Inverse Problems Involving Generalized Axial-Symmetric Helmholtz Equation

MSC 2010: 35J05, 33C10, 45D05

Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates

2000 Mathematics Subject Classification: 35B40, 35L15.

Green's function for paraxial equation

The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Inverse space-dependent force problems for the wave equation

The determination of the displacement and the space-dependent force acting on a vibrating structure from measured final or time-average displacement observation is thoroughly investigated. Several aspects related to the existence and uniqueness of a solution of the linear but ill-posed inverse force problems are highlighted. After that, in order to capture the solution a variational formulation is proposed and the gradient of the least-squares functional that is minimized is rigorously and explicitly derived. Numerical results obtained using the Landweber method and the conjugate gradient method are presented and discussed illustrating the convergence of the iterative procedures for exact input data. Furthermore, for noisy data the semi-convergence phenomenon appears, as expected, and stability is restored by stopping the iterations according to the discrepancy principle criterion once the residual becomes close to the amount of noise. The present investigation will be significant to researchers concerned with wave propagation and control of vibrating structures.

A method of fundamental solutions for theaxisymmetric backward heat equation

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

A coherent master equation for active mode locking in lasers

We present the derivation of a new master equation for active mode locking in lasers that fully takes into account the coherent effects of the light matter interaction through a peculiar adiabatic elimination technique. The coherent effects included in our model could be relevant to describe properly mode-locked semiconductor lasers where the standard Haus' Master Equation predictions show some discrepancy with respect to the experimental results and can be included in the modelling of other mode locking techniques too.

Az ügyfélszerzési mód hatásának vizsgálata a vevőelégedettségre és a lojalitásra a szervezeti piacon (Analyzing the impact of the customer acquisition mode to the customer satisfaction and loyalty on the corporate market – assessing the effect of recommendation with applying structural equation modelling)

A kutatás célja a marketingeszközök hosszú távú hatásának pontosabb megértése szervezetközi viszonylatban a vevőértékelési modellek egyik nehezen számszerűsíthető tényezője, az ajánlás hatásának vizsgálata által. A hatások elemzésére a strukturális egyenlőségek módszerét (Structural Equation Modelling) alkalmazta a szerző. Rámutatott, hogy az ajánlással szerzett ügyfelek elégedettebbek, lojálisabbak és gyakrabban ajánlják a vállalatot a más módon szerzett ügyfeleknél. Az összefüggések feltárása és bizonyítása különösen az ajánlás kumulatív hatása miatt jelentős. Az eredmények gyakorlati alkalmazásával lehetőség nyílik az ügyfélkör differenciáltabb, értékalapú szegmentációjára, amely pontosabb célcsoport-meghatározást lesz lehetővé, és hosszú távon hozzájárul a vállalat optimális ügyfélportfóliójának kialakításához. ______ The research is aimed at more precise understanding of longterm effects of marketing tools in business to business relations by analysing the impacts of recommendation potential, one of the hardly measurable factors of customer value concept. Structural Equation Modelling is applied for conducting effect analysis. The results show that customers acquired with recommendation are more satisfied, more loyal, and make more recommendation that other customer. These results are more interesting if we take the cumulative effect of recommendation in account. They provide bases for a more differentiated segmentation of customers, which results in a more accurate identification of target groups. In the long-run, the application of the customer-value concept considerably contributes to creating an optimal customer portfolio for companies.

Using structural equation modeling to assess two theoretical models that explain dropout

This study evaluated the relative fit of both Finn's (1989) Participation-Identification and Wehlage, Rutter, Smith, Lesko and Fernandez's (1989) School Membership models of high school completion to a sample of 4,597 eighth graders taken from the National Educational Longitudinal Study of 1988, (NELS:88), utilizing structural equation modeling techniques. This study found support for the importance of educational engagement as a factor in understanding academic achievement. The Participation-Identification model was particularly well fitting when applied to the sample of high school completers, dropouts (both overall and White dropouts) and African-American students. This study also confirmed the contribution of school environmental factors (i.e., size, diversity of economic and ethnic status among students) and family resources (i.e., availability of learning resources in the home and parent educational level) to students' educational engagement. Based on these findings, school social workers will need to be more attentive to utilizing macro-level interventions (i.e., community organization, interagency coordination) to achieve the organizational restructuring needed to address future challenges. The support found for the Participation-Identification model supports a shift in school social workers' attention from reactive attempts to improve the affective-interpersonal lives of students to proactive attention to their academic lives. The model concentrates school social work practices on the central mission of schools, which is educational engagement. School social workers guided by this model would be encouraged to seek changes in school policies and organization that would facilitate educational engagement. ^

