Geometric phases in quantum control disturbed by classical stochastic processes

We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as for example classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitely built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally we treat a very simple example of a two level atom driven by a phase modulated laser field with a phase instability described by a gaussian white noise. In particular we compute the average geometric phase issued from the noise

Exotic Smoothness and Physics

The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, ${\bf R^4}$, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures available for space-time models. These can be thought of as source of new global, but not properly topological, features. This paper reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., non-standard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories.Comment: 13 pages, LaTe

Natural extensions and entropy of α\alpha-continued fractions

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$

Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent

One dimensional intermittent maps with stretched exponential separation of nearby trajectories are considered. When time goes infinity the standard Lyapunov exponent is zero. We investigate the distribution of $\lambda_{\alpha}= \sum_{i=0}^{t-1} \ln \left| M'(x_i) \right|/t^{\alpha}$, where $\alpha$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of $\lambda_{\alpha}$ is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that \alpha \left is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that \left and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.Comment: 12 pages, 10 figure

Pesin-type relation for subexponential instability

We address here the problem of extending the Pesin relation among positive Lyapunov exponents and the Kolmogorov-Sinai entropy to the case of dynamical systems exhibiting subexponential instabilities. By using a recent rigorous result due to Zweim\"uller, we show that the usual Pesin relation can be extended straightforwardly for weakly chaotic one-dimensional systems of the Pomeau-Manneville type, provided one introduces a convenient subexponential generalization of the Kolmogorov-Sinai entropy. We show, furthermore, that Zweim\"uller's result provides an efficient prescription for the evaluation of the algorithm complexity for such systems. Our results are confirmed by exhaustive numerical simulations. We also point out and correct a misleading extension of the Pesin relation based on the Krengel entropy that has appeared recently in the literature.Comment: 10 pages, 4 figures. Final version to appear in Journal of Statistical Mechanics (JSTAT

The entropy of alpha-continued fractions: numerical results

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.Comment: 33 pages, 14 figure

Exotic Differentiable Structures and General Relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe

The Standard Model with gravity couplings

In this paper, we examine the coupling of matter fields to gravity within the framework of the Standard Model of particle physics. The coupling is described in terms of Weyl fermions of a definite chirality, and employs only (anti)self-dual or left-handed spin connection fields. It is known from the work of Ashtekar and others that such fields can furnish a complete description of gravity without matter. We show that conditions ensuring the cancellation of perturbative chiral gauge anomalies are not disturbed. We also explore a global anomaly associated with the theory, and argue that its removal requires that the number of fundamental fermions in the theory must be multiples of 16. In addition, we investigate the behavior of the theory under discrete transformations P, C and T; and discuss possible violations of these discrete symmetries, including CPT, in the presence of instantons and the Adler-Bell-Jackiw anomaly.Comment: Extended, and replaced with LaTex file. 25 Page

Inactivation of promoter 1B of APC causes partial gene silencing: evidence for a significant role of the promoter in regulation and causative of familial adenomatous polyposis

Familial adenomatous polyposis (FAP) is caused by germline mutations in the adenomatous polyposis coli (APC) gene. Two promoters, 1A and 1B, have been recognized in APC, and 1B is thought to have a minor role in the regulation of the gene. We have identified a novel deletion encompassing half of this promoter in the largest family (Family 1) of the Swedish Polyposis Registry. The mutation leads to an imbalance in allele-specific expression of APC, and transcription from promoter 1B was highly impaired in both normal colorectal mucosa and blood from mutation carriers. To establish the significance of promoter 1B in normal colorectal mucosa (from controls), expression levels of specific transcripts from each of the promoters, 1A and 1B, were examined, and the expression from 1B was significantly higher compared with 1A. Significant amounts of transcripts generated from promoter 1B were also determined in a panel of 20 various normal tissues examined. In FAP-related tumors, the APC germline mutation is proposed to dictate the second hit. Mutations leaving two or three out of seven 20-amino-acid repeats in the central domain of APC intact seem to be required for tumorigenesis. We examined adenomas from mutation carriers in Family 1 for second hits in the entire gene without any findings, however, loss of the residual expression of the deleterious allele was observed. Three major conclusions of significant importance in relation to the function of APC can be drawn from this study; (i) germline inactivation of promoter 1B is disease causing in FAP; (ii) expression of transcripts from promoter 1B is generated at considerable higher levels compared with 1A, demonstrating a hitherto unknown importance of 1B; (iii) adenoma formation in FAP, caused by impaired function of promoter 1B, does not require homozygous inactivation of APC allowing for alternative genetic models as basis for adenoma formation

Increased Risk of Temporomandibular Joint Closed Lock: A Case-Control Study of ANKH Polymorphisms

Objectives: This study aimed to carry out a histological examination of the temporomandibular joint (TMJ) in ank mutant mice and to identify polymorphisms of the human ANKH gene in order to establish the relationship between the type of temporomandibular disorders (TMD) and ANKH polymorphisms.\ud \ud Materials and Methods: Specimens from the TMJ of ank mutant and wild-type mice were inspected with a haematoxylin and eosin staining method. A sample of 55 TMD patients were selected. Each was examined with standard clinical procedures and genotyping techniques.\ud \ud Results: The major histological finding in ank mutant mice was joint space narrowing. Within TMD patients, closed lock was more prevalent among ANKH-OR homozygotes (p = 0.011, OR = 7.7, 95% CI 1.6–36.5) and the elder (p = 0.005, OR = 2.4, 95% CI 1.3–4.3).\ud \ud Conclusions: Fibrous ankylosis was identified in the TMJ of ank mutant mice. In the human sample, ANKH-OR polymorphism was found to be a genetic marker associated with TMJ closed lock. Future investigations correlating genetic polymorphism to TMD are indicated