Evolution Equation

In this paper, we consider the following second-order abstract semilinear evolution equation with past history and time delay. Under suitable conditions on initial data, we prove the well-posedness by using the semigroup arguments. The stability result is also established defining a suitable Lyapunov functional for a larger class of kernels. An application is also given.

In this paper, we present results of $\omega$-order preserving partial contraction mapping creating a continuous time Markov semigroup. We use Markov and irreducible operators and their integer powers to describe the evolution of a random system whose state changes at integer times, or whose state is only inspected at integer times. We concluded that a linear operator $P:\ell^{1}(X_+)\rightarrow \ell^{1}(X_+)$ is a Markov operator if its matrix satisfies $P_{x,y}\geqslant 0$ and $\sum_{x\in X_+}P_{x,y=1}$ for all $y\in X$.

This paper present results of $\omega$-order preserving partial contraction mapping generating a regular weak*-continuous semigroup. We consider a semigroup on a Banach space $X$ and $B:X^\odot\rightarrow X^*$ is bounded, then the intertwining formula was used to define a semigroup $T^B(t)$ on $X^*$ which extends the perturbed semigroup $T^B_0(t)$ on $X^\odot$ using the variation of constants formula. We also investigated a certain class of weak*-continuous semigroups on dual space $X^*$ which contains both adjoint semigroups and their perturbations by operators $B:X^\odot\rightarrow X^*$.

The evolution of the Universe is considered by means of a nonlinear realization of afˇne and conformal symmetries via MaurerÄCartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type Ia supernova luminosity distances Ä redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefˇcients of the afˇne formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefˇcients of the afˇne formulation. We show that the evolution equations of the afˇne gravitons with respect to the dilaton zeroth mode coincide with the equations of ®squeezed oscillator¯. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators.

Various alternative formulations of the LES equations have been explored in which additional evolution equations for variables such as the acceleration, the subgrid-scale stress tensor, or the subgrid-scale force are explicitly carried. Statistics of the velocity field obtained from the equation for the acceleration are shown to depend strongly on the initial conditions. This feature, which is independent of LES modeling issues, seems to prove that the velocity-acceleration formulation of the Navier-Stokes is not useful for numerical simulation. Equations for the subgrid-scale quantities appear to be much more stable. However, models required by this formulation of the LES problem still require additional study.

Curvature driven surface evolution plays an important role in geometry, applied mathematics and in the natural sciences. In this paper geometric evolution equations such as mean curvature flow and its fourth order analogue motion by surface diffusion are studied as examples of gradient flows of the area functional. Also in many free boundary problems the motion of an interface is given by an evolution law involving curvature quantities. We will introduce the Mullins-Sekerka flow and the Stefan problem with its anisotropic variants and discuss their properties. In phase field models the area functional is replaced by a Ginzburg-Landau functional leading to a diffuse interface model. We derive the Allen-Cahn equation, the Cahn-Hilliard equation and the phase field system as gradient flows and relate them to sharp interface evolution laws.

In this work, we outline the development of a thermodynamically consistent microscopic model for a suspension of aggregating particles under arbitrary, inertia-less deformation. As a proof-of-concept, we show how the combination of a simplified population-balance-based description of the aggregating particle microstructure along with the use of the single-generator bracket description of nonequilibrium thermodynamics, which leads naturally to the formulation of the model equations. Notable elements of the model are a lognormal distribution for the aggregate size population, a population balance-based model of the aggregation and breakup processes and a conformation tensor-based viscoelastic description of the elastic network of the particle aggregates. The resulting example model is evaluated in steady and transient shear forces and elongational flows and shown to offer predictions that are consistent with observed rheological behavior of typical systems of aggregating particles. Ad...

Let Λ ⊂ Rn be a closed and discrete set and let CΛ be the space of all mean periodic functions whose spectrum is simple and contained in Λ. We estimate the behavior at infinity of these mean periodic functions f ∈ CΛ. The tools which are needed to solve this problem will also be used to fill a gap in a preceding paper. 1. Some problems on trigonometric sums Let Λ ⊂ R be a closed and discrete set. Definition 1.1. The vector space of all trigonometric sums P (x) = ∑ λ∈Λ c(λ) exp(2πiλ · x) (1.1) whose spectrum is contained in Λ is denoted by TΛ. Given such a set Λ does there exist a domain of stable uniqueness for TΛ? A domain of stable uniqueness is a compact set K enjoying the following property: For every P ∈ TΛ it suffices to estimate P on K to obtain a global estimate for P . If Λ is a lattice this is obviously true since every P ∈ TΛ is a periodic function. If Γ is the dual lattice, every γ ∈ Γ is a period of P . If K is a fundamental domain of Γ, P is uniquely determined by its ...

This article delves into the realm of double integral transforms (DIT), focusing on the pivotal role played by Fox's H-Function in their theoretical framework. The DIT denoted as ϕ(t) is intricately defined through Fox's H-Function and expressed as a Mellin-Barnes-type contour integral with various parameters and conditions. The study emphasizes chain properties connecting the DIT, presenting a concise representation as ϕ(t) = DT [f(x, y)]. Three theorems are established, leveraging the power series expansions of special functions such as Laplace transforms, Hankel transforms, and specific transforms by Pathak and Narain. These theorems are proven analytically and their results are verified with the help of examples. The Fox H-Function is a powerful mathematical tool, which is explored for its significance in the analysis of DIT. Being a generalization of the hypergeometric series, the H-function finds widespread applications across mathematics, physics, and engineering. Detailed proofs substantiate the three theorems, illustrating the manipulation of the double integral transform under specific conditions. The application of these theorems extends to the evaluation of known and novel integrals involving the product of H-Function and other mathematical functions. This comprehensive exploration unveils the indispensable role of Fox's H-Function in the theoretical landscape of double integral transform and its applications.

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We analyse the so-called small-world network model (originally devised by Strogatz and Watts), treating it, among other things, as a case study of non-linear coupled difference or differential equations. We derive a system of evolution equations containing more of the previously neglected (possibly relevant) non-linear terms. As an exact solution of this entangled system of equations is out of question we develop a (as we think, promising) method of enclosing the "exact" solutions for the expected quantities by upper and lower bounds, which represent solutions of a slightly simpler system of differential equation. Furthermore we discuss the relation between difference and differential equations and scrutinize the limits of the spreading idea for random graphs. We then show that there exists in fact a "broad" (with respect to scaling exponents) crossover zone, smoothly interpolating between linear and logarithmic scaling of the diameter or average distance. We are able to corroborate earlier findings in certain regions of phase or parameter space (as e.g. the finite size scaling ansatz) but find also deviations for other choices of the parameters. Our analysis is supplemented by a variety of numerical calculations, which, among other things, quantify the effect of various approximations being made. With the help of our analytical results we manage to calculate another important network characteristic, the (fractal) dimension, and provide numerical values for the case of the small-world network.

Se demuestra que el problema de valor inicial asociado con el sistema acoplado de ecuaciones de Korteweg-de Vries { 8tU + 8~u + o:8~V + uP8xU + vP8xV = 0 8tV + 8~V + o:8~u + vP8xV + 8x (uvP) = 0 para p 2: 4, tiene única solución global y su comportamiento asintótico satisface sup (1 + t) 113 IIU (t) llw < +oo. tE(O,oo(

We study that a solution of the initial value problem associated for the coupled system of equations of Korteweg -de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has analyticity in time and smoothing effect up to real analyticity if the initial data only has a single point singularity at x = 0.

New nonlinear evolution equations are derived that generalize the system by Matsuno and a terrain-following Boussinesq system by Nachbin . The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters .

In the context of ocean dynamics, a reduced strongly nonlinear one-dimensional model for the evolution of internal waves over an arbitrary seabottom with submerged structures was derived in . This model is a generalization of the one proposed in . The reduced model aims at obtaining an efficient numerical method for a two-dimensional problem with two layers containing inviscid, immiscible, incompressible and irrotational fluids of different densities. The upper layer is shallow compared with the characteristic wavelength at the interface of the twofluid system, while the bottom region's depth is comparable to the characteristic wavelength. The non-linear evolution equations describe the behaviour of η (the internal wave elevation at the interface) and u (the mean upper-velocity) for this water configuration. We intend to use this strongly nonlinear model to study the interaction of large amplitude internal waves with multiscale topography profiles. The dynamics include wave scattering, dispersion and attenuation among other phenomena. The refocusing and stabilization of solitary waves for the large levels of nonlinearity allowed by this kind of models is the goal of current research. To solve our model approximately we considered a numerical scheme based on the method of lines. Two particularities of the model deserve special attention: the dispersive term with a singular integral operator and the variable coefficient accounting for the topography information, which could be in a rapid scale. As used with success in [10], to deal numerically with a dispersive term an auxiliary variable V is introduced. The evolution in time is made on η and V . We implemented a fourth order Runge-Kutta time-integration scheme, which was also the choice in , and a fourth order approximation using a centered five point formula for the spatial derivative. Compared with some predictor-corrector schemes and spectral spatial derivatives, respectively, this is our best choice in terms of stability, up to now. When the nonlinearity is weak and the bottom is flat, we recover u from η and V (after each time step) by going to Fourier space via an FFT . In the presence of rough bottom (variable coefficients) or strong nonlinearity, this efficient strategy does not work and we must solve a linear algebraic system involving spectral matrices. By using a spectral matrix instead of an FFT, we are only paying a price in complexity but not in accuracy. This is an ongoing research. The spectral approach may not be the best for multiple scale problems, and we are currently investigating this. Some preliminary results from the Matlab implementations will be shown, including periodic topography experiments and solitary waves solutions.

In this paper we investigate the relation between discrete and continuous operators. More precisely, we investigate the properties of the semigroup generated by A, and the sequence A n d , n ∈ N, where A d = (I + A)(I -A) -1 . We show that if A and A -1 generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A d is equivalent to the uniform boundedness of the semigroup generated by A.

We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correspondence with general relativity. One obtains in this way a quantum theory on a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form, with a spacetime dependent Lorentz tensor $g_{\mu\nu}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{\mu\nu}$, can be those of general relativity, this theory provides a quantum theory which could be underlying to classical gravitation, and coincides with it in this classical ray approximation. In order to understand the possible origin of the structure of this equation, we appeal to the approach of Nelson in constructing a Schroedinger equati...

We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correspondence with general relativity. One obtains in this way a quantum theory on a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form, with a spacetime dependent Lorentz tensor $g_{\mu\nu}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{\mu\nu}$, can be those of general relativity, this theory provides a quantum theory which could be underlying to classical gravitation, and coincides with it in this classical ray approximation. In order to understand the possible origin of the structure of this equation, we appeal to the approach of Nelson in constructing a Schroedinger equati...

We measured the longitudinal double spin asymmetries A LL for single hadron muoproduction off protons and deuterons at photon virtuality Q 2 < 1(GeV/c) 2 for transverse hadron momenta p T in the range 1 GeV/c to 4 GeV/c. They were determined using COMPASS data taken with a polarised muon beam of 160 GeV/c or 200 GeV/c impinging on polarised 6 LiD or NH 3 targets. The experimental asymmetries are compared to next-to-leading order pQCD calculations, and are sensitive to the gluon polarisation ∆G inside the nucleon in the range of the nucleon momentum fraction carried by gluons 0.05 < x g < 0.2.

Mean-field evolution equations for the exciton and photon populations and polarizations (Bloch-Lamb equations) are written and numerically solved in order to describe the dynamics of electronic states in a quantum dot coupled to the photon field of a microcavity. The equations account for phase space filling effects and Coulomb interactions among carriers, and include also (in a phenomenological way) incoherent pumping of the quantum dot, photon losses through the microcavity mirrors, and electron-hole population decay due to spontaneous emission of the dot. When the dot may support more than one electron-hole pair, asymptotic oscillatory states, with periods between 0.5 and 1.5 ps, are found almost for any values of the system parameters.

We develop theory and algorithms to incorporate image manifold constraints in a level set segmentation algorithm. This provides a framework to simultaneously segment every image of data sets that vary due to two degrees of freedom -such as cardiopulmonary MR images which deform due to patient breathing and heartbeats. We derive two formulations: a 4D level set which loosely couples the level set function between neighbors in the 2D image manifold and a multilayer level set function which uses different levels of the level set function to represent shapes that shrink or grow. We characterize the set of shape manifolds that the multilayer level set function can represent, and derive the evolution equations for both frameworks. We offer results of segmenting the left ventricle in cardiopulmonary MRI; by automatically discovering the 2D manifold structure of the image set then simultaneously segmenting every frame. Both extensions improve on frame-by-frame approaches, and a comparison of the results offers insight into their strengths and weaknesses.

We develop theory and algorithms to incorporate image manifold constraints in a level set segmentation algorithm. This provides a framework to simultaneously segment every image of data sets that vary due to two degrees of freedom -such as cardiopulmonary MR images which deform due to patient breathing and heartbeats. We derive two formulations: a 4D level set which loosely couples the level set function between neighbors in the 2D image manifold and a multilayer level set function which uses different levels of the level set function to represent shapes that shrink or grow. We characterize the set of shape manifolds that the multilayer level set function can represent, and derive the evolution equations for both frameworks. We offer results of segmenting the left ventricle in cardiopulmonary MRI; by automatically discovering the 2D manifold structure of the image set then simultaneously segmenting every frame. Both extensions improve on frame-by-frame approaches, and a comparison of the results offers insight into their strengths and weaknesses.

The Mindlin-type model is used for describing the longitudinal deformation waves in microstructured solids. The evolution equation (one-wave equation) is derived for the hierarchical governing equation (two-wave equation) in the nonlinear case using the asymptotic (reductive perturbation) method. The evolution equation is integrated numerically under harmonic as well as localized initial conditions making use of the pseudospectral method. Analysis of the results demonstrates that the derived evolution equation is able to grasp essential effects of microinertia and elasticity of a microstructure. The influence of these effects can result in the emergence of asymmetric solitary waves.

KdV-type evolution equation, including the third-and the fifth order dispersive and the fourth order nonlinear terms, is used for modelling the wave propagation in microstructured solids like martensitic-austenitic alloys. The character of the dispersion depends on the signs of the third and the fifth order dispersion parameters. In the present paper the model equation is solved numerically under localised initial conditions in the case of mixed dispersion, i.e., the character of dispersion is normal for some wavenumbers and anomalous for others. Two types of solution are defined and discussed. Relatively small solitary waves result in irregular solution. However, if the amplitude exceeds a certain threshold a solution having regular time-space behaviour energes. The latter has tree sub-types: "plaited" solitons, two solitary waves and single solitary wave. Depending on the value of the amplitude of the initial pulse these sub-types can appear alone or in a certain sequence.

We compute the Yang-Mills vacuum wave functional in three dimensions at weak coupling with O(e 2 ) precision. We use two different methods to solve the functional Schroedinger equation. One of them generalizes to O(e 2 ) the method followed by Hatfield at O(e) . The other uses the weak coupling version of the gauge invariant formulation of the Schroedinger equation and the ground state wave functional followed by Karabali, Nair, and Yelnikov [2]. We compare both results and discuss the differences between them.

Quantitative analysis of flows of fiber suspensions in viscoelastic matrices, which is the general situation for thermoplastic composites, requires constitutive equations which incorporate specific features of the system and its constituents. Matrix viscoelasticity, fiber orientation and fiber/matrix interactions are key parameters to model such systems. In this work, the constituents of the system are represented by two second order symmetric tensors: c(r, t) for the viscoelastic matrix and a(r, t) for the fiber orientation. The time evolution equation for c(r, t) is developed in the generalized Poisson bracket framework with a finitely extensible non-linear elastic (FENE-P) and a Hookean Helmholtz energy functions. Several expressions for the mobility tensor including expressions with fiber matrix interactions are used. The time evolution equation for a(r, t) is based on the classical Jeffery equation modified to include fiber/fiber interactions in the case of semi-dilute suspensions. The sensitivity of the model to the choice of the mobility tensor together with the effect of fiber volume fraction on the prediction of material functions in start up and steady shear flows are discussed.

While superhydrophobic surfaces (SHSs) show promise for drag reduction applications, their performance can be compromised by traces of surfactant, which generate Marangoni stresses that increase drag. This question is addressed for soluble surfactant in a three-dimensional laminar channel flow, with periodic SHSs made of long finitelength longitudinal grooves located on both walls. We assume that bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. Exploiting long-wave theory and accounting for the difference between the rapid transverse and slower longitudinal Marangoni flows, we derive a one-dimensional model for surfactant transport from the full three-dimensional transport equations. Our one-dimensional model allows us to predict the drag reduction and surfactant distribution across the parameter space. The system exhibits multiple regimes, involving competition between Marangoni effects, bulk and interfacial diffusion, bulk and interfacial advection, shear dispersion and surfactant exchange between the bulk and the interface. We map out asymptotic regions in the high-dimensional parameter space, and derive explicit closedform approximations of the drag reduction, without any fitting or empirical parameters. The physics underpinning the drag reduction effect and the negative impact of surfactant is discussed through analysis of the velocity field and surfactant concentrations, which show both uniform and non-uniform stress distributions. Our theoretical predictions of the drag reduction compare well with results from the literature solving numerically the full three-dimensional transport problem. Our atlas of maps provides a comprehensive analytical guide for designing surfactant-contaminated channels with SHSs, to maximise the drag reduction in applications.

In this article, we investigate suitable conditions for the existence and uniqueness results of a class of fractional Caputo Volterra-Fredholm integro-differential equations with nonlocal conditions. The findings are based on the Kuratowski measure of non-compactness and Gronwall's inequality. Additionally, we determine the controllability of the given problem using the topological degree approach. We provide an example to support the approach.

The influence of the temperature history on the Mullins effect, its recovery behaviour and the rate dependence is experimentally investigated using NR/BR (NR: natural rubber, BR: polybutadiene rubber) rubber blend. The crystallization which occurs in rubber during long term storage below the melting temperature has been taken into account to interpret the experimental data. To study the influence of low temperatures and large deformations on the Mullins effect, cyclic strain‐controlled processes are applied under different temperatures. The softened specimens are subjected to a sequence of heating, cooling, and conditioning processes in order to study the influence of the temperature history on healing, melting, and recrystallization. The results indicate the existence of a threshold temperature: if the specimen temperature is larger than this threshold, a nearly complete recovery of the material occurs within finite time, while any temperature below this limit will be too small for...

We will report on an ongoing effort towards calculating the N4LO perturbative QCD corrections to the DIS total inclusive cross-section. We are developing a method based on differential equations and series expansion in the inverse Bjorken parameter. As a byproduct our calculation should also deliver analytic or at least precise numerical approximations for the four-loop splitting functions.

The work deals with non-Markov processes and the construction of systems of differential equations with delay that describe the probability vectors of such processes. The generating stochastic operator and properties of stochastic operators are used to construct systems that define non-Markov processes.

The work deals with non-Markov processes and the construction of systems of differential equations with delay that describe the probability vectors of such processes. The generating stochastic operator and properties of stochastic operators are used to construct systems that define non-Markov processes.

We consider a three-species competition-di¤usion system, in order to discuss the problem of competitor-mediated coexistence in situations where one exotic competing species invades a system that already contains two strongly competing species. It is numerically shown that, under some conditions, there exist stable nonconstant equilibrium solutions that indicate the coexistence of two strongly competing species. This result motivates us to develop a semi-exact representation for finding these equilibrium solutions from an analytical viewpoint.

In this paper we study the nonlocal p-Laplacian type diffusion equation, If p > 1, this is the nonlocal analogous problem to the well known local p-Laplacian evolution equation u t = div(|∇u| p-2 ∇u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L ∞ (0, T ; L p (Ω)) to the solution of the p-laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behaviour of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. R N J(x -y) (u(t, y) -u(t, x)) dy,

We compare two Monte Carlo implementations of resummation schemes for the description of parton evolution at small values of Bjorken x. One of them is based on the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation and generates fully differential parton distributions in momentum space making use of reggeized gluons. The other one is based on the Catani-Ciafaloni-Fiorani-Marchesini (CCFM) partonic kernel where QCD coherence effects are introduced. It has been argued that both approaches agree with each other in the x → 0 limit. We show that this is not the case for azimuthal angle dependent quantities since at high energies the BFKL approach is dominated by its zero conformal spin component while the CCFM gluon Green function receives contributions from all conformal spins even at very small x. * Preprint numbers: LPN11-04, IFT-UAM/CSIC-11-03, FTUAM-11-36 1 valid in a certain window of center-of-mass energies and perturbative scales. To extend its range of applicability one should include either higher order corrections, going beyond the 'multi-Regge" kinematics, include non-linear corrections responsible for the restauration of unitarity at very small x 1 or, as we are going to discuss in this paper, to include a more global treatment of collinear regions in phase space using the Catani-Ciafaloni-Fiorani-Marchesini (CCFM) equation, which provides a good matching from BFKL to the x → 1 regime, at least, as we are going to show in this work, as long as anomalous dimensions and k t -diffusion properties are concerned. In this letter we first give a brief introduction to the BFKL and CCFM approaches, in section 2. We then describe our Monte Carlo implementations in sections 3 and 4, where we also compare both resummation schemes in terms of the detailed exclusive information we can obtain from running the Monte Carlo codes. In particular, we show that the UV/IR symmetry in the k t -diffusion present in the BFKL evolution is broken at lower energies in the CCFM equation, to then be slowly restored as x tends to 0. Our main result appears when we study the azimuthal angle dependence of the gluon Green function in both approaches. We show that the CCFM equation generates a stronger azimuthal angle dependence than the BFKL equation. To stress this point we investigate the Fourier components of the solution in this angular sector. We show that the projection on the zero Fourier component, or zero conformal spin in the BFKL context, has the highest growth with energy in both schemes. The differences appear in the non-zero Fourier components, or non-zero conformal spins, which in the BFKL description fall with energy but in the CCFM solution increase. We also discuss the different collinear behaviour of the solutions at different energies. Finally we present our conclusions and scope for further investigations.

Power-law distributions with various exponents are studied. We first introduce a simple and generic model that reproduces Zipf's law. We can regard this model both as the time evolution of the population of cities and that of the asset distribution. We show that our model is very robust against various variations. Next, we explain theoretically why our model reproduces Zipf's law. By considering the time-evolution equation of our model, we see that the essence of Zipf's law is an asymmetric random walk in a logarithmic scale. Finally, we extend our model by introducing an additional asymmetry. We show that the extended model reproduces various power-law exponents. By extending the theoretical argument for Zipf's law, we find a simple equation of the power-law exponent.

We study the existence, uniqueness, asymptotic properties, and continuous dependence upon data of solutions to a class of abstract nonlocal Cauchy problems. The approach we use is based on the theory ofm-accretive operators and related evolution equations in Banach spaces.

This paper is Part I of an integrated experimental/modeling investigation of a procedure to coat nanofibers and core-clad nanostructures with thin-film materials using plasma-enhanced physical vapor deposition. In the experimental effort, electrospun polymer nanofibers are coated with aluminum under varying operating conditions to observe changes in the coating morphology. This procedure begins with the sputtering of the coating material from a target. This paper focuses on the sputtering process and transport of the sputtered material through the reactor. The interrelationships among the processing factors for the sputtering and transport are investigated from a detailed modeling approach that describes the salient physical and chemical phenomena. Solution strategies that couple continuum and atomistic models are used. At the continuum scale, the sheath region and the reactor dynamics near the target surface are described. At the atomic level, molecular-dynamics (MD) simulations ar...

This work is a comparison of modeling and simulation results with experiments for an integrated experimental/modeling investigation of a procedure to coat nanofibers and core-clad nanostructures with thin film materials using plasma enhanced physical vapor deposition. In the experimental effort, electrospun polymer nanofibers are coated with metallic materials under different operating conditions to observe changes in the coating morphology. The modeling effort focuses on linking simple models at the reactor level, nanofiber level and atomic level to form a comprehensive model. The comprehensive model leads to the definition of an evolution equation for the coating free surface around an isolated nanofiber. This evolution equation was previously derived and solved under conditions of a nearly circular coating, with a concentration field that was only radially dependent and that was independent of the location of the coating free surface. These assumptions permitted the development o...

Using the dip coating technique, we fabricate erbia-coated quartz fibers and glass slides. Further we present a thin film model of the dip coating technique applied to the glass slides. The model includes evaporation of the solvent and a bulk reaction term to simulate the creation of the erbium oxide that forms the coating. Evolution equations for the solvent and coating thicknesses are developed and solved numerically. We study how the solvent evaporation rate, the bulk reaction rate, the dynamic viscosity, the angle of the inclined substrate, the characteristic depth of the solution thickness, the initial profile of the solution thickness, and the length of substrate affect the coating thickness. The model is calibrated using the experimental results to provide a tool for predicting the coating thickness.

We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green's functions which we derive in explicit form. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant basis of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green's functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, second order stochastic evolution equations. To probe an image at every location x ∈ R 2 and in every direction e iθ ∈ T, one translates and rotates an anisotropic wavelet ψ by means of a representation g → U g of the Euclidean motion group U g ψ(y) = ψ(R -1 θ (yx)), g = (x, e iθ ). The result of such an image sampling is a function U f ∈ L 2 (G) on the Euclidean motion group manifold G = R 2 T, which is

In the standard scale space approach one obtains a scale space representation u : R d R + → R of an image f ∈ L2(R d ) by means of an evolution equation on the additive group (R d , +). However, it is common to apply a wavelet transform (constructed via a representation U of a Lie-group G and admissible wavelet ψ) to an image which provides a detailed overview of the group structure in an image. The result of such a wavelet transform provides a function g → (Ugψ, f) L 2 (R 2 ) on a group G (rather than (R d , +)), which we call a score. Since the wavelet transform is unitary we have stable reconstruction by its adjoint. This allows us to link operators on images to operators on scores in a robust way. To ensure U-invariance of the corresponding operator on the image the operator on the wavelet transform must be left-invariant. Therefore we focus on leftinvariant evolution equations (and their resolvents) on the Lie-group G generated by a quadratic form Q on left invariant vector fields. These evolution equations correspond to stochastic processes on G and their solution is given by a group convolution with the corresponding Green's function, for which we present an explicit derivation in two particular image analysis applications. In this article we describe a general approach how the concept of scale space can be extended by replacing the additive group R d by a Lie-group with more structure. 1

We prove that arbitrary (nonpolynomial) scalar evolution equations of order m ≥ 7, that are integrable in the sense of admitting the canonical conserved densities ρ(1), ρ(2), and ρ(3) introduced in [1], are polynomial in the derivatives um− i for i = 0, 1, 2. We also introduce a grading in the algebra of polynomials in uk with k ≥ m − 2 over the ring of functions in x, t, u, … , um−3 and show that integrable equations are scale homogeneous with respect to this grading.

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In this article, we analyze a spatio-temporally nonlocal nonlinear parabolic equation. First, we validate the equation by an existence-uniqueness result. Then, we show that blowing-up solutions exist and study their time blow-up profile. Also, a result on the existence of global solutions is presented. Furthermore, we establish necessary conditions for local or global existence.

In this article, we analyze a spatio-temporally nonlocal nonlinear parabolic equation. First, we validate the equation by an existence-uniqueness result. Then, we show that blowing-up solutions exist and study their time blow-up profile. Also, a result on the existence of global solutions is presented. Furthermore, we establish necessary conditions for local or global existence.