Form Of Power Series Expansion Research Articles

Revisiting Stochastic Realization Theory using Functional Itô Calculus

This paper considers the problem of constructing finite-dimensional state space realizations for stochastic processes that can be represented as the outputs of a certain type of a causal system driven by a continuous semimartingale input process. The main assumption is that the output process is infinitely differentiable, where the notion of differentiability comes from the functional Ito calculus introduced by Dupire as a causal (nonanticipative) counterpart to Malliavin's stochastic calculus of variations. The proposed approach builds on the ideas of Hijab, who had considered the case of processes driven by a Brownian motion, and makes contact with the realization theory of deterministic systems based on formal power series and Chen-Fliess functional expansions.

Efficient solutions for fractional Tsunami shallow-water mathematical model: A comparative study via semi analytical techniques

In this study, we delve into the effective solution of the fractional tsunami shallow-water mathematical model. Tsunami propagation and inundation modeling is of paramount importance for hazard assessment and disaster preparedness. By employing two distinct semi-analytical techniques: the residual power series method and the modified Taylor series method, we obtain approximate solutions for the tsunami mathematical model. Solutions for varying coastal inclinations and ocean depths have been acquired for the given model taking into consideration the changes in coastal gradient and ocean depth and how to impact tsunami wave velocity and wave amplification at different values of the fractional order derivative and different steps of time. The utilized semi analytical techniques generate the solution in the form of power series expansion, so the convergence study is implemented. In order to prove the efficiency of the proposed methods, several tables for comparisons between our results and other numerical results from literature are presented. Also graphs of the obtained solutions introduced in two and three dimensions.

De Moivre and Bell polynomials

We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are described, and we discuss the related work of De Moivre, Arbogast and Bell.

Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

Analytical representation and efficient computation of the effective conductivity of two‐phase composite materials

AbstractMany engineered materials display ordered or disordered microstructures. Such materials exhibit transport properties which are unmatched by their single‐phase homogeneous counterparts. These properties are obtained by the mixture of two or more phases typically characterized by a large contrast in their properties. For the development of these materials, it is critical to develop a robust computational framework in order to provide a fundamental understanding of how microstructure affects performance. This hinges on predicting their macroscopic properties, given the constitutive laws and spatial distribution of their constituents. To this end, this work presents a computational framework based on formulating periodic conduction transport problems in terms of boundary integral equations whose kernel is expressed in terms of Weierstrass zeta‐function. The components of the effective conductivity tensor are then sought in the form of power series expansions of a conductivity contrast parameter. To accelerate their convergence, these expansions are transformed into Padé approximants. Presently restricted to the case of two‐dimensional, two‐phase microstructures, this framework is shown to yield accurate results over the entire range of the contrast parameter. Representation of the kernel as a lattice sum allows the use the fast multipole method, thereby making computations significantly more efficient.

Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents Pn(x)/Qn(x) modulo 4, we give the formal power series expansions (modulo 4) of these two continued fractions and prove that they are congruent modulo 4 to algebraic series in Z[[x]]. Therefore, the coefficient sequences of the formal power series expansions are 2-automatic. Write Qn(x)=∑i≥0an,ixi. Then (Qn(x))n≥0 defines a two-dimensional coefficient sequence (an,i)n,i≥0. We prove that the coefficient sequences (an,imod4)n≥0 introduced by both (Qn(x))n≥0 and (Pn(x))n≥0 are 2-automatic for all i≥0. Moreover, the pictures of these two dimensional coefficient sequences modulo 4 present a kind of self-similar phenomenon.

Reexamining the gluon spectrum in the boost-invariant glasma from a semianalytic approach

In high energy heavy-ion collisions, the degrees of freedom at the very early stage can be effectively represented by strong classical gluonic fields within the Color Glass Condensate framework. As the system expands, the strong gluonic fields eventually become weak such that an equivalent description using the gluonic particle degrees of freedom starts to become valid. We revisit the spectrum of these gluonic particles by solving the classical Yang-Mills equations semi-analytically with the solutions having the form of power series expansions in the proper time. We propose a different formula for the gluon spectrum which is consistent with energy density during the whole time evolution. We find that the chromo-electric fields have larger contributions to the gluon spectrum than the chromo-magnetic fields do. Furthermore, the large momentum modes take less time to reach the weak-field regime while smaller momentum modes take more time. The resulting functional form of the gluon spectrum is exponential in nature and the spectrum is close to a thermal distrubtion with effective temperatures around $0.6$ to $0.9\, Q_s$ late in the Glasma evolution. The sensitiveness of the gluon spectrum to the infrared and the ultraviolet cut-offs are discussed.

Non-isothermal Flow of Viscoelastic Elastomer Composites in a Converging Channel

Highly filled polymer composites are very widely used in different processes of modern industry, and therefore the investigation of the flow of such media in different channels of processing equipment is being paid a great deal of attention. A mathematical model of the non-isothermal flow of a Newtonian fluid in a converging plane channel is presented. All the important factors that must be taken into account in the development of the model are noted. Many assumptions are made on the basis of the fact that flow occurs at low Reynolds numbers and high Peclet numbers. As the rheological model, the Maxwell higher convective model is used. The solution is sought using series, and all the stress components are expressed in the form of power series expansion. Thermal boundary conditions of the first kind, the temperature dependence of viscosity, and energy dissipation are taken into account. The problem of finding the temperature profiles is solved by an iteration scheme, at each step of which the collocation method is used. The results of calculations are given.

