Order theory | Articles containing proofs | Perfect graphs | Theorems in combinatorics
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician Robert P. Dilworth. An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain decomposition is its number of chains. The width of the partial order is defined as the common size of the antichain and chain decomposition. A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes of the largest antichain and of the smallest chain decomposition are again equal. (Wikipedia).
Theory of numbers: Congruences: Euler's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim
From playlist Theory of numbers
Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem
In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)
2.6.5 Time versus Processors: Video
MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015
What is the Riemann Hypothesis?
This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation
From playlist Mathematics
Duality from Marriage to Dating: an Exposition on a Theorem of Hall
In this video, we give four incarnations of Hall's famous theorem: Hall’s theorem itself, Dilworth’s theorem, Menger’s theorem, and an introduction to the strong duality theorem of linear programming, along with proofs for each of the first three. The game mentioned in the video is locate
From playlist Summer of Math Exposition 2 videos
Pythagorean Theorem II (visual proof)
This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using a dissection of a square in two different ways. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshort
From playlist Pythagorean Theorem
Multivariable Calculus | Differentiable implies continuous.
We prove the classic result that if a function is differentiable, then it is continuous. To start, we prove this for a two variable function and then repeat for an n-variable function. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcolleg
From playlist Multivariable Calculus
How to Determine if Functions are Linearly Independent or Dependent using the Definition
How to Determine if Functions are Linearly Independent or Dependent using the Definition If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Th
From playlist Zill DE 4.1 Preliminary Theory - Linear Equations
Thanks, Guys! From all of us at PHDtv
**ORDER our new book: http://WeHaveNoIdea.com It's been a crazy 12 months! We just wanted to thank you all for watching, laughing, and participating in our shenanigans in 2012. Here's to another year of PHDtv! All of our videos can be watched on our website: http://www.phdcomics.com/t
From playlist Recent Videos
FAIL LAB Episode Six: Pyrotechnics
Follow us on Facebook where the conversation continues: https://facebook.com/ZoochosisCOM A study of the male homo sapiens' neurological response to explosives. Or: "THAT WAS AWESOME!!!" Fail Lab is an irreverent new science series from filmmaker Patrick Scott, creator of the viral produ
From playlist Fail Lab by Zoochosis
FAIL LAB Episode Seven: Tribalism
Follow us on facebook where the conversation continues: https://www.facebook.com/ZoochosisCOM A study of the sociological and neurological implications of weapons and violence. Or: Real life needs a better rating system. FAIL LAB Episode Seven: Tribalism Series created, directed, and e
From playlist Fail Lab by Zoochosis
In Which Curiosity Goes Gangnam Style
**ORDER our new book: http://WeHaveNoIdea.com The AudioPH[i]D Ep. 1 - 10/16/12: In Which Curiosity Goes Gangnam Style Introducing the latest addition to PHDtv: The AudioPH[i]D (A PHD Podcast)! Join hosts Laurence Yeung, Crystal Dilworth, and Evans Boney as they rifle through this month's
From playlist The Audio Ph.[i]D. - News Podcast
**ORDER our new book: http://WeHaveNoIdea.com Experience the ultimate fan pilgrimage: Ph.Detours host Crystal Dilworth gets her geek on at Comic-Con and finds out what makes someone a superfan! More videos at http://www.phdcomics.com/tv Can't watch now? Read about it at http://www.phdco
From playlist New here? Try these!
In Which a Weight Has Been Lifted
**ORDER our new book: http://WeHaveNoIdea.com In a punny romp through the noir, the AudioPH[i]D team investigates the (hypothetical) theft of one of the world's most precious artifacts: The International Protoype Kilogram, the One True Kilogram against which all other kilograms are measur
From playlist The Audio Ph.[i]D. - News Podcast
Crowdfunding in Science? A conversation with Jai Ranganathan
**ORDER our new book: http://WeHaveNoIdea.com The AudioPHiD's Crystal Dilworth sits down with SciFund co-founder Jai Ranganathan to find out just what's going on with crowdfunding in science. This video is brought to you by PHDtv: http://www.phdcomics.com/tv (more videos!) Can't watch no
From playlist The Audio Ph.[i]D. - News Podcast
I define one of the most important constants in mathematics, the Euler-Mascheroni constant. It intuitively measures how far off the harmonic series 1 + 1/2 + ... + 1/n is from ln(n). In this video, I show that the constant must exist. It is an open problem to figure out if the constant is
From playlist Series
How to Prove a Function is Injective(one-to-one) Using the Definition
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys How to prove a function is injective. Injective functions are also called one-to-one functions. This is a short video focusing on the proof.
From playlist Proofs
Pythagorean Theorem VIII (Bhāskara's visual proof)
This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) following essentially Bhāskara's proof (Behold!). This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #math #manim #
From playlist Pythagorean Theorem
FAIL LAB Episode Five: Rodents
Follow us on Facebook where the conversation continues: https://facebook.com/ZoochosisCOM FAIL LAB Episode Five: Rodents An examination of cross-species empathy in adult homo sapiens. Or: Operation Enduring Freedom. Fail Lab is an irreverent new science series from filmmaker Patrick Sc
From playlist Fail Lab by Zoochosis
Taylor's Theorem with Remainder
This videos shows how to determine the error when approximating a function value with a Taylor polynomial. http://mathispower4u.yolasite.com/
From playlist Infinite Sequences and Series