A hierarchical structure of transformation semigroups with applications to probability limit measures

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels. This kernel hierarchy produces a set of tools that provides direct access to computations of interest in probability limit theorems; in particular, finding certain factors of idempotent limit measures. In addition, when considering transformation semigroups that arise naturally from edge colorings of directed graphs, as in the road-coloring problem, the hierarchy produces simple techniques to determine the rank of the kernel and to decide when a given kernel is a right group. In particular, it is shown that all kernels of rank one less than the number of vertices must be right groups and their structure for the case of two generators is described.Comment: 35 pages, 4 figure

Cooling ultracold bosons in optical lattices by spectral transform

It is shown theoretically how to directly obtain the energy distribution of a weakly interacting gas of bosons confined in an optical lattice in the tight-binding limit. This is accomplished by adding a linear potential to a suitably prepared lattice, and allowing the gas to evolve under the influence of the total potential. After a prescribed time, a spectral transform is effected where each (highly non-local) energy state is transformed into a distinct site of the lattice, thus allowing the energy distribution to be (non-destructively) imaged in real space. Evolving for twice the time returns the atoms to their initial state. The results suggest efficient methods to both measure the temperature in situ, as well as to cool atoms within the lattice: after applying the spectral transform one simply needs to remove atoms from all but a few lattice sites. Using exact numerical calculations, the effects of interactions and errors in the application of the lattice are examined.Comment: 13+e pages, two embedded figures, revte

Multivariate Matching Polynomials of Cyclically Labelled Graphs

We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of t labels {x1, . . ., xt}. We first work with the cyclically labelled path, with first edge label xi, followed by N full cycles of labels {x1, . . ., xt}, and last edge label xj . Let Φi,Nt+j denote the matching polynomial of this path. It satisfies the (τ, Δ)-recurrence: Φi,Nt+j = τΦi,(N−1)t+j−ΔΦi,(N−2)t+j, where τ is the sum of all non-consecutive cyclic monomials in the variables {x1, . . ., xt} and Δ = (−1)t x1 · · ·xt. A combinatorial/algebraic proof and a matrix proof of this fact are given. Let GN denote the first fundamental solution to the (τ, Δ)-recurrence. We express GN (i) as a cyclic binomial using the Symmetric Representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and Δ, and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees

A Combinatorial Interpretation of Lommel Polynomials and Their Derivatives

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis

Fock representation of the renormalized higher powers of white noise and the Virasoro--Zamolodchikov--w∞∗w_{\infty} *--Lie algebra

The identification of the $*$--Lie algebra of the renormalized higher powers of white noise (RHPWN) and the analytic continuation of the second quantized Virasoro--Zamolodchikov--$w_{\infty} *$--Lie algebra of conformal field theory and high-energy physics, was recently established in \cite{id} based on results obtained in [1] and [2]. In the present paper we show how the RHPWN Fock kernels must be truncated in order to be positive definite and we obtain a Fock representation of the two algebras. We show that the truncated renormalized higher powers of white noise (TRHPWN) Fock spaces of order $\geq 2$ host the continuous binomial and beta processes

On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.Comment: 16 pages, latex, no figure

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Symmetries and currents of the ideal and unitary Fermi gases

The maximal algebra of symmetries of the free single-particle Schroedinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schroedinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N) model, a putative bulk dual of the unitary and the ideal Fermi gases is discussed.Comment: 67 pages, 2 figures; references added, minor improvements in the presentation, version accepted for publication in JHE

Sleep study, respiratory mechanics, chemosensitive response and quality of life in morbidly obese patients undergoing bariatric surgery: a prospective, randomized, controlled trial

<p>Abstract</p> <p>Background</p> <p>Obesity is a major public health problem in both developed and developing countries alike and leads to a series of changes in respiratory physiology. There is a strong correlation between obesity and cardiopulmonary sleep disorders. Weight loss among such patients leads to a reduction in these alterations in respiratory physiology, but clinical treatment is not effective for a long period of time. Thus, bariatric surgery is a viable option.</p> <p>Methods/Design</p> <p>The present study involves patients with morbid obesity (BMI of 40 kg/m²or 35 kg/m²to 39.9 kg/m²with comorbidities), candidates for bariatric surgery, screened at the Santa Casa de Misericórdia Hospital in the city of Sao Paulo (Brazil). The inclusion criteria are grade III morbid obesity, an indication for bariatric surgery, agreement to participate in the study and a signed term of informed consent. The exclusion criteria are BMI above 55 kg/m², clinically significant or unstable mental health concerns, an unrealistic postoperative target weight and/or unrealistic expectations of surgical treatment. Bariatric surgery candidates who meet the inclusion criteria will be referred to Santa Casa de Misericórdia Hospital and will be reviewed again 30, 90 and 360 days following surgery. Data collection will involve patient records, personal data collection, objective assessment of HR, BP, neck circumference, chest and abdomen, collection and analysis of clinical preoperative findings, polysomnography, pulmonary function test and a questionnaire on sleepiness.</p> <p>Discussion</p> <p>This paper describes a randomised controlled trial of morbidly obese patients. Polysomnography, respiratory mechanics, chemosensitive response and quality of life will be assessed in patients undergoing or not undergoing bariatric surgery.</p> <p>Trial Registration</p> <p>The protocol for this study is registered with the Brazilian Registry of Clinical Trials - ReBEC (RBR-9k9hhv).</p