Extended Generalized Fibonacci and Tribonacci Polynomials with some Properties

In this paper, bicomplex $k$-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex $k$-Fibonacci quaternions are investigated. For example, the summation formula, generating functions, Binet's formula, the Honsberger identity, the d'Ocagne's identity, Cassini's identity, Catalan's identity for these quaternions are given. In the last part, a different way to find $n-th$ term of the bicomplex $k$-Fibonacci quaternion sequence was given using the determinant of a tridiagonal matrix.

This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

In this paper, bicomplex Pell and bicomplex Pell-Lucas numbers are defined. Also, negabicomplex Pell and negabicomplex Pell-Lucas numbers are given. Some algebraic properties of bicomplex Pell and bicomplex Pell-Lucas numbers which are connected between bicomplex numbers and Pell and Pell-Lucas numbers are investigated. Furthermore, d'Ocagne's identity, Binet's formula, Cassini's identity and Catalan's identity for these numbers are given.

The use of manipulators can improve sustainable energy utilization efficiency and increase sustainable manufacturing practices for solar tracking systems and manufactures, and thus it is significant to guarantee a high tracking accuracy for manipulators. In this paper, an error-tracking adaptive iterative learning control (AILC) method is proposed for a constrained flexible-joint manipulator (FJM) with initial errors. Due to the existence of the repeated positioning drift, the accuracy of the actual manipulator and the sustainable energy utilization efficiency are affected, which motivates the error-tracking approach proposed in this paper to deal with the repeat positioning problem. The desired error trajectory is constructed, such that the tracking error can follow the desired error trajectory without arbitrary initial values and iteration-varying tasks. Then, the system uncertainties are approximated by the capability of fuzzy logic systems (FLSs), and the combined adaptive laws are designed to update the weight and the approximating error of FLSs. Considering the safety operation of the flexible-joint manipulator, both input and output constraints are considered, a quadratic-fractional barrier Lyapunov function (QFBLF) is constructed, such that the system output is always within the constrained region. Therefore, the proposed method can guarantee the output tracking accuracy of manipulators under arbitrary initial values and iteration-varying tasks and keep the system output within the constraints to improve the transient performance, such that the energy utilization and accessory manufacturing efficiency can be improved. Through the Lyapunov synthesis, it is proved that the tracking error can converge to zero as the number of iterations goes to infinity. Finally, comparative simulations are carried out to verify the effectiveness of the proposed method.

Many researchers have been working on recurrence relation sequences of numbers and polynomials which are useful topic not only in mathematics but also in physics, economics and various applications in many other fields. There are many useful identities on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find previous all terms of recurrence relation sequence of numbers and polynomials. There were many important theorems obtained on recurrence relation sequences. In this paper we have given special identity for generalized Fibonacci sequence of number and Fibonacci sequence of polynomials. These identities are very useful to represent Fibonacci generalized sequence of numbers and Fibonacci sequence of polynomials in the form of matrix. Authors define a special formula in this paper by this we can find special representation of Fibonacci generalized sequence of numbers and Fibonacci sequence of polynomials in the form of matrix. So, we can say that this paper is generalization of property of Fibonacci sequence of number and Fibonacci sequence of polynomials

A typical multivariable control problem requires the design of dynamic compensation to guarantee the following desirable behavior of the closed loop system. 1. Each of an assigned set of output variables converges to, or tracks, a corresponding observed reference input from a specified function class. This is the servo problem. 2. Each of an assigned set of output variables converges to zero from arbi-trary initial values, when the system is perturbed by possibly nonmeasur-able disturbances in a specified function class. This is the regulator problem; the convergent variables are said to be regulated; and zero represents the desired or set-point value.

Fluctuating hydrodynamics and concentration fluctuations in ternary mixtures

The holomorphic embedding (HE) method is a novel recursive method for power flow analysis. Using the fixed flat start, the HE guarantees to yield the high-voltage solution and provides unequivocal signals if no solution exists. The fast and flexible HE (FFHE) method is adaptive with flexible initial values. However, the FFHE cannot start from arbitrary initial values because it is not globally convergent. This paper studies the convergence performance of the FFHE and finds their regularity. To further improve the practicality of this method, this paper finds a numerically common domain of convergence for cases with different scales, ranging from 2 to 2383 buses, using the FFHE. The proposed common domain of initial values is applicable for a variety of cases using FFHE.

