Journal articles on the topic '010201 Approximation Theory and Asymptotic Methods'
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1
HEWETT, D. P. "Shadow boundary effects in hybrid numerical-asymptotic methods for high-frequency scattering." European Journal of Applied Mathematics 26, no. 5 (June 30, 2015): 773–93. http://dx.doi.org/10.1017/s0956792515000315.
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The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost of conventional numerical methods for high-frequency wave scattering problems by enriching the numerical approximation space with oscillatory basis functions, chosen based on partial knowledge of the high-frequency solution asymptotics. In this paper, we propose a new methodology for the treatment of shadow boundary effects in HNA boundary element methods, using the classical geometrical theory of diffraction phase functions combined with mesh refinement. We develop our methodology in the context of scattering by a class of sound-soft non-convex polygons, presenting a rigorous numerical analysis (supported by numerical results) which proves the effectiveness of our HNA approximation space at high frequencies. Our analysis is based on a study of certain approximation properties of the Fresnel integral and related functions, which govern the shadow boundary behaviour.
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Lambaré, Gilles, Jean Virieux, Raul Madariaga, and Side Jin. "Iterative asymptotic inversion in the acoustic approximation." GEOPHYSICS 57, no. 9 (September 1992): 1138–54. http://dx.doi.org/10.1190/1.1443328.
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We propose an iterative method for the linearized prestack inversion of seismic profiles based on the asymptotic theory of wave propagation. For this purpose, we designed a very efficient technique for the downward continuation of an acoustic wavefield by ray methods. The different ray quantities required for the computation of the asymptotic inverse operator are estimated at each diffracting point where we want to recover the earth image. In the linearized inversion, we use the background velocity model obtained by velocity analysis. We determine the short wavelength components of the impedance distribution by linearized inversion of the seismograms observed at the surface of the model. Because the inverse operator is not exact, and because the source and station distribution is limited, the first iteration of our asymptotic inversion technique is not exact. We improve the images by an iterative procedure. Since the background velocity does not change between iterations. There is no need to retrace rays, and the same ray quantities are used in the iterations. For this reason our method is very fast and efficient. The results of the inversion demonstrate that iterations improve the spatial resolution of the model images since they mainly contribute to the increase in the short wavelength contents of the final image. A synthetic example with one‐dimensional (1-D) velocity background illustrates the main features of the inversion method. An example with two‐dimensional (2-D) heterogeneous background demonstrates our ability to handle multiple arrivals and a nearly perfect reconstruction of a flat horizon once the perturbations above it are known. Finally, we consider a seismic section taken from the Oseberg oil field in the North Sea off Norway. We show that the iterative asymptotic inversion is a reasonable and accurate alternative to methods based on finite differences. We also demonstrate that we are able to handle an important amount of data with presently available computers.
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Chiappinelli, Raffaele. "Variational Methods for NLEV Approximation Near a Bifurcation Point." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–32. http://dx.doi.org/10.1155/2012/102489.
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We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains ofℝN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.
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Grebenikov, E. A. "Concerning New Perturbation Methods in Solar System Dynamics." International Astronomical Union Colloquium 165 (1997): 399–404. http://dx.doi.org/10.1017/s0252921100046868.
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In this paper a new method of construction of the perturbation motion theory of celestial bodies, based on the averaging principle in view of frequency resonances, is stated. The first approximation of the asymptotic theory is the exact solution of the dynamics averaging equations, in which are included “secular” and “long-periodic” terms. The high-degree approximations are the exact solution of a known Krylov-Bogoliubov generalized equation. It is shown that these iterations are expressed in the analytical form by multiple Fourier series.
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Kal'chuk, I. V., Yu I. Kharkevych, and K. V. Pozharska. "Asymptotics of approximation of functions by conjugate Poisson integrals." Carpathian Mathematical Publications 12, no. 1 (June 12, 2020): 138–47. http://dx.doi.org/10.15330/cmp.12.1.138-147.
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Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.
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Hrabova, U. Z., and I. V. Kal'chuk. "Approximation of the classes $W^{r}_{\beta,\infty}$ by three-harmonic Poisson integrals." Carpathian Mathematical Publications 11, no. 2 (December 31, 2019): 321–34. http://dx.doi.org/10.15330/cmp.11.2.321-334.
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In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.
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Razavy, M., and B. Lenoach. "Reciprocity principle and the approximate solution of the wave equation." Bulletin of the Seismological Society of America 76, no. 6 (December 1, 1986): 1776–89. http://dx.doi.org/10.1785/bssa0760061776.
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Abstract In this paper, we examine the validity of reciprocity when certain approximations are applied to the wave equation. We focus on some of the approximation techniques which are frequently used in investigating seismic waves, particularly in the context of synthetic seismograms, namely, asymptotic ray theory and finite difference methods. The main result is that the reciprocity property is not necessarily preserved when an approximation is used. This conclusion is shown to be valid for both ray theory and finite difference techniques. We give some concrete examples as well as numerical results to illustrate the practical effect of nonreciprocity. We also show that reciprocity breaks down when analytical approximation is required to solve the problem in question.
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Raudenbush, Stephen W., and Anthony S. Bryk. "Examining Correlates of Diversity." Journal of Educational Statistics 12, no. 3 (September 1987): 241–69. http://dx.doi.org/10.3102/10769986012003241.
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Statistical methods are presented for studying “correlates of diversity”: characteristics of educational organizations which predict dispersion on the dependent variable. The conceptual framework for these methods distinguishes between variance heterogeneity that arises from educational program effects and heterogeneity that merely reflects heterogeneity of variance of inputs. The estimation theory is empirical Bayes, requiring probabilistic models both for the data and for the random dispersion parameters from each of many groups. Two strategies are considered, one based on exact distribution theory and the second based on an asymptotic normal approximation. The accuracy of the approximation is evaluated analytically and its use illustrated by an analysis of mathematics achievement data from a random sample of U.S. high schools.
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FEDOROV, A. V., N. D. MALMUTH, and V. G. SOUDAKOV. "Supersonic scattering of a wing-induced incident shock by a slender body of revolution." Journal of Fluid Mechanics 585 (August 7, 2007): 305–22. http://dx.doi.org/10.1017/s0022112007006714.
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The lift force acting on a slender body of revolution that separates from a thin wing in supersonic flow is analysed using Prandtl–Glauert linearized theory, scattering theory and asymptotic methods. It is shown that this lift is associated with multi-scattering of the wing-induced shock wave by the body surface. The local and global lift coefficients are obtained in simple analytical forms. It is shown that the total lift is mainly induced by the first scattering. Contributions from second, third and higher scatterings are zero in the leading-order approximation. This greatly simplifies calculations of the lift force. The theoretical solution for the flow field is compared with numerical solutions of three-dimensional Euler equations and experimental data at free-stream Mach number 2. There is agreement between the theory and the computations for a wide range of shock-wave strength, demonstrating high elasticity of the leading-order asymptotic approximation. Theoretical and experimental distributions of the cross-sectional normal force coefficient agree satisfactorily, showing robustness of the analytical solution. This solution can be applied to the moderate supersonic (Mach numbers from 1.2 to 3) multi-body interaction problem for crosschecking with other computational or engineering methods.
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Prat, V., S. Mathis, K. Augustson, F. Lignières, J. Ballot, L. Alvan, and A. S. Brun. "Asymptotic theory of gravity modes in rotating stars." Astronomy & Astrophysics 615 (July 2018): A106. http://dx.doi.org/10.1051/0004-6361/201832576.
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Context. Differential rotation has a strong influence on stellar internal dynamics and evolution, notably by triggering hydrodynamical instabilities, by interacting with the magnetic field, and more generally by inducing transport of angular momentum and chemical elements. Moreover, it modifies the way waves propagate in stellar interiors and thus the frequency spectrum of these waves, the regions they probe, and the transport they generate. Aims. We investigate the impact of a general differential rotation (both in radius and latitude) on the propagation of axisymmetric gravito-inertial waves. Methods. We use a small-wavelength approximation to obtain a local dispersion relation for these waves. We then describe the propagation of waves thanks to a ray model that follows a Hamiltonian formalism. Finally, we numerically probe the properties of these gravito-inertial rays for different regimes of radial and latitudinal differential rotation. Results. We derive a local dispersion relation that includes the effect of a general differential rotation. Subsequently, considering a polytropic stellar model, we observe that differential rotation allows for a large variety of resonant cavities that can be probed by gravito-inertial waves. We identify that for some regimes of frequency and differential rotation, the properties of gravito-inertial rays are similar to those found in the uniformly rotating case. Furthermore, we also find new regimes specific to differential rotation, where the dynamics of rays is chaotic. Conclusions. As a consequence, we expect modes to follow the same trend. Some parts of oscillation spectra corresponding to regimes similar to those of the uniformly rotating case would exhibit regular patterns, while parts corresponding to the new regimes would be mostly constituted of chaotic modes with a spectrum rather characterised by a generic statistical distribution.
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Zveryaev, Evgeny M. "Extraction of consistent shell theory equations from 3D theory of elasticity." Structural Mechanics of Engineering Constructions and Buildings 15, no. 2 (December 15, 2019): 135–48. http://dx.doi.org/10.22363/1815-5235-2019-15-2-135-148.
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Aims of research. Derivation of consistent equations of the theory of thin elastic shells without hypotheses and stress averaging over the shell thickness. Methods. Using the iterative method of Saint-Venant - Picard - Banach, the three-dimensional problem of the theory of elasticity is solved without any hypotheses. By the principle of compressed mappings, the solution converges asymptotically, regardless of the choice of the values of the initial approximation. Results. A method has been developed for integrating the spatial equations of the theory of elasticity in curvilinear coordinates for a thin shell. The presence of a small parameter allows the integration of the system of equations in such a way that the output data of the first operator is input to the next operator, etc., dividing the original complex operator into a sequence of simple integrable Picard type operators. Each equation contains terms of only one asymptotic order.
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Kiefer, Nicholas M., and Timothy J. Vogelsang. "HETEROSKEDASTICITY-AUTOCORRELATION ROBUST TESTING USING BANDWIDTH EQUAL TO SAMPLE SIZE." Econometric Theory 18, no. 6 (September 24, 2002): 1350–66. http://dx.doi.org/10.1017/s026646660218604x.
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Asymptotic theory for heteroskedasticity autocorrelation consistent (HAC) covariance matrix estimators requires the truncation lag, or bandwidth, to increase more slowly than the sample size. This paper considers an alternative approach covering the case with the asymptotic covariance matrix estimated by kernel methods with truncation lag equal to sample size. Although such estimators are inconsistent, valid tests (asymptotically pivotal) for regression parameters can be constructed. The limiting distributions explicitly capture the truncation lag and choice of kernel. A local asymptotic power analysis shows that the Bartlett kernel delivers the highest power within a group of popular kernels. Finite sample simulations suggest that, regardless of the kernel chosen, the null asymptotic approximation of the new tests is often more accurate than that for conventional HAC estimators and asymptotics. Finite sample results on power show that the new approach is competitive.
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Trigub, V. N., A. B. Blokhin, and I. N. Simakin. "The asymptotic study of dissipation and breakdown of a wing-tip vortex." Journal of Fluid Mechanics 274 (September 10, 1994): 293–337. http://dx.doi.org/10.1017/s0022112094002132.
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The steady axisymmetrical wing-tip vortex is studied in this paper by means of asymptotic methods within the limit of high Reynolds numbers. The smooth regrouping of the vortex under the action of viscous forces is described by a quasicylindrical approximation. The solutions of the quasi-cylindrical approximation are thoroughly analysed numerically and it is shown that a saddle-point bifurcation appears at certain critical values of circulation. At these values the solution may be continued in two ways: as a supercritical branch which approaches the Batchelor limit far downstream; and a subcritical one, which passes the second, nodal-point bifurcation. The parabolic quasi-cylindrical equations past this point allow the downstream disturbances to propagate upstream, like for example, boundary-layer equations in the regime of strong hypersonic interaction. The flow past the second bifurcation point was studied numerically and it was shown that solutions of the quasicylindrical approximation with large reversed-flow regions exist. An asymptotic expansion of such solutions far downstream was constructed, and it turned out that the reversed-flow region expands exponentially. This process is halted by elliptical effects in the external flow. An asymptotic theory of large reversed-flow regions is suggested including viscosity and elliptical effects. Numerical solutions for unbounded vortex breakdown parabolically expanding far downstream are presented. Then the general asymptotic problem statement which describes the flow near the bifurcation points is used to study the asymptotic solutions near the first bifurcation point. The problem is investigated numerically and two kinds of solution, which may be treated as transcritical jumps and marginal vortex breakdown, are found and discussed.
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Long, Jing Fan, Li Qin Duan, and Pei Xin Ye. "The Gel'fand Approximations on Multivariate Functions in the Deterministic and Monte Carlo Settings." Applied Mechanics and Materials 543-547 (March 2014): 1609–12. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1609.
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The concept of Gel'fand width plays an important role in the theory of information complexity and compressed sensing. In this paper, we study the approximation problems on the generalized Besov classesBΩΡ,θ in the norm of Lq by the Gel'fand methods in the deterministic and the Monte Carlo settings. Applying the Maiorov's discretization technique and some properties of pseudo-s-scale, we determine the asymptotic orders of this problem for some values of parameters p, q, θ.
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BALMFORTH, NEIL J. "Stability of vorticity defects in viscous shear." Journal of Fluid Mechanics 357 (February 25, 1998): 199–224. http://dx.doi.org/10.1017/s0022112097008124.
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The stability of viscous shear is studied for flows that consist of predominantly linear shear, but contain localized regions over which the vorticity varies rapidly. Matched asymptotic expansion simplifies the governing equations for the dynamics of such ‘vorticity defects’. The normal modes satisfy explicit dispersion relations. Nyquist methods are used to find and classify the possible instabilities. The defect equations are analysed in the inviscid limit to establish the connection with inviscid theory. Finally, the defect approximation is used to study nonlinear stability using weakly nonlinear techniques, and the initial value problem using Laplace transforms.
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Prat, V., S. Mathis, F. Lignières, J. Ballot, and P. M. Culpin. "Period spacing of gravity modes strongly affected by rotation." Astronomy & Astrophysics 598 (February 2017): A105. http://dx.doi.org/10.1051/0004-6361/201629637.
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Context. As of today, asteroseismology mainly allows us to probe the internal rotation of stars when modes are only weakly affected by rotation using perturbative methods. Such methods cannot be applied to rapidly rotating stars, which exhibit complex oscillation spectra. In this context, the so-called traditional approximation, which neglects the terms associated with the latitudinal component of the rotation vector, describes modes that are strongly affected by rotation. This approximation is sometimes used for interpreting asteroseismic data, however, its domain of validity is not established yet. Aims. We aim at deriving analytical prescriptions for period spacings of low-frequency gravity modes strongly affected by rotation through the full Coriolis acceleration (i.e. without neglecting any component of the rotation vector), which can be used to probe stellar internal structure and rotation. Methods. We approximated the asymptotic theory of gravito-inertial waves in uniformly rotating stars using ray theory described in a previous paper in the low-frequency regime, where waves are trapped near the equatorial plane. We put the equations of ray dynamics into a separable form and used the Einstein-Brillouin-Keller (EBK) quantisation method to compute modes frequencies from rays. Results. Two spectral patterns that depend on stratification and rotation are predicted within this new approximation: one for axisymmetric modes and one for non-axisymmetric modes. Conclusions. The detection of the predicted patterns in observed oscillation spectra would give constraints on internal rotation and chemical stratification of rapidly rotating stars exhibiting gravity modes, such as γ Doradus, SPB, or Be stars. The obtained results have a mathematical form that is similar to that of the traditional approximation, but the new approximation takes the full Coriolis, which allows for propagation near the centre, and centrifugal accelerations into account.
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Besse, Nicolas, Norbert Mauser, and Eric Sonnendrücker. "Numerical Approximation of Self-Consistent Vlasov Models for Low-Frequency Electromagnetic Phenomena." International Journal of Applied Mathematics and Computer Science 17, no. 3 (October 1, 2007): 361–74. http://dx.doi.org/10.2478/v10006-007-0030-3.
