In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a degenerate triangle if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar to the case of a circle, whose dimension shrinks from two to zero as it degenerates into a point. As another example, the solution set of a system of equations that depends on parameters generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the solution set is said to be degenerate. For some classes of composite objects, the degenerate cases depend on the properties that are specifically studied. In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by several different systems of equations, and these different systems of equations may lead to different degenerate cases, while characterizing the same non-degenerate cases. This may be the reason for which there is no general definition of degeneracy, despite the fact that the concept is widely used and defined (if needed) in each specific situation. A degenerate case thus has special features which makes it non-generic or special cases. However, not all non-generic or special cases are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points (e.g., point, line, intersecting lines). (Wikipedia).
Divisibility, Prime Numbers, and Prime Factorization
Now that we understand division, we can talk about divisibility. A number is divisible by another if their quotient is a whole number. The smaller number is a factor of the larger one, but are there numbers with no factors at all? There's some pretty surprising stuff in this one! Watch th
From playlist Mathematics (All Of It)
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From playlist 01. Fundamentals of Science and Astronomy
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From playlist Additional Topics: Generating Functions and Intro to Number Theory (Discrete Math)
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From playlist MATH 1314: College Algebra (depreciated)
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From playlist Differential Equations -- Separation of Variables
Solve the general solution for differentiable equation with trig
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From playlist Differential Equations
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From playlist Differential Equations
How Quantum Mechanics Holds Up a Dead Star - Ask a Spaceman!
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Xie Chen - Foliated Fracton and Beyond - IPAM at UCLA
Recorded 30 August 2021. Xie Chen of the California Institute of Technology presents "Foliated Fracton and Beyond" at IPAM's Graduate Summer School: Mathematics of Topological Phases of Matter. Abstract: This talk will introduce the foliation idea to characterize type I fracton models. We
From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter
When Quantum Physics and Relativity Compete Against Each Other to Keep Stars From Collapsing
Go to Squarespace.com for a free trial, and when you’re ready to launch, go to http://www.squarespace.com/parthg to save 10% off your first purchase of a website or domain. #degeneracypressure #quantum #neutronstar In this video, we'll be looking at degeneracy pressure - a quantum mecha
From playlist Quantum Physics by Parth G
Singular Learning Theory - Seminar 1 - What is learning?
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From playlist Metauni
Vladimiro Benedetti: Orbital degeneracy loci
Abstract: I will present a joint work with Sara Angela Filippini, Laurent Manivel and Fabio Tanturri (arXiv: 1704.01436). We introduce a new class of varieties, called orbital degeneracy loci. The idea is to use any orbit closure in a representation of an algebraic group to generalise the
From playlist Algebraic and Complex Geometry
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From playlist Mathematics
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Program The 2nd Asia Pacific Workshop on Quantum Magnetism ORGANIZERS: Subhro Bhattacharjee, Gang Chen, Zenji Hiroi, Ying-Jer Kao, SungBin Lee, Arnab Sen and Nic Shannon DATE: 29 November 2018 to 07 December 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Frustrated quantum magne
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Dynamics of Complex Systems - 2017 DATES: 10 May 2017 to 08 July 2017 VENUE: Madhava Lecture Hall, ICTS Bangalore This Summer Program on Dynamics of Complex Systems is second in the series. The theme for the program this year is Mathematical Biology. Over the past decades, the focus o
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From playlist Chemistry 131A: Quantum Principles
Cell Learning Theory - Seminar 1 - Introduction to natural programs
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Simplify the Negation of Statements with Quantifiers and Predicates
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