Arbitrary Smooth Function Research Articles

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

Abstract An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.

Energies of knot diagrams

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function $f$ and study the simplest nontrivial case $f(x)=x^2$, obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.

Auxiliary fields representation for modified gravity models

We consider tensor-multiscalar representations for several types of modified gravity actions. The first example is the theory with the action representing an arbitrary smooth function of the scalar curvature R and (Box R), the integrand of the Gauss-Bonnet term and the square of the Weyl tensor. We present a simple procedure leading to an equivalent theory of a space-time metric and four auxiliary scalars and specially discuss calibration of a cosmological constant and the condition of the existence of dS-like solutions in the case of empty universe. The condition for obtaining a smaller number of independent scalar fields is derived. The second example is the Eddington-like gravity action. In this case we show, in particular, the equivalence of the theory to GR with the cosmological constant term, with or without use of the first-order formalism, and also discuss some possible generalizations.

On adapted coordinate systems

The notion of an adapted coordinate system for a given real-analytic function, introduced by V. I. Arnol’d, plays an important role, for instance, in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A. N. Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components. Varchenko’s proof is based on a two-dimensional resolution of singularities result. In this article, we present a more elementary approach to these results, which is based on the Puiseux series expansion of roots of the given function. This approach is inspired by the work of D. H. Phong and E. M. Stein on the Newton polyhedron and oscillatory integral operators. It applies to arbitrary real-analytic functions, and even to arbitrary smooth functions of finite type. In particular, we show that Varchenko’s conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions, even in the smooth, finite type setting. For analytic functions, a construction of adapted coordinates by means of Puiseux series expansions of roots has already been carried out in work by D. H. Phong, E. M. Stein and J. A. Sturm on the growth and stability of real-analytic function, as we learned after the completion of this paper. In contrast to their work, however, our proof more closely follows Varchenko’s algorithm for the construction of an adapted coordinate system, which turns out to be useful for the extension to the smooth setting.

Optimal state-feedback design for non-linear feedback-linearisable systems

This paper addresses the problem of optimal state-feedback design for a class of non-linear systems. The method is applicable to all non-linear systems which can be linearised using the method of state-feedback linearisation. The alternative is to use linear optimisation techniques for the linearised equations, but then there is no guarantee that the original non-linear system behaves optimally. The authors use feedback linearisation technique to linearise the system and then design a state feedback for the feedback-linearised system in such a way that it ensures optimal performance of the original non-linear system. The method cannot ensure global optimality of the solution but the global stability of the non-linear system is ensured. The proposed method can optimise any arbitrary smooth function of states and input, including the conventional quadratic form. The proposed method can also optimise the feedback linearising transformation. The method is successfully applied to control the design of a flexible joint dynamic and the results are discussed. Compared with the conventional linear quadratic regulator (LQR) technique, the minimum value of cost function is significantly reduced by the proposed method.

Generalized linear and additive models for psychophysical data

What do such diverse paradigms as classification images, difference scaling and additive conjoint measurement have in common? We introduce a general framework that permits modeling and evaluating experiments covering a broad range of psychophysical tasks. Psychophysical data are considered within a signal detection model in which a decision variable, d, which is some function, f, of the stimulus conditions, S, is related to the expected probability of response, E[P], through a psychometric function, G: E[P] = G(f(d(S))). In many cases, the function f is linear, in which case the model reduces to E[P] = G(Xb), where X is a design matrix describing the stimulus configuration and b a vector of weights indicating how the observer combines stimulus information in the decision variable. By inverting the psychometric function, we obtain a Generalized Linear Model (GLM). We demonstrate how this model, which has previously been applied to calculation of signal detection theory parameters and fitting the psychometric function, is extended to provide maximum likelihood solutions for three tasks: classification image estimation, difference scaling and additive conjoint measurement. Within the GLM framework, nested hypotheses are easily set-up in a manner resembling classical analysis of variance. In addition, the GLM is easily extended to fitting and evaluating more flexible (nonparametric) models involving arbitrary smooth functions of the stimulus. In particular, this approach permits a principled approach to fitting smooth classification images.