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Numerical Approximation of Self-Consistent Vlasov Models for Low-Frequency Electromagnetic PhenomenaWe present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus "light waves" are somewhat supressed, which in turn allows the numerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian methods and time splitting techniques. We develop a four-dimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a two-dimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beam-plasma instability. For these cases our numerical scheme works very well and is in agreement with analytic kinetic theory.
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Colbrook, Matthew J., and Matthew J. Priddin. "Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions)." IMA Journal of Applied Mathematics 85, no. 5 (July 22, 2020): 790–821. http://dx.doi.org/10.1093/imamat/hxaa021.
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Abstract This paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications ranging from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenized boundary conditions in this case, especially as the homogenized model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods solving the Helmholtz equation: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial) and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for each method. By comparing solutions obtained using homogenized boundary conditions to the problem of collinear plates, we verify that the constant compliance given in previous theoretical research is appropriate to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime. We further investigate tapering the compliance near the extreme endpoints of the screen and find that tapering with $\tanh $ functions reduces the error in the approximation of the far field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber has significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.
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Chernov, Alexey, and Lorenzo Mascotto. "The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains." IMA Journal of Numerical Analysis 39, no. 4 (June 20, 2018): 1787–817. http://dx.doi.org/10.1093/imanum/dry038.
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Abstract We introduce the harmonic virtual element method (VEM) (harmonic VEM), a modification of the VEM (Beirão da Veiga et al. (2013) Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23, 199–214.) for the approximation of the two-dimensional Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an ‘$H^1$-conformisation’ of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM) (Hiptmair et al. (2014) Approximation by harmonic polynomials in starshaped domains and exponential convergence of Trefftz hp-DGFEM. ESAIM Math. Model. Numer. Anal., 48, 727–752.). We address the stabilization of the proposed method and develop an hp version of harmonic VEM for the Laplace equation on polygonal domains. As in TDG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order $\mathscr{O}(\exp (-b\sqrt [2]{N}))$, where $N$ is the number of degrees of freedom. This result overperforms its counterparts in the framework of hp FEM (Schwab, C. (1998)p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press Oxford.) and hp VEM (Beirão da Veiga et al. (2018) Exponential convergence of the hp virtual element method with corner singularity. Numer. Math., 138, 581–613.), where the asymptotic rate of convergence is of order $\mathscr{O}(\exp(-b\sqrt [3]{N}))$.
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Bomba, A., and I. Moroz. "Simulation of the Charge Carriers Distribution in the Active Region of the P-I-N-Diodes by the Perturbation Theory Methods." Mathematical and computer modelling. Series: Technical sciences, no. 22 (November 26, 2021): 20–30. http://dx.doi.org/10.32626/2308-5916.2021-22.20-30.
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A mathematical model of the electron-hole plasma stationary distribution in the active region (i-region) of p-i-n-diodes in the dif-fusion-drift approximation is proposed. The model is represented in the form of a nonlinear singularly perturbed boundary value prob-lem for the system of equations of the electron-hole currents conti-nuity, the Poisson equation and the corresponding boundary condi-tions. The decomposition of the nonlinear boundary value problem of modeling the stationary distribution of charge carriers in the plasma of p-i-n-diodes is carried out on the basis of the solutions asymptotic representation. The model problem is reduced to a se-quence of the linear boundary value problems with a characteristic separation of the main (regular) components of the asymptotics and a boundary corrections. It was found that the formulation of the problem for finding the zero term of the asymptotics regular part coincides with the classical formulation of the p-i-n-diodes charac-teristics modeling problem, which is carried out in the approxima-tion of the ambipolar diffusion (approximation of a self-consistent electrostatic field).The proposed mathematical model and the method of its linearization make it possible to determing the main components in the diffusion-drift process and to study their role. For example, it becomes possible to study (including by analytical methods) the behavior of plasma in the p-i-, n-i-contacts zones. The results of the study are aimed at developing methods for de-signing p-i-n-diode structures, used, in particular, as active ele-ments of the signals switches of a microwave data transmission systems and the corresponding protective devices.
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Bomba, Andrii, Igor Moroz, and Mykhailo Boichura. "Constructing and analyzing mathematical model of plasma characteristics in the active region of integrated p-i-n-structures by the methods of perturbation theory and conformal mappings." Eastern-European Journal of Enterprise Technologies 5, no. 5 (113) (October 31, 2021): 51–61. http://dx.doi.org/10.15587/1729-4061.2021.243097.
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The results of mathematical modeling of stationary physical processes in the electron-hole plasma of the active region (i-region) of integral p-i-n-structures are presented. The mathematical model is written in the framework of the hydrodynamic thermal approximation, taking into account the phenomenological data on the effect on the dynamic characteristics of charge carriers of heating of the electron-hole plasma as a result of the release of Joule heat in the volume of the i-th region and the release of recombination energy. The model is based on a nonlinear boundary value problem on a given spatial domain with curvilinear sections of the boundary for the system of equations for the continuity of the current of charge carriers, Poisson, and thermal conductivity. The statement of the problem contains a naturally formed small parameter, which made it possible to use asymptotic methods for its analytical-numerical solution. A model nonlinear boundary value problem with a small parameter is reduced to a sequence of linear boundary value problems by the methods of perturbation theory, and the physical domain of the problem with curvilinear sections of the boundary is reduced to the canonical form by the method of conformal mappings. Stationary distributions of charge carrier concentrations and the corresponding temperature field in the active region of p-i-n-structures are obtained in the form of asymptotic series in powers of a small parameter. The process of refining solutions is iterative, with the alternate fixation of unknown tasks at different stages of the iterative process. The asymptotic series describing the behavior of the plasma concentration and potential in the region under study, in contrast to the classical ones, contain boundary layer corrections. It was found that boundary functions play a key role in describing the electrostatic plasma field. The proposed approach to solving the corresponding nonlinear problem can significantly save computing resources
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PERCUS, J. K. "CLASSICAL FLUID DYNAMICS UNDER MOLECULAR SCALE CONFINEMENT." Modern Physics Letters B 23, no. 28 (November 10, 2009): 3311–31. http://dx.doi.org/10.1142/s0217984909021478.
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This is a review of some recent developments in the theory of classical nanoscale fluid dynamics, with tagged particle dynamics as the signature of transport properties. A heuristic analysis of the time asymptotic regime in strictly one-dimensional flow is followed by an exact treatment of the same system, with arbitrary non-interacting point particle dynamics. Stretched time and space coordinates allow us to obtain a universal form — the telegrapher's equation — for the self-dynamics after transients have died down. Attention then shifts to one-particle diffusion in a spatially modulated tubular enclosure, which is analyzed by increasingly sophisticated approximation methods. With this background, single-file fluid flow in modulated enclosures is studied, and preliminary analysis of the "passing region" carried out as well.
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Vasco, D. W., and Kurt T. Nihei. "A trajectory mechanics approach for the study of wave propagation in an anisotropic elastic medium." Geophysical Journal International 219, no. 3 (September 9, 2019): 1885–99. http://dx.doi.org/10.1093/gji/ggz406.
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SUMMARY We derive equations describing the path and traveltime of a coherent elastic wave propagating in an anisotropic medium, generalizing expressions from conventional high-frequency asymptotic ray theory. The methodology is valid across a broad range of frequencies and allows for subwavelength variations in the material properties of the medium. The primary difference from current ray methods is the retention of a term that is neglected in the derivation of the eikonal equation. The additional term contains spatial derivatives of the properties of the medium and of the amplitude field, and its presence couples the equations governing the evolution of the amplitude and phase along the trajectory. The magnitude of this term provides a measure of the validity of expressions based upon high-frequency asymptotic methods, such as the eikonal equation, when modelling wave propagation dominated by a band of frequencies. In calculations involving a layer with gradational boundaries, we find that asymptotic estimates do deviate from those of our frequency-dependent approach when the width of the layer boundaries become sufficiently narrow. For example, for a layer with boundaries that vary over tens of meters, the term neglected by a high-frequency asymptotic approximation is significant for frequencies around 10 Hz. The visible differences in the paths of the rays that traverse the layer substantiate this conclusion. For a velocity model derived from an observed well log, the majority of the trajectories calculated using the extended approach, accounting for the frequency-dependence of the rays, are noticeably different from those produced by the eikonal equation. A suite of paths from a source to a specified receiver, calculated for a range of frequencies between 10 and 100 Hz, define a region of sensitivity to velocity variations and may be used for an augmented form of tomographic imaging.
