Search results “Foundations of computational mathematics”

This is a seminar given at the University of Newcastle in April 2013. We explore the possibilities for a new more computational approach to mathematics which replaces the current dubious reliance on `real numbers' with the much more solid and natural rational numbers. The key idea is that of rational trigonometry (RT), whose basic laws are here introduced in a simple way using only very elementary linear algebra. The five main laws of RT are described, and proofs of the Cross law, the Spread law and the Triple spread formula are given.
Paul Miller's simple and elegant spread protractor is described. Some examples from the Zome construction system are illustrated. We discuss the beautiful new spread polynomials that arise from considering composites of spreads. The quadruple quad and quadruple spread formulas are described, and the relation with cyclic quadrilaterals is described.
Then we move to three dimensional applications, introducing projective rational trigonometry, and projective versions of the planar theory described earlier.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 4670
njwildberger

Lecture 1: Introduction and Proofs
Instructor: Tom Leighton
View the complete course: http://ocw.mit.edu/6-042JF10
License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

Views: 778374
MIT OpenCourseWare

Dr. Fariba Fahroo presents an overview of her program - Computational Mathematics - at the 2012 AFOSR Spring Review.

Views: 2027
TheAFOSR

Professor Alex Simpson, Personal Chair in Foundations of Computer Science, delivered his inaugural lecture entitled "The Intertwined Foundations of Mathematics and Computer Science".
Mathematics is commonly perceived as a subject in which there are absolute standards of truth and proof. This perception, however, is not entirely accurate. There are ways in which it is possible to shape mathematics to suit the applications to which it will be put.
In this talk, which is aimed at a general audience, Prof Simpson discusses various ways in which mathematics can be reshaped to take account of concepts arising in computer science.
He also briefly touches upon how such reshapings might even be of use within certain areas of mathematics itself.
Recorded on Thursday 17 May at the Informatics Forum, The University of Edinburgh.

Views: 5712
The University of Edinburgh

The ancient Greeks considered magnitudes independently of numbers, and they needed a way to compare proportions between magnitudes. Eudoxus developed such a theory, and it is the content of Book V of Euclid's Elements. This video describes this important idea.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here: http://www.wildegg.com/store/p100/product-Math-Foundations-A-screenshot-pdf

Views: 13433
njwildberger

Dr. Tiee is a professor in the UCSD math department as well as a researcher at the Center for Computational Math. In this video, Dr. Tiee shared his passion in mathematical modeling and past research projects, illustrating ways of transforming 2D models in to applicable 3D models. His experience brings to light the diverse possibilities available to math majors and the opportunities to work at the cutting edge of mathematical reasoning here at UCSD.

Views: 392
[email protected]

ICME Welcome and Overview
Margot Gerritsen, Director of ICME

Views: 1018
ICMEStudio

Stephen Wolfram, creator of Mathematica and Wolfram Alpha, gives a talk about the future of mathematics and computation.
(rule 30 description here: http://mathworld.wolfram.com/Rule30.html)

Views: 40701
University of Oxford

Univalent Foundations: New Foundations of Mathematics
Vladimir Voevodsky, Professor, School of Mathematics
http://www.ias.edu/people/faculty-and-emeriti/voevodsky
March 26, 2014
In Voevodsky’s experience, the work of a mathematician is 5% creative insight and 95% self-verification. Moreover, the more original the insight, the more one has to pay for it later in self-verification work. The Univalent Foundations project, started at the Institute a few years ago, aims to lower the price by giving mathematicians the ability to verify their constructions with the help of computers. Voevodsky will explain how new ideas that make this goal attainable arise from the meeting of two streams of development—one in constructive mathematics and the theory and practice of programming languages, and the other in pure mathematics. The Institute for Advanced Study is pleased to designate this lecture in honor of the Princeton Adult School’s 75th Anniversary. The Institute supports and shares the Adult School’s mission to promote and foster lifelong learning and exploration in the Princeton community and beyond.
More videos at http://video.ias.edu

