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Computational Mathematics
 
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For more info visit: www.duq.edu/computational-math
Views: 10362 Duquesne Liberal Arts
Towards a more computational mathematics: rational trigonometry and new foundations for geometry
 
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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?... .
Lec 1 | MIT 6.042J Mathematics for Computer Science, Fall 2010
 
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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: 885826 MIT OpenCourseWare
Arrays and matrices I Data structures in Mathematics Math Foundations 164 | NJ Wildberger
 
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We introduce the ideas of arrays and matrices as 2 dimensional data structures. In this video we define arrays as lists of lists, which is standard practice in computer science and popular programming environments. But we will go a bit beyond the usual two dimensional situation, looking also at three dimensional arrays. It should be noted that we are going to distinguish between arrays and matrices. Matrices for us will have quite a different definition: giving a rather novel direction for linear algebra, also one quite in line with a computational orientation! 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
1. Introduction to Computational and Systems Biology
 
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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: 76791 MIT OpenCourseWare
Practice 5 - Using Mathematics and Computational Thinking
 
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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: 24986 Bozeman Science
Four Basic Proof Techniques Used in Mathematics
 
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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: 124656 patrickJMT
Prof. Alex Simpson - The Intertwined Foundations of Mathematics and Computer Science
 
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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.
The successor - limit hierarchy | Data Structures in Mathematics Math Foundations 180
 
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Addition, multiplication and exponentiation are just the first three arithmetical operations on a fascinating ladder of operations which ascends to dizzying heights. Here we introduce this fascinating successor-limit hierarchy using the notions of successor and diagonal limit that we discussed in our last video. We make a rather detailed analysis of one particular higher arithmetical computation here that the viewer might follow carefully. We are moving beyond what we can do computationally here: so there is certainly an air of unreality about such fantastically large scale arithmetical contemplations. But we are still light years away from the complete fantasy of supposing that we can encompass and then transcend ALL natural numbers to obtain infinity! 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
Applied and Computational Mathematics Information Session: Fall 2017
 
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This information session was held October 19, 2017. For more information about our Applied and Computational Mathematics program, please visit https://ep.jhu.edu/acm.
Dr. Fariba Fahroo - Computational Mathematics
 
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Dr. Fariba Fahroo presents an overview of her program - Computational Mathematics - at the 2012 AFOSR Spring Review.
Views: 2092 TheAFOSR
Saul Kato: The Future of Computational Biology - Schrödinger at 75: The Future of Biology
 
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Kato is head of the Foundations of Cognition Laboratory and assistant professor of neurology and physiology at the University of California, San Francisco. Saul and his team study the brain and behavior of the nematode C. elegans in search of basic principles and building blocks of neural computation and cognitive function. Saul has a background in neurobiology, theoretical physics, mathematics, computer science and hardware and software. After studying many-body quantum mechanics with Nobelist Bob Laughlin as an undergraduate at Stanford University, Saul spent a decade as an engineer and tech entrepreneur before returning to science; he founded, financed, built, and sold two technology companies: Sven Technologies, which developed software algorithms and applications for 3D graphics rendering, and WideRay Corp., which pioneered the idea of local ad-hoc wireless content delivery, manufacturing and building a network of several thousand local content delivery points in fifteen countries. After getting his PhD from Columbia University in 2013 – a computational-experimental project determining the signal processing properties of C. elegans neurons, jointly advised by Larry Abbott at the Center for Theoretical Neuroscience at Columbia and Cori Bargmann at Rockefeller University – Saul became an EMBO Long-Term Fellow in the lab of Manuel Zimmer at the Institute of Molecular Pathology in Vienna, Austria. Saul and the rest of the team deciphered a tight relationship between the dynamics of brain-wide activity and behavior in C. elegans. http://www.tcd.ie/
Dedekind cuts and computational difficulties with real numbers | Famous Math Problems 19c
 
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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?... .
F. William Lawvere - What are Foundations of Geometry and Algebra? (2013)
 
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Keynote lecture at the Fifty Years of Functorial Semantics conference, Union College, October 2013. http://www.math.union.edu/~niefiels/13conference/Web/ Transcript: http://www.math.union.edu/~niefiels/13conference/Web/Slides/Fifty_Years_of_Functorial_Semantics.pdf Abstract From observation of, and participation in, the ongoing actual practice of Mathematics, Decisive Abstract General Relations (DAGRs) can be extracted; when they are made explicit, these DAGRs become a guide to further rational practice of mathematics. The worry that these DAGRs may turn out to be as numerous as the specific mathematical facts themselves is overcome by viewing the ensemble of DAGRs as a ’Foundation’, expressed as a single algebraic system whose current description can be finitely-presented. The category of categories (as a cartesian closed category with an object of small discrete categories) aims to serve as such a Foundation. One basic DAGR is the contrast between space and quantity, and especially the relation between the two that is expressed by the role of spaces as domains of variation for intensively and extensively variable quantity; in that way, the foundational aspects of cohesive space and variable quantity inherently includes also the conceptual basis for analysis, both for functional analysis and for the transformation from continuous cohesion to combinatorial semi-discreteness via abstract homotopy theory. Function spaces embody a pervasive DAGR. The year 1960 was a turning point. Kan, Isbell, Grothendieck and Yoneda had further developed the Eilenberg-Mac Lane Theory of Naturality. Their work implicitly pointed towards such a Foundation as a foreseeable goal. Although the work of those four great mathematicians was still unknown to me, I had independently traversed a sufficient fragment of a similar path to encourage me to become a student of Professor Eilenberg. As I slowly became aware of the importance of those earlier developments, I attempted to participate in the realization of a Foundation in the sense described above, first through concentration on the particular docrine known as Universal Algebra, making explicit the fibered category whose base consists of abstract generals (called theories) and whose fibers are concrete generals (known as algebraic categories). The term ’Functorial Semantics’ simply refers to the fact that in such a fibered category, any interpretation T 0 → T of theories induces a map in the opposite direction between the two categories of concrete meanings; this is a direct generalization of the previously observed cases of linear algebra, where the abstract generals are rings and the fibers consist of modules, and of group theory where the dialectic between abstract groups and their actions had long been fundamental in practice. This kind of fibration is special, because the objects T in the base are themselves categories, as I had noticed after first rediscovering the notion of clone, but then rejecting the latter on the basis of the principle that, to compare two things, one must first make sure that they are in the same category; when the two are (a) a theory and (b) a background category in which it is to be interpreted, comparisons being models., the category of categories with products serves. Left adjoints to the re-interpretation functors between fibers exist in this particular doctrine of general concepts, unifying a large number of classical and new constructions of algebra. Isbell conjugacy can provide a first approximation to the general space vs quantity pseudo-duality, because recent developments (KIGY) had shown that also spaces themselves are determined by categories (of figures and incidence relations inside them). My 1963 thesis clearly explains that presentations (having a signature consisting of names for generators and another signature consisting of names for equational axioms) constitute one important source of theories. This syntactical left adjoint directly generalizes the presentations known from elimination theory in linear algebra and from word problems in group theory. No one would confuse rings and groups themselves with their various syntactical presentations, but previous foundations of algebra had underemphasized the existence of another important method for constructing examples, namely the Algebraic Structure functor. Being a left adjoint , it can be calculated as a colimit over finite graphs. Fundamental examples, like cohomology operations as studied by the heroes of the 50’s, show that typically an abstract general (such as an isometry group) arises by naturality; to find a syntactical presentation for it may then be an important question. This extraction, by naturality from a particular family of cases, provides much finer invariants, and as a process bears a profound resemblance to the basic extraction of abstract generals from experience.
Views: 3735 Matt Earnshaw
New: Mathematics Curriculum for the 21st Century Student
 
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This book, published by SIAM, provides the essential foundations of both linear and nonlinear analysis necessary for understanding and working in twenty-first century applied and computational mathematics. In addition to the standard topics, this text includes several key concepts of modern applied mathematical analysis that should be, but are not typically, included in advanced undergraduate and beginning graduate mathematics curricula. This material is the introductory foundation upon which algorithm analysis, optimization, probability, statistics, differential equations, machine learning, and control theory are built. When used in concert with the free supplemental lab materials, this text teaches students both the theory and the computational practice of modern mathematical analysis. Learn more and purchase your copy: http://bookstore.siam.org/ot152/
Mathematical foundation of computer science
 
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Mathematical foundation of computer science for UGC NET, GATE (Set, Relation and Function)
Views: 10179 subhash singh
Univalent Foundations: New Foundations of Mathematics | Vladimir Voevodsky
 
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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
The Mathematics Of Intelligence | Geoff Goodhill | TEDxUQ
 
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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: 10296 TEDx Talks
Bases and dimension for integral linear spaces II | Abstract Algebra Math Foundations 222
 
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We want to tackle the explicit computational question of how to find a convenient basis of an integral linear space. Given some msets, they span an integral linear space, but we would like to find a simpler basis for such a space. The algorithm which we introduce is a variant of the row reduction algorithm from linear algebra, also sometimes called Gaussian elimination. However one big difference is that we are only working with integral combinations, not rational ones. It is somewhat pleasant that such an important algorithm figures prominently in this first foray into abstract algebra!
Difficulties with Dedekind cuts | Real numbers and limits Math Foundations 116 | N J Wildberger
 
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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
13th Workshop in Applied & Computational Math
 
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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: 255 barilanuniversity
Mathematics without real numbers | Real numbers and limits Math Foundations 119 | N J Wildberger
 
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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
CSSE Lecture: Towards a More Computational Mathematics
 
