Euclidean division

In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder exist and are unique, under some conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder i

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (6)

Related publications

Related people

Related units

Related concepts (36)

Related concepts

Related courses (10)

Related courses

Related lectures (23)

Related lectures

Exponential Kleisli Monoids as Eilenberg-Moore Algebras

Dirk Hofmann, Gavin Jay Seal

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Springer2015

Escaping from saddle points on Riemannian manifolds

Nicolas Henri Bernard Flammarion, Yue Sun

We consider minimizing a nonconvex, smooth function f on a Riemannian manifold M. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as 1/epsilon(2) on the accuracy c, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension d, hence is almost dimension -free. The rate also has a polynomial dependence on the parameters describing the curvature of the manifold and the smoothness of the function. While the unconstrained problem (Euclidean setting) is well -studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems.

2019

On sums of eigenvalues of elliptic operators on manifolds

Joachim Stubbe

We use the averaged variational principle introduced in a recent article on graph spectra [10] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kroger's bound for Neumann spectra of Laplacians on Euclidean domains [15]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces ( here extending a result of Strichartz [26] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schrodinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

European Mathematical Soc2017

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic wa

Division (mathematics)

Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the

Fraction

A fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain siz

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, discrete structures & algorithmic thinking.

MATH-310: Algebra

Study basic concepts of modern algebra: groups, rings, fields.

COM-401: Cryptography and security

This course introduces the basics of cryptography. We review several types of cryptographic primitives, when it is safe to use them and how to select the appropriate security parameters. We detail how they work and sketch how they can be implemented.