Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

An algorithm for mean residual life computation of (n − k + 1)-out-of-n systems: An application of exponentiated Weibull distribution

In this paper, we establish an algorithm for the computation of the mean residual life of a (n − k + 1)-out-of-n system in the case of independent but not necessarily identically distributed lifetimes of the components. An application for the exponentiated Weibull distribution is given to study the effect of various parameters on the mean residual life of the system. Also the relationship between the mean residual life for the system and that of its components is investigated.     

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A_j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A₁) + f(A₂) + ? + f(A_m)? ? ? f(A₁ + A₂ + ? + A_m)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A₁) + f(A₂) + ? + f(A_m)? ? f(?A₁ + A₂ + ? + A_m?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.     

On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers

In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C_r(F₀, F₁, …, F_n−1), B = C_r(L₀, L₁, …, L_n−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Symmetric duality with (p, r) − ρ − (η, θ)-invexity

In this paper we introduce a new type of generalized invex function, called (p, r) − ρ − (η, θ)-invex function and study symmetric duality results under these assumptions. In our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars. We establish weak and strong duality theorems under (p, r) − ρ − (η, θ)-invexity assumptions for the symmetric dual problems. We also give many examples to justify our results.     

Recursive algorithm for the reliability of a connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system

The connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system is included by the connected-X-out-of-(m, n):F lattice system defined by Boehme et al. [Boehme, T.K., Kossow, A., Preuss, W., 1992. A generalization of consecutive-k-out-of-n:F system. IEEE Transactions on Reliability 41, 451–457]. This system fails if and only if at least one subset of connected failed components occurs which includes at least a (1, 2)-matrix (that is, a row and two columns) or a (2, 1)-matrix(that is, two rows and a column) of failed components. This system is applied to two-dimensional network problems with adjacent constraints, and various systems, for example, a supervision system, etc.     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix R∈R^n,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ₁, … , λ_n), with λ₁ ? ? ? λ_n ? 0, the symmetric matrix A = RΛR^T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (P, Q), each in R^n,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ^T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ^T (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, , generated in the course of the QR-algorithm when it is applied to A = RΛR^T, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices R_k of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.     

Classification of small (0, 1) matrices

Denote by A_n the set of square (0, 1) matrices of order n. The set A_n, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let D_n, S_n denote the set of absolute determinant values and Smith normal forms of matrices from A_n. Denote by a_n the smallest integer not in D_n. The sets D₉ and S₉ are obtained; especially, a₉ = 103. The lower bounds for a_n, 10 ? n ? 19 (exceeding the known lower bound a_n ? 2f_n − 1, where f_n is nth Fibonacci number) are obtained. Row/permutation equivalence classes of A_n correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.     

A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

The two dimensional diffusion equation of the form is considered in this paper. We try a bi-cubic spline function of the form as its solution. The initial coefficients C_i,j(0) are computed simply by applying a collocation method; C_i,j = f(x_i, y_j) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients C_i,j(t) are computed by X(t) = e^tQX(0) where X(t) = (C_0,1, C_0,1, C_0,2, … , C_0,N, C_1,0, … , C_N,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.     

Sharp distortion theorems for some subclass of close-to-convex functions

For real A, B such that −1 ? B < A ? 1, we denote by R(A,B) the class of functions, such that f′ ? (1 + Az)/(1 + Bz). Sharp distortion results for functions from R_A,B are obtained.     

A dynamic adaptive local search algorithm for the circular packing problem

This paper studies the circular packing problem (CPP) which consists of packing n non-identical circles C_i of known radius r_i, i ∈ N = {1, … , n}, into the smallest containing circle C. The objective is to determine the coordinates (x_i, y_i) of the center of C_i, i ∈ N, as well as the radius r and center (x, y) of C. This problem, which is a variant of the two-dimensional open dimension problem, is solved using a two-step, dynamic, adaptive, local search algorithm. At each iteration, the algorithm identifies the set of potential “best local positions” of a circle C_i, i ∈ N, given the positions of the previously packed circles, and determines for each of these positions the coordinates and radius of the smallest containing circle. The “best local position” minimizes the radius of the current containing circle. That is, every time an additional circle is packed, both the center and the radius of the containing circle are dynamically updated, and the smallest containing circle is known. The experimental results reflect the good performance of the algorithm.     

Exact solutions of nonlinear dispersive K(m, n) model with variable coefficients

The variable-coefficient Korteweg-de Vries (KdV) equation with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condense was presented to describe the dynamics of nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. To understand the role of nonlinear dispersion in this variable-coefficient model, we introduce and study a new variable-coefficient KdV with nonlinear dispersion (called vc-K(m, n) equation). With the aid of symbolic computation, we obtain its compacton-like solutions and solitary pattern-like solutions. Moreover, we also present some conservation laws for both vc-K⁺(n, n) equation and vc-K⁻(n, n) equation.