4,658 research outputs found
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 , let be the Chebyshev polynomial of the first
kind. Expressions for the sequence of numbers , defined as the
coefficients in the expansion of , 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 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
The Cauchy-Schlomilch transformation
The Cauchy-Schl\"omilch transformation states that for a function and , the integral of and over the
interval are the same. This elementary result is used to evaluate
many non-elementary definite integrals, most of which cannot be obtained by
symbolic packages. Applications to probability distributions is also given
A review of the bandwidth and environmental discourses of future energy scenarios:Shades of green and gray
Energy scenarios are often used to investigate various possible energy futures and reduce the uncertainty that surrounds energy transition. However, scenario construction lacks consistent and adequate methodological standards, resulting in limited insight into the actual bandwidth covered by current energy scenarios and whether various perspectives on future energy development pathways are all adequately represented. Our research deployed a non-mathematical clustering approach to identify general trends in future energy scenarios and assess the role of Cornucopian and Malthusian oriented world views therein. We found that the futures communicated in quantified future energy scenarios overlap to a large extent and represent only a narrow bandwidth of moderate world views. We argue that the underrepresentation of extreme representations of world views and environmental discourses in energy scenarios skews the overall outlook on possible energy futures. This implies that scenario-informed policy design and decision-making risks bias towards the status-quo. (C) 2016 Elsevier Ltd. All rights reserved
XNect: Real-time Multi-person 3D Human Pose Estimation with a Single RGB Camera
We present a real-time approach for multi-person 3D motion capture at over 30 fps using a single RGB camera. It operates in generic scenes and is robust to difficult occlusions both by other people and objects. Our method operates in subsequent stages. The first stage is a convolutional neural network (CNN) that estimates 2D and 3D pose features along with identity assignments for all visible joints of all individuals. We contribute a new architecture for this CNN, called SelecSLS Net, that uses novel selective long and short range skip connections to improve the information flow allowing for a drastically faster network without compromising accuracy. In the second stage, a fully-connected neural network turns the possibly partial (on account of occlusion) 2D pose and 3D pose features for each subject into a complete 3D pose estimate per individual. The third stage applies space-time skeletal model fitting to the predicted 2D and 3D pose per subject to further reconcile the 2D and 3D pose, and enforce temporal coherence. Our method returns the full skeletal pose in joint angles for each subject. This is a further key distinction from previous work that neither extracted global body positions nor joint angle results of a coherent skeleton in real time for multi-person scenes. The proposed system runs on consumer hardware at a previously unseen speed of more than 30 fps given 512x320 images as input while achieving state-of-the-art accuracy, which we will demonstrate on a range of challenging real-world scenes
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