Functional Power Series Research Articles

Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative

In the present paper, the synchronization by the Ge-Yao-Chen (GYC) partial region stability theory of chaotic Mathieu-Van der Pol and chaotic Duffing-Van der Pol systems with fractional order-derivative is proposed. Numerical simulations show that this synchronization technique is very effective and it turns out that the fractional order-derivative induces quick synchronization compared to integer order-derivative of these systems. In order to bring out the chaotic behavior of these systems either with fractional or with integer order-derivative, we simulate their phase portraits and the Lyapunov exponent. Moreover, we provide in this work an approximated solution to both systems to show that the solution of such a system can be represented as a simple power-series function. Furthermore, the representation of the error dynamics with respect to the time before and after the control action approves the effectiveness of the control method and proves the possibility of stabilization and controllability of chaotic systems with an appropriate. Furthermore, the synchronization of the fractional Mathieu-Van der Pol system using the fractional Duffing-Van der Pol system is simulated.

Five Points Mono Hybrid Point Linear Multistep Method for Solving Nth Order Ordinary Differential Equations Using Power Series Function

Five Points Mono Hybrid Point Linear Multistep Method for Solving Nth Order Ordinary Differential Equations Using Power Series Function

Concentration dependence of the wings of a dipole-broadened magnetic resonance line in magnetically diluted lattices

The singularities of the time autocorrelation functions (ACFs) of magnetically diluted spin systems with dipole–dipole interaction (DDI), which determine the high-frequency asymptotics of autocorrelation functions and the wings of a magnetic resonance line, are studied. Using the self-consistent fluctuating local field approximation, nonlinear equations are derived for autocorrelation functions averaged over the independent random arrangement of spins (magnetic atoms) in a diamagnetic lattice with different spin concentrations. The equations take into account the specificity of the dipole–dipole interaction. First, due to its axial symmetry in a strong static magnetic field, the autocorrelation functions of longitudinal and transverse spin components are described by different equations. Second, the long-range type of the dipole–dipole interaction is taken into account by separating contributions into the local field from distant and near spins. The recurrent equations are obtained for the expansion coefficients of autocorrelation functions in power series in time. From them, the numerical value of the coordinate of the nearest singularity of the autocorrelation function is found on the imaginary time axis, which is equal to the radius of convergence of these expansions. It is shown that in the strong dilution case, the logarithmic concentration dependence of the coordinate of the singularity is observed, which is caused by the presence of a cluster of near spins whose fraction is small but contribution to the modulation frequency is large. As an example a silicon crystal with different 29Si concentrations in magnetic fields directed along three crystallographic axes is considered.

Regular functions on dual split quaternions in Clifford analysis

This paper shows expressions of a power series for the form of dual split quaternions and provides differential operators in dual split quaternions. The paper also represents a power series of dual split regular functions by using a dual split Cauchy-Riemann system in dual split quaternions.

Symmetric 2-Step 4-Point Hybrid Method for the Solution of General Third Order Differential Equations

This research considers a symmetric hybrid continuous linear multistep method for the solution of general third order ordinary differential equations. The method is generated by interpolation and collocation approach using a combination of power series and exponential function as basis function. The approximate basis function is interpolated at both grid and off-grid points but the collocation of the differential function is only at the grid points. The derived method was found to be symmetric, consistent, zero stable and of order six with low error constant. Accuracy of the method was confirmed by implementing the method on linear and non-linear test problems. The results show better performance over known existing methods solved with the same third order problems. AMS 2010 Subject Classification: 65D05; 65L05; 65L06

Chemical reaction networks for computing logarithm.

Living cells constantly process information from their living environment. It has recently been shown that a number of cell signaling mechanisms (e.g. G protein-coupled receptor and epidermal growth factor) can be interpreted as computing the logarithm of the ligand concentration. This suggests that logarithm is a fundamental computation primitive in cells. There is also an increasing interest in the synthetic biology community to implement analog computation and computing the logarithm is one such example. The aim of this article is to study how the computation of logarithm can be realized using chemical reaction networks (CRNs). CRNs cannot compute logarithm exactly. A standard method is to use power series or rational function approximation to compute logarithm approximately. Although CRNs can realize these polynomial or rational function computations in a straightforward manner, the issue is that in order to be able to compute logarithm accurately over a large input range, it is necessary to use high-order approximation that results in CRNs with a large number of reactions. This article proposes a novel method to compute logarithm accurately in CRNs while keeping the number of reactions in CRNs low. The proposed method can create CRNs that can compute logarithm to different levels of accuracy by adjusting two design parameters. In this article, we present the chemical reactions required to realize the CRNs for computing logarithm. The key contribution of this article is a novel method to create CRNs that can compute logarithm accurately over a wide input range using only a small number of chemical reactions.

