Logarithmic Potential Function Research Articles

Type-based analysis of logarithmic amortised complexity

AbstractWe introduce a novel amortised resource analysis couched in a type-and-effect system. Our analysis is formulated in terms of the physicist’s method of amortised analysis and is potentialbased. The type system makes use of logarithmic potential functions and is the first such system to exhibit logarithmic amortised complexity. With our approach, we target the automated analysis of self-adjusting data structures, like splay trees, which so far have only manually been analysed in the literature. In particular, we have implemented a semi-automated prototype, which successfully analyses the zig-zig case of splaying, once the type annotations are fixed.

Rigid-body attitude stabilization with attitude and angular rate constraints

In this paper, a solution to the problem of rest-to-rest three-axis attitude reorientation of a fully actuated rigid body under multiple attitude-constraint zones and angular velocity limits is presented. Based on the unit-quaternion parameterized attitude-constrained zones, a quadratic potential function is developed with a global minimum locating at the desired attitude and high potential closing to the constrained zones. In addition, to limit the magnitude of the angular velocity, another logarithmic potential function is also designed. Using these two potential functions and sliding mode control technique, a nonlinear attitude control law is obtained to guarantee asymptotic convergence of the closed-loop system with consideration of attitude and angular rate constraints, and external disturbances. The effectiveness of the constrained attitude control method is demonstrated through numerical simulation.

Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn–Hilliard Equation

Abstract An energy stable conservative method is developed for the Cahn–Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach.

The Penalized Analytic Center Estimator

In a linear regression model, the Dantzig selector (Candès and Tao, 2007) minimizes the L1 norm of the regression coefficients subject to a bound λ on the L∞ norm of the covariances between the predictors and the residuals; the resulting estimator is the solution of a linear program, which may be nonunique or unstable. We propose a regularized alternative to the Dantzig selector. These estimators (which depend on λ and an additional tuning parameter r) minimize objective functions that are the sum of the L1 norm of the regression coefficients plus r times the logarithmic potential function of the Dantzig selector constraints, and can be viewed as penalized analytic centers of the latter constraints. The tuning parameter r controls the smoothness of the estimators as functions of λ and, when λ is sufficiently large, the estimators depend approximately on r and λ via r/λ2.

Random point processes and tilings arising from Gaussian analytic functions

Getting a grasp of what aperiodic order really entails is going to require collecting and understanding many diverse examples. Aperiodic crystals are at the top of the largely unknown iceberg beneath. Here we present a recently studied form of random point process in the (complex) plane which arises as the sets of zeros of a specific class of analytic functions given by power series with randomly chosen coefficients: Gaussian analytic functions (GAF). These point sets differ from Poisson processes by having a sort of built in repulsion between points, though the resulting sets almost surely fail both conditions of the Delone property. Remarkably the point sets that arise as the zeros of GAFs determine a random point process which is, in distribution, invariant under rotation and translation. In addition, there is a logarithmic potential function for which the zeros are the attractors, and the resulting basins of attraction produce tilings of the plane by tiles which are, almost surely, all of the same area. We discuss GAFs along with their tilings and diffraction, and as well note briefly their relationship to determinantal point processes, which are also of physical interest.

Thermoelastodynamics in Transversely Isotropic Media With Scalar Potential Functions1

A complete set of potential functions consisting of three scalar functions is presented for coupled displacement-temperature equations of motion and heat equation for an arbitrary x3-convex domain containing a linear thermoelastic transversely isotropic material, where the x3-axis is parallel to the axis of symmetry of the material. The proof of the completeness theorem is based on a retarded logarithmic potential function, retarded Newtonian potential function, repeated wave equation, the extended Boggio's theorem for the transversely isotropic axially convex domain, and the existence of a solution for the heat equation. It is shown that the solution degenerates to a set of complete potential functions for elastodynamics and elastostatics under certain conditions. In a special case, the number of potential functions is reduced to one, and the required conditions are discussed. Another special case involves the rotationally symmetric configuration with respect to the axis of symmetry of the material.

Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman–Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.

The Shooting Method for Solving Eigenvalue Problems

Theshooting methodis a numerically effective approach to solving certain eigenvalue problems, such as that arising from the Schrödinger equation for the two-dimensional hydrogen atom with logarithmic potential function. However, no complete proof of its rationale and correctness has been given until now. This paper gives the proof, in a generalized form.