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SPEICHER, ROLAND, and CARLOS VARGAS. "FREE DETERMINISTIC EQUIVALENTS, RECTANGULAR RANDOM MATRIX MODELS, AND OPERATOR-VALUED FREE PROBABILITY THEORY." Random Matrices: Theory and Applications 01, no. 02 (April 2012): 1150008. http://dx.doi.org/10.1142/s2010326311500080.
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Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can be used to solve our free deterministic equivalents. As an illustration of our methods we show how the free deterministic equivalent of a random matrix model from [6] can be treated and we thus recover in a conceptual way the results from [6]. On a technical level, we generalize a result from scalar-valued free probability, by showing that randomly rotated deterministic matrices of different sizes are asymptotically free from deterministic rectangular matrices, with amalgamation over a certain algebra of projections. In Appendix A, we show how estimates for differences between Cauchy transforms can be extended from a neighborhood of infinity to a region close to the real axis. This is of some relevance if one wants to compare the original random matrix problem with its free deterministic equivalent.
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Okui, Ryo. "Asymptotically Unbiased Estimation of Autocovariances and Autocorrelations with Panel Data in the Presence of Individual and Time Effects." Journal of Time Series Econometrics 6, no. 2 (July 1, 2014): 129–81. http://dx.doi.org/10.1515/jtse-2013-0017.
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AbstractThis article proposes asymptotically unbiased estimators of autocovariances and autocorrelations for panel data with both individual and time effects. We show that the conventional autocovariance estimators suffer from the bias caused by the elimination of individual and time effects. The bias related to individual effects is proportional to the long-run variance, and it related to time effects is proportional to the value of the estimated autocovariance. For the conventional autocorrelation estimators, the elimination of time effects does not cause a bias while the elimination of individual effects does. We develop methods to estimate the long-run variance and propose bias-corrected estimators based on the proposed long-run variance estimator. We also consider the half-panel jackknife estimation for bias correction. The theoretical results are given by employing double asymptotics under which both the number of observations and the length of the time series tend to infinity. Monte Carlo simulations show that the asymptotic theory provides a good approximation to the actual bias and that the proposed bias-correction methods work well.
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Parnell, William J., and I. David Abrahams. "A new integral equation approach to elastodynamic homogenization." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2094 (February 26, 2008): 1461–82. http://dx.doi.org/10.1098/rspa.2007.0254.
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A new theory of elastodynamic homogenization is proposed, which exploits the integral equation form of Navier's equations and relationships between length scales within composite media. The scheme is introduced by focusing on its leading-order approximation for orthotropic, periodic fibre-reinforced media where fibres have arbitrary cross-sectional shape. The methodology is general but here it is shown for horizontally polarized shear (SH) wave propagation for ease of exposition. The resulting effective properties are shown to possess rich structure in that four terms account separately for the physical detail of the composite (associated with fibre cross-sectional shape, elastic properties, lattice geometry and volume fraction). In particular, the appropriate component of Eshelby's tensor arises naturally in order to deal with the shape of the fibre cross section. Results are plotted for circular fibres and compared with extant methods, including the method of asymptotic homogenization. The leading-order scheme is shown to be in excellent agreement even for relatively high volume fractions.
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Tuckwell, Henry C., Roger Rodriguez, and Frederic Y. M. Wan. "Determination of Firing Times for the Stochastic Fitzhugh-Nagumo Neuronal Model." Neural Computation 15, no. 1 (January 1, 2003): 143–59. http://dx.doi.org/10.1162/089976603321043739.
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We present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models. We apply the theory of first exit times for Markov processes to the Fitzhugh-Nagumo system with a constant mean gaussian white noise input, representing stochastic excitation and inhibition. Partial differential equations are obtained for the moments of the time to first spike. The observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one-dimensional approximation. For the moments of the time to reach threshold, this leads to ordinary differential equations that may be easily solved. Several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter. For ranges of the parameters appropriate for these asymptotic methods, the perturbation solutions are used to establish the validity of the one-dimensional approximation for both small and large values of the noise parameter. Additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one-dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations. Such agreement extends to intermediate values of the noise parameter. For the mean time to threshold, we find maxima at small noise values that constitute a form of stochastic resonance. We also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean.
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Budylina, Eugenia, Irina Garkina, and Alexander Danilov. "Approximation of Functions in Multi-criterial Synthesis of Composite Materials." IOP Conference Series: Materials Science and Engineering 1203, no. 2 (November 1, 2021): 022010. http://dx.doi.org/10.1088/1757-899x/1203/2/022010.
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Abstract A prerequisite for the synthesis of composite materials as complex systems is the principles of the control paradigm of Peace and the effectiveness of mathematics (for any reality and any given (not absolute) accuracy, there is a mathematical structure that describes this reality with this accuracy; the converse is also true (homomorphism, arbitrarily close to isomorphism between reality and mathematical structures)).The proposed methodology for managing the identification process (design of composites) includes the process of human choice: the probabilistic nature of the control; the main reason for the inadequacy of a purely analytical research procedure. Here, the optimization of the control of the properties of the composite is carried out experimentally on the model as a result of the approximation of the response function: not the generalized functional is approximated, but the particular criteria of which it consists. The development of composite materials is carried out on the basis of evaluating the parameters of the formation of operational properties. The parameters of each of the kinetic processes of the formation of the physical and mechanical characteristics of the material were taken as particular criteria. Kinetic processes are asymptotic for the composites under study and contain extremum and inflection points. A method is used to approximate multidimensional table-defined functions by generalized polynomials of a particular form. In the parametric identification of kinetic processes, their parameters are considered basic. Approximating models of the main properties are presented. Vector optimization of properties (selection of recipes, technologies and methods of material quality control) is carried out by overcoming ambiguities of goals using linear convolution, introducing benchmarks, building Pareto sets, etc. The expediency of using a systematic approach (the hierarchical structure of properties and the hierarchical structure of the composite proper) to the design of building materials as complex systems is shown. The research results are introduced as prototypes of new identification systems in the development of composite materials with adjustable structure and properties, in contrast to the replication of reference applied developments of identification theory in various industries.
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Dolgov, Sergey, Karim Anaya-Izquierdo, Colin Fox, and Robert Scheichl. "Approximation and sampling of multivariate probability distributions in the tensor train decomposition." Statistics and Computing 30, no. 3 (November 2, 2019): 603–25. http://dx.doi.org/10.1007/s11222-019-09910-z.
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Abstract General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor train format, a methodology that has been exploited for many years for scalable, high-dimensional density function approximation in quantum physics and chemistry. We build upon recent developments of the cross approximation algorithms in linear algebra to construct a tensor train approximation to the target probability density function using a small number of function evaluations. For sufficiently smooth distributions, the storage required for accurate tensor train approximations is moderate, scaling linearly with dimension. In turn, the structure of the tensor train surrogate allows sampling by an efficient conditional distribution method since marginal distributions are computable with linear complexity in dimension. Expected values of non-smooth quantities of interest, with respect to the surrogate distribution, can be estimated using transformed independent uniformly-random seeds that provide Monte Carlo quadrature or transformed points from a quasi-Monte Carlo lattice to give more efficient quasi-Monte Carlo quadrature. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis–Hastings accept/reject step, while the quasi-Monte Carlo quadrature may be corrected either by a control-variate strategy or by importance weighting. We show that the error in the tensor train approximation propagates linearly into the Metropolis–Hastings rejection rate and the integrated autocorrelation time of the resulting Markov chain; thus, the integrated autocorrelation time may be made arbitrarily close to 1, implying that, asymptotic in sample size, the cost per effectively independent sample is one target density evaluation plus the cheap tensor train surrogate proposal that has linear cost with dimension. These methods are demonstrated in three computed examples: fitting failure time of shock absorbers; a PDE-constrained inverse diffusion problem; and sampling from the Rosenbrock distribution. The delayed rejection adaptive Metropolis (DRAM) algorithm is used as a benchmark. In all computed examples, the importance weight-corrected quasi-Monte Carlo quadrature performs best and is more efficient than DRAM by orders of magnitude across a wide range of approximation accuracies and sample sizes. Indeed, all the methods developed here significantly outperform DRAM in all computed examples.
