Reversible skew laurent polynomial rings and deformations of poisson automorphisms

A skew Laurent polynomial ring S = R[x(+/- 1); alpha] is reversible if it has a reversing automorphism, that is, an automorphism theta of period 2 that transposes x and x(-1) and restricts to an automorphism gamma of R with gamma = gamma(-1). We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus, both of which are deformations of Poisson algebras over the base field F. Their reversing automorphisms are deformations of Poisson automorphisms of those Poisson algebras. In each case, the ring of invariants of the Poisson automorphism is the coordinate ring B of a surface in F-3 and the ring of invariants S-theta of the reversing automorphism is a deformation of B and is a factor of a deformation of F[x(1), x(2), x(3)] for a Poisson bracket determined by the appropriate surface

Truncated states obtained by iteration

Quantum states of the electromagnetic field are of considerable importance, finding potential application in various areas of physics, as diverse as solid state physics, quantum communication and cosmology. In this paper we introduce the concept of truncated states obtained via iterative processes (TSI) and study its statistical features, making an analogy with dynamical systems theory (DST). As a specific example, we have studied TSI for the doubling and the logistic functions, which are standard functions in studying chaos. TSI for both the doubling and logistic functions exhibit certain similar patterns when their statistical features are compared from the point of view of DST. A general method to engineer TSI in the running-wave domain is employed, which includes the errors due to the nonidealities of detectors and photocounts.Comment: 10 pages, 22 figure

Nambu-Hamiltonian flows associated with discrete maps

For a differentiable map $(x_1,x_2,..., x_n)\to (X_1,X_2,..., X_n)$ that has an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of the initial value, say $x_n$, of the map plays the role of time variable while the others remain fixed. We present various examples which exhibit the map-flow correspondence.Comment: 19 page

Discrete Dynamical Systems Embedded in Cantor Sets

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with $N$ variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit $N\to\infty$. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error-profile. We made explicit calculations both numerical and analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running top to bottom in figures, to appear in J. Math. Phy

Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua

We compare and contrast two types of deformations inspired by mixing applications -- one from the mixing of fluids (stretching and folding), the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equivalence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker's map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map's Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponentially fast when there is stretching and folding, but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The following article appeared in the American Journal of Physics and may be found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright 2011 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the AAP

Statistical properties of Lorenz like flows, recent developments and perspectives

We comment on mathematical results about the statistical behavior of Lorenz equations an its attractor, and more generally to the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statisitcal behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated to the existence of physical measures: \emph{in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure}. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial conditions, physical measure, singular-hyperbolicity, expansiveness, robust attractor, robust chaotic flow, positive Lyapunov exponent, large deviations, hitting and recurrence times. Minor typos corrected and precise acknowledgments of financial support added. To appear in Int J of Bif and Chaos in App Sciences and Engineerin

Renormalization Group Functional Equations

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling {\sigma} functions, and lead to exact functional relations for the local flow {\beta} functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of {\sigma} are sometimes not true fixed points under continuous changes in scale, and zeroes of {\beta} do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.Comment: A physical model with a limit cycle added as section IV, along with reference

A repurposing strategy for Hsp90 inhibitors demonstrates their potency against filarial nematodes

Novel drugs are required for the elimination of infections caused by filarial worms, as most commonly used drugs largely target the microfilariae or first stage larvae of these infections. Previous studies, conducted in vitro, have shown that inhibition of Hsp90 kills adult Brugia pahangi. As numerous small molecule inhibitors of Hsp90 have been developed for use in cancer chemotherapy, we tested the activity of several novel Hsp90 inhibitors in a fluorescence polarization assay and against microfilariae and adult worms of Brugia in vitro. The results from all three assays correlated reasonably well and one particular compound, NVP-AUY922, was shown to be particularly active, inhibiting Mf output from female worms at concentrations as low as 5.0 nanomolar after 6 days exposure to drug. NVP-AUY922 was also active on adult worms after a short 24 h exposure to drug. Based on these in vitro data, NVP-AUY922 was tested in vivo in a mouse model and was shown to significantly reduce the recovery of both adult worms and microfilariae. These studies provide proof of principle that the repurposing of currently available Hsp90 inhibitors may have potential for the development of novel agents with macrofilaricidal properties

Cryptographic requirements for chaotic secure communications

In recent years, a great amount of secure communications systems based on chaotic synchronization have been published. Most of the proposed schemes fail to explain a number of features of fundamental importance to all cryptosystems, such as key definition, characterization, and generation. As a consequence, the proposed ciphers are difficult to realize in practice with a reasonable degree of security. Likewise, they are seldom accompanied by a security analysis. Thus, it is hard for the reader to have a hint about their security. In this work we provide a set of guidelines that every new cryptosystems would benefit from adhering to. The proposed guidelines address these two main gaps, i.e., correct key management and security analysis, to help new cryptosystems be presented in a more rigorous cryptographic way. Also some recommendations are offered regarding some practical aspects of communications, such as channel noise, limited bandwith, and attenuation.Comment: 13 pages, 3 figure

Invariant varieties of periodic points for some higher dimensional integrable maps

By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}Comment: 24 page