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In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger...
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In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf. Theorem 1), and for C= 1, A, B ≤ 50, with six explicitly given exceptions (A, B, n) (cf. Theorem 3). The equations under consideration are solved by combining powerful techniques, including Prey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree. Along the way, we derive a new result on the solvability of binomial Thue equations (cf. Theorem 6) which is crucial in the proof of our Theorems 1 and 2. Some important applications of our theorems will be given in a forthcoming paper.
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We consider the drying of latex dispersions containing submicrometer-sized particles dispersed in water. It is well known that the consolidation of colloidal particles is influenced by a number of factors such as particle size and...
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We consider the drying of latex dispersions containing submicrometer-sized particles dispersed in water. It is well known that the consolidation of colloidal particles is influenced by a number of factors such as particle size and shape and interparticle potential. In this work, we focus on the effect of surface charge on the consolidation front. Recent experimental and theoretical investigations on the sedimentation of charged colloidal spheres have shown that the large mass difference between noninteracting colloids and ions sets up a macroscopic electric field, thereby enhancing the diffusivity of the particles and resulting in an inflated sedimentation profile. Our experimental measurements of the concentration profile during drying-induced consolidation also reveal similar charge effects. We present a model for the consolidation of charged particles that accounts for the presence of an induced external electric Field. As expected, the predicted particle diffusivity is enhanced by the onset of the electric field at low particle concentration. Fluorescence and bright-field microscopy were used to detect the particle concentration variation in a dispersion dried in a capillary, and the measured profile agrees with the prediction confirming the influence of particle charge on consolidation.
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Quantum transport and other phenomena are typically modeled by coupling the system of interest to an environment, or bath, held at thermal equilibrium. Realistic bath models are at least as challenging to construct as models for t...
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Quantum transport and other phenomena are typically modeled by coupling the system of interest to an environment, or bath, held at thermal equilibrium. Realistic bath models are at least as challenging to construct as models for the quantum systems themselves, since they must incorporate many degrees of freedom that interact with the system on a wide range of timescales. Owing to computational limitations, the environment is often modeled with simple functional forms, with a few parameters fit to experiment to yield semi-quantitative results. Growing computational resources have enabled the construction of more realistic bath models from molecular dynamics (MD) simulations. In this paper, we develop a numerical technique to construct these atomistic bath models with better accuracy and decreased cost. We apply a novel signal processing technique, known as super-resolution, combined with a dictionary of physically motivated bath modes to derive spectral densities from MD simulations. Our approach reduces the required simulation time and provides a more accurate spectral density than can be obtained via standard Fourier transform methods. Moreover, the spectral density is provided as a convenient closed-form expression which yields an analytic time-dependent bath kernel. Exciton dynamics of the Fenna-Matthews-Olson light-harvesting complex are simulated with a second-order time-convolutionless master equation, and spectral densities constructed via super-resolution are shown to reproduce the dynamics using only a quarter of the amount of MD data.
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We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form eq...
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We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (l. Gaal, A. Petho, and M. Pohst, 1993, J. Symbolic Comput. 16, 563-584; 1996, J. Number Theory 57, 90- 104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations determine the generators of power integral bases of K over M up to translation by integers of M and multiplication by unit factors of M. The new method is based on our ability to solve relative Thue equations efficiently by the algorithm in (I. Gaal and M. Pohst, 2000, Math. Comp., to appear). In the case K is an octic field with a quadratic subfield M we can also consider the absolute index of elements of K, having relative index 1 over M. In order to determine all generators of power integral bases of K (over Q) we determine the corresponding translating elements and unit factors properly. This is done by solving an equation similar to an inhomogeneous Thue equation. We illustrate our algorithms with detailed examples.
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By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. It is proved that the solution is equal to t...
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By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. It is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.
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In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree n...
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In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree n) there exists a unique linearly stable soliton, which we call the n-kink. We give numerical evidence that the n-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree n. When the radius of the wormhole throat a is large enough, the convergence to the n-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the one-kink using the Soffer-Weinstein weakly nonlinear perturbation theory.
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We study the solutions of the second Painlevé equation (P II) in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of t...
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We study the solutions of the second Painlevé equation (P II) in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlevé equation (P 34). By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repeller for the dynamics in Okamoto's space. Moreover, we show that the limit set of the solutions exists and is compact and connected.
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We study the asymptotic behavior of solutions of the fourth Painlev, equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We...
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We study the asymptotic behavior of solutions of the fourth Painlev, equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.
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In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential su...
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In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix-vector multiplication, peak finding, and clustering. (C) 2018 Elsevier Inc. All rights reserved.
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Let Y be a pure dimensional analytic variety in С' with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of the present paper is to present a techniq...
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Let Y be a pure dimensional analytic variety in С' with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of the present paper is to present a technique which allows us to determine obstructions to the solvability of the a equation in the L2, respectively L, sense on Y* = Y\{0} in terms of certain cohomology classes on X. More precisely, let t CC Y be a Stein domain with 0 __ Ω, Ω* = Ω\{0}. We give a sufficient condition for the solvability of the α equation in the L~2-sense on Ω*; and in the L__ sense, if Q is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Wilder estimates for solutions of the α equation.
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