Taylor Expansion

We consider an analytic vector fieldẋ = X (x) and study, via a variational approach, whether it may possess analytic first integrals. We assume one solution Γ is known and we study the successive variational equations along Γ. Constructions in [MRRS07] show that Taylor expansion coefficients of first integrals appear as rational solutions of the dual linearized variational equations. We show that they also satisfy linear "filter" conditions. Using this, we adapt the algorithms from [Bar99, vHW97] to design new ones optimized to this effect and demonstrate their use. Part of this work stems from the first author's Ph.D. thesis 1 [AM10].

The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.

In this article, after describing a procedure to construct trajectories for a spacecraft in the four-body model, a method to correct the trajectory violations is presented. To construct the trajectories, periodic orbits as the solutions of the three-body problem are used. On the other hand, the bicircular model based on the Sun-Earth rotating frame governs the dynamics of the spacecraft and other bodies. A periodic orbit around the first libration-point L 1 is the destination of the mission which is one of the equilibrium points in the Sun-Earth/Moon three-body problem. In the way to reach such a far destination, there are a lot of disturbances such as solar radiation and winds that make the plans untrustworthy. However, the solar radiation pressure is considered in the system dynamics. To prevail over these difficulties, considering the whole transfer problem as an optimal control problem makes the designer to be able to correct the unavoidable violations from the pre-designed trajectory and strategies. The optimal control problem is solved by a direct method, transcribing it into a nonlinear programming problem. This transcription gives an unperturbed optimal trajectory and its sensitivities with respect perturbations. Modeling these perturbations as parameters embedded in a parametric optimal control problem, one can take advantage of the parametric sensitivity analysis of nonlinear programming problem to recalculate the optimal trajectory with a very smaller amount of computation costs. This is obtained by evaluating a first-order Taylor expansion of the perturbed solution in an iterative process which is aimed to achieve an admissible solution. At the end, the numerical results show the applicability of the presented method.

Wallis's method of interpolation attracted the attention of the young Euler, who obtained some important results. The problem of interpolation led Euler to formulate the problem of integration, i.e., to express the general term of a series by means of an integral. The latter problem was connected to the question of expressing the sum of a series using an integral. The outcome of this research was Euler's derivation of what would later become known as the Euler–Maclaurin formula. Euler subsequently returned to interpolation and formulated the theory of inexplicable functions including the gamma function.The methods used by Euler illustrate well the principles of 18th-century analysis. Eulerian procedures are based upon the notion of geometric quantity. A function is actually conceived as the expression of a quantity and, for this reason, it intrinsically possesses properties we can term continuity, differentiability, Taylor expansion. These correspond to the usual properties of a curve which has “regular” characteristics (lack of jumps, presence of tangents, curvature radius, etc.). They have a “figural” clarity. Although Eulerian analysis remains rooted in geometry, it dispenses with figural representation: it is substantially nonfigural geometry. Reasoning with figures (which integrates the proof in classical geometry) is replaced by reasoning with analytic symbols. These are general because they do not represent a particular quantity and are not subjected to restrictions, but are an abstract representation of quantity.Copyright 1998 Academic Press.Il problema dell'interpolazione wallisiana attrasse l'attenzione del giovane Euler. Egli ottenne rapidamente risultati di grande interesse e proseguı̀ le sue ricerche formulando il problema dell'integrazione consistente nell'esprimere il termine generale di una serie mediante un integrale. Quest'ultimo problema ben presto si evolse nella questione di esprimere la somma di una serie mediante un integrale che guidò Euler nella derivazione della formula sommatoria oggi detta di Euler-Maclaurin. Euler ritornò in seguito sul problema dell'interpolazione sviluppando la teoria delle funzioni inesplicabili che comprendono come caso particolare la gamma.I metodi usati da Euler sono di grande interesse per una comprensione dell'analisi settecen- tesca. Infatti alla base delle procedure euleriane vi è la nozione di quantità geometrica, la quale comporta che la funzione, intesa come espressione della quantità, abbia intrinsecamente proprietà che, con linguaggio moderno, possiamo chiamare continuità, differenziabilità, sviluppabilità in serie de Taylor, e che corrispondono alle proprietà tipiche di una curva dotata di “regolarità” (non fare salti, esistenza della tangente, del raggio di curvatura, ecc.). Queste proprietà hanno un'immediata evidenza nelle curve comunemente studiate, un'evidenza, per cosi dire, figurale. L'analisi euleriana conserva un forte contenuto geometrico, ma elimina la rappresentazione figurale: sostanzialmente essa appare come una geometria non figurale. Il ragionamento sulla figura (che nella geometria classica integrava la dimostrazione) viene ora sostitutito dal ragionamento sui simboli analitici, i quali sono generali, perché non rappresentano questa o quella particolare quantità e non sone soggetti a limitazioni di sorta, ma sono una rappresentazione astratta della quantità.Copyright 1998 Academic Press.Le problème d'interpolation de Wallis attira l'attention du jeune Euler, qui obtint rapidement des résultats de grand intérêt. Il amena Euler à formuler le problème d'intégration consistant à exprimer le terme général d'une série par une intégrale. Ce dernier problème se transforma rapidement en celui d'exprimer la somme d'une série par une intégrale, menant à la formule appelée de nos jours la formule d'Euler-Mac Laurin. Euler reprit ultérieurement le problème d'interpolation, formulant la théorie des fonctions “inexplicables” qui incluent en particulier la fonction Gamma.Les méthodes utilisées par Euler ont un grand intérêt pour comprendre l'analyse du XVIIIe siècle. En effet, à la base des procédures eulériennes, on trouve la notion de quantité géométrique, qui implique que la fonction, comprise comme une expression de cette quantité, possède intrinsèquement certaines propriétés—en langage moderne la continuité, la différentiabilité, le développement en séries de Taylor—correspondant aux propriétés typiques d'une courbe dotée de régularité—ne pas faire de sauts, avoir une tangente, un rayon de courbure. Ces propriétés étaient évidentes pour les courbes communément étudiées, une évidence pour ainsi dire figurale. L'analyse eulérienne conserve un important contenu géométrique, mais élimine la représentation figurale: concrêtement, elle apparaı̂t comme une géométrie non figurale. Le raisonnement sur la figure, qui, dans la géométrie classique, intégrait la démonstration, est ici remplacé par le raisonnement sur les symboles analytiques: ceux-ci sont généraux parce qu'ils ne représentent pas une quantité particulière et ne sont pas soumis à des restrictions, mais sont une représentation abstraite de la quantité.Copyright 1998 Academic Press.AMS 1991 subject classifications: 01A50

In the rst part we discuss the concept of asymptotic expansion and its importance in applications. We focus our attention on special functions dened through integrals and consider their approximation by means of asymptotic expansions. We explain the general ideas of the theory of asymptotic expansions of integrals and describe two classical methods (Watson's lemma and the saddle point method)

This paper deals with probabilistic upper bounds for the error in functional estimation defined on some interpolation and extrapolation designs, when the function to estimate is supposed to be analytic. The error pertaining to the estimate may depend on various factors: the frequency of observations on the knots, the position and number of the knots, and also on the error committed when approximating the function through its Taylor expansion. When the number of observations is fixed, then all these parameters are determined by the choice of the design and by the choice estimator of the unknown function.

This paper is due to modifying the Method of Successive Linear Problems (MSLP) in solving non-linear eigenvalue problems. The MSLPis a basic method for acquiring roots of non-linear equations with the second rank convergence. In this paper, both the second and the third Taylor expansion are applied for improving the speed of convergence in order to acquire the exact roots of non-linear equations. Numericalexamples are used to illustrate the method results.

The paper deals with the f-divergences of Csiszár generalizing the discrimination information of Kullback, the total variation distance, the Hellinger divergence, and the Pearson divergence. All basic properties of f-divergences including relations to the decision errors are proved in a new manner replacing the classical Jensen inequality by a new generalized Taylor expansion of convex functions. Some new properties are proved too, e.g., relations to the statistical sufficiency and deficiency. The generalized Taylor expansion also shows very easily that all f-divergences are average statistical informations (differences between prior and posterior Bayes errors) mutually differing only in the weights imposed on various prior distributions. The statistical information introduced by De Groot and the classical information of Shannon are shown to be extremal cases corresponding to = 0 and = 1 in the class of the so-called Arimoto -informations introduced in this paper for 0 < < 1 by means of the Arimoto -entropies. Some new examples of f-divergences are introduced as well, namely, the Shannon divergences and the Arimoto -divergences leading for " 1 to the Shannon divergences. Square roots of all these divergences are shown to be metrics satisfying the triangle inequality. The last section introduces statistical tests and estimators based on the minimal f-divergence with the empirical distribution achieved in the families of hypothetic distributions. For the Kullback divergence this leads to the classical likelihood ratio test and estimator.

In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.

A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y = f (x, y), y(x 0 ) = y 0 . The method uses y and higher derivatives y (2) to y (6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y (3) to y (6) . The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

We develop a new solution method for a broad class of discrete-time dynamic portfolio choice problems. The method efficiently approximates conditional expectations of the value function by using (i) a decomposition of the state variables into a component observable by the investor and a stochastic deviation; and (ii) a Taylor expansion of the value function. We illustrate the accuracy of the method in handling several realistic features of portfolio choice problems such as intermediate consumption, multiple assets, multiple state variables, portfolio constraints, non-time-separable preferences, and nonredundant endogenous state variables. We finally use the method to solve a realistic large-scale life-cycle portfolio choice and consumption problem with predictable expected returns and recursive preferences. (JEL G11, G12) Since the seminal work of and , the study of portfolio formation and management for long-lived investors has enjoyed a rich and celebrated history in the financial economics literature. With a few notable exceptions, 1 the vast majority of realistic dynamic portfolio choice problems are analytically intractable, and their solution typically relies on complex numerical procedures. The computationally intensive nature of existing numerical methods restricts the range of potential applications ). 1 Obtaining a fully closed-form solution to a portfolio choice/consumption problem typically requires particular assumptions about preferences, market completeness, and absence of frictions and constraints. Examples of papers that derive closed-form solutions include Kim at University of British Columbia on October 7, 2010 rfs.oxfordjournals.org Downloaded from The Review of Financial Studies / v 23 n 9 2010 4 at University of British Columbia on October 7, 2010 rfs.oxfordjournals.org Downloaded from The Review of Financial Studies / v 23 n 9 2010 9 at University of British Columbia on October 7, 2010 rfs.oxfordjournals.org Downloaded from

The rational fraction polynomial (RFP) modal identification procedure is a well known frequency domain fitting technique. To deal with a linear problem, the RFP procedure does not directly minimize the fitting error, i.e. the difference between the experimental and the analytical frequency response function, but a frequency weighted function of it: this causes bias in the modal parameter estimates. In this paper an iteration procedure is proposed which uses the output of the RFP technique as a starting estimate, and minimizes the true fitting error, expressed as a first order Taylor expansion of the identified parameters. Results are quite satisfactory: the fitting error is notably reduced after few iterations. Moreover, less computational modes with respect to the original RFP method are needed to obtain a good fit in a given frequency band.

We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m&#39;(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi, while that at r = 0 is given by (-4 + gamma + 3*log(2) + Pi/2)*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m(r) gives m = 0 at r = 1/2 and the slope m&#39;(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1...

For the last five years, the variants of the Newton's method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited Taylor expansion. We also prove that they are different forms of the Halley method and are all contractive iterative methods in a common neighbourhood. We extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models.

A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|&lt;1/3. The bound 1/3 is optimal. This result has been extended in a suitable sense by Liu Taishun and Wang Jianfei (2007) to the bounded

We consider second order parabolic equations with coefficients that vary both in space and in time (non-autonomous). We derive closedform approximations to the associated fundamental solution by extending the Dyson-Taylor commutator method that we recently established for autonomous equations. We establish error bounds in Sobolev spaces and show that by including enough terms, our approximation can be proven to be accurate to arbitrary high order in the short-time limit. We show how our method extends to give an approximation of the solution for any fixed time and within any given tolerance. Some applications to option pricing are presented. In particular, we perform several numerical tests, and specifically include results on Stochastic Volatility models.

In this paper, by using SOR-Like method that introduced by Golub, Wu and Yuan and generalized Taylor expansion method for solving linear systems [F. Toutounian, H. Nasabzadeh, A new method based on the generalized Taylor expansion for computing a series solution of linear systems, Appl. Math. Comput. 248 (2014) 602-609], the GTSOR-Like method is proposed for augmented systems. The convergence analysis and the choice of the parameters of the new method are discussed. While there is no guarantee the SOR-Like method converges for the negative parameter, ω additional parameters of the new method can be adjusted for the corresponding GTSOR-Like method to converge. Finally, numerical examples are given to show that the new method is much more efficient than the SOR-Like method.