Use of structural equation modeling to examine the association between breast cancer risk perception and repeat screening mammography among United States women

Breast cancer is the second leading cause of cancer death in United States women, estimated to be diagnosed in 1 out of 8 women in their lifetime. Screening mammography detects breast cancer in its pre-clinical stages when treatment strategies have the greatest chance of success, and is currently the only population-wide prevention method proven to reduce the morbidity and mortality associated with breast cancer. Research has shown that the majority of women are not screened annually, with estimates ranging front 6% - 30% of eligible women receiving all available annual mammograms over a 5-year or greater time frame. Health behavior theorists believe that perception of risk/susceptibility to a disease influences preventive health behavior, in this case, screening mammography The purpose of this dissertation is to examine the association between breast cancer risk perception and repeat screening mammography using a structural equation modeling (SEM) framework. A series of SEM multivariate regressions were conducted using self-reported, nationally representative data from the 2005 National Health Interview Survey. Interaction contrasts were tested to measure the potential moderating effects of variables which have been shown to be predictive of mammography use (physician recommendation, economic barriers, structural barriers, race/ethnicity) on the association between breast cancer risk perception and repeat mammography, while controlling for the covariates of age, income, region, nativity, and educational level. Of the variables tested for moderation, results of the SEM analyses identify physician recommendation as the only moderator of the relationship between risk perception and repeat mammography, thus the potentially most effective point of intervention to increase mammography screening, and decrease the morbidity and mortality associated with breast cancer. These findings expand the role of the physician from recommendation to one of attenuating the effect of risk perception and increasing repeat screening. The long range application of the research is the use of the SEM methodology to identify specific points of intervention most likely to increase preventive behavior in population-wide research, allowing for the most effective use of intervention funds.^

Quantum transition state theory and solving a first order master equation for complex reactions numerically

The field of chemical kinetics is an exciting and active field. The prevailing theories make a number of simplifying assumptions that do not always hold in actual cases. Another current problem concerns a development of efficient numerical algorithms for solving the master equations that arise in the description of complex reactions. The objective of the present work is to furnish a completely general and exact theory of reaction rates, in a form reminiscent of transition state theory, valid for all fluid phases and also to develop a computer program that can solve complex reactions by finding the concentrations of all participating substances as a function of time. To do so, the full quantum scattering theory is used for deriving the exact rate law, and then the resulting cumulative reaction probability is put into several equivalent forms that take into account all relativistic effects if applicable, including one that is strongly reminiscent of transition state theory, but includes corrections from scattering theory. Then two programs, one for solving complex reactions, the other for solving first order linear kinetic master equations to solve them, have been developed and tested for simple applications.

A Look Back To Look Forward: New Patterns In The Supply/Demand Equation In The Lodging Industry

In his dialogue entitled - A Look Back to Look Forward: New Patterns In The Supply/Demand Equation In The Lodging Industry - by Albert J. Gomes, Senior Principal, Pannell Kerr Forster, Washington, D.C. What the author intends for you to know is the following: “Factors which influence the lodging industry in the United States are changing that industry as far as where hotels are being located, what clientele is being served, and what services are being provided at different facilities. The author charts these changes and makes predictions for the future.” Gomes initially alludes to the evolution of transportation – the human, animal, mechanical progression - and how those changes, in the last 100 years or so, have had a significant impact on the hotel industry. “A look back to look forward treats the past as prologue. American hoteliers are in for some startling changes in their business,” Gomes says. “The man who said that the three most important determinants for the success of a hotel were “location, location, location” did a lot of good only in the short run.” Gomes wants to make you aware of the existence of what he calls, “locational obsolescence.” “Locational obsolescence is a fact of life, and at least in the United States bears a direct correlation to evolutionary changes in transportation technology,” he says. “…the primary business of the hospitality industry is to serve travelers or people who are being transported,” Gomes expands the point. Tied to the transportation element, the author also points out an interesting distinction between hotels and motels. In addressing, “…what clientele is being served, and what services are being provided at different facilities,” Gomes suggests that the transportation factor influences these constituents as well. Also coupled with this discussion are oil prices and shifts in transportation habits, with reference to airline travel being an ever increasing method of travel; capturing much of the inter-city travel market. Gomes refers to airline deregulation as an impetus. The point being, it’s a fluid market rather than a static one, and [successful] hospitality properties need to be cognizant of market dynamics and be able to adjust to the variables in their marketplace. Gomes provides many facts and figures to bolster his assertions. Interestingly and perceptively, at the time of this writing, Gomes alludes to America’s deteriorating road and bridge network. As of right now, in 2009, this is a major issue. Gomes rounds out this study by comparing European hospitality trends to those in the U.S.