Geometric progression

Summary

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression's terms is called a geometric series. Elementary properties The n-th term of a geometric sequence with initial value a = a1 and common ratio r is given by :a_n = a,r^{n-1}, and in general :a_n = a_m,r^{n-m}. Such a geometric

Official source

https://en.wikipedia.org/wiki/Geometric_progression

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 (9)

Related publications

Related people

Related units

Related concepts (9)

Related concepts

Related courses (24)

Related courses

Related lectures (66)

Related lectures

Adaptive analysis-aware defeaturing

Ondine Gabrielle Chanon

Removing geometrical details from a complex domain is a classical operation in computer aided design for simulation and manufacturing. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve in the geometrical model of interest, removing some important geometrical features may greatly impact the solution accuracy. For instance, in solid mechanics simulations, such features can be holes or fillets near stress concentration regions. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected, because its analysis is a time-consuming task that is often performed manually, based on the expertise of engineers. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phase.In this thesis, we formalize the process of defeaturing, and we analyze its impact on the accuracy of solutions of some partial differential problems. To achieve this goal, we first precisely define the error between the problem solution defined in the exact geometry, and the one defined in the simplified geometry. Then, we introduce an a posteriori estimator of the energy norm of this error. This allows us to reliably and efficiently control the error coming from the addition or the removal of geometrical features. We subsequently consider a finite element approximation of the defeatured problem, and the induced numerical error is integrated to the proposed defeaturing error estimator. In particular, we address the special case of isogeometric analysis based on (truncated) hierarchical B-splines, in possibly trimmed and multipatch geometries. In this framework, we derive a reliable a posteriori estimator of the overall error, i.e., of the error between the exact solution defined in the exact geometry, and the numerical solution defined in the defeatured geometry.We then propose a two-fold adaptive strategy for analysis-aware defeaturing, which starts by considering a coarse mesh on a fully-defeatured computational domain. On the one hand, the algorithm performs classical finite element mesh refinements in a (partially) defeatured geometry. On the other hand, the strategy also allows for geometrical refinement. That is, at each iteration, the algorithm is able to choose which missing geometrical features should be added to the simplified geometrical model, in order to obtain a more accurate solution.Throughout the thesis, we validate the presented theory, the properties of the aforementioned estimators and the proposed adaptive strategies, thanks to an extensive set of numerical experiments.

EPFL2022

, , ,

Microchannel plates based on amorphous silicon have the inherent capability to overcome the existing spatial and temporal resolution limitations of single photon detectors. A Monte Carlo model-based study is presented here that serves to exploit their full potential. The dynamics of electron multiplication throughout the microchannels in the presence of an electric field is modelled based on electron emission models and experimental measurements. The geometrical limits for suitable amplification and the expected time resolution of amorphous silicon based microchannel plates are presented. Furthermore, the model is implemented in a finite element method simulator to estimate electron multiplication gain and time response for channels with various geometries. With gains above 1000 and timing jitters below 3 ps that are shown for selected geometries, AMCPs enable ultrafast measurements for medical applications and single photon detection.

2022

In this paper we revisit the kernel density estimation problem: given a kernel K(x, y) and a dataset of n points in high dimensional Euclidean space, prepare a data structure that can quickly output, given a query q, a (1 + epsilon)-approximation to mu := 1/vertical bar P vertical bar Sigma p is an element of P K(p, q). First, we give a single data structure based on classical near neighbor search techniques that improves upon or essentially matches the query time and space complexity for all radial kernels considered in the literature so far. We then show how to improve both the query complexity and runtime by using recent advances in data-dependent near neighbor search. We achieve our results by giving an new implementation of the natural importance sampling scheme. Unlike previous approaches, our algorithm first samples the dataset uniformly (considering a geometric sequence of sampling rates), and then uses existing approximate near neighbor search techniques on the resulting smaller dataset to retrieve the sampled points that lie at an appropriate distance from the query. We show that the resulting sampled dataset has strong geometric structure, making approximate near neighbor search return the required samples much more efficiently than for worst case datasets of the same size. As an example application, we show that this approach yields a data structure that achieves query time mu-((1+o(1))/4) and space complexity mu-((1+o(1))) for the Gaussian kernel. Our data dependent approach achieves query time mu(-0.173-o(1)) and space mu-(1+o(1)) for the Gaussian kernel. The data dependent analysis relies on new techniques for tracking the geometric structure of the input datasets in a recursive hashing process that we hope will be of interest in other applications in near neighbor search.

IEEE2020

Adaptive analysis-aware defeaturing

Ondine Gabrielle Chanon

EPFL2022