We investigate the Taylor expansion of the baryon number susceptibility, and hence, pressure, in a series in the baryon chemical potential (µB) through a lattice simulation with light dynamical staggered quarks at a finer lattice cutoff a = 1/6T. We determine the QCD cross over coupling at µB = 0. We find the radius of convergence of the series at various temeperatures, and bound the location of the QCD critical point to be T E /Tc ≈ 0.94 and µ E B /T < 1.8. We also investigate the extrapolation of various susceptibilities and linkages to finite chemical potential.

We present a new projection-based nonlinear model order reduction method, named QLMOR (MOR via quadratic-linear systems). QL-MOR employs two novel ideas: (1) we show that DAEs (differentialalgebraic equations) with many commonly-encountered nonlinear kernels can be re-written equivalently into a special format, QL-DAEs (quadratic-linear differential algebraic equations, i.e., DAEs that are quadratic in their state variables and linear in their inputs); (2) we adapt the moment-matching reduction technique of NORM[1] to reduce these QLDAEs into QLDAEs of much smaller size.

In this article the sum of the series of multivariable Adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable Taylor expansion of an analytic function of the decomposition series of solutions u 1 , u 2 , . . ., u m about the initial solution components u 1,0 , u 2,0 , . . . , u m,0 ; of course the multivariable Adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations. The index matrices and their simplified forms of the multivariable Adomian polynomials are introduced. We obtain the recurrence relations for the simplified index matrices, which provide a convenient algorithm for rapid generation of the multivariable Adomian polynomials. Another alternative algorithm for term recurrence is established. In these algorithms recurrence processes do not require complicated operations such as parametrization, expanding and regrouping, derivatives, etc. as practiced in prior art. The MATHEMATICA program generating the Adomian polynomials based on the algorithm in this article is designed.

By either performing a Taylor expansion or making a polynomial approximation, the Bethe equation for charged particle stopping power in matter can be integrated analytically to obtain the range of charged particles in the continuous deceleration approximation. Ranges match reference data to the expected accuracy of the Bethe model. In the non-relativistic limit, the energy deposition rate was also found analytically. The analytic relations can be used to complement and validate numerical solutions including more detailed physics.

This paper presents evidence on the quality of Taylor series approximations to expected utility. To provide a transparent assessment in a broad setting, we assume that log portfolio returns follow a Gram-Charlier distribution that incorporates skewness and excess kurtosis and consider an investor with CRRA preferences. In this framework, we obtain closed-form approximations to expected utility based on Taylor expansions with respect to gross and log portfolio return. We illustrate the quality of the two approximations across a wide range of scenarios in terms of distribution parameters and levels of risk aversion. The Taylor expansion with respect to log portfolio return is shown to produce reliable approximations.

This paper addresses a nonlinear model predictive controller design for a single-phase PFC rectifier exploiting the Sheppard-Taylor converter. After approximation of the tracking error in the receding horizon by its Taylor-series expansion to any specified order, an analytic solution to the MPC is developed and a closed-form nonlinear predictive controller is introduced. The proposed nonlinear model predictive control (NMPC) derived using approximation can stabilize the original nonlinear systems if certain conditions, which can be met by properly choosing predictive times and the order for Taylor expansion, are satisfied. The main advantage of this digital control method is that unity power factor can be achieved over wide input voltage and load current range. Sheppard-Taylor PFC topology with NMPC shows better and more accurate performance in both steady state and transient state compared to linear control methods. Simulation results show the effectiveness of the proposed NMPC.