Molecular dynamics equation designed for realizing arbitrary density: Application to sampling method utilizing the Tsallis generalized distribution

Several molecular dynamics techniques applying the Tsallis generalized distribution are presented. We have developed a deterministic dynamics to generate an arbitrary smooth density function ρ. It creates a measure-preserving flow with respect to the measure ρdω and realizes the density ρ under the assumption of the ergodicity. It can thus be used to investigate physical systems that obey such distribution density. Using this technique, the Tsallis distribution density based on a full energy function form along with the Tsallis index q ≥ 1 can be created. From the fact that an effective support of the Tsallis distribution in the phase space is broad, compared with that of the conventional Boltzmann-Gibbs (BG) distribution, and the fact that the corresponding energy-surface deformation does not change energy minimum points, the dynamics enhances the physical state sampling, in particular for a rugged energy surface spanned by a complicated system. Other feature of the Tsallis distribution is that it provides more degree of the nonlinearity, compared with the case of the BG distribution, in the deterministic dynamics equation, which is very useful to effectively gain the ergodicity of the dynamical system constructed according to the scheme. Combining such methods with the reconstruction technique of the BG distribution, we can obtain the information consistent with the BG ensemble and create the corresponding free energy surface. We demonstrate several sampling results obtained from the systems typical for benchmark tests in MD and from biomolecular systems.

Bayesian semiparametric zero-inflated Poisson model for longitudinal count data

This paper presents new methods, using a Bayesian approach, for analyzing longitudinal count data with excess zeros and nonlinear effects of continuously valued covariates. In longitudinal count data there are many problems that can make the use of a zero-inflated Poisson (ZIP) model ineffective. These problems are unobserved heterogeneity and nonlinear effects of continuously valued covariates. Our proposed semiparametric model can simultaneously handle these problems in a unified framework. The framework accounts for heterogeneity by incorporating random effects and has two components. The parametric component of the model which deals with the linear effects of time invariant covariates and the non-parametric component which gives an arbitrary smooth function to model the effect of time or time-varying covariates on the logarithm of mean count. The proposed methods are illustrated by analyzing longitudinal count data on the assessment of an efficacy of pesticides in controlling the reproduction of whitefly.

Symmetry group classification for general Burgers’ equation

The present paper solves the problem of the group classification of the general Burgers’ equation ut=f(x,u)ux2+g(x,u)uxx, where f and g are arbitrary smooth functions of the variable x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification that is called preliminary group classification. Looking the adjoint representation of GE on its Lie algebra g5, we will deal with the construction of the optimal system of its one-dimensional subalgebras. The result of the work is a wide class of equations summarized in table form.

A group theoretical identification of integrable cases of the Liénard-type equation ẍ+f(x)ẋ+g(x)=. I. Equations having nonmaximal number of Lie point symmetries

We carry out a detailed Lie point symmetry group classification of the Liénard-type equation, ẍ+f(x)ẋ+g(x)=0, where f(x) and g(x) are arbitrary smooth functions of x. We divide our analysis into two parts. In the present first part we isolate equations that admit lesser parameter Lie point symmetries, namely, one, two, and three parameter symmetries, and in the second part we identify equations that admit maximal (eight) parameter Lie point symmetries. In the former case the invariant equations form a family of integrable equations, and in the latter case they form a class of linearizable equations (under point transformations). Further, we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent Hamiltonians. Several of these equations are being identified for the first time from the group theoretical analysis.

Rough Solutions of the Einstein Constraints on Closed Manifolds without Near-CMC Conditions

We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions for the Hamiltonian constraint, and topological fixed-point arguments. An important new feature of these results is the absense of the near-CMC assumption when the rescaled background metric is in the positive Yamabe class, if the freely specifiable part of the data given by the matter fields (if present) and the traceless-transverse part of the rescaled extrinsic curvature are taken to be sufficiently small. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, giving what are apparently the first non-CMC existence results without the near-CMC assumption. Standard bootstrapping arguments to increase the regularity of the conformal factor are blocked by the use of a weak background metric. In the CMC case, we recover Maxwell's rough solution results as a special case. Our results extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.