Views: 5510
Institute for Advanced Study

Richard Dedekind around 1870 introduced a new way of thinking about what a real number `was'. By analyzing the case of sqrt(2), he concluded that we could associated to a real number a partition of the rational numbers into two subsets A and B, where all the elements of A were less than all the elements of B, and where A had no greatest element. Such partitions are now called Dedekind cuts, and purport to give a logical and substantial foundation for the theory of real numbers.
Does this actually work? Can we really create an arithmetic of real numbers this way? No and no. It does not really work. In this video we raise the difficult issues that believers like to avoid.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
A screenshot PDF which includes MathFoundations80 to 121 can be found at my WildEgg website here: http://www.wildegg.com/store/p102/product-Math-Foundations-B1-screenshot-pdf

Views: 14939
njwildberger

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! A big THANKS to all of those who support me on Patreon! https://www.patreon.com/patrickjmt
Part 1: https://youtu.be/KRLBya7x5ZQ
Extra Proof by Contradiction with some death intrigue (huh?!) https://www.youtube.com/watch?v=rOGqq1O1rzI&feature=youtu.be
New to proving mathematical statements and theorem? I this video I prove the statement 'the sum of two consecutive numbers is odd' using direct proof, proof by contradiction, proof by induction and proof by contrapositive.

Views: 79003
patrickJMT

The 13th Israeli Mini-Workshop in Applied and Computational Mathematics
Prof. Zeev Zalevsky of the Faculty of Engineering, presents novel photonic approaches and means to exceed the limitations of vision science and eventually to allow for super resolved imaging and improved capabilities.

Views: 251
barilanuniversity

Algorithms are the sets of steps necessary to complete computation - they are at the heart of what our devices actually do. And this isn’t a new concept. Since the development of math itself algorithms have been needed to help us complete tasks more efficiently, but today we’re going to take a look a couple modern computing problems like sorting and graph search, and show how we’ve made them more efficient so you can more easily find cheap airfare or map directions to Winterfell... or like a restaurant or something.
Ps. Have you had the chance to play the Grace Hopper game we made in episode 12. Check it out here! http://thoughtcafe.ca/hopper/
CORRECTION:
In the pseudocode for selection sort at 3:09, this line:
swap array items at index and smallest
should be:
swap array items at i and smallest
Produced in collaboration with PBS Digital Studios: http://youtube.com/pbsdigitalstudios
Want to know more about Carrie Anne?
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The Latest from PBS Digital Studios: https://www.youtube.com/playlist?list...
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Views: 517681
CrashCourse

A short trailer about Norman's channel --- Insights into Mathematics --- which aims to reorient mathematics education and research, by looking more carefully at the logical foundations of the subject. We give a quick overview of the main Playlists here, and our general orientation towards a new, more careful mathematics.
The sister channel ---- Wild Egg mathematics courses --- is where the Algebraic Calculus videos are posted, so be sure to check that out too.

Views: 6714
njwildberger

Science and Engineering Practice 5:
Using Mathematics and Computational Thinking
Paul Andersen explains how mathematics and computational thinking can be used by scientists to represent variables and by engineers to improve design. He starts by explaining how mathematics is at the root of all sciences. He then defines computational thinking and gives you a specific example of computational modeling. He finishes the video with a teaching progression for this practice.
Intro Music Atribution
Title: I4dsong_loop_main.wav
Artist: CosmicD
Link to sound: http://www.freesound.org/people/CosmicD/sounds/72556/
Creative Commons Atribution License

Views: 23807
Bozeman Science

Our computational mathematics program will challenge you to explore the world through algorithms, calculations and logical reasoning. You'll find yourself on the cusp of the evolution of many industries.
www.grenfell.mun.ca/math

Views: 578
GrenfellCampus

The mathematics of the coming century is going to look dramatically different. Real numbers will go the way of toaster fish; claims of infinite operations and limits will be recognized as the balder dash they often are; and finite, concrete, write-downable mathematics will enter centre stage.
In this overview video, we look at some of the directions that mathematics will take, once the real number dream is abandoned. You can think of this as a bird's eye view of a lot of the rest of this video series.
It is an exciting time. Mathematics has not had a proper revolution; surely it is long overdue!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
A screenshot PDF which includes MathFoundations80 to 121 can be found at my WildEgg website here: http://www.wildegg.com/store/p102/product-Math-Foundations-B1-screenshot-pdf

Views: 16985
njwildberger

In this final video on the most fundamental and important problem in mathematics [which happens to be: How to model the continuum?] we tackle the seriously unfortunate developments leading to the current misunderstandings about the so-called 'real numbers'. Of course this name is a complete misnomer: they are not 'real' at all; rather they constitute a desperate attempt to enforce the existence in mathematics of objects which are actually unattainable without resorting to an infinite number of computational steps (whatever that might actually mean!)
In this video we give a bird's eye view of the various misguided attempts at establishing 'real numbers' and sketch some of the logical and technical difficulties that students are usually shielded from. The basic construction arises from Stevin's decimal numbers extended, using a dollop of wishful thinking, to arbitrary infinite decimals, not just the repeating decimals encoded by rational numbers. Understanding the difficulties with this approach is not that hard, and in essence the same kinds of problems resurface in the various variants which we also discuss: infinite sequences of nested intervals of rational numbers, monotonic and bounded sequences of rationals, Cauchy sequences of rationals, equivalence classes of Cauchy sequences, and finally the icing on the cake of irrationality: Dedekind cuts.
Students of mathematics! Listen carefully: none of these approaches work. This is the reason why not one of these 'theories' are properly laid out in front of you when you begin work in calculus or even analysis. To those who would try to convince you otherwise, via appeals to authority or numbers, name-calling, or by special pleading on behalf of all those lovely 'results' that supposedly follow from the required beliefs: ask rather for explicit examples and concrete computations.
These are the true coin in the realm of mathematics, and will not lead you astray.
This is perhaps a place to thank my many contributors, subscribers and online friends. We are on our way to a more beautiful and logically coherent mathematics, but there is a long ways to go from here to there! Your support is a big help.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 24413
njwildberger

We have prepared a series of videos that explain the mathematical foundations of Artificial Intelligence (AI), Machine Learning (ML), and Data Science. This first video covers set theory and proofs from first principles.

Views: 941
Thalesians

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Graph Theory - An Introduction! In this video, I discuss some basic terminology and ideas for a graph: vertex set, edge set, cardinality, degree of a vertex, isomorphic graphs, adjacency lists, adjacency matrix, trees and circuits.
There is a MISTAKE on the adjacency matrix; I put a 1 in the v5 row and v5 column, but it should be placed in the v5 row and the v6 column. There are annotations pointing this out along with the corrected matrix!

Views: 415322
patrickJMT

Transcript:
In this video, I will be explaining what Discrete Mathematics is, and why it's important for the field of Computer Science and Programming. Discrete Mathematics is a branch of mathematics that deals with discrete or finite sets of elements rather than continuous or infinite sets of elements. Imagine trying to run a program that requires an infinite number of executions to complete a task. It's obvious to say, that the program would run forever and the task would never be completed because there is an infinite number of executions. In order to avoid this problem, we approximate the continuous sets with discrete sets. Now you may be thinking, I never use math that involves infinite sets, but I promise that you do. The simplest example is with a circle. A circle by definition is an infinite number of points equally distant from a fixed point. The problem with this is that if we try to write a program that prints out all of these points, it will run forever because there is an infinite number of points and therefore an infinite number of executions. So, this is physically impossible, that's why if we zoom in here, you can see that when you come down here, there is all these points, but in reality we should have even more points between these points. And if we zoom in on those, we should have more points between those points, and we can never complete the task. Now we've all seen circles on computers, how is this possible, because we just established that it's impossible. The answer is, is that there is approximations. For example, consider regular polygons. Regular Polygons, like a triangle, or a square, or a pentagon. They don't really look like circles. However, if you keep increasing the number of vertices. Eventually you will get hexagons, octagons, decagons, hexadecagons, icosagons. You can see that these regular polygons, the more and more you increase the number of vertices, which the vertices are equally distant from a fixed point, they will eventually approximate a circle, and eventually they will be indistinguishable to the naked eye and will look identical to a circle.
We're busy people who learn to code, then practice by building projects for nonprofits. Learn Full-stack JavaScript, build a portfolio, and get great references with our open source community.
Join our community at https://freecodecamp.com
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Follow Quincy on Quora: https://www.quora.com/Quincy-Larson

Views: 64167
freeCodeCamp.org

The integers are introduced as pairs of natural numbers, representing differences. The standard arithmetical operations are also defined. Often these important mathematical objects are defined only loosely, by `negating' somehow the usual natural numbers. However this makes proving the laws of arithmetic more painful, as then we need to worry about different cases. This procedure here, developed in the 19th century, provides a more uniform approach with distinct theoretical advantages.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here: http://www.wildegg.com/store/p100/product-Math-Foundations-A-screenshot-pdf

Views: 11939
njwildberger

MIT 7.91J Foundations of Computational and Systems Biology, Spring 2014
View the complete course: http://ocw.mit.edu/7-91JS14
Instructor: Christopher Burge, David Gifford, Ernest Fraenkel
In this lecture, Professors Burge, Gifford, and Fraenkel give an historical overview of the field of computational and systems biology, as well as outline the material they plan to cover throughout the semester.
License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

Views: 69877
MIT OpenCourseWare

Want to know more about studying at Oxford University? Watch this short film to hear tutors and students talk about this undergraduate degree. For more information on this course, please visit our website at https://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics-and-computer-science

Views: 15970
University of Oxford

A revised version of this video is available at https://youtu.be/2PSqWBIrn90 .
This video gives an accessible introduction to persistent homology, which is a popular tool in topological data analysis and also a subject of my research.
Citations and links to papers referenced in this video:
Robert Ghrist, "Barcodes: The Persistent Topology of Data"
http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01191-3/
Herbert Edelsbrunner and John Harer, "Persistent Homology -- A Survey", in Twenty Years After, AMS (2007).
http://cygnus-x1.cs.duke.edu/~edels/Papers/2008-B-02-PersistentHomology.pdf
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, "Stability of Persistence Diagrams", in Discrete and Computational Geometry, vol. 37 (2007), p. 103-120.
ftp://ftp-sop.inria.fr/prisme/dcohen/Papers/Stability.pdf
Afra Zomorodian and Gunnar Carlsson, "Computing Persistent Homology", in Discrete and Computational Geometry, vol. 33 (2005), p. 249-274.
http://www.cs.dartmouth.edu/~afra/papers/socg04/persistence.pdf
Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian, "On the Local Behavior of Spaces of Natural Images", in International Journal of Computer Vision, vol. 76 (2008), p. 1-12.
http://pages.pomona.edu/~vds04747/public/papers/CIdSZ_natural.pdf
Jose Perea and Gunnar Carlsson, "A Klein-Bottle-Based Dictionary for Texture Representation", in International Journal of Computer Vision, vol. 107 (2014), p. 75-97.
https://fds.duke.edu/db/attachment/2638
Jose Perea and John Harer, "Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis", in Foundations of Computational Mathematics, vol. 15 (2015), p. 799-838.
http://arxiv.org/abs/1307.6188

Views: 17371
Matthew Wright

Views: 39
Ann Reid

This video gives an accessible introduction to persistent homology, which is a popular tool in topological data analysis and also a subject of my research.
This is a revised version of the video that first appeared at https://youtu.be/h0bnG1Wavag .
Citations and links to papers referenced in this video:
Robert Ghrist, "Barcodes: The Persistent Topology of Data"
http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01191-3/
Herbert Edelsbrunner and John Harer, "Persistent Homology -- A Survey", in Twenty Years After, AMS (2007).
http://cygnus-x1.cs.duke.edu/~edels/Papers/2008-B-02-PersistentHomology.pdf
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, "Stability of Persistence Diagrams", in Discrete and Computational Geometry, vol. 37 (2007), p. 103-120.
ftp://ftp-sop.inria.fr/prisme/dcohen/Papers/Stability.pdf
Afra Zomorodian and Gunnar Carlsson, "Computing Persistent Homology", in Discrete and Computational Geometry, vol. 33 (2005), p. 249-274.
http://www.cs.dartmouth.edu/~afra/papers/socg04/persistence.pdf
Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian, "On the Local Behavior of Spaces of Natural Images", in International Journal of Computer Vision, vol. 76 (2008), p. 1-12.
http://pages.pomona.edu/~vds04747/public/papers/CIdSZ_natural.pdf
Jose Perea and Gunnar Carlsson, "A Klein-Bottle-Based Dictionary for Texture Representation", in International Journal of Computer Vision, vol. 107 (2014), p. 75-97.
https://fds.duke.edu/db/attachment/2638
Jose Perea and John Harer, "Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis", in Foundations of Computational Mathematics, vol. 15 (2015), p. 799-838.
http://arxiv.org/abs/1307.6188

