Showing papers on "Auxiliary function published in 2020"

A Note on Extended Z-Contraction

Abstract: In this article, we aim to evaluate and merge the as-scattered-as-possible results in fixed point theory from a general framework. In particular, we considered a common fixed point theorem via extended Z-contraction with respect to ψ -simulation function over an auxiliary function ξ in the setting of b-metric space. We investigated both the existence and uniqueness of common fixed points of such mappings. We used an example to illustrate the main result observed. Our main results cover several existing results in the corresponding literature.

The distribution of Gaussian multiplicative chaos on the unit interval

Abstract: We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval $[0,1]$ and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in $0$ and $1$, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides nontrivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly, we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.

Fixed point theorems for nonlinear contractive mappings in ordered b-metric space with auxiliary function.

Abstract: In this paper we present some fixed point theorems for self mappings satisfying generalized $$(\phi , \psi )$$ -weak contraction condition in partially ordered complete b-metric spaces. The results presented over here generalize and extend some existing results in the literature. Finally, we illustrate two examples to support our results. We obtained a unique fixed point of a self mapping satisfying certain contraction condition which is involving an auxiliary function. Also, the results are presented for the existence of a common fixed point and a coincidence point for generalized $$(\phi , \psi )$$ -weak contraction mappings in partially ordered complete b-metric space.

Optimal Auxiliary Functions Method for a Pendulum Wrapping on Two Cylinders

Abstract: In the present work, the nonlinear oscillations of a pendulum wrapping on two cylinders is studied by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM). The equation of motion is derived from the Lagrange’s equation. Analytical solutions and natural frequency of the system are calculated. Our results obtained through this new procedure are compared with numerical ones and a very good agreement was found, which proves the accuracy of the method. The presented numerical examples show that the proposed approach is simple, easy to implement and very accurate.

Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder

Abstract: Based on a new kind of analytical approach, namely the Optimal Auxiliary Functions Method (OAFM), a new analytical procedure is proposed to solve the problem of the annular axisymmetric stagnation flow and heat transfer on a moving cylinder with finite radius. As a novelty, explicit analytical solutions were obtained for the considered complex problem. First, the Navier–Stokes equations were simplified by means of similarity transformations that depended on different parameters and some combinations of these parameters, and the problem under study was reduced to six nonlinear ordinary differential equations with six unknowns. The OAFM proves to be a powerful tool for finding an accurate analytical solution for nonlinear problems, ensuring a fast convergence after the first iteration, even if the small or large parameters are absent, since the determination of the convergence-control parameters is independent of the magnitude of the coefficients that appear in the nonlinear differential equations. Concerning the main novelties of the proposed approach, it is worth mentioning the presence of some auxiliary functions, the involvement of the convergence-control parameters, the construction of the first iteration and much freedom to select the procedure for determining the optimal values of the convergence-control parameters.

Some variants of Cauchy's mean value theorem

Abstract: In this note, some variants of Cauchy's mean value theorem are proved. The main tools to prove these results are some elementary auxiliary functions.

Convergence of adaptive discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations

Abstract: We prove the convergence of adaptive discontinuous Galerkin and $C^0$-interior penalty methods for fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincare and trace inequalities, a proof of density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results.

Analytical Solution of System of Volterra Integral Equations Using OHAM

Abstract: In this work, a reliable technique is used for the solution of a system of Volterra integral equations (VIEs), called optimal homotopy asymptotic method (OHAM). The proposed technique is successfully applied for the solution of different problems, and comparison is made with the relaxed Monto Carlo method (RMCM) and hat basis function method (HBFM). The comparisons show that the present technique is more suitable and reliable for the solution of a system of VIEs. The presented technique uses auxiliary function containing auxiliary constants, which control the convergence. Moreover, OHAM does not require discretization like other numerical methods and is also free from small or large parameter.

Optimal minimax bounds for time and ensemble averages of dissipative infinite-dimensional systems with applications to the incompressible Navier-Stokes equations

Abstract: Obtaining sharp estimates for quantities involved in a given model is an integral part of a modeling process. For dynamical systems whose orbits display a complicated, perhaps chaotic, behaviour, the aim is usually to estimate time or ensemble averages of given quantities. This is the case, for instance, in turbulent flows. In this work, the aim is to present a minimax optimization formula that yields optimal bounds for time and ensemble averages of dissipative infinite-dimensional systems, including the two- and three-dimensional Navier-Stokes equations. The results presented here are extensions to the infinite-dimensional setting of a recent result on the finite-dimensional case given by Tobasco, Goluskin, and Doering in 2017. The optimal result occurs in the form of a minimax optimization problem and does not require knowledge of the solutions, only the law of the system. The minimax optimization problem appears in the form of a maximization over a portion of the phase space of the system and a minimization over a family of auxiliary functions made of cylindrical test functionals defined on the phase space. The function to be optimized is the desired quantity plus the duality product between the law of the system and the derivative of the auxiliary function.

Stability of stochastic functional differential systems with semi-Markovian switching and Lévy noise by functional Itô’s formula and its applications

Abstract: This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Levy noise (SFDEs-SMS-LN). Based on functional Ito’s formula, multiple degenerate Lyapunov functionals and nonnegative semi-martingale convergence theorem, new pth moment and almost surely stability criteria with general decay rate for SFDEs-SMS-LN are established. Meanwhile, the diffusion operators are allowed to be controlled by multiple auxiliary functions with time-varying coefficients, which can be more adaptable to the non-autonomous SFDEs-SMS-LN with high-order nonlinear coefficients. Furthermore, as applications of the presented stability criteria, new delay-dependent sufficient conditions for general decay stability of the stochastic delayed neural network with semi-Markovian switching and Levy noise (SDNN-SMS-LN) and the scalar non-autonomous SFDE-SMS-LN with non-global Lipschitz condition are respectively obtained in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the effectiveness of the proposed results.

Iteration Methods with an Auxiliary Function for Nonlinear Equations

Abstract: Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function to solve the nonlinear equation . In this paper, we introduce the expansion of with the inclusion of weights such that and knots in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

A Dichotomy for Real Boolean Holant Problems

Abstract: We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions as in previous results. It is proved that for every set $\mathcal{F}$ of real-valued constraint functions, Holant$(\mathcal{F})$ is either P-time computable or #P-hard. The classification has an explicit criterion. This is the culmination of much research on this problem, and it uses previous results and techniques from many researchers. Some particularly intriguing concrete functions $f_6$, $f_8$ and their associated families with extraordinary closure properties related to Bell states in quantum information theory play an important role in this proof.

A general stability criterion for multidimensional fractional-order network systems based on whole oscillation principle for small fractional-order operators

Abstract: The research on the fractional-order network system (FONS, The derivative model of network system is fractional-order) has seen fruitful achievements, but ignores whether the fractional-order operator (The order of fractional derivatives α) in the FONS will affect its stability and dynamic characteristics. To tackle this problem, this paper adopts a new method to study the effect of fractional-operators in gamma functions on the dynamic state of the gamma function. This new method helps us to derive a novel dynamic principle of multidimensional FONS. We define it as the Whole Oscillation Principle. According to this principle, the choice of fractional operator reflects the dynamic oscillation characteristic of the multidimensional FONS in two dimensions of time and system state, thus better optimizing the complex case of the fractional-order system research process in the future. Furthermore, based on the auxiliary function-based integral inequality, the paper derives a new stability criterion for all dimensional FONS in the general form. Finally, the validity and correctness of the above theories are verified through numerical simulation to its good effect.

General Optimal Stopping with Linear Costs

Abstract: This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of threshold type. The optimal threshold is then characterized as the root of that function. For random walks our results condense in the fact that all combinations of concave increasing pay-off functions and convex cost functions lead to a one-sided solution. For Levy processes an explicit way to obtain the auxiliary function and the threshold is given by use of the ladder height processes. Lastly, the connection from discrete and continuous problem and possible approximation of the latter one via the former one is discussed.

Peak Estimation and Recovery with Occupation Measures

Abstract: Peak Estimation aims to find the maximum value of a state function achieved by a dynamical system. This problem is non-convex when considering standard Barrier and Density methods for invariant sets, and has been treated heuristically by using auxiliary functions. A convex formulation based on occupation measures is proposed in this paper to solve peak estimation. This method is dual to the auxiliary function approach. Our method will converge to the optimal solution and can recover trajectories even from approximate solutions. This framework is extended to safety analysis by maximizing the minimum of a set of costs along trajectories.

Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

Abstract: This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bagci and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.