Euclidean geometry | Linear algebra
In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2n orthants in n-dimensional space. More specifically, a closed orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: ε1x1 ≥ 0 ε2x2 ≥ 0 · · · εnxn ≥ 0, where each εi is +1 or −1. Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities ε1x1 > 0 ε2x2 > 0 · · · εnxn > 0, where each εi is +1 or −1. By dimension: * In one dimension, an orthant is a ray. * In two dimensions, an orthant is a quadrant. * In three dimensions, an orthant is an octant. John Conway defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2n simplex facets, one per orthant. The nonnegative orthant is the generalization of the first quadrant to n-dimensions and is important in many constrained optimization problems. (Wikipedia).
Linear Algebra: Orthonormal Basis
Learn the basics of Linear Algebra with this series from the Worldwide Center of Mathematics. Find more math tutoring and lecture videos on our channel or at http://centerofmath.org/ More on unit vectors: https://www.youtube.com/watch?v=C6EYJVBYXIo
From playlist Basics: Linear Algebra
Orthogonality and Orthonormality
We know that the word orthogonal is kind of like the word perpendicular. It implies that two vectors have an angle of ninety degrees or half pi radians between them. But this term means much more than this, as we can have orthogonal matrices, or entire subspaces that are orthogonal to one
From playlist Mathematics (All Of It)
Math 060 Fall 2017 111317C Orthonormal Bases
Motivation: how to obtain the coordinate vector with respect to a given basis? Definition: orthogonal set. Example. Orthogonal implies linearly independent. Orthonormal sets. Example of an orthonormal set. Definition: orthonormal basis. Properties of orthonormal bases. Example: Fou
From playlist Course 4: Linear Algebra (Fall 2017)
Orthonormal bases. The Gram-Schmidt Procedure. Schuur's Theorem on upper-triangular matrix with respect to an orthonormal basis. The Riesz Representation Theorem.
From playlist Linear Algebra Done Right
Linear Algebra - Lecture 39 - Orthonormal Sets
In this lecture, we discuss orthonormal sets of vectors. We investigate matrices with orthonormal columns. We also define an orthogonal matrix.
From playlist Linear Algebra Lectures
Simon Telen - Likelihood Equations and Scattering Amplitudes
We identify the scattering equations from particle physics as the likelihood equations for a particular statistical model. The scattering potential plays the role of the log-likelihood function. We employ recent methods from numerical nonlinear algebra to solve challenging instances of the
From playlist Research Spotlight
Cynthia Vinzant: Log concave polynomials and matroids
Strong log concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients and features in the polynomials defining several common conic programs. Recent work by several independent authors shows that the multivariate basisgener
From playlist Workshop: Tropical geometry and the geometry of linear programming
Anthea Monod (2/11/21): Tropical geometry of phylogenetic tree spaces
Title: Tropical geometry of phylogenetic tree spaces Abstract: BHV (Billera Holmes Vogtmann) space is a well-studied moduli space of phylogenetic trees that appears in many scientific disciplines, including computational biology, computer vision, combinatorics, and category theory. Speye
From playlist AATRN 2021
Marco Di Summa: Gomory mixed integer cuts are optimal
The lecture was held within the framework of the follow-up workshop to the Hausdorff Trimester Program: Combinatorial Optimization. Abstract: Among many families of cutting planes for (mixed) integer programming proposed in the literature, Gomory mixed-integer cuts seem to stand out for
From playlist Follow-Up-Workshop "Combinatorial Optimization"
Orthocenters exist! | Universal Hyperbolic Geometry 10 | NJ Wildberger
In classical hyperbolic geometry, orthocenters of triangles do not in general exist. Here in universal hyperbolic geometry, they do. This is a crucial building block for triangle geometry in this subject. The dual of an orthocenter is called an ortholine---also not seen in classical hyperb
From playlist Universal Hyperbolic Geometry
LANTHANIDES - a quick definition
A quick definition of the lanthanides. Chem Fairy: Louise McCartney Director: Michael Harrison Written and Produced by Kimberly Hatch Harrison ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : https://www.patreon.com/socratica ► Make a one-time PayPal donation: https://ww
From playlist Chemistry glossary
Lecture 7 | Convex Optimization I
Professor Stephen Boyd, of the Stanford University Electrical Engineering department, expands upon his previous lectures on convex optimization problems for the course, Convex Optimization I (EE 364A). Convex Optimization I concentrates on recognizing and solving convex optimization pro
From playlist Lecture Collection | Convex Optimization
Log-concavity, matroids and expanders - Cynthia Vinzant
Members' Seminar Topic: Log-concavity, matroids and expanders Speaker: Cynthia Vinzant Affiliation: North Carolina State University; von Neumann Fellow, School of Mathematics Date: October 19, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Log-concave polynomials in theory and applications - Part 2 - Cynthia Vinzant
Computer Science/Discrete Mathematics Seminar II Topic: Log-concave polynomials in theory and applications - Part 2 Speaker: Cynthia Vinzant Affiliation: Member, School of Mathematics Date: February 02, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Linear Algebra for Computer Scientists. 7. Linear Combinations of Vectors
This computer science video is one of a series on linear algebra for computer scientists. In this video you will learn about linear combinations of vectors, that is, you will learn how to create new vectors by scaling then adding other vectors together. You will also learn that some sets
From playlist Linear Algebra for Computer Scientists
Aaron Sidford: Introduction to interior point methods for discrete optimization, lecture II
Over the past decade interior point methods (IPMs) have played a pivotal role in mul- tiple algorithmic advances. IPMs have been leveraged to obtain improved running times for solving a growing list of both continuous and combinatorial optimization problems including maximum flow, bipartit
From playlist Summer School on modern directions in discrete optimization
RNT1.4. Ideals and Quotient Rings
Ring Theory: We define ideals in rings as an analogue of normal subgroups in group theory. We give a correspondence between (two-sided) ideals and kernels of homomorphisms using quotient rings. We also state the First Isomorphism Theorem for Rings and give examples.
From playlist Abstract Algebra
Mahesh Kakde: Brumer-Stark units and a conjecture of Gross
The existence of Brumer-Stark unit is guaranteed by the Brumer-Stark conjecture. A conjecture of Dasgupta gives an explicit p-adic analytic formula for these units. An approach to this explicit formula is given by the tower of fields conjecture of Gross. After recalling these conjecture an
From playlist Number Theory