In this paper, we study the iterated application of the Binomial transform, the k–Binomial transform, the Rising k–Binomial transform, and the Falling k–Binomial transform to the k–Fibonacci sequence. In particular, we obtain the recurrence relation between the terms of the sequences obtained from these transforms and prove that they are all generalized Fibonacci sequences. As a consequence of this result, we obtain the Generating Function of these sequences, and Binnet Identity and Combinatorial Formula for the general term of each of them. We can consider the iterated application of the Binomial transforms as a new way to get integer sequences. But the way we have done, we have also found the recurrence relation between the terms of these sequences and how to find the general term of the same, either by the Binet Identity and the Combinatorial formula.
A new approach of the Mickens’ extended iteration method has been presented to obtain approximate analytic solutions of some nonlinear jerk equations. There has been shown that the partial derivatives of integral functions are valid for modified extended iteration method in each step of iteration. Also the solutions give more accurate result than other existing methods and show a good agreement with its exact solution.
A general system of ordinary differential equations that appear in neural network theory is studied. We consider nonlinear activation functions which have not been treated in the literature so far. Namely, we assume that the nonlinear activation functions are continuous functions but not necessarily Lipschitz continuous.
In this note, the quasicontinuity of a dcpo and its principal ideals/filters are considered. It is shown that every principal ideal, as a subposet, in a quasicontinuous domain is also a quasicontinuous domain. A dcpo is quasicontinuous if every principal ideal is quasicontinuous and it has finitely many maximal elements. For a bc-domain being quasicontinuous, the requirement of directedness of the approximation family of each element can be omitted.
The dispersive long wave (DLW) equations are very important coupled nonlinear partial differential equations that appear for describing nonlinear water wave model in harbor and coastal design. In this paper, the exp(-Φ(ξ))-expansion method has been implemented to find the explicit solutions of the coupled (1+1)-dimensional DLW equations. The efficiency of the method for finding exact solutions has been demonstrated. With the help of symbolic computation, it has been shown that the method is direct, effective and can be used for many other nonlinear partial differential equations (NPDEs) in mathematical physics and engineering.
In this paper, we propose an ecient modication of a New Homotopy Perturbation Method (NHPM) to obtain approximate and exact analytical solutions of Partial Dierential-Algebraic Equations (PDAEs). The NHPM is rst applied to the PDAE to obtain the exact solution in convergent series form. To improve the solution obtained from NHPM's truncated series, a post-treatment combining Laplace transform and Pade approximant is proposed. This modied Laplace-Pade new homotopy perturbation method is shown to be eective and greatly improves NHPM's truncated series solutions in convergence rate, and often leads to the exact solution. Two problems are solved to demonstrate the eciency of the method; the rst one is a nonlinear index-one system with an integral term and the second one is a linear index-three system with variable coecients.
This work is devoted to study a doubly non linear elliptic-parabolic problem with quadratic gradient term by Rothe's method. We investigate the long time behavior of the solution to the discrete problem and prove the existence of compact global attractor. Our method relays on semi-discretization with respect to the time variable.
We use tensor product to introduce a new approach to the theory of integration. Such an approach will strengthen the existing various classical concepts of integral and will provide a continuous thread tying the subject matter together. The integral of vector-valued functions with respect to vector-valued additive measures will be covered without any assumption of measurability. As applications, we state and prove extensions of the Lebesgue fundamental theorems of convergence in a more general setting.
Incidence response and handling has become quite a crucial, indispensible constituent of information technology security management, as it provides an organised way of handling the aftermaths of a security breach. It presents an organisation’s reaction to illegitimate and unacceptable exploits on its assets or infrastructure. The goal must be to successfully neutralise the incident, such that damages are significantly reduced with attendant reduction in recovery time and costs. To achieve this, several approaches and methodologies proposed have been reviewed with a view to identifying essential processes. What is needed is referred to as incident capability mingled with collaborations. This defines a shift from response to management of computer security incidents in anointer relationship manner that foster collaboration through the exchange and sharing of incidence management details among several distinct organizations. Key step-up aspects centre on issues of enforcing and assuring trust and privacy. A viable collaborative incident response approach must be able to proffer both proactive and reactive mechanisms that are management-oriented and incorporating all required techniques and procedures.