Infinite Terms Research Articles

An analytical treatment for the exact solutions of MHD flow and heat over two–three dimensional deforming bodies

The present paper is dedicated to the mathematical analysis of MHD permeable heat and fluid flow fields induced by stretching or shrinking two–three dimensional objects. In contrast to a variety of existing numerical solutions in the open literature on the ongoing topic, obtaining exact analytical solutions is targeted either in general or in restrictive forms. Particularly, exponential type series solutions are shown to exist, having finite or infinite terms in the expansion depending on the prevailing parameters of the fields of flow and heat. Special closed-form solutions are shown to be possible when the series terminates. In the case of infinite terms are present in the series, an algorithm is presented to non-iteratively evaluate the coefficients while resolving parameter domain of the problem. The heat domain is specifically resolved without any numerical means. The occurrence of algebraic solutions corresponding to linear/nonlinear/exponential stretching/shrinking of the surface is also accounted for. Interesting algebraic solutions are found pointing to multiple solutions; even for the non-MHD stretching plate problem. These solutions are much valuable since they represent a rare class of exact solutions to boundary layer equations and they will shed light upon further studies on the physical phenomenon.

A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model)

In this paper, for a Lotka-Volterra system with infinite delays and patch structure related to a multi-group SI epidemic model, applying Lyapunov functional techniques without using the form of diagonal dominance of the instantaneous negative terms over the infinite delay terms, we establish the complete global dynamics by a threshold parameter $s(M(0))$, that is, the trivial equilibrium is globally asymptotically stable if $s(M(0)) \leq 0$ and the positive equilibrium is globally asymptotically stable if $s(M(0))>0$, respectively. This offer new type condition of global stability for Lotka-Volterra systems with patch structure.

Performance of Restarted Homotopy Perturbation Method for TV-Based Image Denoising Problem

We first propose a restarted homotopy perturbation method (RHPM) for solving a nonlinear PDE problem which repeats HPM process by computing only the first few terms instead of computing infinite terms, and then we present an application of RHPM to TV- (Total Variation-) based image denoising problem. The main difficulty in applying RHPM to the nonlinear denoising problem is settled by using binomial series techniques. We also provide finite difference schemes for numerical implementation of RHPM. Lastly, numerical experiments for several test images are carried out to demonstrate the feasibility, efficiency, and reliability of RHPM by comparing the performance of RHPM with that of existing TM and recently proposed RHAM methods.

Modelling Study of Magnetic Field’s Effects on Solar Cell’s Transient Decay

Experimental setup of transient decay which occurs between two steady state operating points is recalled. The continuity equation is resolved using both the junction dynamic velocity (Sf) and back side recombination velocity (Sb). The transient excess minority carriers density appears as the sum of infinite terms. Influence of magnetic field on the transient excess minority carriers density and transient photo voltage is studied and it is demonstrated that the use of this technique is valid only when the magnetic field is lower than 0.001 T.

Global robust exponential synchronization of BAM recurrent FNNs with infinite distributed delays and diffusion terms on time scales

In this article, the global robust exponential synchronization of reaction-diffusion BAM recurrent fuzzy neural networks (FNNs) with infinite distributed delays on time scales is investigated. Applied Lyapunov functional and inequality skills, some sufficient criteria are established to guarantee the global robust exponential synchronization of reaction-diffusion BAM recurrent FNNs with infinite distributed delays on time scales. One example is given to illustrate the effectiveness of our results.

Infinite Boundary Terms of Ewald Sums and Pairwise Interactions for Electrostatics in Bulk and at Interfaces.

We present a unified derivation of the Ewald sum for electrostatics in a three-dimensional infinite system that is periodic in one, two, or three dimensions. The derivation leads to the Ewald3D sum being expressed as a sum of a real space contribution and a reciprocal space contribution, as in previous work. However, the k → 0 term in the reciprocal space contribution is analyzed further and found to give an additional contribution that is not part of previous reciprocal space contributions. The transparent derivation provides a unified view of the existing conducting infinite boundary term, the vacuum spherical infinite boundary term and the vacuum planar infinite boundary term for the Ewald3D sum. The derivation further explains that the infinite boundary term is conditional for the Ewald3D sum because it depends on the asymptotic behavior that the system approaches the infinite in 3D but it becomes a definite term for the Ewald2D or Ewald1D sum irrespective of the asymptotic behavior in the reduced dimensions. Moreover, the unified derivation yields two formulas for the Ewald sum in one-dimensional periodicity, and we rigorously prove that the two formulas are equivalent. These formulas might be useful for simulations of organic crystals with wirelike shapes or liquids confined in uniform cylinders. More importantly, the Ewald3D, Ewald2D, and Ewald1D sums are further written as sums of well-defined pairwise potentials overcoming the difficulty in splitting the total Coulomb potential energy into contributions from each individual group of charges. The pairwise interactions with their clear physical meaning of the explicit presence of the periodic images thus can be used to consistently perform analysis based on the trajectories from computer simulations of bulk or interfaces.