The dynamics and internal geometry of the three-city noxious location problem

A logarithmic potential function is used to model the noxious facility location problem and a closed-form solution (in terms of radicals) is given for locating a noxious facility within the convex hull of three cities. The location of the facility is chosen to coincide with the critical (or saddle-) points of the logarithmic potential function. Since the logarithmic potential function is harmonic, the maximum and minimum solutions will lie on the boundary of the region under consideration and will not coincide with the critical points of the function. It is shown that only in the three-city case can an exact formula be obtained for the critical points; i.e., determining the critical points within four or more cities will necessarily lead to an approximate solution. Algorithmic procedures to solve for the n-city problem, however, are available and are discussed. The geometry of the critical points associated with the logarithmic potential is examined in detail for the three-city case. Using a classical result on the geometry of complex zeros, a constructive approach is used to parameterize the conic sections associated with these points. A computer implementation of the parameterization is given, and by varying certain “coefficients of toxicity” associated with each city point, the behavior of the complex zeros are examined.

CENTER POINTS, EQUILIBRIUM POSITIONS, AND THE OBNOXIOUS LOCATION PROBLEM

ABSTRACT. Real variable analysis has een used to great benefit in a variety of classical problems in location theory. In this paper we explore basic complex variable techniques in one formulation of the obnoxious location problem. A general definition of center points is first given and used to formulate several alternate versions of the obnoxious location problem. A logarithmic transformation is then used to demonstrate some equivalences between these families of distinct location problems (defined via center points). A prototype logarithmic potential function which results from this formulation is then investigated, and it is demonstrated that the extremal solutions with this objective reside on the boundary of its domain of definition. An application using zero‐ and one‐dimensional centers is discussed, and a generalization to the spatial obnoxious problem is also briefly examined. We define a zero‐dimensional center as a critical point of the logarithmic potential function, and it is shown that these centers are equivalent to the solutions of the Complex Moment Problem.

Conical projection algorithms for linear programming

The Linear Programming Problem is manipulated to be stated as a Non-Linear Programming Problem in which Karmarkar's logarithmic potential function is minimized in the positive cone generated by the original feasible set. The resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non-linear programming algorithm. Several different procedures for the internal algorithm are proposed, giving priority either to the reduction of the potential function or of the actual cost. We show that Karmarkar's algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration. All variants of the new algorithm have the same complexity as Karmarkar's method, but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm.

The Anderson-Grüneisen parameter for cubic crystals

A new, simple relation for δ the Anderson-Grüneisen parameter, developed by Sharma and Tripathi (M. N. Sharma and S. R. Tripathi, Phys. Status Solidi (b) 59, K1 (1973)), which involves fewer approximations than other complex, cumbersome equations available in literature, has been used to compute values of δ employing three different potentials viz the Varshni-Shukla, the Logarithmic, and the Rydberg's function. The comparison of these values with experimental ones indicates the suitability of Varshni-Shukla and Logarithmic potential functions for the study of the property δ.

A new characterization of the 𝐹 set of a rational function

In the early part of this century G. Julia and P. Fatou extensively studied the iteration of functions on the complex plane. More recently Hans Brolin reopened the investigation. In this paper, we are interested in the F set which is the set of points at which the family of iterates of a given rational function R is not normal and in a measure which is in some sense naturally imposed on the F set by the iterates of R. We construct a sequence of probability measures via the inverse functions of the iterates of R and almost any starting point. The measure of primary interest is the weak limit of such sequences. These weak limits are supported by F and have certain invariance properties. We establish that this weak limit measure is unique and is ergodic with respect to the transformation R on the F set for a large class of rational functions. In the course of the proof of uniqueness we develop expressions for the logarithmic potential function and for the energy integral of F. We also establish inequalities for the capacity of the F set which become equalities for the polynomial case.

A fluxintegral theorem for functions which have harmonic support

1. Introduction. This paper deals with logarithmic potential functions defined in plane domains and with the problem of the computation of mass distributions by means of the Gauss flux integral formula. For potentials which have harmonic support a formulation is possible wherein the flux integral is taken over boundaries consisting of a finite set of rectifiable Jordan curves. As defined in [10] a real valued function u is said to have harmonic support in a domain D if and only if u is subuniformly bounded(2) in D and has at each point of D a support function which is harmonic and single valued in D. The latter statement is understood to mean that for each point ? in D there is a harmonic function h such that h(r) = u(v) and h<? u in D. Denote by (hs) the class of such functions for a domain D. The result may be considered as an extension, for the class of potentials under consideration, of the formulations by Evans [7] and F. Riesz [18] in the following sense. Evans described a generalized flux integral valid on curves which support no mass with a subsequent determination of the mass on a general region as a limit of integrals on the boundaries of a sequence of expanding regions. Riesz' method for the class of subharmonic functions is to obtain the mass as a limit of flux integrals of best harmonic majorants. The present result allows a direct evaluation of a flux integral on an arbitrary rectifiable boundary without requiring the use of approximating regions. To obtain such a theorem one is first confronted with the possible lack of a normal derivative in the flux integral. There are (hs)-functions for which the normal derivative fails to exist on quite simple curves, for example