Functional Power Series Research Articles

Spectrally Simple Zeros of Zeon Polynomials

Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons. In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods.

Impact of non-thermal electrons on spatial damping: a kinetic model for the parallel propagating modes

Abstract A sequence of in situ measurements points the presence of non-thermal species in the profile of particle distributions. This study highlights the role of such energetic electrons on the wave-spectrum. Using Vlasov–Maxwell’s model, the dispersion relations of the parallel propagating modes along with the space scale of damping are discussed using non-relativistic bi-Maxwellian and bi-Kappa distribution functions under the weak field approximation, i.e., ω − k . v > Ω 0 $\left\vert \omega -\mathbf{k}.\mathbf{v}\right\vert { >}{{\Omega}}_{0}$ . Power series and asymptotic expansions of plasma dispersion functions are performed to derive the modes and spatial damping of waves, respectively. The role of these highly energetic electrons is illustrated on real frequency and anomalous damping of R and L-modes which is in fact controlled by the parameter κ in the dispersion. Further, we uncovered the effect of external magnetic field and thermal anisotropy on such spatial attenuation. In global perspective of the kinetic model, it may be another step.

A Speedy New Proof of the Riemann's Hypothesis

In this paper we show that Riemann's function <i>(xi)</i>, involving the Riemann’s (<i>zeta)</i> function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the <i>xi</i> function we deduce a relation between each zero of the function <i>xi</i> and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function.

A unified power series method for vibration analysis of composite laminate conical, cylindrical shell and annular plate

In this article, the power series method is adopted to investigate the free vibration characteristics of the composite laminated conical, cylindrical shell, and annular plate with arbitrary boundary conditions by the wave based matrix. By the first-order shear deformation shell theory, the governing equation of the composite laminated structure is established. The displacement and moment variables are transformed as the power series function form. Through the introduction of the boundary conditions and the continuous condition between the adjacent segments, the total matrix of the composite structure is established. Searching the zero locations of the total matrix determinant by the bisection method, the natural frequencies of the composite laminated structure under arbitrary boundary conditions are obtained. To further verify the correctness of the solutions by the presented method, some numerical examples are established. From the comparison of the solutions by the presented method and the results in some reported literature and experimental studies, the calculation accuracy is verified. Furthermore, the effect of the geometric parameters and material constants, such are elastic restrained spring stiffness constants, the angle between the shell surface and axis, length to the radius ratios, and modulus ratios, on the free vibration characteristics of the composite laminated structure, have been discussed. The advantage of the presented method is high calculation accuracy, simple matrix assembly, and convenient replacement of boundary conditions. Furthermore, some new numerical solutions are presented in this paper to provide the basic foundation for subsequent research.

Explicit Properties of q-Cosine and q-Sine Euler Polynomials Containing Symmetric Structures

In this paper, we introduce q-cosine and q-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. We investigate the approximate roots of q-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of q-cosine Euler polynomials in a complex plane using the Newton method.

Kronecker function rings and power series rings

Let D be an integral domain with quotient field K , X be an indeterminate over D , D [ [ X ] ] be the power series ring over D , and c ( f ) be the ideal of D generated by the coefficients of f ∈ D [ [ X ] ] . We will say that a star operation ∗ on D is a c-star operation if (i) c ( f g ) ∗ = ( c ( f ) c ( g ) ) ∗ for all 0 ≠ f , g ∈ D [ [ X ] ] and (ii) ( A B ) ∗ ⊆ ( A C ) ∗ implies B ∗ ⊆ C ∗ for all nonzero fractional ideals A , B , C of D . Assume that D admits a c-star operation ∗ , and let Kr ( ( D , ∗ ) ) = { f ∕ g ∣ f , g ∈ D [ [ X ] ] , g ≠ 0 , and c ( f ) ⊆ c ( g ) ∗ } . Among other things, we show that Kr ( ( D , ∗ ) ) is a Bezout domain, D is completely integrally closed, the v -operation on D is a c-star operation, and Kr ( ( D , v ) ) is a completely integrally closed Bezout domain. We also show that if V is a rank-one valuation domain, then the v -operation on V is a c-star operation, Kr ( ( V , v ) ) is a rank-one valuation domain, and Kr ( ( V , v ) ) is a DVR if and only if V is a DVR. Using this result, we show that if D is a generalized Krull domain, then Kr ( ( D , v ) ) is a one-dimensional generalized Krull domain.

