Charles Leonard Bouton

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The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature

The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these `curved' harmonic oscillators simultaneously on any such configuration space, using a Cayley-Klein (CK) type approach, with two free parameters \ki, \kii which altogether correspond to the possible values for curvature and signature type: the generic Riemannian and Lorentzian spaces of constant curvature (sphere ${\bf S}^2$, hyperbolic plane ${\bf H}^2$, AntiDeSitter sphere {\bf AdS}^{\unomasuno} and DeSitter sphere {\bf dS}^{\unomasuno}) appear in this family, with the Euclidean and Minkowski spaces as flat limits. We solve the equations of motion for the `curved' harmonic oscillator and obtain explicit expressions for the orbits by using three different methods: first by direct integration, second by obtaining the general CK version of the Binet's equation and third, as a consequence of its superintegrable character. The orbits are conics with centre at the potential origin in any CK space, thereby extending this well known Euclidean property to any constant curvature configuration space. The final part of the article, that has a more geometric character, presents those results of the theory of conics on spaces of constant curvature which are pertinent.Comment: 29 pages, 6 figure

Counting and computing regions of DD-decomposition: algebro-geometric approach

New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying $D$-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure

Geometric descriptions of entangled states by auxiliaries varieties

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting we describe well-known classifications of multipartite entanglement such as $2\times 2\times(n+1)$, for $n\geq 1$, quantum systems and a new example with the $2\times 3\times 3$ quantum system. Our description completes the approach of Miyake and makes stronger connections with recent work of algebraic geometers. Moreover for the quantum systems detailed in this paper we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.Comment: 32 pages, 15 Tables, 5 Figures. References and remarks adde

Interacting Preformed Cooper Pairs in Resonant Fermi Gases

We consider the normal phase of a strongly interacting Fermi gas, which can have either an equal or an unequal number of atoms in its two accessible spin states. Due to the unitarity-limited attractive interaction between particles with different spin, noncondensed Cooper pairs are formed. The starting point in treating preformed pairs is the Nozi\`{e}res-Schmitt-Rink (NSR) theory, which approximates the pairs as being noninteracting. Here, we consider the effects of the interactions between the Cooper pairs in a Wilsonian renormalization-group scheme. Starting from the exact bosonic action for the pairs, we calculate the Cooper-pair self-energy by combining the NSR formalism with the Wilsonian approach. We compare our findings with the recent experiments by Harikoshi {\it et al.} [Science {\bf 327}, 442 (2010)] and Nascimb\`{e}ne {\it et al.} [Nature {\bf 463}, 1057 (2010)], and find very good agreement. We also make predictions for the population-imbalanced case, that can be tested in experiments.Comment: 10 pages, 6 figures, accepted version for PRA, discussion of the imbalanced Fermi gas added, new figure and references adde

Additional psychometric data for the Spanish Modified Dental Anxiety Scale, and psychometric data for a Spanish version of the Revised Dental Beliefs Survey

<p>Abstract</p> <p>Background</p> <p>Hispanics comprise the largest ethnic minority group in the United States. Previous work with the Spanish Modified Dental Anxiety Scale (MDAS) yielded good validity, but lower test-retest reliability. We report the performance of the Spanish MDAS in a new sample, as well as the performance of the Spanish Revised Dental Beliefs Survey (R-DBS).</p> <p>Methods</p> <p>One hundred sixty two Spanish-speaking adults attending Spanish-language church services or an Hispanic cultural festival completed questionnaires containing the Spanish MDAS, Spanish R-DBS, and dental attendance questions, and underwent a brief oral examination. Church attendees completed the questionnaire a second time, for test-retest purposes.</p> <p>Results</p> <p>The Spanish MDAS and R-DBS were completed by 156 and 136 adults, respectively. The test-retest reliability of the Spanish MDAS was 0.83 (95% CI = 0.60-0.92). The internal reliability of the Spanish R-DBS was 0.96 (95% CI = 0.94-0.97), and the test-retest reliability was 0.86 (95% CI = 0.64-0.94). The two measures were significantly correlated (Spearman's rho = 0.38, p < 0.001). Participants who do not currently go to a dentist had significantly higher MDAS scores (t = 3.40, df = 106, p = 0.003) as well as significantly higher R-DBS scores (t = 2.21, df = 131, p = 0.029). Participants whose most recent dental visit was for pain or a problem, rather than for a check-up, scored significantly higher on both the MDAS (t = 3.00, df = 106, p = 0.003) and the R-DBS (t = 2.85, df = 92, p = 0.005). Those with high dental fear (MDAS score 19 or greater) were significantly more likely to have severe caries (Chi square = 6.644, df = 2, p = 0.036). Higher scores on the R-DBS were significantly related to having more missing teeth (Spearman's rho = 0.23, p = 0.009).</p> <p>Conclusion</p> <p>In this sample, the test-retest reliability of the Spanish MDAS was higher. The significant relationships between dental attendance and questionnaire scores, as well as the difference in caries severity seen in those with high fear, add to the evidence of this scale's construct validity in Hispanic samples. Our results also provide evidence for the internal and test-retest reliabilities, as well as the construct validity, of the Spanish R-DBS.</p

Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.Comment: 46 pages, LaTe

Vector Continued Fractions using a Generalised Inverse

A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically.Comment: Published form - minor change

Visuospatial Integration: Paleoanthropological and Archaeological Perspectives

The visuospatial system integrates inner and outer functional processes, organizing spatial, temporal, and social interactions between the brain, body, and environment. These processes involve sensorimotor networks like the eye–hand circuit, which is especially important to primates, given their reliance on vision and touch as primary sensory modalities and the use of the hands in social and environmental interactions. At the same time, visuospatial cognition is intimately connected with memory, self-awareness, and simulation capacity. In the present article, we review issues associated with investigating visuospatial integration in extinct human groups through the use of anatomical and behavioral data gleaned from the paleontological and archaeological records. In modern humans, paleoneurological analyses have demonstrated noticeable and unique morphological changes in the parietal cortex, a region crucial to visuospatial management. Archaeological data provides information on hand–tool interaction, the spatial behavior of past populations, and their interaction with the environment. Visuospatial integration may represent a critical bridge between extended cognition, self-awareness, and social perception. As such, visuospatial functions are relevant to the hypothesis that human evolution is characterized by changes in brain–body–environment interactions and relations, which enhance integration between internal and external cognitive components through neural plasticity and the development of a specialized embodiment capacity. We therefore advocate the investigation of visuospatial functions in past populations through the paleoneurological study of anatomical elements and archaeological analysis of visuospatial behaviors