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Paninski, Liam. "Estimation of Entropy and Mutual Information." Neural Computation 15, no. 6 (June 1, 2003): 1191–253. http://dx.doi.org/10.1162/089976603321780272.
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We present some new results on the nonparametric estimation of entropy and mutual information. First, we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators. The setup is related to Grenander's method of sieves and places no assumptions on the underlying probability measure generating the data. Second, we prove a converse to these consistency theorems, demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information, even given unlimited data. This “inconsistency” theorem leads to an analytical approximation of the bias, valid in surprisingly small sample regimes and more accurate than the usual [Formula: see text] formula of Miller and Madow over a large region of parameter space. The two most practical implications of these results are negative: (1) information estimates in a certain data regime are likely contaminated by bias, even if “bias-corrected” estimators are used, and (2) confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods. Finally, we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem. By casting bias calculation problems in this approximation theory framework, we obtain the best possible generalization of known asymptotic bias results. More interesting, this framework leads to an estimator with some nice properties: the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions, and this maximum error turns out to be surprisingly small. We demonstrate the application of this new estimator on both real and simulated data.
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Cui, Zihan, Yuhang Liu, Jinfeng Zhang, and Xing Qiu. "Super-delta2: an enhanced differential expression analysis procedure for multi-group comparisons of RNA-seq data." Bioinformatics 37, no. 17 (March 8, 2021): 2627–36. http://dx.doi.org/10.1093/bioinformatics/btab155.
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Abstract Motivation We developed super-delta2, a differential gene expression analysis pipeline designed for multi-group comparisons for RNA-seq data. It includes a customized one-way ANOVA F-test and a post-hoc test for pairwise group comparisons; both are designed to work with a multivariate normalization procedure to reduce technical noise. It also includes a trimming procedure with bias-correction to obtain robust and approximately unbiased summary statistics used in these tests. We demonstrated the asymptotic applicability of super-delta2 to log-transformed read counts in RNA-seq data by large sample theory based on Negative Binomial Poisson (NBP) distribution. Results We compared super-delta2 with three commonly used RNA-seq data analysis methods: limma/voom, edgeR and DESeq2 using both simulated and real datasets. In all three simulation settings, super-delta2 not only achieved the best overall statistical power, but also was the only method that controlled type I error at the nominal level. When applied to a breast cancer dataset to identify differential expression pattern associated with multiple pathologic stages, super-delta2 selected more enriched pathways than other methods, which are directly linked to the underlying biological condition (breast cancer). Conclusions In conclusion, by incorporating trimming and bias-correction in the normalization step, super-delta2 was able to achieve tight control of type I error. Because the hypothesis tests are based on asymptotic normal approximation of the NBP distribution, super-delta2 does not require computationally expensive iterative optimization procedures used by methods such as edgeR and DESeq2, which occasionally have convergence issues. Availability and implementation Our method is implemented in a R-package, ‘superdelta2’, freely available at: https://github.com/fhlsjs/superdelta2. Supplementary information Supplementary data are available at Bioinformatics online.
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Bieg, Bohdan, and Janusz Chrzanowski. "Electromagnetic wave polarization state evolution in weakly anisotropic and nonuniform media with dissipation." Photonics Letters of Poland 9, no. 3 (September 30, 2017): 94. http://dx.doi.org/10.4302/plp.v9i3.761.
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The change of the polarization state of electromagnetic beam propagating in weakly anisotropic and smoothly inhomogeneous media with dissipation is analysed. On the basis of a quasi-isotropic approximation, which provides the consequent asymptotic solution of Maxwell's equation, the differential equation for the evolution of four component Stokes vector is derived. Obtained equation generalizes previous results for the nonadsorbing media and is written in terms of the dielectric tensor of birefringent media with dissipation. The formalism is illustrated by an example of magnetised plasma with dissipation due to the electron collisions. Full Text: PDF ReferencesK.G.Budden, Radio Waves in the Ionosphere (Cambridge U. Press 1961).V.I.Ginzburg, Propagation of Electromagnetic Waves in Plasma (Gordon & Breach 1970).Yu.A.Kravtsov, ""Quasiisotropic" Approximation to Geometrical Optics", Sov. Phys. Dokl. 13, 1125 (1969).A.A. Fuki, Yu.A. Kravtsov, and O.N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, Lond., N.Y. 1997).Yu.A. Kravtsov and Yu.I. Orlov, Geometrical optics of inhomogeneous media (Springer Verlag, Berlin, Heidelberg 1990). CrossRef Yu.A. Kravtsov et al., "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics", Physics-Uspekhi 39, 129(1996). CrossRef F.De Marco, S.E.Segre, "The polarization of an e.m. wave propagating in a plasma with magnetic shear. The measurement of poloidal magnetic field in a Tokamak", Plasma Phys. 14, 245 (1972). DirectLink S.E.Segre, "A review of plasma polarimetry - theory and methods", Plasma Phys. Control. Fusion 41, R57 (1999). CrossRef M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford 1980). CrossRef B.Bieg et al., "Quasi-Isotropic Approximation of Geometrical Optics Method as Adequate Electrodynamical Basis for Tokamak Plasma Polarimetry", Physics Procedia 62, 102 (2015). CrossRef B.Bieg et al., "Two approaches to plasma polarimetry: Angular variables technique and Stokes vector formalism", Nucl. Instr. Meth. Phys. Res. Sect. A 720, 157 (2013). CrossRef S.E.Segre, "New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma", J. Opt. Soc. Am. A 18, 2601 (2001). CrossRef
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Joyce, Duncan, William J. Parnell, Raphaël C. Assier, and I. David Abrahams. "An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2201 (May 2017): 20170080. http://dx.doi.org/10.1098/rspa.2017.0080.
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In Parnell & Abrahams (2008 Proc. R. Soc. A 464 , 1461–1482. ( doi:10.1098/rspa.2007.0254 )), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.
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Sokil, B. I., P. Ya Pukach, A. P. Senyk, M. B. Sokil, A. I. Andrukhiv, and M. I. Vovk. "Asymptotic method and wave theory of motion in studying the effect of periodic impulse forces on systems characterized by longitudinal motion velocity." Mathematical Modeling and Computing 9, no. 4 (2022): 909–20. http://dx.doi.org/10.23939/mmc2022.04.909.
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A methodology for researching dynamic processes of one-dimensional systems with distributed parameters that are characterized by longitudinal component of motion velocity and are under the effect of periodic impulse forces has been developed. The boundary problem for the generalized non-linear differential Klein–Gordon equation is the mathematical model of dynamics of the systems under study in Euler variables. Its specific feature is that the unexcited analogue does not allow applying the known classical Fourier and D'Alembert methods for building a solution. Non-regularity of the right part for the excited non-linear analogue is another problem. This study shows that the dynamic process of the respective unexcited motion can be treated as overlapping of the direct and reflected waves of different lengths but equal frequencies. Analytical dependencies have been obtained for describing the aforesaid parameters of the waves. They show that the dynamic process in such mechanical systems depends not only on their main physical and mechanical parameters and boundary conditions, but also on the longitudinal motion velocity (relative momentum). As relative momentum increases, the frequency of the process decreases. To describe excited motion, we use the principle of single frequency of oscillations in non-linear systems with concentrated masses and distributed parameters as well as regularization of periodic impulse excitations. The main idea of asymptotic integration of systems with small non-linearity into the class of dynamic systems under study has been generalized. A standard equation for the resonance and non-resonance cases has been obtained. It has been established that for the first approximation, in the non-resonance case, impulse excitation affects only the partial change of the form of oscillations. Resonance processes are possible at a specific relation between the impulse excitation period, the motion velocity of the medium, and physical-mechanical features of the body. The amplitude of transition through resonance becomes higher when impulse actions are applied closer to the middle of the body. As the longitudinal motion velocity increases, it initially increases and then decreases.
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Christensen, Erik, and Allan M. Sinclair. "Completely bounded isomorphisms of injective von Neumann algebras." Proceedings of the Edinburgh Mathematical Society 32, no. 2 (June 1989): 317–27. http://dx.doi.org/10.1017/s0013091500028716.
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Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).
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Geloni, Gianluca, Vitali Kocharyan, and Evgeni Saldin. "Brightness of synchrotron radiation from undulators and bending magnets." Journal of Synchrotron Radiation 22, no. 2 (February 3, 2015): 288–316. http://dx.doi.org/10.1107/s1600577514026071.
