Smooth Potential Function Research Articles

On the smooth unfolding of bifurcations in quantal-response equilibria

We report on the topological structure of logit quantal response equilibria for asymmetric, two-player, two-choice games in normal form, via catastrophe theory. We present three main outcomes: first, a novel, smooth potential function for the underlying dynamics, relative to which logit equilibria arise as stationary points comprising a parameterised hyper-surface; secondly, a proof that all catastrophes within this manifold are of order at most three, i.e. “folds and cusps”; thirdly, discovery of a new topological phenomenon, the “pleated loop”, corresponding to a pair of opposed pitchfork bifurcations. This we exhibit for games, such as Prisoner's Dilemma, that have a unique Nash equilibrium, with multiple logit equilibria for finite precision parameters. These results extend work at the intersection of stochastic decision theory and catastrophe theory with economic games. Their application to the problem of bifurcations in the tracing procedure for logit solutions, proposed by McKelvey and Palfrey, is discussed throughout.

Sign-changing solutions for a modified quasilinear Kirchhoff–Schrödinger–Poisson system via perturbation method

In this paper, we consider the following quasilinear Kirchhoff–Schrödinger–Poisson system: where a, b are positive constants, is a smooth potential function and g is an appropriate nonlinear function. To overcome the technical difficulties caused by the quasilinear term, the perturbation method of adding 4-Laplacian operator is adapted to consider the perturbation problem, so that the corresponding functional has both smoothness and compactness in the appropriate space. Moreover, when g satisfies the appropriate hypotheses, a sign-changing solution of above problem can be obtained by taking advantage of constraint variational method, the quantitative deformation lemma and approximation technique, which has precisely two nodal domains.

On sign-changing solutions for quasilinear Schrödinger-Poisson system with critical growth

In this paper, we consider the following quasilinear Schrödinger-Poisson system with critical growth: where , is a smooth potential function and g is a appropriate nonlinear function. For the sake of overcoming the technical difficulties caused by the quasilinear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. Moreover, when g satisfying suitable assumptions and sufficiently large μ, we take advantage of constraint variational method, the quantitative deformation lemma, Moser iteration and approximation technique to obtain a least-energy sign-changing solution , which has precisely two nodal domains.

A potential generalization of some canonical Riemannian metrics

The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases, we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper, we also describe the “nongradient” version of this construction.

Image Regularity and Fidelity Measure with a Two-Modality Potential Function

We define a strictly convex smooth potential function and use it to measure the data fidelity as well as the regularity for image denoising and cartoon-texture decomposition. The new model has several advantages over the well-known ROF or TV-L2 and the TV-L1 model. First, due to the two-modality property of the new potential function, the new regularity has strong regularizing properties in all directions and thus encourages removing noise in smooth areas, while, near edges, it smoothes the edge mainly along the tangent direction and thus can well preserve the edges. Second, the new potential function is very close to the L1 norm; thus using it to measure the data fidelity makes the new model perform very well in removing impulse noise and preserving the contrast. Lastly, the proposed fidelity and regularization term is strictly convex and smooth and thus allows a unique global minimizer and it can be solved by using the steepest descent method. Numerical experiments show that the proposed model outperforms TV-L2 and TV-L1 in removing impulse noise and mixed noise. It also outperforms some state-of-the-art methods specially designed for impulse noise. Tests on cartoon-texture decomposition show that our method is effective and performs better than TV-L1.

Interparticle interactions between water molecules

We determined many functional representations of interparticle interactions between water molecules, all of which reproduce the experimentally measured density-temperature relation at 1 bar with an accuracy better than obtained by previous models. Numerous similar descriptions of pair interactions will be discovered increasingly in the coming years, which will help us to understand why solid water has polymorphic structures and why liquid water has a large number of anomalies. We used a self-consistent Ornstein-Zernike approximation (SCOZA) with a potential given by multi-Yukawa terms. Because any smooth potential function can be fitted by multi-Yukawa terms, the method can be applied to various types of fluids. We also present a new simple fitting technique that makes the application of the SCOZA to any type of liquid much easier compared to a conventional Yukawa fit. Our new SCOZA fitting technique is among the most useful methods for determining the pair interaction between molecules of any liquid, and the potential will be helpful in improving realistic models.

On [formula omitted]-spectrum and τ-quasi-Einstein metric

In this paper, we study L f p -spectrum estimates for some smooth potential function f on complete noncompact Riemannian manifolds via the τ-Bakry–Émery curvature. As its applications, we give the upper bound estimate of the L f 2 -spectrum for the complete noncompact τ-quasi-Einstein metric, we give an example to show the sharpness of our estimate; we also get a lower bound estimate when λ < 0 , τ > 1 and μ > 0 .

Deployment of Mobile Sensor Networks with Discontinuous Dynamics

In this paper, we analyze the stability of a network that uses piecewise smooth potential functions. A gravitation-like force is applied to deploy a group of agents and to form a certain configuration. We use a nonsmooth version of the Lyapunov stability theory and LaSalles invariance principle to show asymptotic stability of the network which is governed by discontinuous dynamics. Hexagonal deployment using such a force is shown through simulation.

Improved dual algorithm for constrained optimization problems

One class of effective methods for the optimization problem with inequality constraints are to transform the problem to a unconstrained optimization problem by constructing a smooth potential function. In this paper, we modifies a dual algorithm for constrained optimization problems and establishes a corresponding improved dual algorithm; It is proved that the improved dual algorithm has the local Q-superlinear convergence; Finally, we performed numerical experimentation using the improved dual algorithm for many constrained optimization problems, the numerical results are reported to show that it is valid in practical computation.

Constructing smooth potential functions for protein folding

A protein folding potential function ideally has several properties: it favors the native conformations for a number of protein sequences over a variety of nonnative folds; it can guide the search over conformations for the native state; it reflects changes in stability of the native fold due to changes in sequence; and it is relatively insensitive to small changes in conformation. While these are not mutually incompatible goals, attaining one property does not ensure that the others are satisfied. Examples are given of simple potentials having one property but lacking others. A new functional form of a folding potential is described where interactions between all nonhydrogen atoms are used to estimate interresidue interactions and implicit solvation. Its parameters can be adjusted to satisfy the above properties at least for barnase and a few other proteins. © 2001 by Elsevier Science Inc.

The convergence of a dual algorithm for nonlinear programming

A dual algorithm based on the smooth function proposed by Polyak (1988) is constructed for solving nonlinear programming problems with inequality constraints. It generates a sequence of points converging locally to a Kuhn-Tucker point by solving an unconstrained minimizer of a smooth potential function with a parameter. We study the relationship between eigenvalues of the Hessian of this smooth potential function and the parameter, which is useful for analyzing the effectiveness of the dual algorithm.

Constructing smooth potential functions for protein folding

Constructing smooth potential functions for protein folding

Kählerian metrics given by certain smooth potential functions

Kählerian metrics given by certain smooth potential functions

Potential Functions in Mathematical Pattern Recognition

This paper discusses a class of methods for pattern classification using a set of samples. They may also be used in reconstructing a probability density from samples. The methods discussed are potential function methods of a type directly derived from concepts related to superposition. The characteristics required of a potential function are examined, and it is shown that smooth potential functions exist that will separate arbitrary sets of sample points. Ideas suggested by Specht in regard to polynomial potential functions are extended.

On the universality of potential well dynamics

Given a smooth potential function $V : \mathbf{R}^m \to \mathbf{R}$, one can consider the ODE $\partial_t^2 u = -(\nabla V)(u)$ describing the trajectory of a particle $t \mapsto u(t)$ in the potential well $V$. We consider the question of whether the dynamics of this family of ODE are \emph{universal} in the sense that they contain (as embedded copies) any first-order ODE $\partial_t u = X(u)$ arising from a smooth vector field $X$ on a manifold $M$. Assuming that $X$ is nonsingular and $M$ is compact, we show (using the Nash embedding theorem) that this is possible precisely when the flow $(M,X)$ supports a geometric structure which we call a \emph{strongly adapted $1$-form}; many smooth flows do have such a $1$-form, but we give an example (due to Bryant) of a flow which does not, and hence cannot be modeled by the dynamics of a potential well. As one consequence of this embeddability criterion, we construct an example of a (coercive) potential well system which is \emph{Turing complete} in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a certain open set. In particular, this system contains trajectories for which it is undecidable whether that trajectory enters such a set. Remarkably, the above results also hold if one works instead with the nonlinear wave equation $\partial_t^2 u - \Delta u = -(\nabla V)(u)$ on a torus instead of a particle in a potential well, or if one replaces the target domain $\mathbf{R}^m$ by a more general Riemannian manifold.