Granular core phenomenon induced by convection in a vertically vibrated cylindrical container.

A mixture of 13X molecular sieve (13XMS) particles and glass particles with identical diameters is placed in a cylindrical container. Under vertical vibration, heavier glass particles tend to cluster and are wrapped inside the convection of 13XMS particles, resulting in the granular core phenomenon. The vibration frequency f strongly influences particle convection and particle cluster modes. By contrast, the effect of the dimensionless acceleration amplitude Γ can be neglected. For different f ranges, the granular core is classified as center-type and ring-type cores. For the center-type core, heavy particles are distributed as an approximate zeroth-order Bessel function of the first kind in the radial direction and an exponential function in the height direction. For the ring-type core, the concentration of heavy particles follows the power-series function in the radial direction. A granular transport model is then established based on heavy-particle movements under steady state to analyze the effect of vibration parameters and granular convection on density segregation.

Symmetric Hybrid Linear Multistep Method for General Third Order Differential Equations

A symmetric hybrid linear multistep method for direct solution of general third order ordinary differential equations is considered in this paper. The method is developed by interpolation and collocation approach using a combination of power series and exponential function as basis function. The consistency, stability, order and error constant of the method were determined. The results showed that the method is consistent, zero stable and of order five with low error constant. The accuracy compared favorably over existing methods with higher order of accuracy.

固定辺を含む薄肉シェルパネルの面内・面外連成線形振動解析

In this study, a new analytical method on natural frequencies and vibration modes of a thin shell-panel with clamped edge is presented. At the boundaries of two opposite edges, the shell panel is simply supported, and at the other two edges the panel is simply supported or clamped. Coordinate function of deflection is assumed with the product of power series and trigonometric function. Neglecting the effect of in-plane inertia force, the Donnell type equations are applied as the governing equation of the shell-panel. Stress function is obtained which satisfies exactly the compatibility equation with the inhomogeneous term due to the coupling with deflection and the in-plane boundary conditions. Applying Galerkin method to equation of motion, natural frequencies and vibration modes are calculated and compared with the exact result.

An analytical approach for buckling analysis of generally laminated conical shells under axial compression

In the present investigation, the buckling of generally laminated conical shells with various boundary conditions subjected to axial pressure is studied using an analytical approach. The governing equations are obtained using classical shell theory with Donnell assumptions in strain–deformation relations and the principle of minimum potential energy. The differential equations are solved using trigonometric functions in circumferential and power series in longitudinal directions. All types of boundary conditions can be applied in this method. The results are compared and validated with the results available in the literature, and good agreement is observed. Finally, the effects of the length, semi-vertex angle, and lamination sequences on the buckling load and mode shapes of generally laminated conical shells are presented.

Volterra representation enables modeling of complex synaptic nonlinear dynamics in large-scale simulations.

Chemical synapses are comprised of a wide collection of intricate signaling pathways involving complex dynamics. These mechanisms are often reduced to simple spikes or exponential representations in order to enable computer simulations at higher spatial levels of complexity. However, these representations cannot capture important nonlinear dynamics found in synaptic transmission. Here, we propose an input-output (IO) synapse model capable of generating complex nonlinear dynamics while maintaining low computational complexity. This IO synapse model is an extension of a detailed mechanistic glutamatergic synapse model capable of capturing the input-output relationships of the mechanistic model using the Volterra functional power series. We demonstrate that the IO synapse model is able to successfully track the nonlinear dynamics of the synapse up to the third order with high accuracy. We also evaluate the accuracy of the IO synapse model at different input frequencies and compared its performance with that of kinetic models in compartmental neuron models. Our results demonstrate that the IO synapse model is capable of efficiently replicating complex nonlinear dynamics that were represented in the original mechanistic model and provide a method to replicate complex and diverse synaptic transmission within neuron network simulations.

Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series S(z) with coefficients in Qp to have coefficients in Zp. We establish theorems on the p-adic valuation of the coefficients of the exponential of S(z), assuming weaker conditions on the coefficients of S(z) than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the p-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian p-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers modulo p for free products of finite Abelian p-groups, results on p-divisibility of permutation numbers with restrictions on their cycle structure, and a curious “supercongruence” for a certain binomial sum.

Normalisation of Hamiltonian in photogravitational elliptic restricted three body problem with Poynting-Robertson drag

In this paper, we have performed first order normalization in the photogravitational elliptic restricted three body problem  with Poynting-Robertson drag. We suppose that bigger primary as radiating and smaller primary is an oblate spheroid. We have found the Lagrangian and the Hamiltonian of the problem. Then, we have expanded the Lagrangian function in power series of x and y, where (x, y) are the coordinates of the triangular equilibrium points. Using Whittaker (1965) method, we have found that the second order part H2 of the Hamiltonian is transformed into the normal form.

Differentiation formulas of some hypergeometric functions with respect to all parameters

In this work we present two methods to derive some differentiation formulas of the generalized hypergeometric function mFn(a1,…,am;b1,…,bn;z), including the most commonly used Gauss hypergeometric function 2F1(μ,ν;λ;z) and Kummer confluent hypergeometric function 1F1(μ;ν;z) as special cases, with respect to all parameters. We first briefly describe the direct derivative method for the convergent power series of hypergeometric functions. Secondly, we mainly focus on the differential equation method, which is based on differentiating the generalized hypergeometric differential equation with respect to parameters. Particularly, by using the differential equation method, some general analytical expressions of any sth derivatives with respect to single parameter can be deduced by induction in s. Moreover, we can obtain all the mixed derivatives of higher order very conveniently. Finally, examples are given to illustrate the usefulness of these derivatives in mathematics, physics and other related fields. Numerical examples for computing those singular oscillatory integrals presented in Kang et al. (2013) and Kang and Ling (in press), in turn verify that the approximation value of the required derivatives can be of great precision, and show the correctness of differentiation formulas obtained by the proposed methods.

Universality properties of the quaternionic power series and entire functions

In this paper we establish the existence of “almost universal” quaternionic power series and entire functions. Denoting by B(0, 1) the open unit ball in , this means that there exists a quaternionic power series with radius of convergence 1 such that, denoting by the n‐th partial sum of S, for every , for every axially symmetric open subset Ω of containing K and every f slice regular on Ω, there exists a subsequence of the partial sums of S such that uniformly on K, as . The symbol denotes the set of axially symmetric compact sets in such that is connected for some . This is a slightly weaker property than the classical universal power series phenomenon obtained for analytic only on the interior of K and continuous on K. We also generalize a result originally proven by Birkhoff and finally we show that there exists an entire quaternionic function whose set of derivatives is dense in the class of entire quaternionic functions.

Modeling of Wave Propagation in Degenerate Hyperbolic Systems

Degeneration in a small spatial coordinate of the original initial---boundary-value problem for hyperbolic equations is considered in order to construct simpler models. The problem is investigated in a domain whose one spatial scale is much less than the other scales. This makes it possible to expand the desired functions in power series and hence to reduce the dimension of the problem. However, this comes at the cost of degeneration of the spectrum of the original three-dimensional problem.

A NEW NUMERICAL INTEGRATOR FOR THE SOLUTION OF GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper considered the development of numerical integrator for the solution of second order initial value problems. The method was derived through the interpolation and collocation of the basis polynomial which is com- bination of power series and exponential function to derive a continuous linear multistep method. The method was implemented in block method which gave solution at a non overlapping interval. The method was found to be conver- gence and A stable. The efficiency of the method was tested on some numerical examples.

Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator

This paper presents the derivation and implementation of a block integrator for the solution of stiff and oscillatory first-order initial value problems of Ordinary Differential Equations (ODEs). The integrator was derived by collocation and interpolation of the combination of power series and exponential function to generate a continuous implicit Linear Multistep Method (LMM). The basic properties of the derived integrator were investigated and the integrator was implemented on some sampled stiff and oscillatory problems. From the results obtained, it is obvious that the block integrator gives better approximation than some existing ones.

The Number of Roots of Polynomials of Large Degree in a Prime Field

We establish asymptotic upper bounds on the number of roots modulo p of some polynomials with rational coefficients, with p an arbitrarily large prime numbers, using a variant of Stepanov's method. The polynomials we consider have degree of size close to p and are obtained by truncating power series of polylogarithms, polyexponentials, and Bessel functions. This answers a question of Heath-Brown.

Theory of Attenuated Total Reflection Including Effects of Roughness

We have developed a theory of attenuated total reflection applicable to multi-layered samples with roughness at surfaces and/or interfaces. Angle dependence of reflectivity for incident light with any polarization are expressed by a power series of profile functions of roughness, and the terms up to the second order polynomials of the profile functions are presented. Numerical analyses for Au films in the Kretschmann configuration reveal that differences between the predictions taking into account roughness and the ones assuming flat interfaces depend on both lateral and longitudinal sizes of roughness, and that the differences becomes not negligible for roughness greater than a certain level, e.g., roughness whose correlation length a and root-mean-square δ of the amplitude are 20 and 2 nm, respectively.