Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be uniformly discrete if there exists a r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left{2^{-n} : n \in \N_0\right}. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology o

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Convolution Operators And Homomorphisms Of Locally Compact Groups

Let 1 < p < infinity, let G and H be locally compact groups and let c) be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp (G) of the p-convolution operators on G into CVp (H) which extends the usual definition of the image of a bounded measure by omega. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let G(d) denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, vertical bar parallel to mu vertical bar parallel to CVp(G)

Cambridge University Press2008

Isogeometric Analysis for an earthquake simulation

Christoph Jäggli

We consider Isogeometric Analysis to simulate an earthquake in a two dimensional model of a sinusoidal shaped valley. We analyze two different time integration schemes, namely the Generalized- and the Leap-Frog methods. Solving a wave propagation problem leads to a numerical dispersion of the wave velocities which depends on the discrete function space, as well as on the direction of propagation. In this work we also analyze the numerical dispersion with respect to the number of quadrature nodes per wavelength and the direction of wave propagation.

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A Space-Time Adaptive Algorithm to Illustrate the Lack of Collision of a Rigid Disk Falling in an Incompressible Fluid

Samuel Dubuis, Marco Picasso

A space-time adaptive algorithm to solve the motion of a rigid disk in an incompressible Newtonian fluid is presented, which allows collision or quasi-collision processes to be computed with high accuracy. In particular, we recover the theoretical result proven in [M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1345-1371], that the disk will never touch the boundary of the domain in finite time. Anisotropic, continuous piecewise linear finite elements are used for the space discretization, the Euler scheme for the time discretization. The adaptive criteria are based on a posteriori error estimates for simpler problems.

WALTER DE GRUYTER GMBH2021

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological