve innovation similarly suggested that the needs and experiences of the potential adopter were significantly affecting the research findings. Such analysis being supported by both sociological and psychological perceptual research. The present study sought to identify the presence of differential perception of innovation and explore the nature of the process. It was decided to base the research in an organisational context and to concentrate upon manufacturing innovation. It has been recognised that such adoption of technological innovation is commonly the product of a collective decision-making process, involving individuals from a variety of occupational backgrounds, both in terms of occupational speciality and level within the hierarchy. Such roles appeared likely to significantly influence perception of technological innovation, as gathered through an appropriate measure and were readily identifiable. Data vas collected by means of a face-to-face card presentation technique, a questionnaire and through case study material. Differential perception of innovation effects were apparent In the results, many similarities and differences of perception being related to the needs and experiences of the individuals studied. Phenomenological analysis, which recognises the total nature of experience in infiuencing behaviour, offered the best means of explaining the findings. It was also clear that the bureaucratic model of role definition was not applicable to the area studied, it seeming likely that such definitions are weaker under conditions of uncertainty, such as encountered in innovative decision-making.

Differential patterns of activity and functional connectivity in emotion processing neural circuitry to angry and happy faces in adolescents with and without suicide attempt

Background - Neural substrates of emotion dysregulation in adolescent suicide attempters remain unexamined. Method - We used functional magnetic resonance imaging to measure neural activity to neutral, mild or intense (i.e. 0%, 50% or 100% intensity) emotion face morphs in two separate emotion-processing runs (angry and happy) in three adolescent groups: (1) history of suicide attempt and depression (ATT, n = 14); (2) history of depression alone (NAT, n = 15); and (3) healthy controls (HC, n = 15). Post-hoc analyses were conducted on interactions from 3 group × 3 condition (intensities) whole-brain analyses (p < 0.05, corrected) for each emotion run. Results - To 50% intensity angry faces, ATT showed significantly greater activity than NAT in anterior cingulate gyral–dorsolateral prefrontal cortical attentional control circuitry, primary sensory and temporal cortices; and significantly greater activity than HC in the primary sensory cortex, while NAT had significantly lower activity than HC in the anterior cingulate gyrus and ventromedial prefrontal cortex. To neutral faces during the angry emotion-processing run, ATT had significantly lower activity than NAT in the fusiform gyrus. ATT also showed significantly lower activity than HC to 100% intensity happy faces in the primary sensory cortex, and to neutral faces in the happy run in the anterior cingulate and left medial frontal gyri (all p < 0.006,corrected). Psychophysiological interaction analyses revealed significantly reduced anterior cingulate gyral–insula functional connectivity to 50% intensity angry faces in ATT v. NAT or HC. Conclusions - Elevated activity in attention control circuitry, and reduced anterior cingulate gyral–insula functional connectivity, to 50% intensity angry faces in ATT than other groups suggest that ATT may show inefficient recruitment of attentional control neural circuitry when regulating attention to mild intensity angry faces, which may represent a potential biological marker for suicide risk.

Surface- and deep-level dissimilarity effects on social integration and individual effectiveness related outcomes in work groups:a meta-analytic integration

Prior research linking demographic (e.g., age, ethnicity/race, gender, and tenure) and underlying psychological (e.g., personality, attitudes, and values) dissimilarity variables to individual group member's work-related outcomes produced mixed and contradictory results. To account for these findings, this study develops a contingency framework and tests it using meta-analytic and structural equation modelling techniques. In line with this framework, results showed different effects of surface-level (i.e., demographic) dissimilarity and deep-level (i.e., underlying psychological) dissimilarity on social integration, and ultimately on individual effectiveness related outcomes (i.e., turnover, task, and contextual performance). Specifically, surface-level dissimilarity had a negative effect on social integration under low but not under high team interdependence. In return, social integration fully mediated the negative relationship between surface-level dissimilarity and individual effectiveness related outcomes under low interdependence. In contrast, deep-level dissimilarity had a negative effect on social integration, which was stronger under high and weaker under low team interdependence. Contrary to our predictions, social integration did not mediate the negative relationship between deep-level dissimilarity and individual effectiveness related outcomes but suppressed positive direct effects of deep-level dissimilarity on individual effectiveness related outcomes. Possible explanations for these counterintuitive findings are discussed. © 2011 The British Psychological Society.