Algebra | Real algebraic geometry
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set S of polynomials in Rn, a cylindrical algebraic decomposition is a decomposition of Rn into connected semialgebraic sets called cells, on which each polynomial has constant sign, either +, − or 0. To be cylindrical, this decomposition must satisfy the following condition: If 1 ≤ k < n and π is the projection from Rn onto Rn−k consisting in removing the last k coordinates, then for every pair of cells c and d, one has either π(c) = π(d) or π(c) ∩ π(d) = ∅. This implies that the images by π of the cells define a cylindrical decomposition of Rn−k. The notion was introduced by George E. Collins in 1975, together with an algorithm for computing it. Collins' algorithm has a computational complexity that is double exponential in n. This is an upper bound, which is reached on most entries. There are also examples for which the minimal number of cells is doubly exponential, showing that every general algorithm for cylindrical algebraic decomposition has a double exponential complexity. CAD provides an effective version of quantifier elimination over the reals that has a much better computational complexity than that resulting from the original proof of Tarski–Seidenberg theorem. It is efficient enough to be implemented on a computer. It is one of the most important algorithms of computational real algebraic geometry. Searching to improve Collins' algorithm, or to provide algorithms that have a better complexity for subproblems of general interest, is an active field of research. (Wikipedia).
How to integrate by partial fractions
Free ebook http://bookboon.com/en/learn-calculus-2-on-your-mobile-device-ebook How to integrate by the method of partial fraction decomposition. In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is a fraction such that the numerator
From playlist A second course in university calculus.
In LU decomposition we decompose a matrix into two matrices, that, when multiplied in a certain order, gives rise to the original matrix. L is a lower triangular matrix, and U is an upper triangular matrix. An upper triangular matrix has entries equaling zero below the main diagonal and a
From playlist Introducing linear algebra
Linear Algebra 13e: The LU Decomposition
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 1 Linear Algebra: An In-Depth Introduction with a Focus on Applications
Integration Using Partial Fraction Decomposition Part 1
This video shows how partial fraction decomposition can be used to simplify and integral. This video only shows linear factors. Part 1 of 2 Site: http://mathispower4u.com
From playlist Integration Using Partial Fractions
Linear Algebra 18a: Introduction to the Eigenvalue Decomposition
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 3 Linear Algebra: Linear Transformations
Live CEOing Ep 396: Calculus & Algebra Features Design Review for Wolfram Language
In this episode of Live CEOing, Stephen Wolfram reviews the design of upcoming algebra and calculus functionality for the Wolfram Language. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twitch chan
From playlist Behind the Scenes in Real-Life Software Design
Solve a System of Linear Equations Using LU Decomposition
This video explains how to use LU Decomposition to solve a system of linear equations. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com
From playlist Matrix Equations
Review of Decomposition by the Dot Product
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 4 Linear Algebra: Inner Products
Quantitative bounds on the topology of semi-algebraic and (...) - S. Basu - Workshop 1 - CEB T1 2018
Saugata Basu (Purdue) / 02.02.2018 Quantitative bounds on the topology of semi-algebraic and definable sets I will survey some old and new results on bounding the topology of semi-algebraic and definable sets in terms of various parameters of their defining formulas, and indicate how som
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Find the Partial Fraction Decomposition 3x/((x + 1)(x^2 + 1))
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Find the Partial Fraction Decomposition 3x/((x + 1)(x^2 + 1))
From playlist Partial Fraction Decomposition
(New Version Available) Partial Fraction Decomposition - Part 1 of 2
New Version Available: https://youtu.be/c2oLHtPA03U This video explain how to perform partial fraction decomposition with linear factors. http://mathispower4u.yolasite.com/
From playlist Integration Using Partial Fraction Decomposition
James Davenport - How to prove a calculation correct? - IPAM at UCLA
Recorded 16 February 2023. James Davenport of the University of Bath presents "How to prove a calculation correct?" at IPAM's Machine Assisted Proofs Workshop. Abstract: How might one prove a calculation correct. The usual approach is to prove the algorithm correct, prove that the implemen
From playlist 2023 Machine Assisted Proofs Workshop
Zlil Sela - Automorphisms of groups and a higher rank JSJ decomposition
The JSJ (for groups) was originally constructed to study the automorphisms and the cyclic splittings of a (torsion-free) hyperbolic group. Such a structure theory was needed to complete the solution of the isomorphism problem for (torsion-free) hyperbolic groups. Later, the JSJ was genera
From playlist Geometry in non-positive curvature and Kähler groups
Ole Warnaar: Cylindric partitions and character identities
Abstract: As was shown in the 1980s by Kac, Peterson and Wakimoto, the characters of infinite dimensional Lie algebras provide a rich source of modular forms. Finding manifestly positive expressions for such characters remains, however, a difficult open problem. In this talk I will describ
From playlist Number Theory Down Under 9
Denis Osin: Acylindrically hyperbolic groups (part 1)
The lecture was held within the framework of Follow-up Workshop TP Rigidity. 28.4.2015
From playlist HIM Lectures 2015
In this talk, Adam Strzebonski shows some examples of Wolfram Language optimization functions and discusses the algorithms used to implement them. Minimize, Maximize, MinValue, MaxValue, ArgMin and ArgMax compute exact global extrema of univariate or multivariate functions, constrained by
From playlist Wolfram Technology Conference 2020
Tensor Calculus 4e: Decomposition by Dot Product in Tensor Notation
This course will eventually continue on Patreon at http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Te
From playlist Introduction to Tensor Calculus
Planarity in Higher Codimension Mean Curvature Flow - Keaton Naff
Analysis Seminar Topic: Planarity in Higher Codimension Mean Curvature Flow Speaker: Keaton Naff Affiliation: Columbia University Date: February 08, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Partial Fraction Decomposition Part 1 (Linear)
This video introduces partial fraction decomposition.
From playlist Integration Using Partial Fractions
Getting the Most from Algebraic Solvers in Mathematica
This talk by Adam Strzebonski at the Wolfram Technology Conference 2011 gives a survey of Mathematica functions related to solving algebraic equations and inequalities. It also discusses the choice of the most appropriate solvers for various types of problems and the ways of formulating th
From playlist Wolfram Technology Conference 2011