On the denominators of the Taylor coefficients of G-functions

Let $\sum\_{n=0}^\infty a\_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $\delta\_n\ge 1$ denote the least integer such that $\delta\_n a\_0, \delta\_n a\_1, ..., \delta\_n a\_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ such that $\delta\_n\le c^{n+1}$ for all $n\ge 0$. In practice, it is observed that $\delta\_n$ always divides $D\_{bn}^{s} C^{n+1}$ where $D\_n=lcm\{1,2, ..., n\}$, $b, C$ are positive integers and $s\ge 0$ is an integer. We prove that this observation holds for any $G$-function provided the following conjecture is assumed: {\em Let $\mathbb{K}$ be a number field, and $L\in \mathbb{K}[z,\frac{d }{d z}]$ be a $G$-operator; then the generic radius of solvability $R\_v(L)$ is equal to 1, for all finite places $v$ of $\mathbb{K}$ except a finite number.} The proof makes use of very precise estimates in the theory of $p$-adic differential equations, in particular the Christol-Dwork Theorem. Our result becomes unconditional when $L$ is a geometric differential operator, a special type of $G$-operators for which the conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts that any $G$-operator is of geometric type, hence it implies the above conjecture

Absorption cross section in de Sitter space

We study the wave equation for a minimally coupled massive scalar in three-dimensional de Sitter space. We compute the absorption cross section to investigate its cosmological horizon in the southern diamond. Although the absorption cross section is not defined exactly, we can be determined it from the fact that the low-energy $s(j=0)$-wave absorption cross section for a massless scalar is given by the area of the cosmological horizon. On the other hand, the low-temperature limit of $j\not=0$-mode absorption cross section is useful for extracting information surrounding the cosmological horizon. Finally we mention a computation of the absorption cross section on the CFT-side using the dS/CFT correspondence.Comment: 13 pages, version to appear in MPL

Dark Radiation Emerging After Big Bang Nucleosynthesis?

We show how recent data from observations of the cosmic microwave background may suggest the presence of additional radiation density which appeared after big bang nucleosynthesis. We propose a general scheme by which this radiation could be produced from the decay of non-relativistic matter, we place constraints on the properties of such matter, and we give specific examples of scenarios in which this general scheme may be realized.Comment: v3: 5 pages, 1 figure. References added, typos corrected, notation changed throughout. v2: 5 pages, 1 figure. Reformatted, references added, acknowledgments updated, effect of radiation on CMB clarified. v1: 11 pages, 1 figur

On Siegel's problem for E-functions

In this new version, a similar problem for G-functions is considered in Section 6.Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of confluent hypergeometric series with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively "small" ring H generated by algebraic numbers, $1/\pi$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove that the coefficients of the asymptotic expansion of a confluent hypergeometric series with rational parameters are in H. Finally, we prove a similar result for G-functions

Towards Semantic Fast-Forward and Stabilized Egocentric Videos

The emergence of low-cost personal mobiles devices and wearable cameras and the increasing storage capacity of video-sharing websites have pushed forward a growing interest towards first-person videos. Since most of the recorded videos compose long-running streams with unedited content, they are tedious and unpleasant to watch. The fast-forward state-of-the-art methods are facing challenges of balancing the smoothness of the video and the emphasis in the relevant frames given a speed-up rate. In this work, we present a methodology capable of summarizing and stabilizing egocentric videos by extracting the semantic information from the frames. This paper also describes a dataset collection with several semantically labeled videos and introduces a new smoothness evaluation metric for egocentric videos that is used to test our method.Comment: Accepted for publication and presented in the First International Workshop on Egocentric Perception, Interaction and Computing at European Conference on Computer Vision (EPIC@ECCV) 201

Effective Potential on Fuzzy Sphere

The effective potential of quantized scalar field on fuzzy sphere is evaluated to the two-loop level. We see that one-loop potential behaves like that in the commutative sphere and the Coleman-Weinberg mechanism of the radiatively symmetry breaking could be also shown in the fuzzy sphere system. In the two-loop level, we use the heavy-mass approximation and the high-temperature approximation to perform the evaluations. The results show that both of the planar and nonplanar Feynman diagrams have inclinations to restore the symmetry breaking in the tree level. However, the contributions from planar diagrams will dominate over those from nonplanar diagrams by a factor N^2. Thus, at heavy-mass limit or high-temperature system the quantum field on the fuzzy sphere will behave like those on the commutative sphere. We also see that there is a drastic reduction of the degrees of freedom in the nonplanar diagrams when the particle wavelength is smaller than the noncommutativity scale.Comment: Latex 18 pages, some typos correcte

Scattering of Closed String States from a Quantized D-Particle

By developing an appropriate path-integral formalism, we compute, in bosonic string theory, the disk amplitude for the scattering of closed string states from a D-particle, in which the collective coordinate of the D-particle is fully quantized. As a consequence, the recoil of the D-particle is naturally taken into account. Our result can be readily factorized in the closed string channel to yield the boundary state describing the recoiling D-particle. This turned out to agree with the BRST invariant vertex recently proposed by Ishibashi to the leading order in the derivative expansion, but it will receive corrections in subsequent orders. The advantage of our formalism is that it is extendable to deal with more general processes involving multiple D-particles. A viewpoint regarding our work as describing a dynamical transition of CFT's is also discussed.Comment: 24 pages, LaTeX, no figures. Improvements are made on explanations of the approximation scheme and the handling of the divergenc

The String Coupling Accelerates the Expansion of the Universe

Generic cosmological models in non-critical string theory have a time-dependent dilaton background at a late epoch. The cosmological deceleration parameter Q_0 is given by the square of the string coupling, g_s^2, up to a negative sign. Hence the expansion of the Universe must accelerate eventually, and the observed value of Q_0 coresponds to g_s^2 ~ 0.6. In this scenario, the string coupling is asymptotically free at large times, but its present rate of change is imperceptibly small.Comment: 7 page