Compressible flow through solar chimneys with variable cross section - an exact solution

Buoyancy driven, adiabatic and compressible flow in relatively high solar chimneys is treated in the paper analytically by using one-dimensional model of flow. General equations written suitably in a non-dimensional form are used for a qualitative discussion pertaining to the mutual effects of gravity, viscosity and varying cross section of the chimney. It is shown that in case of low Mach number flow these equations possess exact solutions obtainable by ordinary mathematical methods for any given chimney shape. Also shown, and demonstrated on an example, is the procedure of evaluation of the chimney shape that satisfies a condition imposed beforehand upon the flow. For better insight into the role of various parameters the solutions are presented in the form of power series expansions.

The regular state in higher order gravity

We consider the higher-order gravity theory derived from the quadratic Lagrangian [Formula: see text] in vacuum as a first-order (ADM-type) system with constraints, and build time developments of solutions of an initial value formulation of the theory. We show that all such solutions, if analytic, contain the right number of free functions to qualify as general solutions of the theory. We further show that any regular analytic solution which satisfies the constraints and the evolution equations can be given in the form of an asymptotic formal power series expansion.

A formal power series expansion–regularization approach for Lévy stable distributions: the symmetric case with (M positive integer)

In practice, the Lévy α-stable distribution is usually expressed in terms of the Fourier integral of its characteristic function. Indeed, known closed form expressions are relatively scarce given the huge parameters space: , and . Hence, systematic efforts have been made towards the development of proper methods for analytically solving the mentioned integral. As a further contribution in this direction, here we propose a new way to tackle the problem. We consider an approach in which one first solves the Fourier integral through a formal (thus not necessarily convergent) series representation. Then, one uses (if necessary) a pertinent sum-regularization procedure to the resulting divergent series, so as to obtain an exact formula for the distribution, which is amenable to direct numerical calculations. As a concrete study, we address the centered, symmetric, unshifted and unscaled distribution (, , ), with , . Conceivably, the present protocol could be applied to other sets of parameter values.

Power series with positive coefficients arising from the characteristic polynomials of positive matrices

Let A be an $$n \times n$$ (entrywise) positive matrix and let $$f(t)=\det (I-t A)$$ . We prove the surprising result that there always exists a positive integer N such that the formal power series expansion of $$1-f(t)^{1/N}$$ around $$t=0$$ has positive coefficients.

Congruences for the Fishburn numbers

The Fishburn numbers, ξ(n), are defined by a formal power series expansion∑n=0∞ξ(n)qn=1+∑n=1∞∏j=1n(1−(1−q)j). For half of the primes p, there is a non-empty set of numbers T(p) lying in [0,p−1] such that if j∈T(p), then for all n≥0,ξ(pn+j)≡0(modp).

Quicksilver Solutions of a q‐Difference First Painlevé Equation

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (qPI), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q‐domain. The method, while demonstrated for qPI, is also applicable to other q‐difference Painlevé equations.

A Method of Nonlinear Modal Superposition for Weakly Nonlinear Autonomous Systems

For a weakly nonlinear autonomous systems, the nonlinear coordinate transformation is provided in the form of power series expansions and the mathematical transformation from the physics system coordinate to the modal coordinate is presented. And then, according to the relations between physics system coordinate and modal coordinate, a method of nonlinear modal superposition is obtained. Using this method, the general solution of the vibration under any initial condition can be answered. In order to explain the applications of the method, numerical examples are presented in this paper.

On [formula omitted]-series identities related to interval orders

We prove several power series identities involving the refined generating function of interval orders, as well as the refined generating function of the self-dual interval orders. These identities may be expressed as ∑n≥0(1p;1q)n=∑n≥0pqn(p;q)n(q;q)n and ∑n≥0(−1)n(1p;1q)n=∑n≥0pqn(p;q)n(−q;q)n=∑n≥0(qp)n(p;q2)n, where the equalities apply to the (purely formal) power series expansions of the above expressions at p=q=1, as well as at other suitable roots of unity.

Asymptotic integration of Navier–Stokes equations with potential forces. II. An explicit Poincaré–Dulac normal form

We study the incompressible Navier–Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier–Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1–47], produces a Poincaré–Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces.

Measures on Banach Manifolds, Random Surfaces, and Nonperturbative String Field Theory with Cut-offs

We construct a cut-off version of nonpertubative closed Bosonic string field theory in the light-cone gauge with imaginary string coupling constant. We show that the partition function is a continuous function of the string coupling constant, and conjecture a relation between the formal power series expansion of this partition function and Riemann Surfaces.

The structure of the hydrodynamic plasma flow near the heliopause stagnation point

The plasma flow in the vicinity of the heliopause stagnation point in the presence of the H atom flow is studied. The plasma at both sides of the heliopause is considered to be a single fluid. The back reaction of the plasma flow on the H atom flow is neglected, and the density, temperature and velocity of the H atom flow are taken to be constant. The solution describing the plasma flow is obtained in the form of power series expansions with respect to the radial distance from the symmetry axis. The main conclusion made on the basis of the obtained solution is that the heliopause is not the surface of discontinuity anymore. Rather, it is the surface separating the flows of the solar wind and interstellar medium with all plasma parameters continuous at this surface.

Q-Derivative Operators and Basic Hypergeometric Series

By means of q-derivative operators, we investigate formal power series expansions. Two main expansion formulae in terms of q-derivative operators are established which can be considered as extensions of the corresponding results due to Carlitz (1973) and Liu (2002). Their applications to basic hypergeometric evaluations and transformations are discussed through series compositions and their q-derivative operations. Direct verification of the two main theorems are also presented.