Related papers: An optimal first-order Taylor-like formula with a …
This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce…
In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function $f$ defined on the interval $[a,b]$, this formula is derived by introducing a linear combination of $f'$ computed at $n+1$…
The aim of this paper is to derive a refined first-order expansion formula in Rn, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor's formula. For a given function, the formula we derived is…
We prove the neo-classical inequality with the optimal constant, which was conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For the proof, we introduce the fractional order Taylor's series with residual terms. Their…
We prove some new results on existence of solutions to first--order ordinary differential equations with deviating arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the…
First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…
In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the $P_1$-interpolation error…
Multi-task learning (MTL) has emerged as a pivotal paradigm in machine learning by leveraging shared structures across multiple related tasks. Despite its empirical success, the development of likelihood-based efficiently solvable…
In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the…
In this paper, a new calculus on sequences is defined. Also, the $\lambda$-derivative and the $\lambda$-integration are investigated. The fundamental theorem of $\lambda$-calculus is included. A suitable function basis for the…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
We will investigate proof-theoretic and linguistic aspects of first-order linear logic. We will show that adding partial order constraints in such a way that each sequent defines a unique linear order on the antecedent formulas of a sequent…
In large-scale modern data analysis, first-order optimization methods are usually favored to obtain sparse estimators in high dimensions. This paper performs theoretical analysis of a class of iterative thresholding based estimators defined…
A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that…
Real world scenarios can be captured with lifted probability distributions. However, distributions are usually encoded in a table or list, requiring an exponential number of values. Hence, we propose a method for extracting first-order…
Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate…
In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise,…
This paper offers a matrix-free first-order numerical method to solve large-scale conic optimization problems. Solving systems of linear equations pose the most computationally challenging part in both first-order and second-order numerical…
We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error…
Based on a new Taylor-like formula, we derived an improved interpolation error estimate in $W^{1,p}$. We compare it with the classical error estimates based on the standard Taylor formula, and also with the corresponding interpolation error…