A note on primes of the form a2 + 1

In this note I prove using an algebraic identity and Wilson's Theorem

Efficient harmonic oscillator chain energy harvester driven by colored noise

We study the performance of an electromechanical harmonic oscillator chain as an energy harvester to extract power from finite-bandwidth ambient random vibrations, which are modelled by colored noise. The proposed device is numerically simulated and its performance assessed by means of the net electrical power generated and its efficiency in converting the external noise-supplied power into electrical power. Our main result is a much enhanced performance, both in the net electrical power delivered and in efficiency, of the harmonic chain with respect to the popular single oscillator resonator. Our numerical findings are explained by means of an analytical approximation, in excellent agreement with numerics

A HALF-GRAPH DEPTH FOR FUNCTIONAL DATA

A recent and highly attractive area of research in statistics is the analysis of functional data. In this paper a new definition of depth for functional observations is introduced based on the notion of “half-graph” of a curve. It has computational advantages with respect to other concepts of depth previously proposed. The half-graph depth provides a natural criterion to measure the centrality of a function within a sample of curves. Based on this depth a sample of curves can be ordered from the center outward and L-statistics are defined. The properties of the half-graph depth, such as the consistency and uniform convergence, are established. A simulation study shows the robustness of this new definition of depth when the curves are contaminated. Finally real data examples are analyzed.

DEPTH-BASED CLASSIFICATION FOR FUNCTIONAL DATA

Classification is an important task when data are curves. Recently, the notion of statistical depth has been extended to deal with functional observations. In this paper, we propose robust procedures based on the concept of depth to classify curves. These techniques are applied to a real data example. An extensive simulation study with contaminated models illustrates the good robustness properties of these depth-based classification methods.

Asymptotically Exact Approximations for the Symmetric Difference of Generalized Marcum-Q Functions

(c) 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. DOI: 10.1109/TVT.2014.2337263In this paper, we derive two simple and asymptotically exact approximations for the function defined as ΔQm(a, b) =Δ Qm(a, b) - Qm(b, a). The generalized Marcum Q-function Qm(a, b) appears in many scenarios in communications in this particular form and is referred to as the symmetric difference of generalized Marcum Q-functions or the difference of generalized Marcum Q-functions with reversed arguments. We show that the symmetric difference of Marcum Q-functions can be expressed in terms of a single Gaussian Q-function for large and even moderate values of the arguments a and b. A second approximation for ΔQm(a, b) is also given in terms of the exponential function. We illustrate the applicability of these new approximations in different scenarios: 1) statistical characterization of Hoyt fading; 2) performance analysis of communication systems; 3) level crossing statistics of a sampled Rayleigh envelope; and 4) asymptotic approximation of the Rice Ie-function.Universidad de Málaga. Campus de Excelencia Internacional. Andalucía Tech