1-Soliton solution of the coupled KdV equation and Gear–Grimshaw model

This paper carries out the integration of the coupled KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The domain restrictions of the coefficients of nonlinear and dispersion terms fall out. The results are then supplemented by numerical simulations.     

On formulas of Ramanujan and Evans

We give short elementary proofs of a formula of Ramanujan as interpreted by Bradley, and a companion formula originally proved by W. Chu. We also give an elementary proof of a generalization of an identity originally proved using modular functions and used to study a generating function for the number of partitions with specified crank. Research partially supported by NSF grant DMS 99-70865. 2000 Mathematics Subject Classification Primary—39A70, 11P83; Secondary—33C20     

Classification of Some Solvable Leibniz Algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension ≤ 8 with one dimensional derived subalgebra. We use the canonical forms for the congruence classes of matrices of bilinear forms to obtain our result. Our approach can easily be extended to classify these algebras of higher dimensions. We also revisit the classification of three dimensional non-Lie solvable (left) Leibniz algebras.     

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two‐weight Hardy, Hardy–Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.     

Equivalence relations among inequalities for some relative operator entropies

Positivity - The relative operator entropy has properties like operator means. In addition, the relative operator entropy has entropy-like properties. In this paper, we prove a Loewner–Heinz...     

Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery

This paper addresses a location-routing problem with simultaneous pickup and delivery (LRPSPD) which is a general case of the location-routing problem. The LRPSPD is defined as finding locations of the depots and designing vehicle routes in such a way that pickup and delivery demands of each customer must be performed with same vehicle and the overall cost is minimized. We propose an effective branch-and-cut algorithm for solving the LRPSPD. The proposed algorithm implements several valid inequalities adapted from the literature for the problem and a local search based on simulated annealing algorithm to obtain upper bounds. Computational results, for a large number of instances derived from the literature, show that some instances with up to 88 customers and 8 potential depots can be solved in a reasonable computation time.     

Stabilized mixed triangular finite elements at large deformations using area bubble functions

In this paper stabilized mixed triangular finite elements are presented in order to avoid volume locking and to damp stress oscillations. Geometrically non-linear elastic problems are addressed. The mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases. As a key idea, volume and area bubble functions are used for the method of incompatible modes and the enhanced strain method [1], thus giving both the interpretation of a mixed finite element method with stabilization terms. Concerning non-linear problems these are non-linearly dependent on the current deformation state, however, linearly dependent stabilization terms are used [1]. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step is completely avoided. The variational formulation for the standard two-field method, the method of incompatible modes and the enhanced strain method in finite deformation problems is derived for a hyper elastic Neo-Hookean material. In the representative example Cook's membrane problem illustrates the good performance of the presented approaches compared to existing finite element formulations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Soc-Injective Rings and Modules*

If M and N are right R-modules, M is called Socle-N-injective (Soc-N-injective) if every R-homomorphism from the socle of N into M extends to N. Equivalently, for every semisimple submodule K of N, any R-homomorphism f : K → M extends to N. In this article, we investigate the notion of soc-injectivity.