On Generalized co-Narayana Numbers

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

In this paper, we investigate the generalized Guglielmo sequences and we deal with, in detail, four special cases, namely, triangular, triangular-Lucas, oblong and pentagonal sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

In this paper, closed forms of the summation formulas for generalized Tribonacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third order linear recurrance sequences. All the summing formulas of well known recurrence sequences which we deal with are linear except the cases Pell-Padovan and Padovan-Perrin.

We introduce a new generalization of Gaussian Pell polynomials. We call it [Formula: see text]-Gaussian Pell polynomials. Then we present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representation of [Formula: see text]-Gaussian Pell polynomials. We introduce the matrix as binary representations according to the Riordan matrix representation. Using the Riordan method, we obtain the factorizations of the Pascal matrix involving the polynomial. In addition, we present the inverse of matrices of the polynomial.

Polynomials and matrices have played a very important role in the development of different branches of mathematics. Indeed, several mathematicians have introduced classical polynomials very useful for the scientific community such as the Lagrange’s interpolation polynomials, the Chebyshev’s polynomials and the Bernstein’s polynomials [1,2]. Also, there is a strong link between polynomials and matrices through the notions of the determinant, the characteristic polynomial and the minimal polynomial. Similarly, we will introduce in this article two polynomials which we will call twin polynomials as well as a matrix called kernels matrix. Finally, we will present some applications such as the resolution of recurrent sequences with second member and the establishment of several sum formulas.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

ObjectivesTo study the problem of determining the degree of stress at the apex of a wedge-shaped area in cases where the sides (or one of them) are covered with a thin flexible coating.MethodIt is assumed that the coating is not stretchable. On the other side of the wedge-shaped area, the same coating is assumed to be present; it is either fixed, stress-free or in smooth contact with a rigid base. Mathematically, the problem is reduced to the task of determining the roots of characteristic transcendental equations arising from the existence of a nontrivial solution to the system of linear homogeneous equations.ResultsValues for the specific characteristics of the radial component of a stress tensor are determined for different combinations of boundary conditions and solution angles. In particular, the angles at which the singular behaviour of stresses occurs are determined. The case is considered when a special boundary condition is given on the edge surface, simulating the overlay. Characteristic equations are obtained to determine the index of the degree dependency of the asymptotic solution in its vicinity for four variants of boundary conditions. In two cases, transcendental equations are obtained, which are solved numerically.ConclusionCalculations of the first positive roots of the equations depending on the angle of the edge solution and Poisson's ratio are presented. The values of the angles, at which the singular behaviour of stresses occurs, are determined. In the case of a combination of boundary conditions (III – IV), the singular stress behaviour is observed for the angle ???? = ????/8, while in the case of (III – III) this value is equal to ????/4.

The article deals with transport schemes of cargo delivery from manufacturer to consumer. They consist of linear or intersect logistic chains. It is shown that when the input tact of stream of this scheme is stationary then static delays of material flow are present there. There was given the dependence of the delay from tact of initial flow. It was found that the static minimum of time delay reaches the optimal mode of delivery schemes. The problem of evaluating difficulties of the most probable delay of dynamic material flow in straight-way transportation and technological schemes or those with branching logistics chains was partly solved. Delays of material flow in this case are caused by the changes of input flows parameters. Options of material flows are recorded as discrete relationships. It was used the method of finite difference equations to evaluate the stability of transport and technological schemes with external disturbances. The order of the equation depends on the structure of logistic chains of the transport and technological scheme. The equation can be reduced to a uniform appearance. Ones can concluded about the stability of transport and technological scheme to external disturbances with the roots of the characteristic equation. An example of a unstable linear circuits was given. The selection of parameters of logistic operations are not sufficient conditions for stability evaluation. According to given models the delivery schemes should have additional input supply flow regulation.

The borderlines of the area in which the circular waveguide, entirely filled with azimuthally magnetized ferrite, works as a phase shifter for the normal TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">01</sub> mode, are determined. For the purpose the special features of the cut-off regime of the configuration and of its limiting one, observed at higher frequencies in which the propagation stops, provided the magnetic bias is negative, are used. For a fixed numerical equivalent of the normalized in an appropriate way guide radius, the off-diagonal ferrite permeability tensor element and the phase constant of the wave are counted by the roots of the structure's characteristic equation, derived through the Kummer confluent hypergeometric function. This is done repeatedly for a varying negative (positive) value of the imaginary part of its complex first parameter. The process continues, until the computed value of the element coincides with that of the same, corresponding to the cut-off state for the radius chosen (to the limiting one, if the latter is greater than certain number) with a prescribed accuracy. The relevant calculated numerical equivalent of the phase constant is accepted as the one, searched for. The differential phase shift equals the constant mentioned in the first case, whereas in the second one, it is found from it by a simple arithmetic. Changing the normalized radius, the boundaries of the domain studied are traced. The outcomes are presented graphically in a normalized form.

IN AN important paper, Theil and Boot [4] developed a test for the stability of a dynamic econometric model. The test statistic is the (estimated) absolute value of the largest root of the model's characteristic equation. Theil and Boot made the test operational by providing a formula for the asymptotic variance of the test statistic. This formula involved the asymptotic covariance matrix of the estimated reduced form parameters.1 More recently, Oberhofer and Kmenta [3] provided a formula for the asymptotic variance of the same test statistic, which involved the asymptotic covariance matrix of the estimated structural parameters. This is, of course, more readily available than the covariance matrix of the reduced form estimates. This note shows that the two approaches are algebraically equivalent. That is, starting from a set of structural estimates, it makes no difference whether one applies the Oberhofer-Kmenta formula, or whether one calculates the derived reduced form and then uses the Theil-Boot formula; the results are exactly the same either way. 2. THE OBERHOFER-KMENTA FORMULA Let p represent the absolute value of the largest characteristic root, and p its estimate. Let 0 be the vector of structural coefficients, say of dimension N. Suppose that one has a consistent estimator 0 (say, the three-stage least squares estimate) and its (estimated) asymptotic covariance matrix. Then the Oberhofer-Kmenta result is just that (1) i=a.v. (( a.c. i=, 0j), where a.v. (p) represents the (estimated) asymptotic variance of p and a.c. (0i, 0) represents the

In the previous two chapters, the classical control theory was discussed and the concept of transfer function was used to describe the dynamic behavior of various systems. In practice, the transfer function of higher order than three or four becomes tedious and it is better to use state variable control theory. It was stated that with PID control, three parameters can be adjusted to design control systems. With three parameters, all roots of characteristic equation can not be adjusted to achieve desirable transient and steady state behavior. In practice, it is only possible to make one or two roots to become dominant in response and a compromise between the transient response and steady state error has to be made. The derivative term always amplifies the noise in the practical systems and is not recommended. Instead a lead-lag network produces a better response.

The stability of multidimensional hamiltonian systems

CHAPTER XIII - Longitudinal Transients and Control Response

A three-dimensional system is presented with unknown parameters that employs two nonlinearities terms. The basic characteristics of the system are studied. The stability is measured by Characteristic equation roots, Routh stability criteria, Hurwitz stability criteria and Lapiynov function, all show that the system unstable. Then, Chaoticity is measured by maximum Lapiynov exponent of (Lmax=2.509426) and “Kaplan-Yorke” dimension (DL=2.22349544). The system is controlled effectively and synchronized by designed adaptive controllers. Furthermore, the theoretical and graphic results of the system before and after control are compared.

Hyperchaos–chaos–hyperchaos transition in modified Rössler systems