The effects of cubic potentials on discrete breathers in a mixed Klein－Gordon /Fermi－Pasta－Ulam chain

Nonlinearity has a crucial impact on the symmetry properties of dynamical systems. This paper studies a one-dimensional mixed Klein--Gordon/Fermi--Pasta--Ulam diatomic chain using the expanded rotating plane-wave approximation and numerical calculations to determine the effect of cubic potentials on the symmetry properties of discrete breathers in this system. The results will be very useful to researchers in the field of numerical calculations on discrete breathers.     

On congruences of groupoids closely connected with quasigroups

Conditions when a congruence of a left (right) division groupoid and a left (right) cancellation groupoid is closed (“normal”) are given. Conditions for the simplicity of the above-mentioned groupoids are obtained.     

The Geometry of Systems of Third Order Differential Equations Induced by Second Order Regular Lagrangians

A system of third order differential equations, whose coefficients do not depend explicitly on time, can be viewed as a third order vector field, which is called a semispray, and lives on the second order tangent bundle. We prove that a regular second order Lagrangian induces such a semispray, which is uniquely determined by two associated Poincaré-Cartan one-forms. To study the geometry of this semispray, we construct a horizontal distribution, which is a Lagrangian subbundle for an associated Poincaré-Cartan two-form. Using this semispray and the associated nonlinear connection we define dynamical covariant derivatives of first and second order. With respect to this, the second order dynamical derivative of the Lagrangian metric tensor vanishes.     

Counterexamples to a triality theorem in “Canonical dual least square method”

In this short note we give two counterexamples to the Triality Theorem concerning the optimization problem presented in Ruan et al., Comput. Optim. Appl. (2008). In order to understand the problem first we introduce the framework of this paper and quote the theorem we have in view. Then, while trying to verify this theorem we encounter several difficulties and note their nature. We end by revealing our counterexamples.     

Limiting the Number of Potential Binding Modes by Introducing Symmetry into Ligands: Structure‐Based Design of Inhibitors for Trypsin‐Like Serine Proteases

In the absence of X‐ray data, the exploration of compound binding modes continues to be a challenging task. For structure‐based design, specific features of active sites in different targets play a major role in rationalizing ligand binding characteristics. For example, dibasic compounds have been reported as potent inhibitors of various trypsin‐like serine proteases, the active sites of which contain several binding pockets that can be targeted by cationic moieties. This results in several possible orientations within the active site, complicating the binding mode prediction of such compounds by docking tools. Therefore, we introduced symmetry in bi‐ and tribasic compounds to reduce conformational space in docking calculations and to simplify binding mode selection by limiting the number of possible pocket occupations. Asymmetric bisbenzamidines were used as starting points for a multistage and structure‐guided optimization. A series of 24 final compounds with either two or three benzamidine substructures was ultimately synthesized and evaluated as inhibitors of five serine proteases, leading to potent symmetric inhibitors for the pharmaceutical drug targets matriptase, matriptase‐2, thrombin and factor Xa. This study underlines the relevance of ligand symmetry for chemical biology.     

Hamilton-Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque

We give sufficient conditions for the rigid body in the presence of an axisymmetric force field and a gyroscopic torque to admit a Hamilton-Poisson formulation. Even if by adding a gyroscopic torque we initially lose one of the conserved Casimirs, we recover another conservation law as a Casimir function for a modified Poisson structure. We apply this frame to several well-known results in the literature.     

On the Superlinear Convergence of the Successive Approximations Method

The Ostrowski theorem is a classical result which ensures the attraction of all the successive approximations x _k+1 = G(x _k ) near a fixed point x*. Different conditions [ultimately on the magnitude of G(x*)] provide lower bounds for the convergence order of the process as a whole. In this paper, we consider only one such sequence and we characterize its high convergence orders in terms of some spectral elements of G(x*); we obtain that the set of trajectories with high convergence orders is restricted to some affine subspaces, regardless of the nonlinearity of G. We analyze also the stability of the successive approximations under perturbation assumptions.