Symmetry group analysis and similarity solutions of the CBS equation in (2+1) dimensions

AbstractWe consider the (2+1)—dimensional integrable Calogero—Bogoyavlenskii—Schiff (CBS) written in a potential form. By using classical Lie symmetries, we consider travelling‐wave reductions with variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)‐dimensional equation involve arbitrary smooth functions. Consequently the solutions exhibit a rich variety of qualitative behaviours. Indeed by making adequate choices for the arbitrary functions, we exhibit solitary waves and bound states. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Suboptimal Controller Design Methodology for Input-Output Feedback-Linearizable Systems

This paper addresses suboptimal control of nonlinear systems which can be feedback-linearized from input to output. The case of input-to-state linearizable systems is also covered as a special case. The method is thus applicable to all nonlinear systems which can be partially linearized using the method of output-feedback linearization while having a stable internal (or zero) dynamics. The well-known LQR technique applied to the linearized system does not guarantee the suboptimality of the nonlinear system. This paper uses output feedback linearization technique to partially linearize the system and then designs an output-feedback for the feedback-linearized system in such a way that it ensures suboptimal performance of the original nonlinear system. The proposed method can optimize any arbitrary smooth function of states and input. The proposed controller is, however, suboptimal due to the facts that (1) the form of the controller is a linear static feedback of the linearized state, (2) the search algorithm may fall into a local extremum rather than a global, and (3) the calculated controller depends on the initial conditions. The method is successfully applied to control design of the longitudinal subsystem of a laboratory double-rotor helicopter and the results are discussed and compared with those of the LQR method.

Exact solutions of diffusion-convection equations

(1) where f = f(x), g = g(x), h = h(x), A = A(u) and B = B(u) are arbitrary smooth functions of their variables, f(x)g(x)A(u) 6. Our aim is not to give a physical interpretation of the solution of diffusion equations (that is too huge and cannot be reached in the scope of a short paper), but to list the already known exact solutions of equations from the class under consideration. However, in some cases we give a short discussion of the nature of the listed solutions. The majority of the listed solutions have been obtained by means of different symmetry methods, such as reduction with respect to Lie and non-Lie symmetries, separation of variables, equivalence transformations, etc. Let us note that the constant coefficient diffusion equations ( = g = 1, B = 0) are well

A generalization of the Lindeberg principle

We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.

Paley–Wiener theorems for the Dunkl transform

We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley–Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl’s intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.

On the Erlang Loss Model with Time Dependent Input

We consider the M ( t )/ M ( t )/ m / m queue, where the arrival rate ( t ) and service rate ( t ) are arbitrary (smooth) functions of time. Letting p n ( t ) be the probability that n servers are ...

On the Calogero–Degasperis–Fokas equation in [formula omitted] dimensions

In this paper we study a ( 2 + 1 ) -dimensional integrable Calogero–Degasperis–Fokas equation derivable by using a method proposed by Calogero. A catalogue of classical symmetry reductions are given. These reductions to partial differential equations in (1+1) admit symmetries which lead to further reductions, i.e., to second-order ordinary differential equations. These ODEs provide several classes of solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the second and third Painleve trascendents. The corresponding solutions of the ( 2 + 1 ) -dimensional equation, involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviour. Indeed by making appropriate choices for the arbitrary functions, we exhibit solitary waves, coherent structures and bound states.

The [formula omitted]-dimensional integrable coupling of KdV equation: Auto-Bäcklund transformation and new non-traveling wave profiles

The ( 2 + 1 ) -dimensional integrable coupling of the KdV equation, which was first presented by Ma and Fussteiner from the celebrated KdV equation using the perturbation method of multiple scales u = u 1 + ε u 2 , y = ε x , is investigated. With the aid of symbolic computation, a new auto-Bäcklund transformation is gained and used to seek new types of non-traveling wave solutions involving an arbitrary function of y. Moreover a non-traveling wave similarity variable transformation reduces this system to a system of non-linear ordinary differential equations with constant coefficient, which is solved to get non-traveling wave Jacobi elliptic function solutions and Weierstrass elliptic function solutions involving an arbitrary smooth function of y. When the arbitrary function are taken as some special functions, these obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the rules of the wave propagation related to ( 2 + 1 ) -dimensional integrable coupling of the KdV equation.

Traveling-Wave Solutions of the Calogero-Degasperis-Fokas Equation in 2+1 Dimensions

Soliton solutions are among the more interesting solutions of the (2+1)-dimensional integrable Calogero-Degasperis-Fokas (CDF) equation. We previously derived a complete group classiffication for the CDF equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. The solutions consequently exhibit a rich variety of qualitative behaviors. Choosing the arbitrary functions appropriately, we exhibit solitary waves and bound states.