Views: 4630
Matthew Wright

Happy New Year everyone, and I wish you all the best for 2015!
In this video we introduce some basic orientation to the problem of how we represent, and think about, space in mathematics. One key idea is the fundamental duality between the affine and projective views: two sides of the same coin.
We explain how the Cartesian revolution of the 17th century built geometry from a prior theory of arithmetic: for us that of the rational numbers- of course!
And we introduce some useful notations for points and proportions, and give a geometrical view of the relation between the affine line and the projective line.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
A screenshot PDF which includes MathFoundations122 to 149 can be found at my WildEgg website here: http://www.wildegg.com/store/p103/product-math-foundations-B2-screenshot-pdfs

Views: 7929
njwildberger

Computational Mathematics in the Department of Bioengineering, Imperial College London

Views: 1152
Liam Madden

Hey Swarms. Today I have a video about a Ferrari Owners Club Malaysia gathering that took place in Sunway Hotel near Sunway Lagoon. This video was taken about a few months ago. Enjoy!

Views: 1657
ImpexBee

Professor Goodhill's research aims to discover the computational rules underlying brain development and function. He originally trained in the UK in maths, physics and artificial intelligence, and then spent 10 years researching in the USA, including 8 as a professor of neuroscience at Georgetown University. He moved to the University of Queensland in 2005, where he holds a joint appointment between the Queensland Brain Institute and School of Mathematics and Physics. His lab uses experimental, mathematical and computational techniques to understand the brain as a computational device.
Professor Goodhill did a Joint Honours BSc in Mathematics and Physics at Bristol University (UK), followed by an MSc in Artificial Intelligence at Edinburgh University and a PhD in Cognitive Science at Sussex University. Following a postdoc at Edinburgh University he moved to the USA in 1994, where he did further postdoctoral study in Computational Neuroscience at Baylor College of Medicine and the Salk Institute. Professor Goodhill formed his own lab at Georgetown University in 1996, where he was awarded tenure in the Department of Neuroscience in 2001. In 2005 he moved to a joint appointment between the Queensland Brain Institute and the School of Mathematical and Physical Sciences at the University of Queensland.
This talk was given at a TEDx event using the TED conference format but independently organized by a local community. Learn more at http://ted.com/tedx

Views: 9189
TEDx Talks

Oregon Programming Languages Summer School
Parallelism and Concurrency
July 3-21, 2018
University of Oregon
https://www.cs.uoregon.edu/research/summerschool/summer18/
Title: Computational Type Theory [1/5]
Speaker: Robert Harper, Carnegie Mellon University
Date: Monday, 16 July 2018, Session 1
Topics:
- fundamental premise of constructivism
- type theory as basis for all mathematics
- type theory as a programming language
- theory of truth vs. theory of formal proof
- deterministic operational semantics
- abstract syntax with binding, scope, and substitution
- forms of expression
- judgment forms: values and transition rules
- derived notion
- binary decision diagrams
- types are specifications of program behavior
- judgments as forms of expression
- algorithms as communication
- behavioral vs. structural expressions
- types and values are programs
- families of types
- type-indexed families of types (a.k.a., dependent types)
- hypothetical/general judgments
- functionality: respecting equality of indices
- what is equality of types?
- equi-satisfaction
- preview of higher-dimensional/cubical type theory
- meanings of judgments, a.k.a., meaning explanations, a.k.a., computational semantics
- equality of canonical types, a.k.a., type-values
- head expansion lemma, a.k.a., reverse execution
© 2018, University of Oregon

Views: 368
OPLSS

MIT 7.91J Foundations of Computational and Systems Biology, Spring 2014
View the complete course: http://ocw.mit.edu/7-91JS14
Instructor: David Gifford
Prof. Gifford talks about library complexity as it relates to genome sequencing. He explains how to create a full-text minute-size (FM) index, which involves a Burrows-Wheeler transform (BWT). He ends with how to deal with the problem of mismatching.
License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

Views: 12355
MIT OpenCourseWare

Today we introduce propositional logic. We talk about what statements are and how we can determine truth values.
Visit my website: http://bit.ly/1zBPlvm
Subscribe on YouTube: http://bit.ly/1vWiRxW
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

Views: 100421
TheTrevTutor

Cool Math, Math is Understanding, the Continuum hypothesis and computational proof
http://youtu.be/ekZrJ9jjNXQ
Math is understanding
Math relies on understanding to move forward.
If you don't understand 1+1=2 then how could you get to the point where you understand 2+2=4?
If you did not understand 1+1=2 you simply could not move forward. Unless someone has stated that it is true by way of proof. You should be able to take it as fact and deduce with logic that 1+2 would = 3. and from there you could move further.
Well, maths has moved a lot further than simple addition. But it is still not totally understood.
Usually, something is either true or false and that is it.
But, there is 1 problem that as far as we can tell has another answer. Undecided.
This is the Continuum hypothesis
without getting into it too deeply, the basic idea is this,
We have a number line, that runs to infinity in either direction 1234 etc
and -1-2-3-4-5 etc. these are natural numbers.
Lets call the infinity that the numbers go to 'Infinity set A'.
Now when we look between 1+2 we see ½ and 1/4 and thirds etc etc these are called real numbers there is also an infinite number of these numbers just between 1 and 2.
so lets call the sets between natural numbers infinity set B.
It is agreed that infinity B is bigger than infinity A.
Now the question.
Is there an infinity set that is smaller than B but bigger than A?
Its a tough question to answer. Infact it is so hard that the only proofs we have are
that it has been proven that we cannot prove it to be false.
and we can not prove it to be true.
Because we cant properly comprehend infinities we just cant get an answer either way.
But, what if a computer could comprehend infinities and actually answer the question for us?
and say Yes. there is an infinity between A and B.
Well, then we would have an answer. But how can we tell if the answer is right or wrong? Maybe get another computer to tell us, but then, we go down a never ending spiral of 'But how do we know for sure?'
Well, math is heading in this direction. We are looking at computers to solve problems that are so deep and complex that we can hardly understand them. Take the information it spits out and then go on to use it in another equation. for deeper and more complex answers.
But What's the value of an answer without the understanding?
Well, with these answers we will start to be able to do some pretty crazy things.
Who knows, maybe one day we will meet aliens. and the alien will ask us, how the heck did you get all the way out here.
and our answer will be...I dont know.
This video is completely open source

Views: 25761
Cool Math

There is one mathematical problem which is much more important and fundamental, both historically and practically, than any other, and which impinges on almost all areas of pure mathematics.
This is the first of four videos on this most fundamental and important problem.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
If you are interested in supporting my production of high quality math videos, why not consider becoming a Patron of this channel? Here is the link to my Patreon page: https://www.patreon.com/njwildberger?ty=h

Views: 20720
njwildberger

We introduce the two basic operations on natural numbers: addition and multiplication. Then we state the main laws that they satisfy. This is a basic and fundamental fact about natural numbers; that we can combine them in these two different ways. A lot of arithmetic, and later algebra, comes down to the interaction between addition and multiplication!
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here: http://www.wildegg.com/store/p100/product-Math-Foundations-A-screenshot-pdf

Views: 32284
njwildberger

This film discusses the techniques, and outlines the vision of the future computerization of pure mathematics through interviews and talk segments from renowned mathematicians, meta-mathematicians, computational mathematicians, and theorem provers.
The interviews and talks were conducted at the Sloan Foundation sponsored "Semantic Representation of Mathematical Knowledge Workshop", held at the Fields Institute (Toronto) in February 2016.
Scripted, produced, and edited by Amy Young and Michael Trott.

Views: 8400
Wolfram

Avi Wigderson is a professor of Mathematics at the Institute for Advanced Study in Princeton. After studying Computer Science at Technion in Haifa, he obtained his PhD in 1983 from Princeton University. He held then various visiting positions including IBM Research at San Jose, MSRI Berkeley, and IAS Princeton. From 1986 to 2003 he was associate professor at the Hebrew University in Jerusalem. Wigderson has been for two decades a leading figure in the field of Mathematics of Computer Science, with fundamental contributions, in particular in Complexity Theory, Randomness, and Cryptography. He has been invited speaker at ICM in Tokyo (1990), and Zurich (1994), and plenary speaker in Madrid (2006). Among many awards he received both the Nevanlinna Prize (1994), and the Gödel Prize (2009).
This lecture about a computational theory of randomness was hold on 10 May 2012 at ETH Zurich, when Avi Wigderson was invited as guest speaker of the Wolfgang Pauli Lectures. The Wolfgang Pauli Lectures are an annual lecture series that is devoted alternately to physics, mathematics and biology. They are named after the great theoretical physicist and Nobel laureate Wolfgang Pauli, who was professor at ETH Zurich from 1928 until his death in 1958.

Views: 15216
ETH Zürich

Anders Hansen: What is the Solvability Complexity Index (SCI) of your problem? - On the SCI Hierarchy and the foundations of computational mathematics
Abstract: This talk addresses some of the fundamental barriers in the theory of computations. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence (think about computing eigenvalues of a matrix for example). However, as we demonstrate, for several basic problems in computations such as computing spectra of operators, solutions to inverse problems, roots of polynomials using rational maps, solutions to convex optimization problems, imaging problems etc. such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in order to compute a desired quantity. The SCI phenomenon is independent of the axiomatic setup and hence any theory aiming at establishing the foundations of computational mathematics will have to include the so called SCI Hierarchy. We will specifically discuss the vast amount of classification problems in this non-collapsing complexity/computability hierarchy that occur in inverse problems, compressed sensing problems, l1 and TV optimization problems, spectral problems, PDEs and computational mathematics in general.
The lecture was held within the framework of the Hausdorff Trimester Program Mathematics of Signal Processing. (17.02.2016)

Views: 129
Hausdorff Center for Mathematics

Rational trigonometry works over the rational numbers, and allows us a more elementary and logical approach to the basics of trigonometry. This video illustrates the Spread law, the Cross law and the Triple spread formula. These are among the most important formulas in geometry, indeed in all of mathematics, and they allow us to recast trigonometry into a simpler and more computationally elegant subject. See the WildTrig YouTube series for lots of applications of these laws.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger. I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?ty=h .
A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here: http://www.wildegg.com/store/p100/product-Math-Foundations-A-screenshot-pdf

Views: 7428
njwildberger

Maxel algebra is a large scale extension of matrix algebra. To see ordinary matrix algebras inside as subalgebras, it is useful to focus on particular diagonal maxels determined by sets J of natural numbers. These are idempotent elements in the maxel world, and we will see how they project maxels onto the rows and columns of J, and restrict a maxel to J x J.
This gives us a way of seeing the entire nested family or matrix algebras M(nxn) imbedded one inside another.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
Please consider supporting this Channel bringing you high quality mathematics lectures by becoming a Patron at https://www.patreon.com/njwildberger?
Screenshot pdf's for the lectures are available at http://www.wildegg.com/divineproportions-rationaltrig.html
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/
A screenshot PDF which includes MathFoundations150 to 183 can be found at my WildEgg website here: http://www.wildegg.com/store/p104/product-Math-Foundations-C-screenshot-pdf

Views: 1642
njwildberger

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Current Dividend Preference. Participating Preferred Stock. Convertible Preferred Stock. Cumulative preferred stock includes a provision that requires the company to pay preferred shareholders all dividends, including those that were omitted in the past, before the common shareholders are able to receive their dividend payments. Non-cumulative preferred stock does not issue any omitted or unpaid dividends. If the company chooses not to pay dividends in any given year, the shareholders of the non-cumulative preferred stock have no right or power to claim such forgone dividends at any time in the future. Participating preferred stock provides its shareholders with the right to be paid dividends in an amount equal to the generally specified rate of preferred dividends, plus an additional dividend based on a predetermined condition. This additional dividend is typically designed to be paid out only if the amount of dividends received by common shareholders is greater than a predetermined per-share amount. If the company is liquidated, participating preferred shareholders may also have the right to be paid back the purchasing price of the stock as well as a pro-rata share of remaining proceeds received by common shareholders. Significance to Investors. Shareholder. Preferred Stock.