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This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. These approximations are obtained using the techniques called \emph{generalized simplex Hessian} and…
It is known that rational approximations of elementary analytic functions (exp, log, trigonometric, and hyperbolic functions, and their inverse functions) are computable in the weak complexity class $\mathrm{TC}^0$. We show how to formalize…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
We study the best approximation problem: \[ \displaystyle \min_{\alpha\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m \alpha_j \Gamma_j ({\bf x}_i) \right|. \] Here: $\Gamma:=\left\{\Gamma_1,...,\Gamma_m\right\}$ is a list of…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
Hedges' unbiased estimator g* has been broadly used in statistics. We propose a sequence of polynomials to better approximate the multiplicative correction factor of g* by incorporating analytic estimations to the ratio of gamma functions.
This expository article proves some results of Ferguson, on the approximation of continuous functions on a compact subset of R by polynomials with integral coefficients.
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…
The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the GW approximation. However, current implementations are incomplete since they ignore terms that lead to partial…
A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are…
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…
In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More…
We investigate training and using Gaussian kernel SVMs by approximating the kernel with an explicit finite- dimensional polynomial feature representation based on the Taylor expansion of the exponential. Although not as efficient as the…
A mathematical proposition with a trainable pair, operator and quantum circuit, are introduced to approximate functions expressed as cubic Taylor polynomials, numerical simulations illustrate three cases.
Derivative-matching approximations are constructed as power series built from functions. The method assumes the knowledge of special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented…
This paper addresses the problem of evaluating a subset of the range of a vector-valued function. It is based on a work by Gold- sztejn and Jaulin which provides methods based on interval analysis to address this problem when the dimension…