Laplace transformation

The principal objective of the current work is to investigate the combined effects of thermophoretic diffusion and chemical reactions on the dynamics of Williamson fluid, as well as heat and mass transmission, over a continuously magnetized Riga surface. In this analysis, variable thermal properties are also considered. To solve the model dimensionless nonlinear system of equations, an innovative scheme is introduced, the Morlet wavelet neural network algorithm. This scheme validates unparalleled performance and accuracy. Moreover, the statistical procedures such as mean, standard deviation, MSE, MAD, and ENSE validate the exceptional precision of the proposed scheme. The MSE ranges from 10 0 to 10-10 ; MAD is within the interval of 10-01 and 10-10 ; ENSE ranges from 10 0 to 10-15 , while fitness curves fall within 10 0 and 10-10 for all four cases across multiple runs. Additionally, using visual illustrations of the outcome improves the overall understanding of fundamental phenomena. This research work emphasizes the potential practical applications in fluid dynamics and encourages further research to expose new insights and advancements in the field. Combining ANN techniques with numerical simulations opens new research avenues and promotes progress in engineering and fluid dynamics.

This paper presents a exploration of the Tensor Model of Discrete Dynamics (TMDD), a framework that derives physical reality—spacetime, gravity, and matter—from fundamental principles of discrete information dynamics. The model is built on core ontological ideas: discreteness, bounded information, emergence, and universality. At its heart lies the θ-network, a discrete structure of interconnected θ-cells whose collective behavior gives rise to the familiar laws of physics.
Key Results
1. Emergent Spacetime and Gravity:
o The study demonstrates how the dynamics of the θ-network, governed by minimizing information tension, naturally leads to the emergence of spacetime and gravity.
o In the continuum limit, the model reduces to the equations of General Relativity (GR), showing that GR is not fundamental but an approximation of deeper discrete dynamics.
2. Spectral Representation and Stability:
o The action functional of the model is analyzed in terms of spectral modes, revealing how coherence, geometry, and entropy arise from the network’s structure.
o Stability conditions are derived, defining the boundaries where phase transitions occur, leading to phenomena like decoherence or the formation of topological defects.
3. Phase Transitions and Bifurcations:
o The θ-network undergoes phase transitions, analogous to quantum-to-classical transitions or the emergence of matter-like structures.
o Bifurcations in parameter space explain how the network "chooses" between different macroscopic states, such as coherent quantum regimes or classical geometries.
4. Dark Matter and Dark Energy as Residual Effects:
o Deviations from classical GR, termed the "tail" of the action, provide natural candidates for dark matter (linked to topological defects) and dark energy (arising from nonlocal information consensus).
o These effects are not ad hoc additions but emerge from the model’s discrete foundation.
5. Variable Fundamental Constants:
o Gravitational constants like G, Λ, and *c* are not fixed but depend on the network’s state, offering a potential explanation for observed anomalies in cosmology.
Implications
The TMDD framework shifts the paradigm: spacetime and gravity are not pre-existing entities but emergent properties of a deeper informational structure. This work bridges discrete ontology with classical physics, providing a foundation for understanding quantum gravity, dark matter, and the evolution of the universe.

The microbending losses induced by axial strain in tightly-jacketed double-coated optical fibers are investigated. The lateral pressure in the glass fiber is derived, which is determined by the axial strain, material's properties of polymeric coatings and their thicknesses. The lateral pressure in the glass fiber would produce microbending loss. To minimize such a microbending loss, the Young's moduli of the secondary coating and jacket, and the Poisson's ratios of the primary and secondary coatings should be increased. On the other hand, the Young's modulus of the primary coating and the Poisson's ratio of the jacket should be decreased. However, based on the mechanical strength of polymeric coatings, there exist optimum values for the thicknesses of the primary and secondary coatings.

The microbending losses induced by axial strain in tightly-jacketed double-coated optical fibers are investigated. The lateral pressure in the glass fiber is derived, which is determined by the axial strain, material's properties of polymeric coatings and their thicknesses. The lateral pressure in the glass fiber would produce microbending loss. To minimize such a microbending loss, the Young's moduli of the secondary coating and jacket, and the Poisson's ratios of the primary and secondary coatings should be increased. On the other hand, the Young's modulus of the primary coating and the Poisson's ratio of the jacket should be decreased. However, based on the mechanical strength of polymeric coatings, there exist optimum values for the thicknesses of the primary and secondary coatings.

We study an invariant solution of the equations of thermodiffusion motion and of a viscous heatconducting fluid, which is treated as an unidirectional motion in a circular pipe with a common interface under the action of an unsteady pressure gradient. A priori estimates of the velocity, temperature and concentration are obtained. The steady state is determined and it is shown (under some conditions on the pressure gradient) that, at larger times, this state is the limiting one. On the other hand, if the pressure gradient of binary mixture tends to zero sufficiently fast, then the motion in pipe is slowed down by the viscous friction.

The Heisenberg equation of motion for the surface spin operator in a semi-infinite S = -XY chain is exactly solved at infinite temperatures via the recurrence-relations method of Lee. It is found that the time evolution of the surface spin shows Bessel-like on-site relaxation that is drastically different from the corresponding gaussian relaxation of a bulk spin. This difference is shown to be related to broken translational symmetry in the former case. The memory function and the random force for the surface spin is exactly determined. It is found that the surface spin dynamics is, in some sense, "harmonic" in nature.

The problems of systems identification, analysis and optimal control have been recently studied using orthogonal functions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, Haar and many other functions. In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convolution product and expanded to the Riemann-Liouville fractional integral of matrices. For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations, the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, several systems are solved by the new and other approaches and illustrative examples are also considered.

discussed. The identification of two MR damper models as well as their cross validation emphasize the importance of the persistence of experimental inputs and the combinations of rod displacement and electric current sequences for better modeling. New findings in Design of Experiments for model identification are presented.

The finite difference (FD) technique is used to investigate the air yap effect on the coupling between two planar microstrip lines. The strip lines are embedded between a dielectric substrate and a dielectric overlay. The air gap between the substrate and the overlay is present due to the thickness of the strips and~or the roughness of the planar surfaces. A quasi-static FD technique based on the solution of Laplace's equation is used to solve the potential distribution in the region around the strips. The FD mesh is truncated by an artificial boundary to allow for a numerical solution with the available computer resources. At this artificial boundary an approximate boundary condition is used to truncate the FD mesh. From the potential distribution, the charge and the capacitance and inductance matrices of the stripline system are evaluated. The coupling or crosstalk between the lines is then defined and computed. The effects of the conducting strips thickness and the properties of the materials between the strips on the coupling are illustrated through numerical examples.

A unified analysis is derived for the treatment of the laminar unsteady boundary layer equations with heat conductive mass transfer to establish conditions under which similarity solutions are possible. The method of new similarity technique is applied to derive the various conditions under which similarity variables for the unsteady thermal boundary layer flows is exist. Controlled similarity equations reduce to some well known flow equations. Numerical solution is discussed in detail.

In this paper, the Laplace decomposition method is employed to obtain approximate analytical solutions of the linear and nonlinear fractional diffusion-wave equations. This method is a combined form of the Laplace transform method and the Adomian decomposition method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The fractional derivative described here is in the Caputo sense. Some illustrative examples are presented and the results show that the solutions obtained by using this technique have close agreement with series solutions obtained with the help of the Adomian decomposition method.

This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on-off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in This work has been carried out partly in the Dutch BSIK/BRICKS project.

A review, study, and quick reference guide to partial differential equations from a science and engineering perspective. Included are fundamental concepts, descriptions of common solution techniques (separation of variables, integral transforms, etc.), and other supporting mathematical concepts.

We propose a new method for estimating the probability mass function (pmf) of a discrete and finite random variable from a small sample. We focus on the observed counts-the number of times each value appears in the sample-and define the maximum likelihood set (MLS) as the set of pmfs that put more mass on the observed counts than on any other set of counts possible for the same sample size. We characterize the MLS in detail in this article. We show that the MLS is a diamond-shaped subset of the probability simplex [0, 1] k bounded by at most k × (k -1) hyperplanes, where k is the number of possible values of the random variable. The MLS always contains the empirical distribution, as well as a family of Bayesian estimators based on a Dirichlet prior, particularly the wellknown Laplace estimator. We propose to select from the MLS the pmf that is closest to a fixed pmf that encodes prior knowledge. When using Kullback-Leibler distance for this selection, the optimization problem comprises finding the minimum of a convex function over a domain defined by linear inequalities, for which standard numerical procedures are available. We apply this estimate to language modeling using Zipf's law to encode prior knowledge and show that this method permits obtaining state-of-the-art results while being conceptually simpler than most competing methods.

Thermal stratification of magneto-hydrodynamic self-similar Casson-Walters-B fluids past a porous vertical plate is explored in this paper. The occurrence of this nature is plausible in industrial processes such as polymer industries. The numerical investigation in this paper is a binary solution which consists of Casson and Walters-B type of fluids. The flow equations that govern the study are nonlinear coupled partial differential equations (PDEs). These sets of PDEs were evaluated using suitable variables to obtain nonlinear ordinary differential equations (ODEs). The set of transformed ODEs were solved numerically by employing the spectral homotopy analysis method (SHAM). SHAM combines the Chebyshev pseudo-spectral approach with the homotopy analysis method in solving set of differential equations. From the physical point of view, the significance of each pertinent flow parameters is discussed with the help of graphical results. It was found that the plastic dynamic viscosity greatly affects the velocity of Casson fluid within the boundary layer. In the thermally-stratified boundary layer, the magnetic parameter was found to slow down the motion of an electrically conducting fluid due to Lorentz force. The rate of heat transport was determined by pertinent flow parameters such as thermal radiation, viscous dissipation and heat generation. The distribution of thermal radiation becomes very significant at a very high temperature and viscous dissipation in frictional heating of fluid particles. The validation conducted in this paper shows the accuracy of SHAM. The comparison of the present results and previously published works are in good agreement.

An exact analysis of unsteady free convection flow of fractionalized viscous fluid over an oscillating vertically inclined plate is obtained. The phenomenon of exponential heating is added into account for thermal aspects of an inclined plate. Moreover, in the model of problem, additional effects like thermo diffusion and slip are also used. Caputo fractional derivative is used in the model. The novelty of present study is to analyze the effect of angle of inclination on the flow phenomena and the model is generalized by using Fourier's and Fick's laws. The governing dimensionless equations for velocity, concentration, and temperature profiles are solved using Laplace transform method and compared graphically. The effects of different parameters like fractional parameter, thermo diffusion parameter, and slip parameter are discussed through numerous graphs. From figures, it is observed that Prandtl and Smith numbers have decreasing effect on velocity profile, whereas thermo diffusion and mass Grashof numbers have increasing effect on velocity of fluid.

The potential of diffusion-ordered 2D NMR spectroscopy (DOSY) for the analysis of solutions of polymer mixtures and polymers with complex molecular mass distributions is investigated. Diffusion coefficient labeling in NMR is generally achieved by stepwise ramping up of the amplitudes of pairs of pulsed field gradients (PFGs). After Fourier transformation in the acquisition dimension and an inverse Laplace transform (ILT) with respect to the square of the gradient strength, 2D spectra are obtained that show the chemical shift along one dimension and the translational diffusion coefficient along the other. Since polymers may have broad, nonsymmetric or complex (bi-and multimodal) molecular weight distributions (MWDs), the diffusion coefficient distribution should follow the main features of the MWD. However, the calculation of the diffusion coefficient distribution involves a numerically unstable data inversion (ILT), which limits the resolution in the diffusion dimension. The applications of DOSY NMR techniques to the study of polymer blends and for the determination of MWDs of polymers with PFG NMR are assessed. DOSY spectra recorded from industrial polypropylene and polystyrene samples and mixtures are shown and interpreted to illustrate the features of this technique.

Nos centramos en los siguientes apartados en la probabilidad de que las reservas alcancen un determinado nivel b , y finalmente trabajamos con el modelo modificado con una barrera de dividendos constantes, planteando el cálculo de la esperanza del valor actual de los dividendos. PALABRAS CLAVE: transformadas de Laplace, probabilidad de supervivencia, barrera constante, esperanza del valor actual de los dividendos.