Stability analysis and robust synchronisation of fractional-order modified Colpitts oscillators

Based on the stability theory of the fractional order system, the dynamic behaviours of the uncertain Colpitts oscillator with fractional order-derivative is studied. Furthermore, based on the extended bounded real lemma, the robust controller is obtained using the drive-response synchronisation concept together with the Lyapunov stability theory formulated using the fractional Lyapunov direct method where the fractional-order q belongs to 0 < q < 1. In order to bring out the dynamic behaviour of this system, their phase portraits, the bifurcation diagrams and the Lyapunov exponent are simulated. Moreover, in this work, an approximated solution for both systems to show that the solution of such a system can be represented as a simple power-series function is provided. This study equally provides a systematic procedure to highlight the simplicity and flexibility of the suggested control approach. Simulations with both parameters uncertainty and external disturbance show the applicability and the efficiency of the proposed scheme.

Дифференцирование функций кватернионной переменной

В данной работе рассматривается определение дифференцируемости и регулярностипо Фютеру [1–2] и примеры регулярных по Фютеру функций, приводится и определениеС-регулярности и С-производной или производной Куллена [3], на основе которой строитсяновая теория регулярных функций в [4], которая уже включает полиномы и сходящиеся ря-ды гиперкомплексной переменной как дифференцируемые функции. Затем предлагаетсяновое определение дифференцируемости, имеющее классический вид, но со специфиче-ской сходимостью, которое позволяет доказать теоремы о дифференцируемости суммы ипроизведения дифференцируемых функций, о дифференцируемости “частного” дифферен-цируемых функций. Далее выводится производная степени и доказывается дифференци-руемость полиномов и степенных рядов, что позволяет строить обобщения элементарныхфункций для кватернионных аргументов. Приводится пример, показывающий, что без спе-цифической сходимости приведенное определение дифференцируемости теряет смысл. Спомощью степенных рядов задаются функции, которые являются решениями дифферен-циальных уравнений с постоянными кватернионными коэффициентами. Рассматриваетсязадача отыскания корней квадратного уравнения с кватернионными коэффициентами, ко-торая возникает при решении дифференциальных уравнений

Stability analysis and robust synchronisation of fractional-order modified Colpitts oscillators

Based on the stability theory of the fractional order system, the dynamic behaviours of the uncertain Colpitts oscillator with fractional order-derivative is studied. Furthermore, based on the extended bounded real lemma, the robust controller is obtained using the drive-response synchronisation concept together with the Lyapunov stability theory formulated using the fractional Lyapunov direct method where the fractional-order q belongs to 0 < q < 1. In order to bring out the dynamic behaviour of this system, their phase portraits, the bifurcation diagrams and the Lyapunov exponent are simulated. Moreover, in this work, an approximated solution for both systems to show that the solution of such a system can be represented as a simple power-series function is provided. This study equally provides a systematic procedure to highlight the simplicity and flexibility of the suggested control approach. Simulations with both parameters uncertainty and external disturbance show the applicability and the efficiency of the proposed scheme.

A sage implementation for DD-finite functions

We present here the Sage package dd_functions which provides many features to compute with DD-finite functions, a natural extension of the class of holonomic or D-finite functions. The package, focused on a functional approach, allows the user to compute closure properties, extract coefficient sequences and compute the composition (as formal power series) of functions treated in this package. All these operations reduce the problem to linear algebra computations where classical division-free algorithms are used.

