Rational function

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. A function is called a rational function if and only if it can be written in the form where and are polynomial functions of and is not the zero function. The domain of is the set of all values of for which the denominator is not zero. However, if and have a non-constant polynomial greatest common divisor , then setting and produces a rational function which may have a larger domain than , and is equal to on the domain of It is a common usage to identify and , that is to extend "by continuity" the domain of to that of Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions and are considered equivalent if . In this case is equivalent to . A proper rational function is a rational function in which the degree of is less than the degree of and both are real polynomials, named by analogy to a proper fraction in . There are several non equivalent definitions of the degree of a rational function. Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q, when the fraction is reduced to lowest terms. If the degree of f is d, then the equation has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator).

Efficient Continual Finite-Sum Minimization

Volkan Cevher, Efstratios Panteleimon Skoulakis, Leello Tadesse Dadi

Given a sequence of functions

f_1,\ldots,f_n

with

f_i:\mathcal{D}\mapsto \mathbb{R}

, finite-sum minimization seeks a point

{x}^\star \in \mathcal{D}

minimizing

\sum_{j=1}^nf_j(x)/n

. In this work, we propose a key twist into the finite-sum minimizat ...

2024

Efficient Continual Finite-Sum Minimization

Ultra long turbulent eddies, magnetic topology, and the triggering of internal transport barriers in tokamaks

Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions

Ultra long turbulent eddies, magnetic topology, and the triggering of internal transport barriers in tokamaks

Efficient Continual Finite-Sum Minimization

Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions