Green's functions on finite lattices and their connection to the infinite lattice limit

It is shown that the Green's function on a finite lattice in arbitrary space dimension can be obtained from that of an infinite lattice by means of translation operator. Explicit examples are given for one- and two-dimensional lattices

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

[EN] Parabolic heat equation based on Fourier's theory (FHE), and hyperbolic heat equation (HHE), has been used to mathematically model the temperature distributions of biological tissue during thermal ablation. However, both equations have certain theoretical limitations. The FHE assumes an infinite thermal energy propagation speed, whereas the HHE might possibly be in breach of the second law of thermodynamics. The relativistic heat equation (RHE) is a hyperbolic-like equation, whose theoretical model is based on the theory of relativity and which was designed to overcome these theoretical impediments. In this study, the three heat equations for modelling of thermal ablation of biological tissues (FHE, HHE and RHE) were solved analytically and the temperature distributions compared. We found that RHE temperature values were always lower than those of the FHE, while the HHE values were higher than the FHE, except for the early stages of heating and at points away from the electrode. Although both HHE and RHE are mathematically hyperbolic, peaks were only found in the HHE temperature profiles. The three solutions converged for infinite time or infinite distance from the electrode. The percentage differences between the FHE and the other equations were larger for higher values of thermal relaxation time in HHE.This work received financial support from the Spanish Government (Ministerio de Ciencia e Innovacion, Ref. TEC2011-27133-C02-01).López Molina, JA.; Rivera Ortun, MJ.; Berjano, E. (2014). Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 470:1-16. https://doi.org/10.1098/rspa.2014.0547S11647

Differential Subordinations Involving Generalized Bessel Functions

In this paper our aim is to present some subordination and superordination results, by using an operator, which involves the normalized form of the generalized Bessel functions of first kind. These results are obtained by investigating some appropriate classes of admissible functions. We obtain also some sandwich-type results and we point out various known or new special cases of our main results.Comment: 15 pages, accepted in Bulletin of the Malaysian Mathematical Sciences Societ

The Off-Shell Electromagnetic T-matrix: momentum-dependent scattering from spherical inclusions with both dielectric and magnetic contrast

The momentum- and frequency-dependent T-matrix operator for the scattering of electromagnetic waves by a dielectric/conducting and para- or diamagnetic sphere is derived as a Mie-type series, and presented in a compact form emphasizing various symmetry properties, notably the unitarity identity. This result extends to magnetic properties one previously obtained for purely dielectric contrasts by other authors. Several situations useful to spatially-dispersive effective-medium approximations to one-body order are examined. Partial summation of the Mie series is achieved in the case of elastic scattering.Comment: 22 pages. Preprint of a paper to appear in `Waves in Complex And Random Media' ((c) Taylor and Francis, 2011

Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

Harnack inequality for fractional sub-Laplacians in Carnot groups

In this paper we prove an invariant Harnack inequality on Carnot-Carath\'eodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups

A Repeated Measures Experiment of Green Exercise to Improve Self-Esteem in UK School Children

Exercising in natural, green environments creates greater improvements in adult's self-esteem than exercise undertaken in urban or indoor settings. No comparable data are available for children. The aim of this study was to determine whether so called 'green exercise' affected changes in self-esteem; enjoyment and perceived exertion in children differently to urban exercise. We assessed cardiorespiratory fitness (20 m shuttle-run) and self-reported physical activity (PAQ-A) in 11 and 12 year olds (n = 75). Each pupil completed two 1.5 mile timed runs, one in an urban and another in a rural environment. Trials were completed one week apart during scheduled physical education lessons allocated using a repeated measures design. Self-esteem was measured before and after each trial, ratings of perceived exertion (RPE) and enjoyment were assessed after completing each trial. We found a significant main effect (F (1,74), = 12.2, p<0.001), for the increase in self-esteem following exercise but there was no condition by exercise interaction (F (1,74), = 0.13, p = 0.72). There were no significant differences in perceived exertion or enjoyment between conditions. There was a negative correlation (r = -0.26, p = 0.04) between habitual physical activity and RPE during the control condition, which was not evident in the green exercise condition (r = -0.07, p = 0.55). Contrary to previous studies in adults, green exercise did not produce significantly greater increases in self-esteem than the urban exercise condition. Green exercise was enjoyed more equally by children with differing levels of habitual physical activity and has the potential to engage less active children in exercise. © 2013 Reed et al

Comments on Non-holomorphic Modular Forms and Non-compact Superconformal Field Theories

We extend our previous work arXiv:1012.5721 [hep-th] on the non-compact N=2 SCFT_2 defined as the supersymmetric SL(2,R)/U(1)-gauged WZW model. Starting from path-integral calculations of torus partition functions of both the axial-type (`cigar') and the vector-type (`trumpet') models, we study general models of the Z_M-orbifolds and M-fold covers with an arbitrary integer M. We then extract contributions of the degenerate representations (`discrete characters') in such a way that good modular properties are preserved. The `modular completion' of the extended discrete characters introduced in arXiv:1012.5721 [hep-th] are found to play a central role as suitable building blocks in every model of orbifolds or covering spaces. We further examine a large M-limit (the `continuum limit'), which `deconstructs' the spectral flow orbits while keeping a suitable modular behavior. The discrete part of partition function as well as the elliptic genus is then expanded by the modular completions of irreducible discrete characters, which are parameterized by both continuous and discrete quantum numbers modular transformed in a mixed way. This limit is naturally identified with the universal cover of trumpet model. We finally discuss a classification of general modular invariants based on the modular completions of irreducible characters constructed above.Comment: 1+40 pages, no figure; v2 some points are clarified with respect to the `continuum limit', typos corrected, to appear in JHEP; v3 footnotes added in pages 18, 23 for the relation with arXiv:1407.7721[hep-th

Warped Riemannian metrics for location-scale models

The present paper shows that warped Riemannian metrics, a class of Riemannian metrics which play a prominent role in Riemannian geometry, are also of fundamental importance in information geometry. Precisely, the paper features a new theorem, which states that the Rao-Fisher information metric of any location-scale model, defined on a Riemannian manifold, is a warped Riemannian metric, whenever this model is invariant under the action of some Lie group. This theorem is a valuable tool in finding the expression of the Rao-Fisher information metric of location-scale models defined on high-dimensional Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by only two functions of a single variable, irrespective of the dimension of the underlying Riemannian manifold. Starting from this theorem, several original contributions are made. The expression of the Rao-Fisher information metric of the Riemannian Gaussian model is provided, for the first time in the literature. A generalised definition of the Mahalanobis distance is introduced, which is applicable to any location-scale model defined on a Riemannian manifold. The solution of the geodesic equation is obtained, for any Rao-Fisher information metric defined in terms of warped Riemannian metrics. Finally, using a mixture of analytical and numerical computations, it is shown that the parameter space of the von Mises-Fisher model of $n$-dimensional directional data, when equipped with its Rao-Fisher information metric, becomes a Hadamard manifold, a simply-connected complete Riemannian manifold of negative sectional curvature, for $n = 2,\ldots,8$. Hopefully, in upcoming work, this will be proved for any value of $n$.Comment: first version, before submissio