Symmetry reduction, exact solutions and conservation laws of the Bogoyavlenskii equation

In this paper, by applying the direct symmetry method, we obtain the symmetry reductions, group invariant solutions and some new exact solutions of the Bogoyavlenskii equation, which include hyperbolic function solutions, trigonometric function solutions and power series solutions. We also give the conservation laws of the Bogoyavlenskii equation.

Aiding and opposing mixed convection of water with density inversion about a wall of varying temperature

Mixed convection of water over a vertical surface of varying temperature with density inversion is studied for aiding and opposing convection configurations. The temperature of the surface is assumed to be an arbitrary function of vertical distance. The governing equations are transformed using dimensionless stream function and temperature. The temperature differentials of varying wall temperature are used as perturbation functions. The dimensionless stream function and temperature are expanded in power series of perturbation variables and coefficient functions. The obtained coefficient functions are valid for any arbitrary wall temperature function, and hence they are ’universal functions’. Power law wall temperature variation is chosen to show the usefulness of universal functions. The results are presented for velocity and temperature distributions in the boundary layer, velocity and thermal boundary layer thicknesses, skin friction coefficient and heat transfer rates for various values of governing parameter and wall temperature power law index for both aiding and opposing flows. It is found that for the range of \(Gr_y/Re_y^2\) values considered in the study, the skin friction coefficient and heat transfer rates vary almost linearly with wall temperature power law index value for a given \(Gr_y/Re_y^2\) value for both aiding and opposing flows. For special wall temperature cases, the present results are compared to benchmark solutions available in literature and good agreement is found.

Motion of a rolling sphere on an azimuthally symmetric surface

This paper analyzes the translational motion that a sphere rolling over an azimuthally symmetric surface, under the presence of a constant gravitational field, and with the rolling-without-slipping condition, exhibits in two different situations: with and without friction with air, where the latter is expressed as a power-series function of the sphere’s translational speed. In order to achieve this, the equations of motion for each case are obtained through the use of Lagrangian Mechanics and are subsequently solved by numerical computation in Wolfram Mathematica. For the frictionless case, periodic behavior and a conservation law for the angular coordinate have been found, along with the condition under which an effective potential energy can be approximated as well as the relationships between initial conditions that produce gravitational-like trajectories for the motion of the sphere. The equations of motion derived for the case with friction are found to predict the energy loss and general decay of the sphere’s motion. Likewise, the normal force over the sphere as a function of time is obtained through the method of Lagrange's Undetermined Multipliers, and thus, the general conditions that the motion must satisfy in order to be described by the obtained models. Overall, this research provides insight into the type and characteristics of the motion performed by the system in these two cases, both through equations and their numerical solutions for different surfaces and initial conditions.

A New Class of Power Function Distribution: Properties and Applications

A new class of lifetime distributions called the power function power series (PFPS) is introduced. This new class is obtained by compounding power function and power series distributions. Structural properties of PFPS class are studied, including; quantile function, order statistics, moments, Bonferroni and Lorenz curves, probability weighted moments, mean residual life and entropy measure. Two sub-models in the new class of PFPS are provided. Maximum likelihood method is employed to obtain parameter estimator of the class. Analysis of two real data sets is conducted to demonstrate the usefulness of two sub-models from the new class.

A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections

The study of thin-bed seismic phenomena is important in crustal, exploration and engineering seismology. Presently, seismic reflectivity theories based on single-interface assumption are widely used though they are only suitable for thick deposits. Thin-bed reflectivity theories are established on complex propagator matrices and are difficult to be applied to reveal thin-bed properties directly. Therefore, an approximation of thin-bed PP-wave reflection coefficients (RPP) is derived in this paper. First, the relationship between thin-bed RPP and incidence angles is analyzed through series expansion method. For PP-wave, its reflection coefficients are even power series functions of sine incidence angles. Then, for small incidence, RPP of the thin bed is further simplified into a second-order series approximation with respect to the sine incidence angles. Simulations and accuracy analyses of the approximate formula show that approximation errors are smaller than 5% as the incidence angles smaller than 20 degrees. Based on this approximate formula, an approach is given for estimating thin-bed properties including P-wave impedance ratios and thickness. The estimation approach is applied in properties estimation of a thin bed model. Perfect performances of the model example show the future potentiality of the approximate formula in thin-bed Amplitude-Versus-Offset (AVO) analysis and inversion.

