Related papers: Polynomial time computable functions over the real…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
The class of Basic Feasible Functionals BFF$_2$ is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a…
Natural laws are often described through differential equations yet finding a differential equation that describes the governing law underlying observed data is a challenging and still mostly manual task. In this paper we make a step…
This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the…
Recently new solvable systems of nonlinear evolution equations -- including ODEs, PDEs and systems with discrete time -- have been introduced. These findings are based on certain convenient formulas expressing the $k$-th time-derivative of…
Working over an algebraically closed base field $k$ of characteristic 2, the ring of invariants $R^G$ is studied, where $G$ is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring $R$ of…
This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every $m$ the first $m$ digits of a real number $\alpha\ge…
We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The…
Moment estimation for stochastic differential equations (SDEs) is fundamental to the formal reasoning and verification of stochastic dynamical systems, yet remains challenging and is rarely available in closed form. In this paper, we study…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…
A famous result due to Ko and Friedman (1982) asserts that the problems of integration and maximisation of a univariate real function are computationally hard in a well-defined sense. Yet, both functionals are routinely computed at great…
In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice polynomial…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$…
We show (almost) separation between certain important classes of Boolean functions. The technique that we use is to show that the total influence of functions in one class is less than the total influence of functions in the other class. In…
We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$) of functions from some "simple" class ${\cal C}$. In particular, given ${\cal C}$ we are interested in finding…
This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…