As key electrical devices, large reactors and converter transformers play an important role in the operation of ultra high voltage direct current (UHVDC) transmission [1]. Meanwhile, the existence of rich harmonics in the exciting current or driving voltage leads to distorted flux densities in the saturated iron core. Consequently the magnetic force due to the leakage flux not only exhibits special spectrum in the frequency domain but also threatens the normal operation of electrical devices e.g. vibration and deformation of windings [2–3]. Therefore it is important to investigate the influence of magnetic force on iron core and windings when electrical devices such as reactors work under harmonic magnetizations. The time-stepping method starting from arbitrary initial value is usually used for the computation of time-periodic nonlinear problems. However, the main drawback is that many periods are required to be stepped through for steady state solution, especially when a much larger time constant exists in the nonlinear system to be analyzed. Numerical methods in time domain and frequency domain have been presented and developed to solve the steady state nonlinear magnetic field efficiently [4–5] by combining the fixed point technique with the finite element method. In this paper, the harmonic-balance system of equations is established first by approximating all time-periodic variables by complex series. Thereupon, decomposition of the system of equations is achieved according to diagonal dominant characteristic of the reluctivity matrix so that harmonic solutions can be decoupled and computed separately and in parallel. This method is used to compute the magnetic force in a gapped-core model under harmonic magnetizations based on the harmonic solutions of magnetic field.

The stationary, periodic solution to the problem of oscillating flow of a conducting fluid, in a duct closed at one end and with periodical mass flow rate at the other end, is known to exhibit a singularity in the mean temperature at the closed end when transverse conduction in the fluid and the duct wall is taken into account, but longitudinal conduction is neglected. For instance, a solution of that type was originally suggested by Gifford and Longsworth as a prototype for pulse-tube refrigeration. Whether these stationary singular solutions are physical or mere theoretical curiosities depend upon the existence of a scenario leading to such a limit cycle. To address that question, a transient theory is formulated, using the narrow duct approximation. The results show that at least for constant fluid thermal conductivity, all singular profiles generated by the mechanism under study are linearly stable. For round tubes, the temperature profile is shown to evolve, from an arbitrary initial value, toward an equilibrium profile which results in balanced energy fluxes.

We consider a solution u(p, g, a, b) to an initial value–boundary value problem for a wave equation: and we discuss an inverse problem of determining a coefficient p(x) and a, b by observations of u(p, g, a, b)(x, t) in a neighbourhood ω of ∂Ω over a time interval (0, T) and ∂itu(p, g, a, b)(x, T0), x ∊ Ω, i = 0, 1, with T0 < T. We prove that if T − T0 and T0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g1, …, gN, so that the mapping is uniformly Lipschitz continuous with suitable Sobolev norms provided that {p, aj, bj}1 ⩽ j ⩽ N remains in some bounded set in a suitable Sobolev space. In our inverse problem, initial values are also unknown, and we do not assume any positivity of the initial values. Our key is a Carleman estimate and the exact controllability in a Sobolev space of higher order.

The state vector equation for lateral buckling in finite displacement theory is formulated using only the hypothesis of the Bernoulli–Euler beam. By using an appropriate orthogonalization of the warping functions, the normally complicated calculation has been processed systematically using only matrix notation. As a numerical analysis, the lateral buckling load on the cantilever receiving a concentrated end load on the upper flange was calculated using the coefficient matrix of the first-order increment; the post-buckling behaviour was investigated with increasing load. Since the state vector equation is a higher order nonlinear equation, the original coefficient matrix was fixed with an arbitrary initial value and the solution was provided by the Runge–Kutta transfer matrix method. Subsequent calculations were pursued in the same way with the solution obtained via Runge–Kutta methods as a new initial value and then shifted to the next load condition. This theory and analysis method does not employ an assumed displacement function, such as the Ritz's method; it is therefore useful for the finite displacement analysis of a beam with arbitrary boundary conditions and intermediate support conditions.

The aim of this paper is to investigate the behavior of quotients of sequences w n defined by the linear homogeneous recurrence relation $$w_{n + 2} = pw_{n + 1} - qw_n$$ (1) with arbitrary initial values w 0 = a, w 1 = b belonging to a normed field K, and p, q ∈ K. We shall write w n (a,b; p,q) when it swill be necessary to specify the initial data and parameters. In [3], Horadam has studied the properties of these sequences. If w n ≠ 0, ∀n ∈ N, then the sequence of quotien1ts $$Q_n (a,b;p,q) = \frac{{w_{n + 1} }} {{w_n }}$$ (2.1) is defined in K. If the field K is normed and α, β are the roots of the polynomial x 2 – px + q, with ∥ α ∥ > ∥ β ∥ then the sequences of Q n ’s, where it is defined, converges, and $$ \mathop{{\lim }}\limits_{{n \to \infty }} {Q_n} = \alpha. $$ (3)

The existing Adaptive Notch Filter (ANF)-based frequency estimation methods have problems of slow convergence speed, unstable error and limited selection of iterative initial values. In this paper, a frequency estimation method based on the indirect average plain gradient algorithm is proposed. It uses a new error function applied to the second-order adaptive Infinite Impulse Response (IIR) notch filter. The proposed error function is the mean value of the weighted average of the squares of the two signals outputted by Finite Impulse Response (FIR) and IIR sections of the ANF. It has better gradient characteristics, and arbitrary initial value can be selected for the following iterative calculation. The theory and simulation show that the proposed algorithm, compared with other indirect gradient algorithm, improves the convergence speed and estimation accuracy with little addition of the computation, while it is superior in varying-frequency signal tracking performance.