Penyelesaian Terus Persamaan Pembezaan Biasa Peringkat Tiga menggunakan Kaedah Blok Hibrid Kolokasi Diperbaiki

The improved block hybrid collocation method (KBHK(B)) with two off-step points and four collocation points is proposed for the direct solution of general third order ordinary differential equations. Modification is done by adding first derivative of third order function into the general form of block hybrid collocation method to yield KBHK(B). Both main and additional methods are derived via interpolation (power series function) and collocation of the basic polynomial. An improved block method is derived to provide the approximation at five points concurrently. Zero stability, consistency, convergence and absolute stability region of KBHK(B) are investigated. Some numerical examples with exact solution are tested to illustrate the efficiency of the method. KBHK(B) is compared by other block hibrid collocation method in term of global error and step number. It is shown that KBHK(B) generates minimum global error with minimum step number.

Optimization of nonlinear wave energy converters

This paper presents an optimization approach for the nonlinear control of wave energy converters (WECs). The proposed optimization method also presents the option of optimizing the system nonlinearities, such as those due to the buoy shape, such that the harvested energy is maximized. For the sake of control design, the control force and the system optimizable nonlinear force, each is expressed as a truncated power series function of the system states. The power series coefficients in both the control and system forces are optimized. A hidden genes genetic algorithm is used for optimization. The optimized system's nonlinear force is assumed to drive the design of the WEC. The numerical test cases presented in this paper show that it is possible to attain multiple fold higher harvested energy when using nonlinear control optimization. The advantage of being able to optimize the WEC design simultaneously with the control is the potential of harvesting this multiple fold higher energy without causing large WEC motion and with less dependence on reactive power. While this paper focuses on the optimization part of the problem, the implementation of the obtained control in realtime is discussed at the end of the paper.

Skin effect in the time and frequency domain—comparison of power series and Bessel function solutions

In a time dependent solution of electromagnetic field penetration into a conductor, it turns out that the impedance power series solution diverges sharply if the ratio conductor radius to skin depth a/δ exceeds 2.7, close to the mathematical irrational number e. This problem has previously been considered to be due to the role of inversion. However, in this paper we show that the power series solution may be derived analytically from Bessel function solutions where the latter does not show the divergence problem. Experimental results are presented and we find that the logarithmic slope of the ac resistance and reactance change close to the value a/δ = e in both models. Since both models are based on the solution of a diffusion-like equation then the divergence effect at e may be expected to occur in other diffusion processes.

Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic. It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many special functions. The core number types are designed for versatility and speed in a range of scenarios, allowing performance that is competitive with non-interval arbitrary-precision types such as MPFR and MPC floating-point numbers. We discuss the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients.

Thermal dispersion effects on forced convection in a porous-saturated pipe

Thermal dispersion effects on fully developed forced convection inside a porous-saturated pipe are investigated. The pipe wall is assumed to be kept at a uniform and constant heat flux. Having the fully developed velocity field furnished by an arbitrary power series function, the energy equation is solved using asymptotic techniques for the limiting case when thermal conductivity, as a result of thermal dispersion, weakly changes with the Péclet number. A numerical solution, valid for the entire range of thermal dispersion conductivity, is also presented. This latter solution is presented to check the accuracy of the former. The two solutions are then cross-validated in the limits. Besides, results are found to be in good agreement with those previously reported in the literature.