Some open questions in "wave chaos"

The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent

Extended metal-organic solids based on benzenepolycarboxylic and aminobenzoic acids

This article describes the recent results obtained in our laboratory on the interaction of polyfunctional ligands with divalent alkaline earth metal ions and a few divalent transition metal ions. Treatment of MC12·nH2O (M = Mg, Ca, Sr or Ba) with 2-amino benzoic acid leads to the formation of complexes [Mg(2-aba)2] (1), [Ca(2-aba)2(OH2)3]∞ (2), [{Sr(2-aba)2(OH2)2}2·H2O)]∞ (3), [Ba(2-aba)2(OH2)]∞ (4), respectively. While the calcium ions in 2 are hepta-coordinated, the strontium and barium ions in 3 and 4 reveal a coordination number of nine apart from additional metal-metal interactions. Apart from the carboxylate functionality, the amino group also binds to the metal centres in the case of strontium and barium complexes 3 and 4. Complexes [{Mg(H2O)6}(4-aba)2·2H2O] (5), [Ca(4-aba)2(H2O)2] (6) prepared from 4-aminobenzoic acid reveal more open or layered structures. Interaction of 2-mercaptobenzoic acid with MCl2·6H2O (M = Mg, Ca), however, leads to the oxidation of the thiol group resulting in the disulphide 2,2' -dithiobis(benzoic acid). New metal-organic framework based hydrogen-bonded porous solids [{M(btec) (OH2)4}n·n(C4H12N2)·4nH2O] (btec = 1,2,4,5-benzene tetracarboxylate) (M = Co9; Ni10; Zn11) have been synthesized from 1,2,4,5-benzene tetracarboxylic acid in the presence of piperazine. These compounds are made up of extensively hydrogen-bonded alternating layers of anionic M-btec co-ordination polymer and piperazinium cations. Compounds 2- 11 described herein form polymeric networks in the solid-state with the aid of different coordinating capabilities of the carboxylate anions hydrogen bonding interactions

Experiences with Mycobacterium leprae soluble antigens in a leprosy endemic population

Rees and Convit antigens prepared from armadillo-derived Mycobacterium leprae were used for skin testing in two leprosy endemic villages to understand their use in the epidemiology of leprosy. In all, 2602 individuals comprising 202 patients with leprosy detected in a prevalence survey, 476 household contacts and 1924 persons residing in non-case households were tested with two antigens. There was a strong and positive correlation ( r = 0.85) between reactions to the Rees and Convit antigens. The distribution of reactions was bimodal and considering reactions of 12 mm or more as ‘positive’, the positivity rate steeply increased with the increase in age. However. the distributions of reactions to these antigens in patients with leprosy. their household contacts and persons living in non-case households were very similar. These results indicate that Rees and Convit antigens are not useful in the identification of M. leprae infection or in the confirmation of leprosy diagnosis in a leprosy endemic population with a high prevalence of nonspecific sensitivity

Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact

Synthesis of surfactant free stable nanofluids based on barium hexaferrite by pulsed laser ablation in liquid

Barium hexaferrite nanofluids based on five different solvents have been prepared by employing Pulsed Laser Ablation in Liquid (PLAL) at two different wavelengths of 532 nm and 1064 nm. They were then characterized using Transmission Electron Microscopy (TEM), Scanning Electron Microscopy (SEM), X-ray Photoelectron Spectroscopy (XPS), UV-Vis spectroscopy, and Vibrating Sample Magnetometry (VSM). The chemical states of the ablated nanoparticles were identified from XPS analysis and found to be matching with that of the target. The crystallinity of the nanoparticles were confirmed from high resolution TEM (HRTEM) images and SAED patterns. It is found that different liquid environments lead to the formation of barium ferrite nanoparticles with different particle diameters. The plausible mechanism involved in this process is discussed. This study can pave way for the synthesis of stable magnetic nanofluids of permanent magnets. Further, this technique could be utilized for tailoring the morphology of nanoparticles with a judicious choice of the solvents and other ablation parameter

Anatomy of quantum chaotic eigenstates

The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The levels of description are macroscopic (one wants to understand the quantum averages of smooth observables), and microscopic (one wants informations on maxima of eigenfunctions, "scars" of periodic orbits, structure of the nodal sets and domains, local correlations), and often focusses on statistical results. Various models of "random wavefunctions" have been introduced to understand these statistical properties, with usually good agreement with the numerical data. We also discuss some specific systems (like arithmetic ones) which depart from these random models.Comment: Corrected typos, added a few references and updated some result

Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map

We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving for many states of the quantum baker's map. These new transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal) symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title; corrected minor error

Hypersensitivity and chaos signatures in the quantum baker's maps

Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.Comment: 32 pages, 6 figure