Six-point conformal blocks in the snowflake channel

We compute d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3F2-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

Numerical analysis of the indoor climate in a Zenith-type greenhouse in the Valle de Santiago region, Guanajuato

The objective of this project was to carry out an analysis of the indoor climate of the greenhouse located in the Valle de Santiago region, Guanajuato. It is a zenith-type greenhouse with two wings with a symmetrical face. An orthogonal mesh was made of 50 nodes (25 nodes for each height) Total Den located in the cultivable area and taking the value at the midpoint of each rectangle considering two heights 0.25m and 1.30m with respect to the ground. Humidity and temperature readings were taken in each of the nodes for three weeks and subsequently a data analysis was made and a comparison with the data collected in the different situations; also the temperature was analyzed with the double sum of Riemann and the rule of the middle point. In conclusion, it was determined that the greenhouse yields heat on warm days, while it receives heat on cold days. This behavior coincides with previous studies; however, it occurs that with the hydroponic method there is a greater growth of the crop.

A triple integral analog of a multiple zeta value

We establish the triple integral evaluation [Formula: see text] as well as the equivalent polylogarithmic double sum [Formula: see text] This double sum is related to, but less approachable than, similar sums studied by Ramanujan. It is also reminiscent of Euler’s formula [Formula: see text], which is the simplest instance of duality of multiple polylogarithms. We review this duality and apply it to derive a companion identity. We also discuss approaches based on computer algebra. All of our approaches ultimately require the introduction of polylogarithms and nontrivial relations between them. It remains an open challenge to relate the triple integral or the double sum to [Formula: see text] directly.

Asymptotics of certain double series in Eulerian form

We examine the asymptotic behavior of [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are rational numbers satisfying some conditions while [Formula: see text] is not necessarily positive-definite. As an application, we obtain non-modularities of certain double-sum Eulerian series.

Positivity and recursion formula of the linearization coefficients of Bessel polynomials

Positivity of the linearization coefficients for Bessel polynomials is proved in a more general case. The proof is based not only on a recursion formula (a formula similar to one given by Berg and Vignat) but also on giving an explicit triple sum formula. Moreover, this triple sum is simplified and a double sum formula for these linearization coefficients is given. In two general cases, this formula reduces indeed to either Atia and Zeng’s formula (M. J. Atia and J. Zeng, An explicit formula for the linearization coefficients of Bessel polynomials, Ramanujan J. 28(2) (2012) 211–221, doi: 10.1007/s11139-011-9348-4) or Berg and Vignat’s formulas in their proof of the positivity results about these coefficients (C. Berg and C. Vignat, Linearization coefficients of Bessel polynomials and properties of student [Formula: see text]-distributions, Constr. Approx. 27 (2008) 15–32).

Remarks on Pickands’ theorem

REMARKS ON PICKANDS’ THEOREMIn this article we present the Pickands theorem and his double sum method. We follow Piterbarg’s proof of this theorem. Since his proof relies on general lemmas, we present a complete proof of Pickands’ theorem using the Borell inequality and Slepian lemma. The original Pickands’ proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant. Moreover, we review equivalent definitions, simulations and bounds of Pickands constant.