C++ string handling

Abstract: In this paper we introduce the notion of string stability of a countably infinite interconnection of a class of nonlinear systems. Intuitively, string stability implies uniform boundedness of all the states of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded. It is well known that the I/O gain of all the subsystems less than unity guarantees that the interconnected system is I/O stable. We derive sufficient ("weak coupling") conditions which guarantee asymptotic string stability of a class of interconnected systems. Under the same "weak coupling" conditions, string stable interconnected systems remain string stable in the presence of small structural/singular perturbations. In the presence of parameter mismatch, these "weak coupling" conditions ensure that the states of all the subsystems are all uniformly bounded when gradient based parameter adaptation law is used and that they go to zero asymptotically.

Abstract: Motivated by attempts to extend AdS/CFT duality to non-BPS states we consider classical closed string solutions with several angular momenta in different directions of AdS_5 and S^5. We find a novel solution describing a circular closed string located at a fixed value of AdS_5 radius while rotating simultaneously in two planes in AdS_5 with equal spins S. This solution is a direct generalization of a two-spin flat-space solution where the string rotates in two orthogonal planes while always lying on a 3-sphere. Similar solution exists for a string rotating in S^5: it is parametrized by the angular momentum J of the center of mass and the two equal SO(6) angular momenta J_2=J_3=J' in the two rotation planes. The remarkably simple case is of J=0 where the energy depends on J' as E=[(2J')^2 + {\l}]^{1/2} with {\l}being the string tension or `t Hooft coupling. We discuss interpolation of the E(J') relation to weak coupling by identifying the N=4 SYM theory operator that should be dual to the corresponding semiclassical string state and utilizing existing results for its perturbative anomalous dimension. This opens up a possibility of studying AdS/CFT duality in this new non-BPS sector. We also investigate stability of these classical solutions under small perturbations and comment on several generalizations.

Abstract: Interpolation methods such as the nudged elastic band and string methods are widely used for calculating minimum energy pathways and transition states for chemical reactions. Both methods require an initial guess for the reaction pathway. A poorly chosen initial guess can cause slow convergence, convergence to an incorrect pathway, or even failed electronic structure force calculations along the guessed pathway. This paper presents a growing string method that can find minimum energy pathways and transition states without the requirement of an initial guess for the pathway. The growing string begins as two string fragments, one associated with the reactants and the other with the products. Each string fragment is grown separately until the fragments converge. Once the two fragments join, the full string moves toward the minimum energy pathway according to the algorithm for the string method. This paper compares the growing string method to the string method and to the nudged elastic band method using the alanine dipeptide rearrangement as an example. In this example, for which the linearly interpolated guess is far from the minimum energy pathway, the growing string method finds the saddle point with significantly fewer electronic structure force calculations than the string method or the nudged elastic band method.