In this paper we analyze the time of ruin in a risk process with the interclaim times being Erlang(n) distributed and a constant dividend barrier. We obtain an integro-differential equation for the Laplace Transform of the time of ruin. Explicit solutions for the moments of the time of ruin are presented when the individual claim amounts have a distribution with rational Laplace transform. Finally, some numerical results and a compare son with the classical risk model, with interclaim times following an exponential distribution, are given

Consideramos el proceso clásico del riesgo modificado con la introducción de una barrera de dividendos constante, de tal forma que cuando el proceso de reservas alcanza la barrera se pagan dividendos hasta la ocurrencia del siguiente siniestro. En la literatura actuarial se plantea el cálculo de W(u,b) definida como la esperanza del valor actual, a un tanto constante, de los dividendos repartidos hasta el momento de ruina en un modelo con barrera constante b(t)=b. Se calcula el valor de la barrera que maximiza dicha esperanza. En este trabajo se realizan dos contribuciones en este tema. En primer lugar se profundiza en el análisis de W(u,b), proponiéndose combinaciones de las variables de control que proporcionan resultados económicamente óptimos. En segundo lugar se definen nuevas medidas relacionadas con W(u,b) que la complementan y pueden ayudar al decisor en el proceso de definición de las variables de control.

This paper presents a class of approximations to a type of wave field for which the dispersion relation is transcendental. The approximations have two defining characteristics: (i) they give the field shape exactly when the frequency and wavenumber lie on a grid of points in the (frequency, wavenumber) plane and (ii) the approximate dispersion relations are polynomials that pass exactly through points on this grid. Thus, the method is interpolatory in nature, but the interpolation takes place in (frequency, wavenumber) space, rather than in physical space. Full details are presented for a non-trivial example, that of antisymmetric elastic waves in a layer. The method is related to partial fraction expansions and barycentric representations of functions. An asymptotic analysis is presented, involving Stirling's approximation to the psi function, and a logarithmic correction to the polynomial dispersion relation.

The quasi-TEM characteristics of a microstrip line on a multilayered cylindrical dielectric substrate partially embedded in a perfectly conducting ground plane are rigorously determined. Exact expressions for the potential distributions inside and outside the substrate, charge distributions on the strip, and capacitance are derived.

The transient elastodynamic response of a transversely isotropic material containing a semi-infinite crack under uniform impact loading on the faces is examined. The crack lies in a principle plane of the material, but the crack front does not coincide with a principle direction. Rather, the crack front is at an angle to a principle direction and thus the problem becomes more three-dimensional in nature. Three loading modes are considered, i.e., opening, in-plane shear and anti-plane shear. The solutions for the stress intensity factor history around the crack tip are found. Laplace and Fourier transforms together with the Wiener-Hopf technique are employed to solve the equations of motion directly. The asymptotic expression of stress near the crack tip leads to a closed-form solution for the dynamic stress intensity factor for each loading mode. It is found that the stress intensity factors are proportional to the square root of time as expected. Results given here converge to know...

A linear advectionediffusion equation with variable coefficients in a one-dimensional semi-infinite medium is solved analytically using a Laplace transformation technique, for two dispersion problems: temporally dependent dispersion along a uniform flow and spatially dependent dispersion along a non-uniform flow. Uniform and varying pulse type input conditions are considered. The variable coefficients in the advectionediffusion equation are reduced into constant coefficients with the help of two transformations which introduce new space and time variables, respectively. It is observed that the temporal dependence of increasing nature causes faster solute transport through the medium than that of decreasing nature. Similarly the effect of inhomogeneity of the medium on the solute transport is studied with the help of a function linearly interpolated in a finite space domain.

Influence des conditions de Robin sur le spectre du Laplacien magnétique Résumé : Cette thèse concerne l'étude spectrale dans le régime semi-classique de l'opérateur de Schrödinger en présence d'un champ magnétique sur un domaine borné et régulier avec condition de Robin au bord. L'objectif est de décrire l'influence des conditions de Robin sur le spectre magnétique lorsque le paramètre semi-classique tend vers zéro et lorsque le champ magnétique joue un rôle majeur en concurrence avec la condition de Robin. La description précise des éléments propres passe par la compréhension de différents opérateurs modèles. Dans la première partie, lorsque le champ magnétique est d'intensité modérée et dans la limite du couplage fort, on montre que les valeurs propres du Laplacien de Robin sont bien approchées par celles d'un opérateur de Schrödinger périodique 1D sur la frontière. Lorsque la courbure a un unique maximum non dégénéré, on estime l'écart spectral entre les valeurs propres successives. Dans le cas du disque, on révèle explicitement la contribution du champ magnétique sur la première valeur propre.