A general decay result of a viscoelastic equation with past history and boundary feedback

In this paper, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and in the presence of infinite memory term. In the domain as well as on a part of the boundary, we use the multiplier method and some properties of the convex functions to prove an explicit and general decay result.

An Intensional Concurrent Faithful Encoding of Turing Machines

The benchmark for computation is typically given as Turing computability; the ability for a computation to be performed by a Turing Machine. Many languages exploit (indirect) encodings of Turing Machines to demonstrate their ability to support arbitrary computation. However, these encodings are usually by simulating the entire Turing Machine within the language, or by encoding a language that does an encoding or simulation itself. This second category is typical for process calculi that show an encoding of lambda-calculus (often with restrictions) that in turn simulates a Turing Machine. Such approaches lead to indirect encodings of Turing Machines that are complex, unclear, and only weakly equivalent after computation. This paper presents an approach to encoding Turing Machines into intensional process calculi that is faithful, reduction preserving, and structurally equivalent. The encoding is demonstrated in a simple asymmetric concurrent pattern calculus before generalised to simplify infinite terms, and to show encodings into Concurrent Pattern Calculus and Psi Calculi.

Complex period-1 motions of a periodically forced Duffing oscillator with a time-delay feedback

In this paper, complex period-1 motions in a periodically forced Duffing oscillator with a time-delay feedback are investigated, and the symmetric and asymmetric, complex period-1 motions exist in lower excitation frequency. The analytical solutions of complex period-1 motions in such a Duffing oscillator are obtained through the finite Fourier series, and the corresponding stability and bifurcations of complex period-1 motions are discussed by eigenvalue analysis. The frequency–amplitude characteristics of complex period-1 motions in the periodically forced Duffing oscillator with a time-delay feedback are discussed. Complex period-1 motions generated numerically and analytically are illustrated. As excitation frequency is close to zero, the complex period-1 motions need almost infinite harmonic terms in the Fourier series to express the analytical solutions. From this study, the initial time-delay in the time-delayed, nonlinear systems should be uniquely determined to achieve a specific periodic motion.

A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations

The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives alongy-direction are removed to the derivatives alongx-direction iteratively using information given by the original differential equation (similarly fromx-direction toy-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite terms which constitute convergent series are kept on deriving the exponential coefficients of the scheme. From the construction process one may gain more insight into the relations among the stencil coefficients. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of second-order. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable, and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones. The idea of the method can be applied to a wide variety of differential equations.

On D-brane anti D-brane effective actions and their corrections to all orders in alpha-prime

Based on a four point function, the S-matrix elements at disk level of the scattering amplitude of one closed string Ramond-Ramond field (C) and two tachyons and one scalar field, we find out new couplings in brane anti brane effective actions for p = n, p+2 = n cases. Using the infinite corrections of the vertex of one RR, one gauge and one scalar field and applying the correct expansion, it is investigated in detail how we produce the infinite gauge poles of the amplitude for p = n case. By discovering new higher derivative corrections of two tachyon-two scalar couplings in brane anti brane systems to all orders in α′, we also obtain the infinite scalar poles in (t′+s′+u)-channel in field theory. Working with the complete form of the amplitude with the closed form of the expansion and comparing all the infinite contact terms of this amplitude, we derive several new Wess-Zumino couplings with all their infinite higher derivative corrections in the world volume of brane anti brane systems. In particular, in producing all the infinite scalar poles of < VCVϕVTVT > , one has to consider the fact that scalar's vertex operator in (-1)-picture must carry the internal σ3 Chan-Paton matrix. The symmetric trace effective action has a non-zero coupling between Dϕ(1)i and Dϕ(2)i while this coupling does not exist in ordinary trace effective action.

Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay

In this paper, we consider a second-order abstract linear equation with infinite memory and time delay terms. Under appropriate assumptions on the convolution kernel and on the weight of the delay, we prove the well-posedness and the exponential stability of the system. Our stability estimate proves that the unique dissipation given by the memory term is strong enough to stabilize exponentially the system in presence of delay. Some applications are also given.

On the H∞ Controller Design for a Magnetic Suspension System Model

This paper deals with the H∞ optimal controller design for a magnetic suspension system model derived in Knospe and Zhu [2011], with added input/output delay. The plant is a fractional order system with time delay i.e., the transfer function of the plant involves infinite dimensional terms including a rational function of √s and e–hs, where h > 0 represents the delay. The H∞ optimal controller is designed by using the recent formulation given in özbay [2012] for the mixed sensitivity minimization problem for unstable infinite dimensional plants with low order weights. The effect of time delay on the achievable performance level is illustrated.

Integral Sequences of Infinite Length Whose Terms Are Relatively Prime

It is given in Weil and Rosenlicht ([1], p. 15) that (resp. 2) for all non-negative integers m and n with m≠n if c is any even (resp. odd) integer. In the present paper we generalize this. Our purpose is to give other integral sequences such that G.C.D.(ym,yn)=1 for all positive integers m and n with m≠n. Roughly speaking we show the following 1) and 2). 1) There are infinitely many polynomial sequences such that G.C.D.(fm(a),fn(a))=1 for all positive integers m and n with with m≠n and infinitely many rational integers a. 2) There are polynomial sequences such that G.C.D.(gm(a,b),gn(a,b))=1 for all positive integers m and n with m≠n and arbitrary (rational or odd) integers a and b with G.C.D.(a,b)=1. Main results of the present paper are Theorems 1 and 2, and Corollaries 3, 4 and 5.

Natural Theory of Relativity

I have started this theory by deriving a different time dilation formula in an attempt to make the concepts of relativity more clear. I used the two postulates of special relativity i.e. the speed of light is constant for all inertial observers in free space i.e. vacuum and the same in all directions and the laws of physics are the same in all inertial frames together with the well-known fact that light takes a definite amount of time to travel between two points in space. I have then been able to get rid of the distortion, caused by the characteristics of the speed of light namely its constancy in all directions and the definite amount of time it takes to travel when it brings information from one point to another in space in combination with relative motion, in the form of infinite series terms. The distortion occurs symmetrically in the form of infinite series and leaves no skewness behind when got rid of. If we approximate the distorted value of a physical quantity to the first order, we get a distorted value. If we get rid of the distortion, in the form of infinite series, we get the actual value of the physical quantity. In the course of completing this theory I rejuvenated the concept of relative inertial kinetic energy and introduced relative gravitational acceleration at constant velocity in uniform circular motion. I have also been able to introduce the concept of gravitational shift in the dimensions of matter. My theory is very consistent.

Infinitary rewriting: closure operators, equivalences and models

Infinitary Term Rewriting allows to express infinite terms and transfinite reductions that converge to those terms. Underpinning the machinery of infinitary rewriting are closure operators on relations that facilitate the formation of transfinite reductions and transfinite equivalence proofs. The literature on infinitary rewriting has largely neglected to single out such closure operators, leaving them implicit in definitions of (transfinite) rewrite reductions, or equivalence relations. This paper unpicks some of those definitions, extracting the underlying closure principles used, as well as constructing alternative operators that lead to alternative notions of reduction and equivalence. A consequence of this unpicking is an insight into the abstraction level at which these operators can be defined. Some of the material in this paper already appeared in Kahrs (2010). The paper also generalises the notion of equational model for infinitary rewriting. This leads to semantics-based notions of equivalence that tie in with the equivalences constructed from the closure operators.

Pure type systems with corecursion on streams

In this paper, we use types for ensuring that programs involving streams are well-behaved. We extend pure type systems with a type constructor for streams , a modal operator next and a fixed point operator for expressing corecursion . This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calculus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS. CoPTSs allow us to study a wide range of typed lambda calculi extended with corecursion using only one framework. In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq. We use the machinery of infinitary rewriting and formalise the idea of well-behaved programs using the concept of infinitary normalisation. The set of finite and infinite terms is defined as a metric completion. We establish a precise connection between the modal operator (• A ) and the metric at a syntactic level by relating a variable of type (• A ) with the depth of all its occurrences in a term. This syntactic connection between the modal operator and the depth is the key to the proofs of infinitary weak and strong normalisation.

Another Approach to the Extended Stokes’ Problems for the Oldroyd-B Fluid

The extended Stokes problems, which study the flow suddenly driven by relatively moving half-planes, are reexamined for the Oldroyd-B fluid. This topic has been studied (Liu, 2011) by applying the series expansion to calculate the inverse Laplace transform. The derived solution was correct but tough to perform the calculation due to the series expansion of infinite terms. Herein another approach, the contour integration, is applied to calculate the inversion. Moreover, the Heaviside unit step function is included into the boundary condition to ensure the consistence between boundary and initial conditions. Mathematical methods used herein can be applied to other fluids for the extended Stokes’ problems.

Productivity of Non-Orthogonal Term Rewrite Systems

Productivity is the property that finite prefixes of an infinite constructor term can be computed using a given term rewrite system. Hitherto, productivity has only been considered for orthogonal systems, where non-determinism is not allowed. This paper presents techniques to also prove productivity of non-orthogonal term rewrite systems. For such systems, it is desired that one does not have to guess the reduction steps to perform, instead any outermost-fair reduction should compute an infinite constructor term in the limit. As a main result, it is shown that for possibly non-orthogonal term rewrite systems this kind of productivity can be concluded from context-sensitive termination. This result can be applied to prove stabilization of digital circuits, as will be illustrated by means of an example.

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach (SA) is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations (DDEs), the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach.