Search results for "polynomials"
showing 10 items of 144 documents
Bayesian inference for the extremal dependence
2016
A simple approach for modeling multivariate extremes is to consider the vector of component-wise maxima and their max-stable distributions. The extremal dependence can be inferred by estimating the angular measure or, alternatively, the Pickands dependence function. We propose a nonparametric Bayesian model that allows, in the bivariate case, the simultaneous estimation of both functional representations through the use of polynomials in the Bernstein form. The constraints required to provide a valid extremal dependence are addressed in a straightforward manner, by placing a prior on the coefficients of the Bernstein polynomials which gives probability one to the set of valid functions. The…
Using the Hermite Regression Formula to Design a Neural Architecture with Automatic Learning of the “Hidden” Activation Functions
2000
The value of the output function gradient of a neural network, calculated in the training points, plays an essential role for its generalization capability. In this paper a feed forward neural architecture (αNet) that can learn the activation function of its hidden units during the training phase is presented. The automatic learning is obtained through the joint use of the Hermite regression formula and the CGD optimization algorithm with the Powell restart conditions. This technique leads to a smooth output function of αNet in the nearby of the training points, achieving an improvement of the generalization capability and the flexibility of the neural architecture. Experimental results, ob…
Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method
2021
AbstractIn this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the prop…
An optimal Poincaré-Wirtinger inequality in Gauss space
2013
International audience; Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$% -dimensional Gaussian measure.
D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization
2015
The D-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group GL(2, C) of invertible 2 × 2 matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions…
Third-order accurate monotone cubic Hermite interpolants
2019
Abstract Monotonicity-preserving interpolants are used in several applications as engineering or computer aided design. In last years some new techniques have been developed. In particular, in Arandiga (2013) some new methods to design monotone cubic Hermite interpolants for uniform and non-uniform grids are presented and analyzed. They consist on calculating the derivative values introducing the weighted harmonic mean and a non-linear variation. With these changes, the methods obtained are third-order accurate, except in extreme situations. In this paper, a new general mean is used and a third-order interpolant for all cases is gained. We perform several experiments comparing the known tec…
A nonlinear algorithm for monotone piecewise bicubic interpolation
2016
We present an algorithm for monotone interpolation on a rectangular mesh.We use the sufficient conditions for monotonicity of Carlton and Fritsch.We use nonlinear techniques to approximate the partial derivatives at the grid points.We develop piecewise bicubic Hermite interpolants with these approximations.We present some numerical examples where we compare different results. In this paper we present an algorithm for monotone interpolation of monotone data on a rectangular mesh by piecewise bicubic functions. Carlton and Fritsch (1985) develop conditions on the Hermite derivatives that are sufficient for such a function to be monotone. Here we extend our results of Arandiga (2013) to obtain…
Using the Hermite Regression Algorithm to Improve the Generalization Capability of a Neural Network
1999
In this paper it is shown that the ability of classification and the ability of approximating a function are correlated to the value (in the training points) of the gradient of the output function learned by the network.
Multiwavelet Frames Originated From Hermite Splines
2015
The chapter presents a method for the construction of multiwavelet frame transform for manipulation of discrete-time signals. The frames are generated by three-channel perfect reconstruction oversampled multifilter banks. The design of the multifilter bank starts from a pair of interpolating multifilters, which originate from the cubic Hermite splines. The remaining multifilters are designed by factoring polyphase matrices. Input to the oversampled analysis multifilter bank is a vector-signal, which is derived from an initial scalar signal by one out of three pre-processing algorithms. The post-processing algorithms convert the vector output from the synthesis multifilter banks into a scala…
Products of Bessel functions and associated polynomials
2013
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.