Related papers: Generalized Taylor's Theorem
In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{\alpha,\rho}\left(x^\rho-a^\rho\right)^{j\alpha} + e_m(x)}$ \noindent with $m\in \mathbb{N}$,…
Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin…
We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an…
The original k-means clustering method works only if the exact vectors representing the data points are known. Therefore calculating the distances from the centroids needs vector operations, since the average of abstract data points is…
We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.
We describe an improved version of the quantum simulation method based on the implementation of a truncated Taylor series of the evolution operator. The idea is to add an extra step to the previously known algorithm which implements an…
The generalized number-theoretic transformation (NPT) is formulated on the basis of the exponential function theorem, which allows us to replace operations modulo the expression as a whole by modulo operations on the exponent of this…
The method of Taylor series expansion is used to develop a numerical solution to the reactor point kinetics equations. It is shown that taking a first order expansion of the neutron density and precursor concentrations at each time step…
We show a statistical version of Taylor's theorem and apply this result to non-parametric density estimation from truncated samples, which is a classical challenge in Statistics \cite{woodroofe1985estimating, stute1993almost}. The…
The transfer property for the generalized Browder's theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the $B$-Weyl spectrum inclusion. In addition, the isolated points of…
As a rigorous statistical approach, statistical Taylor expansion extends the conventional Taylor expansion by replacing precise input variables with random variables of known distributions and sample counts to compute the mean, the…
In this short note, we give an elementary proof of a universal approximation theorem for neural networks with three hidden layers and increasing, continuous, bounded activation function. The result is weaker than the best known results, but…
The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the…
Starting with the recursive extended Euclid's algorithm, we apply a systematic approach using matrix notation to transform it into an iterative algorithm. The partial correctness proof derived from the transformation turns out to be very…
By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators…
We are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean field wave functions by using the extension of the statistical Wick's theorem (Gaudin's theorem) in the appropriate limits.
D. Hensley showed in 1994 that the number of steps taken by the Euclidean algorithm to find the greatest common divisor of two natural numbers less than or equal to n follows a normal distribution in the limit as n tends to infinity. V.…
In this article we present a generalization of a Leibniz's geometrical theorem and an application of it.
We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or…