Inverse Laplace transform. A spectral reconstruction method, based on the inverse Laplace transform of the attenuation curve, was implemented to dental X-ray units. The validity of the method is verified and dental X-ray spectra are studied.

Forward-backward solute dispersion with an intermediate point source in one-dimensional semi-infinite homogeneous porous media is studied in this paper. Solute transport under sorption conditions, firstorder decay and zero-order production terms are included. The first type of boundary condition is taken as a constant point source at an intermediate point from where forward and backward solute dispersion is examined. The Laplace transform method is adopted to solve the governing equation analytically. All the analytical results are obtained in graphical form to investigate the forward-backward solute transport in porous media for various hydrological input data. The graphical nature of the analytical solution is compared with numerical data taken from existing literature and similar results are obtained. Also, numerical solution of the governing equation is obtained by the Crank-Nicolson finite difference scheme and validated with the analytical solution, which demonstrates good agreement between them. Accuracy of the solution is also observed by using RMSE.

An analytical integral transformation of the thermal wave propagation problem in a finite slab is obtained through the generalized integral transform technique (GITT). The use of the GITT approach in the analysis of the hyperbolic heat conduction equation leads to a coupled system of second order ordinary differential equations in the time variable. The resulting transformed ODE system is then numerically solved by Gear's method for stiff initial value problems. Numerical results are presented for the local and average temperatures with different Biot numbers and dimensionless thermal relaxation times, permitting a critical evaluation of the technique performance. A comparison is also performed with previously reported results in the literature for special cases and with those produced through the application of the Laplace transform method (LTM), and the finite volume-Gear method (FVGM).

Inverse Laplace transform. A spectral reconstruction method, based on the inverse Laplace transform of the attenuation curve, was implemented to dental X-ray units. The validity of the method is verified and dental X-ray spectra are studied.

In this paper, the Laplace decomposition method is employed to obtain approximate analytical solutions of the linear and nonlinear fractional diffusion-wave equations. This method is a combined form of the Laplace transform method and the Adomian decomposition method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The fractional derivative described here is in the Caputo sense. Some illustrative examples are presented and the results show that the solutions obtained by using this technique have close agreement with series solutions obtained with the help of the Adomian decomposition method.

The bubble departure diameters and bubble emission frequency have been ^^°^ S^dytveals that the ^^"™™^ bubble emission frequency is the strong function of heat flux. 1 he^«^ & size increase in pressure, however, not as strong-t-c-ses ^^^^^ frequencv of the bubble makes the heat transfer process sluggish. An increase in tnec, the heat leads to higher heat transfer rates due to enhanced turbulence in the process. Therefore then t an f r coefficients are lower at lower pressures and higher at higher pressures and heat fluxes, 'work supported by Army Research Office and Office of Naval Research

Abstract: The bubble departure diameters and bubble emission frequency have been calculated for the nucleate pool boiling data of Engelhom for many refrigerants over a wide range of heat flux and pressure. The pressure ranged from 0.019 bar to 10.55 bar and the ...

This paper presents the reliability and mean time to failure estimation of a deteriorating system that deteriorates with time. The system is two unit active parallel system where the units operates simultaneously. In this study maintenance is not allowed to the system. Laplace transforms is being utilize to solve the mathematical model developed. Expressions for reliability and Mean time to failure (MTTF) have been computed. The objective is to see the effect of deterioration rates on system reliability and MTTF of a non maintained system. It also the objective of this study to see the effect of time on system reliability. Graphical study of the system through reliability and MTTF is obtained to highlight important results. The results have shown that reliability and MTTF of non maintained system decrease as time and deterioration rates increase. Models developed in this paper are important to engineers, maintenance managers, and plant management for proper maintenance decision and for safety of the system as a whole.

