Quasi-Homogeneous Thermodynamics and Black Holes

We propose a generalized thermodynamics in which quasi-homogeneity of the thermodynamic potentials plays a fundamental role. This thermodynamic formalism arises from a generalization of the approach presented in paper [1], and it is based on the requirement that quasi-homogeneity is a non-trivial symmetry for the Pfaffian form $\delta Q_{rev}$. It is shown that quasi-homogeneous thermodynamics fits the thermodynamic features of at least some self-gravitating systems. We analyze how quasi-homogeneous thermodynamics is suggested by black hole thermodynamics. Then, some existing results involving self-gravitating systems are also shortly discussed in the light of this thermodynamic framework. The consequences of the lack of extensivity are also recalled. We show that generalized Gibbs-Duhem equations arise as a consequence of quasi-homogeneity of the thermodynamic potentials. An heuristic link between this generalized thermodynamic formalism and the thermodynamic limit is also discussed.Comment: 39 pages, uses RevteX. Published version (minor changes w.r.t. the original one

Treating some solid state problems with the Dirac equation

The ambiguity involved in the definition of effective-mass Hamiltonians for nonrelativistic models is resolved using the Dirac equation. The multistep approximation is extended for relativistic cases allowing the treatment of arbitrary potential and effective-mass profiles without ordering problems. On the other hand, if the Schrodinger equation is supposed to be used, our relativistic approach demonstrate that both results are coincidents if the BenDaniel and Duke prescription for the kinetic-energy operator is implemented. Applications for semiconductor heterostructures are discussed.Comment: 06 pages, 5 figure

Universal restrictions to the conversion of heat into work derived from the analysis of the Nernst theorem as a uniform limit

We revisit the relationship between the Nernst theorem and the Kelvin-Planck statement of the second law. We propose that the exchange of entropy uniformly vanishes as the temperature goes to zero. The analysis of this assumption shows that is equivalent to the fact that the compensation of a Carnot engine scales with the absorbed heat so that the Nernst theorem should be embedded in the statement of the second law. ----- Se analiza la relaci{\'o}n entre el teorema de Nernst y el enunciado de Kelvin-Planck del segundo principio de la termodin{\'a}mica. Se{\~n}alamos el hecho de que el cambio de entrop{\'\i}a tiende uniformemente a cero cuando la temperatura tiende a cero. El an{\'a}lisis de esta hip{\'o}tesis muestra que es equivalente al hecho de que la compensaci{\'o}n de una m{\'a}quina de Carnot escala con el calor absorbido del foco caliente, de forma que el teorema de Nernst puede derivarse del enunciado del segundo principio.Comment: 8pp, 4 ff. Original in english. Also available translation into spanish. Twocolumn format. RevTe

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

Discovering New Physics in the Decays of Black Holes

If the scale of quantum gravity is near a TeV, the LHC will be producing one black hole (BH) about every second, thus qualifying as a BH factory. With the Hawking temperature of a few hundred GeV, these rapidly evaporating BHs may produce new, undiscovered particles with masses ~100 GeV. The probability of producing a heavy particle in the decay depends on its mass only weakly, in contrast with the exponentially suppressed direct production. Furthemore, BH decays with at least one prompt charged lepton or photon correspond to the final states with low background. Using the Higgs boson as an example, we show that it may be found at the LHC on the first day of its operation, even with incomplete detectors.Comment: 4 pages, 3 figure

Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator

The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the totally antisymmetrised n-th powers up to n=9, identifying (see Tables 3 and 6) families of representations with integer eigenvalues 5,...,9 for the quadratic Casimir operator, in each case providing a formula (see eq. (11) to (15)) for the dimensions of the representations in the family as a function of D=dim g. This generalises previous results for powers j and Casimir eigenvalues j, j<=4. Many intriguing, perhaps puzzling, features of the dimension formulas are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered.Comment: 16 pages, LaTeX, 1 figure, 9 tables; v2: presentation improved, typos correcte

Thermodynamic fluctuation relation for temperature and energy

The present work extends the well-known thermodynamic relation $C=\beta ^{2}< \delta {E^{2}}>$ for the canonical ensemble. We start from the general situation of the thermodynamic equilibrium between a large but finite system of interest and a generalized thermostat, which we define in the course of the paper. The resulting identity $=1+< \delta {E^{2}}> \partial ^{2}S(E) /\partial {E^{2}}$ can account for thermodynamic states with a negative heat capacity $C<0$; at the same time, it represents a thermodynamic fluctuation relation that imposes some restrictions on the determination of the microcanonical caloric curve $\beta (E) =\partial S(E) /\partial E$. Finally, we comment briefly on the implications of the present result for the development of new Monte Carlo methods and an apparent analogy with quantum mechanics.Comment: Version accepted for publication in J. Phys. A: Math and The

Is it still worth searching for lepton flavor violation in rare kaon decays?

Prospective searches for lepton flavor violation (LFV) in rare kaon decays at the existing and future intermediate-energy accelerators are considered. The proposed studies are complementary to LFV searches in muon-decay experiments and offer a unique opportunity to probe models with approximately conserved fermion-generation quantum number with sensitivity superior to that in other processes. Consequently, new searches for LFV in kaon decays are an important and independent part of the general program of searches for lepton flavor violation in the final states with charged leptons.Comment: 30 pages, 10 figures. An extended version of the talk given at the Chicago Flavor Seminar, February 27, 2004. In the new version some misprints were corrected and some new data for LFV-processes were added. The main content of the paper was not changed. The paper is published in Yad. Fiz. 68, 1272 (2005

Lines on projective varieties and applications

The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined by quadratic equations the base locus of the projective second fundamental form at a general point coincides, as a scheme, with the variety of lines. The second part concerns the problem of extending embedded projective manifolds, using the geometry of the variety of lines. Some applications to the case of homogeneous manifolds are included.Comment: 15 pages. One example removed; one remark and some references added; typos correcte

Decomposition of homogeneous polynomials with low rank

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset \mathbb{P}^{{m+d\choose d}-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1, ..., M_r$ is $F=M_1^d+... + M_r^d$ with $r>s$. We show that if $s+r\leq 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.Comment: final version. Math. Z. (to appear