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An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…
We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function,…
The ability to simulate and predict the outcome of contacts is paramount to the successful execution of many robotic tasks. Simulators are powerful tools for the design of robots and their behaviors, yet the discrepancy between their…
We present an algorithm for generating approximations for the logarithm of Barnes $G$-function in the half-plane $Re(z)\ge 3/2$. These approximations involve only elementary functions and are easy to implement. The algorithm is based on a…
We study the problem of computing the value function from a discretely-observed trajectory of a continuous-time diffusion process. We develop a new class of algorithms based on easily implementable numerical schemes that are compatible with…
In this paper we use Lagrange-Poincare reduction to understand the coupling between a fluid and a set of Lagrangian particles that are supposed to simulate it. In particular, we reinterpret the work of Cendra et al. by substituting velocity…
Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out…
The Lambert W function has utility for solving various exponential and logarithmic equations arranged in the form of $g(x)e^{g(x)}$. Using the Lambert W function and tetration, a variety of categorized inversion formulas are presented.…
It is well-known that the Riemann zeta function does not satisfy any exact polynomial differential equation. Here we present numerical evidence for the existence of approximate polynomial dependencies between the values of the alternating…
In the present paper we use the expansion formula of the polylogarithm function to construct approximations of the Caputo derivative which are related to the midpoint approximation of the integral in the definition of the Caputo derivative.…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…
In this paper we study the consequences of overinterpolation, i.e., the situation when a function can be interpolated by polynomial, or rational, or algebraic functions in more points that normally expected. We show that in many cases such…
The derivative expansion approach to the calculation of the interaction between two surfaces, is a generalization of the proximity force approximation, a technique of widespread use in different areas of physics. The derivative expansion…
We study approximation of functions by algebraic polynomials in the H\"older spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of…
A first encounter with $A(\lambda) := \int_0^{+\infty}\{\lambda t\}\{t\}t^{-2}dt.$ This text contains the detailed calculations leading to the results of our paper "Etude de l'autocorrelation multiplicative de la fonction "partie…
We propose a rational version of the classic Rodrigues' rotation formula, which leads to a more accurate and efficient modelling of rotations and their derivatives in finite precision arithmetic. We explain how the rational Rodrigues'…
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations)…
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is…