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The maximum of the Wigner distribution (WD) of synchrotron radiation (SR) fields is considered as a possible definition of SR source brightness. Such a figure of merit was originally introduced in the SR community by Kim [(1986),Nucl. Instrum. Methods Phys. Res. A,246, 71–76]. The brightness defined in this way is always positive and, in the geometrical optics limit, can be interpreted as the maximum density of photon flux in phase space. For undulator and bending magnet radiation from a single electron, the WD function can be explicitly calculated. In the case of an electron beam with a finite emittance the brightness is given by the maximum of the convolution of a single electron WD function and the probability distribution of the electrons in phase space. In the particular case when both electron beam size and electron beam divergence dominate over the diffraction size and the diffraction angle, one can use a geometrical optics approach. However, there are intermediate regimes when only the electron beam size or the electron beam divergence dominate. In these asymptotic cases the geometrical optics approach is still applicable, and the brightness definition used here yields back once more to the maximum photon flux density in phase space. In these intermediate regimes a significant numerical disagreement is found between exact calculations and the approximation for undulator brightness currently used in the literature. The WD formalism is extended to a satisfactory theory for the brightness of a bending magnet. It is found that in the intermediate regimes the usually accepted approximation for bending magnet brightness turns out to be inconsistent even parametrically.
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Bernasconi, Giancarlo, and Giuseppe Drufuca. "3-D traveltimes and amplitudes by gridded rays." GEOPHYSICS 66, no. 1 (January 2001): 277–82. http://dx.doi.org/10.1190/1.1444906.
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Seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium. They can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite‐difference methods. Unfortunately, this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used. We propose a new method for computing traveltimes and amplitudes in 3-D media that is simple, fast, unconditionally stable, and robust. Defining the slowness vector as [Formula: see text] and assuming an isotropic medium, the ray velocity v is related to the slowness vector by the relation [Formula: see text]. Rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane. The components of the slowness vector are then interpolated to gridpoints on this next horizontal plane. This is termed regridding; the process of downward propagation of rays, one vertical grid step at a time, is continued until some prescribed depth is reached. Computation of amplitudes is achieved using a method similar to that for obtaining the zero‐order approximation in asymptotic ray theory. We show comparisons with a full‐wave method on readily accessible 3-D velocity models.
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Jia, Lin, Yaonan Wang, Changfan Zhang, Kaihui Zhao, Li Liu, and Xuan Quynh Nguyen. "A Robust Adaptive Trajectory Tracking Algorithm Using SMC and Machine Learning for FFSGRs with Actuator Dead Zones." Applied Sciences 9, no. 18 (September 12, 2019): 3837. http://dx.doi.org/10.3390/app9183837.
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The actuator dead zone of free-form surface grinding robots (FFSGRs) is very common in the grinding process and has a great impact on the grinding quality of a workpiece. In this paper, an improved trajectory tracking algorithm for an FFSGR with an asymmetric actuator dead zone was proposed with consideration of friction forces, model uncertainties, and external disturbances. The presented control algorithm was based on the machine learning and sliding mode control (SMC) methods. The control compensator used neural networks to estimate the actuator’s dead zone and eliminate its effects. The robust SMC compensator acted as an auxiliary controller to guarantee the system’s stability and robustness under circumstances with model uncertainties, approximation errors, and friction forces. The stability of the closed-loop system and the asymptotic convergence of tracking errors were evaluated using Lyapunov theory. The simulation results showed that the dead zone’s non-linearity can be estimated correctly, and satisfactory trajectory tracking performance can be obtained in this way, since the influences of the actuator’s dead zone were eliminated. The convergence time of the system was reduced from 1.1 to 0.8 s, and the maximum steady-state error was reduced from 0.06 to 0.015 rad. In the grinding experiment, the joint steady-state error decreased by 21%, which proves the feasibility and effectiveness of the proposed control method.
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Xie, Jiaojiao, and Qiaohong Li. "A Model for Teaching English Translation in Universities Based on the Random Matrix Optimization Bell Translation Model." Mathematical Problems in Engineering 2022 (June 28, 2022): 1–11. http://dx.doi.org/10.1155/2022/4421228.
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This paper adopts the random matrix optimization fusion of Bell’s translation model to conduct in-depth research and analysis on the teaching model of English translation in colleges and universities. Bell’s translation model theory explicates the mental model of the translation process to a certain extent based on psycholinguistic research results. Using the results of random matrix theory on the eigenvalues of the sample covariance matrix, the energy of each subspace is estimated, and the estimated energy is then used to construct the subspace-weighting matrix. The statistical properties of the sample covariance matrix eigenvectors were analyzed using the first-order perturbation approximation, and then asymptotic results from random matrix theory on the projection of the sample covariance matrix signal subspaces to the real signal parametrization were used to obtain the weighting matrix based on the random matrix eigenvectors. The analysis shows that the teacher can help students understand the teaching content and master the knowledge points by using guided meta-discoursal to indicate the logical relationships inherent in linguistic expressions and make the discourse transitions natural, coherent, and organized. The use of interactive meta-discoursal attracts students’ attention and treats them as participants in the discourse, making them feel a sense of belonging and satisfying their need for psychological belonging, which is beneficial to the students’ main position in the classroom. In teaching methods, there is a principle of comprehensible input, which is to provide students with a lot of interesting and understandable listening and reading input. In the translation classroom, the translation teachers code-switched infrequently and mainly used three types of code-switching: inter-sentence code-switching, intra-sentence code-switching, and additional code-switching to carry out translation teaching. Among them, intra-sentence code-switching accounts for 57.46%, inter-sentence code-switching accounts for 41.53%, and the least frequent is additional code-switching, accounting for 1.01%.
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Mossaiby, Farshid, Majid Bazrpach, and Arman Shojaei. "Extending the method of exponential basis functions to problems with singularities." Engineering Computations 32, no. 2 (April 20, 2015): 406–23. http://dx.doi.org/10.1108/ec-01-2014-0019.
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Purpose – The purpose of this paper is to aim at extending the method of exponential basis functions (EBF) to solve a class of problems with singularities. Design/methodology/approach – In the procedure of EBF a summation of EBF satisfying the governing differential equation with unknown constant coefficients is considered for the solution. These coefficients are determined by the satisfaction of prescribed boundary conditions through a collocation approach. The applied basis functions are available in the case of linear partial differential equations (PDEs) with constant coefficients. Moreover, the method contributes to yield highly accurate results with ultra convergence rates for problems with smooth solution. This leads EBF to offer many advantages for a variety of engineering problems. However, owing to the global and smooth nature of the bases, the performance of EBF deteriorates in problems with singularities. In the present study, some exponential-like influence functions are developed, and a few of them are added to original bases. Findings – The new bases are capable of forming the constitutive terms of the asymptotic solution near the singularity points and alleviate the aforementioned limitation. The appealing feature of this method is that all the advantages of EBF such as its simplicity and efficiency are completely preserved. Research limitations/implications – In its current form, EBF can only solve PDEs with constant coefficients. Originality/value – Application of the method to some benchmark problems demonstrates its robustness over some other boundary approximation methods. This research may pave the road for future investigations corresponding to a wide range of practical engineering problems.
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Dobson, John F. "Towards efficient description of type-C London dispersion forces between low-dimensional metallic nanostructures." Electronic Structure 3, no. 4 (November 30, 2021): 044001. http://dx.doi.org/10.1088/2516-1075/ac3807.