Inflation is now an accepted paradigm in standard cosmology, with its predictions consistent with observations of the cosmic microwave background. It lacks, however, a firm physical theory, with many possible theoretical origins beyond the simplest, canonical, slow-roll inflation, including Dirac-Born-Infeld inflation and k-inflation. We discuss how a hierarchy of Hubble flow parameters, extended to include the evolution of the inflationary sound speed, can be applied to compare a general, single field inflationary action with cosmological observational data. We show that it is important to calculate the precise scalar and tensor primordial power spectra by integrating the full flow and perturbation equations, since values of observables can deviate appreciably from those obtained using typical second-order Taylor expanded approximations in flow parameters. As part of this, we find that a commonly applied approximation for the tensor to scalar ratio, r \approx 16 c_s\epsilon, becomes poor (deviating by as much as 50%) as c_s deviates from 1 and hence the Taylor expansion including next-to-leading order contribution terms involving c_s is required. By integrating the full flow equations, we use a Monte-Carlo-Markov-Chain approach to impose constraints on the parameter space of general single field inflation, and reconstruct the properties of such an underlying theory in light of recent cosmic microwave background and large-scale structure observations.

An algorithm, which reduces to velocity Verlet in the limit of zero friction, is obtained for the generalized Langevin equation. The formulation presented is unique in that the velocities are based on a direct second order Taylor expansion of the inertial forces. The resulting finite difference equation is compared with a previous formulation of Verlet-based Langevin dynamics. The equations are implemented to study the physical properties of dense neon and liquid water at constant temperatures as a function of the friction rate g. The results show canonical ensembles can be obtained at moderate friction rates without recurring to switching functions to control energy losses, or overdamping. The LD methodology is further applied to a protein in solution and compared to MD simulations which use a more conventional scaling function for temperature control. Analyses of the MD trajectories show the protein remains at either lower or higher temperature compared to the solvent, depending upon the treatment of the potential function at the system's boundary. On the contrary, LD simulations show a rapid and even equilibration between protein and solvent in all cases. q 1998 Elsevier Science B.V. All rights reserved.

Following Talo and Ba³ar [Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Comput. Math. Appl. 58(2009), 717 733], we essentially deal with the power series of fuzzy numbers with real or fuzzy coecients. We consider the dierent cases of power series of fuzzy numbers to be convergent and give the theorem on the term by term dierentiation of power series of fuzzy numbers. Finally, we present a result on the Taylor expansion of a fuzzy valued function.

One of the bottlenecks in molecular simulations is to treat large systems involving electrostatic interactions. Computational time in conventional molecular simulation methods scales with O(N 2 ), where N is the number of atoms. With the emergence of new simulations methodologies, such as the cell multipole method ͑CMM͒, and massively parallel supercomputers, simulations of 10-million atoms or more have been performed. In this work, the optimal hierarchical cell level and the algorithm for Taylor expansion were recommended for fast and efficient molecular dynamics simulations of three-dimensional ͑3D͒ systems. CMM was then extended to treat quasi-two-dimensional ͑2D͒ systems, which is very important for condensed matter physics problems. In addition, CMM was applied to grand canonical ensemble Monte Carlo simulations for both 3D and 2D systems. Under the optimal conditions, our results show that computational time is approximately linear with N for large systems, average error in total potential energy is about 0.05% for 3D and 0.32% for 2D systems, and the RMS force error is 0.27% for 3D and 0.43% for 2D systems when compared with the Ewald summation.

Absfruct-Block-diagonalization of a singularly perturbed system requires the solution of the Riccati equation and the Lyapunov equation. A new approach is suggested for both equations, using Taylor expansions. The convergence is studied in detail; it is shown that it is ensured under restrictions which are less restrictive than the case of previously suggested methods. Y.-M. Diao et al., "Solution of state equations in terms of separated modes with applications to synchronous machines," IEEE Trans. Energy Convers., vol. EC-I, pp. 211-216, Sept. 1986. T. Grodt and Z. Gajic, "The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation," IEEE Trans. Automat. Contr., vol. 33, Aug. 1988. N. Derbel et al., "Two step three time scale reduction of doubly fed synchronous machine models," ZEEE Trans. Energy Convers., vol. 9, Mar. 1994. and if (38) is replaced into (39),