We find the distributions in R n for the independent random variables X and Y such that E(X|X + Y) = a(X + Y) and E(q(X)|X + Y) = bq(X + Y) where q runs through the set of all quadratic forms on R n orthogonal to a given quadratic form v. The essential part of this class is provided by distributions with Laplace transforms (1 − 2 c, s + v(s)) −p that we describe completely, obtaining a generalization of a Gindikin theorem. This leads to the classification of natural exponential families with the variance function of type 1 p m ⊗ m − ϕ(m)M v , where M v is the symmetric matrix associated to the quadratic form v and m → ϕ(m) is a real function. These natural exponential families extend the classical Wishart distributions on Lorentz cones already considered by Jensen, and later on by Faraut and Korányi.

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v  ∈  L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estima...

Description of integrated mode profile by determine of κ, γ, δ parameters as functions of the propagation constant (β) and effective refractive index (neff). The profile can be seen from E (x) formula for each guide TE (Transverse Electric) modes. Assumptions given in this slab waveguide is used for wavelength (λ) 1.55 μm, the thickness (d) of the core is 0.9 μm with a type of symmetric step-index slab waveguide, refractive index of n1 is 3.5 and refractive index of n2 is 3, also n3= n1. The results of analysis are presented in ...

Availability and profit of an industrial system are becoming an increasingly important issue. Where the availability of a system increases, the related profit will also increase. The objective of this paper is to present the effect of minor and major maintenance rates, and perfect repair rate on system availability and profit generated. We analyzed the system by using Kolmogorov's forward equations method. Results have shown that perfect repair increases system availability and profit generated than minor and major maintenance. Through combine action of minor maintenance, major maintenance and perfect repair action the system availability and profit generated are improved. Models presented in this paper are important to engineers, maintenance managers, and plant management for proper maintenance analysis, decision and for safety of the system as a whole.

This paper presents the reliability and mean time to failure estimation of a deteriorating system that deteriorates with time. The system is two unit active parallel system where the units operates simultaneously. In this study maintenance is not allowed to the system. Laplace transforms is being utilize to solve the mathematical model developed. Expressions for reliability and Mean time to failure (MTTF) have been computed. The objective is to see the effect of deterioration rates on system reliability and MTTF of a non maintained system. It also the objective of this study to see the effect of time on system reliability. Graphical study of the system through reliability and MTTF is obtained to highlight important results. The results have shown that reliability and MTTF of non maintained system decrease as time and deterioration rates increase. Models developed in this paper are important to engineers, maintenance managers, and plant management for proper maintenance decision and for safety of the system as a whole.

We give new representations of solutions for the periodic linear di¤erence equation of the type xðn þ 1Þ ¼ BðnÞxðnÞ þ bðnÞ, where complex nonsingular matrices BðnÞ and vectors bðnÞ are r-periodic. These are based on the Floquet multipliers and the Floquet exponents, respectively. By using these representations, asymptotic behavior of solutions is characterized by initial values. In particular, we can characterize necessary and su‰cient conditions that the equation has a bounded solution (or a rperiodic solution), and the Massera type theorem by initial values.

We consider the problem of minimizing a fractional quadratic problem involving the ratio of two indefinite quadratic functions, subject to a two-sided quadratic form constraint. This formulation is motivated by the so-called regularized total least squares (RTLS) problem. A key difficulty with this problem is its nonconvexity, and all current known methods to solve it are guaranteed only to converge to a point satisfying first order necessary optimality conditions. We prove that a global optimal solution to this problem can be found by solving a sequence of very simple convex minimization problems parameterized by a single parameter. As a result, we derive an efficient algorithm that produces an-global optimal solution in a computational effort of O(n 3 log −1). The algorithm is tested on problems arising from the inverse Laplace transform and image deblurring. Comparison to other well-known RTLS solvers illustrates the attractiveness of our new method.

In this work, the problem of illuminating a thermoelastic half space by a laser beam is solved by utilizing the fractional order theory of thermoelasticity. The assumptions that the illuminated surface is exposed to a cooling effect and free from traction are considered. The problem is solved using Laplace transform techniques. The inverse Laplace transform has been calculated in numerical fashion. The obtained results are presented graphically.