On polynomials connected to powers of Bessel functions

The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional recurrences. Connections to the umbral formalism on Bessel functions introduced by Cholewinski are established

Generalized Bernoulli numbers and a formula of Lucas

An overlooked formula of E. Lucas for the generalized Bernoulli numbers is proved using generating functions. This is then used to provide a new proof and a new form of a sum involving classical Bernoulli numbers studied by K. Dilcher. The value of this sum is then given in terms of the Meixner-Pollaczek polynomials

Identities for generalized Euler polynomials

For $N \in \mathbb{N}$, let $T_{N}$ be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers $p_{\ell}^{(N)}$, defined as the coefficients in the expansion of $1/T_{N}(1/z)$, are provided. These coefficients give formulas for the classical Euler polynomials in terms of the so-called generalized Euler polynomials. The proofs are based on a probabilistic interpretation of the generalized Euler polynomials recently given by Klebanov et al. Asymptotics of $p_{\ell}^{(N)}$ are also provided

A probabilistic interpretation of a sequence related to Narayana polynomials

A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M. Lasalle is then obtained from a classical convolution identity. Some arithmetical properties of the generalized coefficients are also established

Double-diffusive erosion of the core of Jupiter

We present Direct Numerical Simulations of the transport of heat and heavy elements across a double-diffusive interface or a double-diffusive staircase, in conditions that are close to those one may expect to find near the boundary between the heavy-element rich core and the hydrogen-helium envelope of giant planets such as Jupiter. We find that the non-dimensional ratio of the buoyancy flux associated with heavy element transport to the buoyancy flux associated with heat transport lies roughly between 0.5 and 1, which is much larger than previous estimates derived by analogy with geophysical double-diffusive convection. Using these results in combination with a core-erosion model proposed by Guillot et al. (2004), we find that the entire core of Jupiter would be eroded within less than 1Myr assuming that the core-envelope boundary is composed of a single interface. We also propose an alternative model that is more appropriate in the presence of a well-established double-diffusive staircase, and find that in this limit a large fraction of the core could be preserved. These findings are interesting in the context of Juno's recent results, but call for further modeling efforts to better understand the process of core erosion from first principles.Comment: Accepted for publication in Ap

Tex2Shape: Detailed Full Human Body Geometry From a Single Image

We present a simple yet effective method to infer detailed full human body shape from only a single photograph. Our model can infer full-body shape including face, hair, and clothing including wrinkles at interactive frame-rates. Results feature details even on parts that are occluded in the input image. Our main idea is to turn shape regression into an aligned image-to-image translation problem. The input to our method is a partial texture map of the visible region obtained from off-the-shelf methods. From a partial texture, we estimate detailed normal and vector displacement maps, which can be applied to a low-resolution smooth body model to add detail and clothing. Despite being trained purely with synthetic data, our model generalizes well to real-world photographs. Numerous results demonstrate the versatility and robustness of our method

Negotiating the Korea-United States Free Trade Agreement

Difficult and sensitive issues will command the attention of US and Korean officials as they negotiate a bilateral free trade agreement (FTA). The United States will have to put long-standing US barriers to Korean exports of textiles, apparel, and steel on the table and resolve problems with South Korean access to the US visa waiver program. In turn, South Korea will have to open new opportunities for US goods and services, including autos, beef, and rice. Such a deal will pose a stiff political challenge for Korean officials. However, they will be under pressure in any event to reform their farm programs--either in the context of a final deal in the WTO talks or in response to Chinese initiatives in the region, which Korea will need to match. Both Korea and the United States also have important foreign policy interests in the FTA, particularly enhanced security on the Korean peninsula. South Korea would like the FTA to promote the policy of constructive engagement with North Korea, which the former has been pursuing by extending trade preferences to goods produced in the Kaesong industrial complex in North Korea. However, such a request would put the entire negotiation in jeopardy since the US Congress would reject preferences for the North Korean regime.

Learning to Reconstruct People in Clothing from a Single RGB Camera

We present a learning-based model to infer the personalized 3D shape of people from a few frames (1-8) of a monocular video in which the person is moving, in less than 10 seconds with a reconstruction accuracy of 5mm. Our model learns to predict the parameters of a statistical body model and instance displacements that add clothing and hair to the shape. The model achieves fast and accurate predictions based on two key design choices. First, by predicting shape in a canonical T-pose space, the network learns to encode the images of the person into pose-invariant latent codes, where the information is fused. Second, based on the observation that feed-forward predictions are fast but do not always align with the input images, we predict using both, bottom-up and top-down streams (one per view) allowing information to flow in both directions. Learning relies only on synthetic 3D data. Once learned, the model can take a variable number of frames as input, and is able to reconstruct shapes even from a single image with an accuracy of 6mm. Results on 3 different datasets demonstrate the efficacy and accuracy of our approach