d-dimensional homogeneous isotropic incompressible turbulence is defined, for arbitrary nonintegral d, by analytically continuing the Taylor expansion in time of the energy spectrum Ek(t), assuming Gaussian initial conditions. If d & 2, the positivity of the energy spectrum is not necessarily preserved in time. For d)2 all steady-state and initial-value calculations have been made with a realizable second-order closure, the eddydamped quasinormal Markovian approximation. Near two dimensions the enstrophy (mean square vorticity) conservation law is weakly broken, enough to allow ultraviolet singularities to develop in a finite time but not enough to prevent energy from cascading in the infrared direction. A systematic investigation is made of zero-transfer (inertial) steady-state scaling solutions Et,~k and of their stability. Energy-inertial solutions with m = 5/3 exist for arbitrary d; the direction of the energy cascade reverses at d = d, 2.05. For d & d', 2.06 there are in addition, as in the cascade model studied by Bell and Nelkin, inertial solutions with zero eriergy flux; their exponents m(d) are given by a roughly parabolic curve in the (m, d) plane, linking enstrophy cascade (m = 3, d = 2) to enstrophy equipartition (m = 1, d = 2). For any point in the (m, d) plane such that the transfer integral is finite and negative, a steady-state scaling solution Ek~k is obtained when the fluid is subject to random forces with spectrum F" fx: k ' ' ". A special case is the "model B" tm =-1+ 3 e + 0(&), d = 4cj obtained by Forster, Nelson, and Stephen using a dynamical renormalization-group procedure. Forced steady-state solutions are actually not resticted to the neighborhood. of m =-1, d = 4; they are amenable to renormalization-group calculations on the primitive equations for arbitrary d & 2 when m is close to the crossover-1 and, perhaps, also near the crossover +3.

This paper proposes a strategy for estimating the DA (domain of attraction) for non-polynomial systems via LFs (Lyapunov functions). The idea consists of converting the non-polynomial optimization arising for a chosen LF in a polynomial one, which can be solved via LMI optimizations. This is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst-case remainders in truncated Taylor expansions. Moreover, a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen LF, and another condition is provided for establishing whether such an estimate has been found. The proposed strategy can readily be exploited with variable LFs in order to search for optimal estimates. Lastly, it is worth to remark that no other method is available to estimate the DA for non-polynomial systems via LMIs.

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model [7]; here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spinorbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.

In this paper we present a variational approach to accurately estimate the motion vector field in a image sequence introducing a second order Taylor expansion of the flow in the energy function to be minimized. This feature allows us to simultaneously obtain, in addition, an estimation of the partial derivatives of the motion vector field. The performance of our approach is illustrated with the estimation of the displacement vector field on the well known Yosemite sequence and compared to other techniques from the state of the art.

This work proposes a framework to the numerical identification of nonlinear fluid film bearing parameters from large journal orbital motion (20-60% of the bearing clearance). Nonlinear coefficients are defined by a third order Taylor expansion of bearing reaction forces and are evaluated through a least mean square in time domain technique. The journal response is obtained from a computational fluid dynamic (CFD) model of a plain journal bearing on high dynamic loading conditions. The model considers fluid-structure interaction between the fluid flow and the journal. The case in study considers a laboratory test rig. Results indicate that nonlinear coefficients have an important effect on stiffness and damping. It was found a change on nonlinear behavior occurred when the Oil Whirl phenomenon starts, which it is not seen in classical linear models.

By taking an appropriate zero-curvature limit, we obtain the spherical functions on flat symmetric spaces G0/K as limits of Harish-Chandra's spherical functions. New and explicit formulas for the spherical functions on G0/K are given. For symmetric spaces with root system of type An, we find the Taylor expansion of the spherical functions on G0/K in a series of Jack polynomials.

In seismic data processing it is common to use a non-hyperbolic travel-time approximation assuming weak anisotropy in a transversely isotropic medium with vertical symmetry axis. It has the correct short-spread (normal) moveout velocity, but higher order terms in offset squared are only approximations. Using Taylor expansions for the squared vertical slowness for P-and SV-waves result in new approximations for the phase velocities which may be used in pre-stack migration algorithms. These approximations are combined with expressions for travel time and offset as functions of horizontal slowness, giving three new approximations for travel time squared as functions of offset squared, without the assumption of weak anisotropy. Numerical examples show that these new approximations all have better accuracy than the standard weak-anisotropy approximation. Based on the numerical results, we propose a new travel-time approximation for reflected PP-, SS-and PS-waves to be used in seismic data processing. They have almost the same functional form as the weak-anisotropy approximations, with the same normal moveout velocities but with a different heterogeneity factor which is non-linear in the anisotropy parameters.