Interval stability for complex systems

Stability of dynamical systems against strong perturbations is an important problem of nonlinear dynamics relevant to many applications in various areas. Here, we develop a novel concept of interval stability, referring to the behavior of the perturbed system during a finite time interval. Based on this concept, we suggest new measures of stability, namely interval basin stability (IBS) and interval stability threshold (IST). IBS characterizes the likelihood that the perturbed system returns to the stable regime (attractor) in a given time. IST provides the minimal magnitude of the perturbation capable to disrupt the stable regime for a given interval of time. The suggested measures provide important information about the system susceptibility to external perturbations which may be useful for practical applications. Moreover, from a theoretical viewpoint the interval stability measures are shown to bridge the gap between linear and asymptotic stability. We also suggest numerical algorithms for quantification of the interval stability characteristics and demonstrate their potential for several dynamical systems of various nature, such as power grids and neural networks

Dilogarithm Identities in Conformal Field Theory and Group Homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all $2 \times 2$ real matrices viewed as a {\it discrete} group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic $K$-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all $2 \times 2$ real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with the summary of a number of open conjectures on the mathematical side.Comment: 20 pages, 2 figures not include

Analytic Bethe Ansatz for Fundamental Representations of Yangians

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian $Y(X_r)$ or its quantum affine analogue $U_q(X^{(1)}_r)$ for $X_r = B_r, C_r$ and $D_r$. Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of $Y(X_r)$. Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the $T$-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.Comment: 45 pages, Plain Te

Relativistic instant-form approach to the structure of two-body composite systems

A new approach to the electroweak properties of two-particle composite systems is developed. The approach is based on the use of the instant form of relativistic Hamiltonian dynamics. The main novel feature of this approach is the new method of construction of the matrix element of the electroweak current operator. The electroweak current matrix element satisfies the relativistic covariance conditions and in the case of the electromagnetic current also the conservation law automatically. The properties of the system as well as the approximations are formulated in terms of form factors. The approach makes it possible to formulate relativistic impulse approximation in such a way that the Lorentz-covariance of the current is ensured. In the electromagnetic case the current conservation law is ensured, too. The results of the calculations are unambiguous: they do not depend on the choice of the coordinate frame and on the choice of "good" components of the current as it takes place in the standard form of light--front dynamics. Our approach gives good results for the pion electromagnetic form factor in the whole range of momentum transfers available for experiments at present time, as well as for lepton decay constant of pion.Comment: 26 pages, Revtex, 5 figure

A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions

We define and discuss classical and quantum gravity in 2+1 dimensions in the Galilean limit. Although there are no Newtonian forces between massive objects in (2+1)-dimensional gravity, the Galilean limit is not trivial. Depending on the topology of spacetime there are typically finitely many topological degrees of freedom as well as topological interactions of Aharonov-Bohm type between massive objects. In order to capture these topological aspects we consider a two-fold central extension of the Galilei group whose Lie algebra possesses an invariant and non-degenerate inner product. Using this inner product we define Galilean gravity as a Chern-Simons theory of the doubly-extended Galilei group. The particular extension of the Galilei group we consider is the classical double of a much studied group, the extended homogeneous Galilei group, which is also often called Nappi-Witten group. We exhibit the Poisson-Lie structure of the doubly extended Galilei group, and quantise the Chern-Simons theory using a Hamiltonian approach. Many aspects of the quantum theory are determined by the quantum double of the extended homogenous Galilei group, or Galilei double for short. We study the representation theory of the Galilei double, explain how associated braid group representations account for the topological interactions in the theory, and briefly comment on an associated non-commutative Galilean spacetime.Comment: 38 pages, 1 figure, references update

Spin chains with dynamical lattice supersymmetry

Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N=(2,2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension admits a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.Comment: Revised version, 52 pages, 2 figure

The eight-vertex model and lattice supersymmetry

We show that the XYZ spin chain along the special line of couplings J_xJ_y+J_xJ_z+J_yJ_z=0 possesses a hidden N=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain.Comment: 40 pages, 6 figures, Tik

Modifying Lax equations and the second Hamiltonian structure

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46607/1/222_2005_Article_BF01394252.pd

Results of measurements of the analyzing powers for polarized neutrons on C, CH <inf>2</inf> and Cu targets for momenta between 3 and 4.2 GeV/c

The analyzing powers for neutron charge exchange nA → pX reactions on nuclei have been measured on C, CH2 and Cu targets at incident neutron momenta 3.0 - 4.2 GeV/c by detecting one charged particle in forward direction. The polarized neutron measurements are the first of their kind. The experiment was performed using the Nuclotron accelerator in JINR Dubna, where polarized neutrons and protons were obtained from breakup of a polarized deuteron beam which has a maximum momentum of 13 GeV/c. The polarimeter ALPOM2 was used to obtain the analyzing power dependence on the transverse momentum of the final-state nucleon. These data have been used to estimate the figure of merit of a proposed experiment at Jefferson Laboratory to measure the recoiling neutron polarization in the quasi-elastic 2H(e, e'n) reaction, which yields information on the charge and magnetic elastic form factors of the neutron

Uncertainty and analyticity

We describe a connection between minimal uncertainty states and holomorphy-type conditions on the images of the respective wavelet transforms. The most familiar example is the Fock–Segal–Bargmann transform generated by the Gaussian, however, this also occurs under more general assumptions