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Abstract This paper concerns one of the remaining difficulties encountered in efficient modelling schemes for dispersion forces between nanostructures. Dispersion (van der Waals, vdW) interactions between molecules and nanostructures can be reliably described in principle by computationally intensive high-level theory, but this is often not computationally feasible in practice, so more efficient methods are continually being developed. Progress has been made with nonlocal density functionals (vdW-DFs) and with atom-based schemes (D4, MBD, uMBD). In such efficient schemes, the effects beyond additive two-atom terms have been categorized as ‘type A’, ‘type B’ and ‘type C’ non-additivity (see Dobson (2014 Int. J. Quantum Chem. 114 1157)). Atom-based models using coupled-harmonic-oscillator theory (MBD) now deal adequately with type A and type B non-additivity, but type-C effects, related to gapless collective electronic excitations, can occur in low-dimensional metals, and these are not correctly described by the efficient schemes mentioned above. From analytic work within the direct random phase approximation (dRPA), type-C effects have long been known to cause the vdW interaction between well-separated low-d metals to fall off much more slowly with separation than is predicted by the above-mentioned efficient schemes. The slower decay means that type-C effects dominate in this asymptotic large-separation regime. It has not been clear, however, whether type-C physics contributes significantly to the vdW interaction of low-d metals near to contact, where the forces are much larger. The present work uses recent semi-analytic dRPA results to provide some evidence that type-C effects are indeed significant near to contact between metallic carbon nanotubes, and between doped graphene sheets. Some guidelines are therefore suggested for ways to combine the semi-analytic dRPA approach, here termed ‘SAM-dRPA’, with the existing efficient vdW algorithms described above.
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GAGNAIRE-RENOU, ELODIE, MICHEL BENOIT, and SERGEI I. BADULIN. "On weakly turbulent scaling of wind sea in simulations of fetch-limited growth." Journal of Fluid Mechanics 669 (January 5, 2011): 178–213. http://dx.doi.org/10.1017/s0022112010004921.
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Extensive numerical simulations of fetch-limited growth of wind-driven waves are analysed within two approaches: a ‘traditional’ wind-speed scaling first proposed by Kitaigorodskii (Bull. Acad. Sci. USSR, Geophys. Ser., Engl. Transl., vol. N1, 1962, p. 105) in the early 1960s and an alternative weakly turbulent scaling developed recently by Badulin et al. (J. Fluid Mech.591, 2007, 339–378). The latter one uses spectral fluxes of wave energy, momentum and action as physical scales of the problem and allows for advanced qualitative and quantitative analysis of wind-wave growth and features of air–sea interaction. In contrast, the traditional approach is shown to be descriptive rather than proactive. Numerical simulations are conducted on the basis of the Hasselmann kinetic equation for deep-water waves in a wide range of wind speeds from 5 to 30 m s −1 and for the ideal case of fetch-limited growth: permanent wind blowing perpendicularly to a straight coastline. Two different wave input functions, Sin, and two methods for calculating the nonlinear transfer term Snl (Gaussian quadrature method, or GQM, a quasi-exact method based on the use of Gaussian quadratures, and the discrete interaction approximation, or DIA) are used in the simulations. Comparison of the corresponding results firstly shows the relevance of the analysis of wind-wave growth in terms of the proposed weakly turbulent scaling, and secondly, allows us to highlight some critical points in the modelling of wind-generated waves. Three stages of wind-wave development corresponding to qualitatively different balance of the source terms, Sin, Sdiss and Snl, are identified: initial growth, growing sea and fully developed sea. Validity of the asymptotic weakly turbulent approach for the stage of growing wind sea is determined by the dominance of nonlinear transfers, which results in a rigid link between spectral fluxes and wave energy. This stage of self-similar growth is investigated in detail and presented as a consequence of three sub-stages of qualitatively different coupling of air flow and growing wind waves. The key self-similarity parameter of the asymptotic theory is estimated to be αss = 0.68 ± 0.1.Further prospects of wind-wave modelling in the context of the presented weakly turbulent scaling are discussed.
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Wargnier, Q., A. Alvarez Laguna, J. B. Scoggins, N. N. Mansour, M. Massot, and T. E. Magin. "Consistent transport properties in multicomponent two-temperature magnetized plasmas." Astronomy & Astrophysics 635 (March 2020): A87. http://dx.doi.org/10.1051/0004-6361/201834686.
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Aims. We present a fluid model that has been developed for multicomponent two-temperature magnetized plasmas in chemical non-equilibrium for the partially to fully ionized collisional regimes. We focus on transport phenomena with the aim of representing the atmosphere of the Sun. Methods. This study is based on an asymptotic fluid model for multicomponent plasmas derived from kinetic theory, yielding a rigorous description of the dissipative effects. The governing equations and consistent transport properties are obtained using a multiscale Chapman-Enskog perturbative solution to the Boltzmann equation based on a dimensional analysis. The mass disparity between free electrons and heavy particles is accounted for, as well as the influence of the electromagnetic field. We couple this model to the Maxwell equations for the electromagnetic field and derive the generalized Ohm’s law for multicomponent plasmas. The model inherits a well-identified mathematical structure leading to an extended range of validity for the Sun’s atmospheric conditions. We compute consistent transport properties by means of a spectral Galerkin method using the Laguerre-Sonine polynomial approximation. Two non-vanishing polynomial terms are used when deriving the transport systems for electrons, whereas only one term is retained for heavy particles. Results. In a simplified framework where the plasma is fully ionized, we compare the transport properties for the lower solar atmosphere to conventional expressions for magnetized plasmas attributed to Braginskii, showing a good agreement between both results. For more general partially ionized conditions, representative of the lower solar atmosphere, we compute the muticomponent transport properties corresponding to the species diffusion velocities, heavy-particle and electron heat fluxes, and viscous stress tensor of the model for a helium-hydrogen mixture in local thermodynamic equilibrium. The model is assessed for the 3D radiative magnetohydrodynamic simulation of a pore at the Sun photosphere. The resistive term is found to dominate mainly the dynamics of the electric field at the pore location. The battery term for heavy particles appears to be higher at the pore location and at some intergranulation boundaries.
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Park, J., V. Prat, and S. Mathis. "Horizontal shear instabilities in rotating stellar radiation zones." Astronomy & Astrophysics 635 (March 2020): A133. http://dx.doi.org/10.1051/0004-6361/201936863.
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Context. Rotational mixing transports angular momentum and chemical elements in stellar radiative zones. It is one of the key processes for modern stellar evolution. In the past two decades, an emphasis has been placed on the turbulent transport induced by the vertical shear instability. However, instabilities arising from horizontal shear and the strength of the anisotropic turbulent transport that they may trigger remain relatively unexplored. The weakest point of this hydrodynamical theory of rotational mixing is the assumption that anisotropic turbulent transport is stronger in horizontal directions than in the vertical one. Aims. This paper investigates the combined effects of stable stratification, rotation, and thermal diffusion on the horizontal shear instabilities that are obtained and discussed in the context of stellar radiative zones. Methods. The eigenvalue problem describing linear instabilities of a flow with a hyperbolic-tangent horizontal shear profile was solved numerically for a wide range of parameters. When possible, the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation was applied to provide analytical asymptotic dispersion relations in both the nondiffusive and highly diffusive limits. As a first step, we consider a polar f-plane where the gravity and rotation vector are aligned. Results. Two types of instabilities are identified: the inflectional and inertial instabilities. The inflectional instability that arises from the inflection point (i.e., the zero second derivative of the shear flow) is the most unstable when at a zero vertical wavenumber and a finite wavenumber in the streamwise direction along the imposed-flow direction. While the maximum two-dimensional growth rate is independent of the stratification, rotation rate, and thermal diffusivity, the three-dimensional inflectional instability is destabilized by stable stratification, while it is stabilized by thermal diffusion. The inertial instability is rotationally driven, and a WKBJ analysis reveals that its growth rate reaches the maximum value of √f(1 − f) in the inviscid limit as the vertical wavenumber goes to infinity, where f is the dimensionless Coriolis parameter. The inertial instability for a finite vertical wavenumber is stabilized as the stratification increases, whereas it is destabilized by the thermal diffusion. Furthermore, we found a selfsimilarity in both the inflectional and inertial instabilities based on the rescaled parameter PeN2 with the Péclet number Pe and the Brunt–Väisälä frequency N.
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Fendick, Kerry W., and Ward Whitt. "Verifying cell loss requirements in high-speed communication networks." Journal of Applied Mathematics and Stochastic Analysis 11, no. 3 (January 1, 1998): 319–38. http://dx.doi.org/10.1155/s1048953398000276.
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In high-speed communication networks it is common to have requirements of very small cell loss probabilities due to buffer overflow. Losses are measured to verify that the cell loss requirements are being met, but it is not clear how to interpret such measurements. We propose methods for determining whether or not cell loss requirements are being met. A key idea is to look at the stream of losses as successive clusters of losses. Often clusters of losses, rather than individual losses, should be regarded as the important loss events. Thus we propose modeling the cell loss process by a batch Poisson stochastic process. Successive clusters of losses are assumed to arrive according to a Poisson process. Within each cluster, cell losses do not occur at a single time, but the distance between losses within a cluster should be negligible compared to the distance between clusters. Thus, for the purpose of estimating the cell loss probability, we ignore the spaces between successive cell losses in a cluster of losses. Asymptotic theory suggests that the counting process of losses initiating clusters often should be approximately a Poisson process even though the cell arrival process is not nearly Poisson. The batch Poisson model is relatively easy to test statistically and fit; e.g., the batch-size distribution and the batch arrival rate can readily be estimated from cell loss data. Since batch (cluster) sizes may be highly variable, it may be useful to focus on the number of batches instead of the number of cells in a measurement interval. We also propose a method for approximately determining the parameters of a special batch Poisson cell loss with geometric batch-size distribution from a queueing model of the buffer content. For this step, we use a reflected Brownian motion (RBM) approximation of a G/D/1/C queueing model. We also use the RBM model to estimate the input burstiness given the cell loss rate. In addition, we use the RBM model to determine whether the presence of losses should significantly affect the estimation of server utilization when both losses and utilizations are estimated from data. Overall, our analysis can serve as a basis for determining required observation intervals in order to reach conclusions with a solid statistical basis. Thus our analysis can help plan simulations as well as computer system measurements.
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Stinson, Charles P., Saleh S. Almuthaybiri, and Christopher C. Tisdell. "A note regarding extensions of fixed point theorems involving two metrics via an analysis of iterated functions." ANZIAM Journal 61 (June 13, 2020): C15—C30. http://dx.doi.org/10.21914/anziamj.v61i0.15048.
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The purpose of this work is to advance the current state of mathematical knowledge regarding fixed point theorems of functions. Such ideas have historically enjoyed many applications, for example, to the qualitative and quantitative understanding of differential, difference and integral equations. Herein, we extend an established result due to Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42] that involves two metrics to ensure wider classes of functions admit a unique fixed point. In contrast to the literature, a key strategy herein involves placing assumptions on the iterations of the function under consideration, rather than on the function itself. In taking this approach we form new advances in fixed point theory under two metrics and establish interesting connections between previously distinct theorems, including those of Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42], Caccioppoli [Rend. Acad. Naz. Linzei. 11, 1930, 31–49] and Bryant [Am. Math. Month. 75, 1968, 399–400]. Our results make progress towards a fuller theory of fixed points of functions under two metrics. Our work lays the foundations for others to potentially explore applications of our new results to form existence and uniqueness of solutions to boundary value problems, integral equations and initial value problems. References Almuthaybiri, S. S. and C. C. Tisdell. ``Global existence theory for fractional differential equations: New advances via continuation methods for contractive maps''. Analysis, 39(4):117–128, 2019. doi:10.1515/anly-2019-0027 Almuthaybiri, S. S. and C. C. Tisdell. ``Sharper existence and uniqueness results for solutions to third-order boundary value problems, mathematical modelling and analysis''. Math. Model. Anal. 25(3):409–420, 2020. doi:10.3846/mma.2020.11043 Banach, S. ``Sur les operations dans les ensembles abstraits et leur application aux equations integrales''. Fund. Math., 3:133–181 1922. doi:10.4064/fm-3-1-133-181 Brouwer, L. E. J. ``Ueber Abbildungen von Mannigfaltigkeiten''. Math. Ann. 71:598, 1912. doi:10.1007/BF01456812 Bryant, V. W. ``A remark on a fixed point theorem for iterated mappings'' Am. Math. Month. 75: 399–400, 1968. doi:10.2307/2313440 Caccioppoli, R. ``Un teorema generale sullesistenza de elemente uniti in una transformazione funzionale''. Rend. Acad. Naz. Linzei. 11:31–49, 1930. Goebel, K., and W. A. Kirk. Topics in metric fixed point theory. Cambridge University Press, 1990, doi:10.1017/CBO9780511526152 Leray, J., and J. Schauder. ``Topologie et equations fonctionnelles''. Ann. Sci. Ecole Norm. Sup. 51:45–78, 1934. doi:10.24033/asens.836 O'Regan, D. and R. Precup. Theorems of Leray–Schauder type and applications, Series in Mathematical Analysis and Applications, Vol. 3. CRC Press, London, 2002. doi:10.1201/9781420022209 Rus, I. A. ``On a fixed point theorem of Maia''. Studia Univ. Babes-Bolyai Math. 22:40–42, 1977. Schaefer, H. H. ``Ueber die Methode der a priori-Schranken''. Math. Ann. 129:415–416, 1955. doi:10.1007/bf01362380 Tisdell, C. C. ``When do fractional differential equations have solutions that are bounded by the Mittag-Leffler function?'' Fract. Calc. Appl. Anal. 18(3):642–650, 2015. doi:10.1515/fca-2015-0039 Tisdell, C. C. ``A note on improved contraction methods for discrete boundary value problems.'' J. Diff. Eq. Appl. 18(10):1773–1777, 2012. doi:10.1080/10236198.2012.681781 Tisdell, C. C. ``On the application of sequential and fixed-point methods to fractional differential equations of arbitrary order.'' J. Int. Eq. Appl. 24(2):283–319, 2012. doi:10.1216/JIE-2012-24-2-283 Ehrnstrom, M., Tisdell, C. C. and E. Wahlen. ``Asymptotic integration of second-order nonlinear difference equations.'' Glasg. Math. J. 53(2):223–243, 2011. doi:10.1017/S0017089510000650 Erbe, L., A. Peterson and C. C. Tisdell. ``Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales.'' Nonlin. Anal. 69(7):2303–2317, 2008. doi:10.1016/j.na.2007.08.010 Tisdell, C. C. and A. Zaidi. ``Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling.'' Nonlin. Anal. 68(11):3504–3524, 2008. doi:10.1016/j.na.2007.03.043 Tisdell, C. C. ``Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions.'' Int.. J. Math. Ed. Sci. Tech. 48(7):1087–1095, 2017. doi:10.1080/0020739X.2017.1298856 Tisdell, C. C. ``On Picard's iteration method to solve differential equations and a pedagogical space for otherness.'' Int. J. Math. Ed. Sci. Tech. 50(5):788–799, 2019. doi:10.1080/0020739X.2018.1507051 Zeidler, E. Nonlinear functional analysis and its applications. Springer-Verlag, New York, 1986. doi:10.1007/978-1-4612-4838-5
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Huang, Xiao-Min, Yu Lin, and Yu-Qiu Zhao. "Asymptotics of the Charlier polynomials via difference equation methods." Analysis and Applications, August 18, 2020, 1–35. http://dx.doi.org/10.1142/s021953052050013x.
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We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained, respectively, in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.
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Karageorge, Panos Demetrios, and George Makrakis. "Asymptotic Approximations for the phase space Schrödinger equation." Journal of Physics A: Mathematical and Theoretical, June 8, 2022. http://dx.doi.org/10.1088/1751-8121/ac76f6.
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Abstract We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the Anisotropic Gaussian Approximation, related to the Nearby Orbit Approximation, by which we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the Complex WKB Theory developed by Maslov, on the basis of a canonical system in double phase space related to the Berezin-Shubin-Marinov Hamilton-Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in R.
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"Numerical and asymptotic methods for certain viscous free-surface flows." Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 340, no. 1656 (July 15, 1992): 1–45. http://dx.doi.org/10.1098/rsta.1992.0054.
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This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.
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Uimezawa, Naoto, and Susumu Saito. "Excitation Spectra in the Time-Dependent Density-Functional Theory with Gradient Correction." MRS Proceedings 579 (1999). http://dx.doi.org/10.1557/proc-579-75.
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ABSTRACTWe study tile optical absorption spectra of Na clusters using the time-dependent density-functional theory with gradient correction. A jellium-sphere background model, which is free from basis-set incompleteness error and is suitable for the comparison of various theoretical methods, is adopted. For energies of surface-plasinon excitations governing profiles of photoabsorption spectra with huge oscillator strengths., the gradient correction by van Leeiiwen and Baerends with correct asymptotic behavior of the effective potential is found to show considerable improvement over the time-dependent local-density approximation.