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Guest Lecturer A/Prof Norman Wildberger from UNSW presents "Towards a More Computational Mathematics: Rational Trigonometry and New Foundations For Geometry" - a novel approach to fundamental trigonometric mathematics.
Views: 556 UON FEBE
Difficulties with real numbers as infinite decimals (II) | Real numbers + limits Math Foundations 92
 
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This lecture introduces some painful realities which cast a long shadow over the foundations of modern analysis. We study the problem of trying to define real numbers via infinite decimals from an algorithmic/constructive/computational point of view. There are many advantages of trying to do this: historically this was the point of view towards decimals like sqrt(2) or pi or e, and this provides us with tools to define and evaluate infinite series, functions and integrals. However in reality the idea bumps against seemingly unsurmountable technical obstacles: the difficulties in defining algorithms and implementing arithmetical operations at this level, non-uniqueness of algorithms and corresponding ambiguity in recognizing equality of real numbers, and a vagueness or tautological aspect to arithmetic with these objects. 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. 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 MathFoundations80 to 121 can be found at my WildEgg website here: http://www.wildegg.com/store/p102/product-Math-Foundations-B1-screenshot-pdf
Towards a Semantic Language of Mathematics
 
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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: 8830 Wolfram
Meet Christopher Tiee, Researcher in Computational Mathematics
 
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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: 416 [email protected]
The mostly absent theory of real numbers|Real numbers + limits Math Foundations 115 | N J Wildberger
 
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In this video we ask the question: how do standard treatments of calculus and analysis deal with the vexatious issue of defining real numbers and their supposed arithmetic?? We pull out a selection of popular Calculus and Analysis texts, and go through them with a view of finding out: what exactly is a real number? All the books I examine are excellent books---aside from their treatment of foundational issues, where we see that they mostly fall clearly short. We look at Calculus texts by Steward, Sallas Hille and Etgen, Rogawski, Courant, Spivak, Caunt, Apostol, Keisler and Adams, and Analysis texts by Spiegel, Apostol, Royden, Kolmogorov and Fomin, and Rudin. This video really should be an eye opener to students of mathematics. Yes, it is possible to challenge the standard thinking, and the mathematical world need not collapse. Admitting current weaknesses, and the lack of acknowledgement of them by the Academy, is an important step in moving forward to a newer, better mathematics. 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?... . 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
Real numbers as Cauchy sequences don't work! | Real numbers and limits Math Foundations 114
 
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This longish video lays out the various reasons why Cauchy sequences---as a basis for the theory of real numbers---don't work. Necessary viewing for all maths students! We really need to start addressing the logical weaknesses, rather than pretending that they are not there! 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
Anders Hansen: What is the Solvability Complexity Index SCI....
 
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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)
Reorienting pure mathematics education and research | Insights into Mathematics | N J Wildberger
 
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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.
Graph Theory - An Introduction!
 
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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: 448173 patrickJMT
Intro to Algorithms: Crash Course Computer Science #13
 
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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? https://about.me/carrieannephilbin The Latest from PBS Digital Studios: https://www.youtube.com/playlist?list... Want to find Crash Course elsewhere on the internet? Facebook - https://www.facebook.com/YouTubeCrash... Twitter - http://www.twitter.com/TheCrashCourse Tumblr - http://thecrashcourse.tumblr.com Support Crash Course on Patreon: http://patreon.com/crashcourse CC Kids: http://www.youtube.com/crashcoursekids
Views: 659478 CrashCourse
Maths for Programmers: Introduction (What Is Discrete Mathematics?)
 
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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 Follow us on twitter: https://twitter.com/freecodecamp Like us on Facebook: https://www.facebook.com/freecodecamp Follow Quincy on Quora: https://www.quora.com/Quincy-Larson
Views: 82423 freeCodeCamp.org
DLS: Image Processing and Computational Mathematics
 
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Tony Chan, President The Hong Kong University of Science and Technology (HKUST) October 7th, 2015 - Davis Centre, University of Waterloo Image processing has traditionally been studied as a form of signal processing, and a subfield of electrical engineering. Recently, with the advent of inexpensive and integrated image capturing devices, leading to massive data and novel applications, the field has seen tremendous growth. Within computational mathematics, image processing has emerged not only as an application domain where computational mathematics provide ideas and solutions, but also in spurring new research directions (a “new Computational Fluid Dynamics”) in geometry (Total Variation regularization and Level Set methods), optimization (primal-dual, Bregman and Augmented Lagrangian methods, L1 convexification), inverse problems (inpainting, compressive sensing), and graph algorithms (high-dimensional data analysis). This talk gives an overview of these developments.
Views: 1915 uwaterloo
Seminar - Introduction to Structured Deformations: Foundations and Applications (1/2)
 
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Seminar offered by the Chair of Computational Mathematics in DeustoTech. Author: Marco Morandotti, Politecnico di Torino (Torino, Italy).
Foundations of Computing
 
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By Aidan McCarthy 20046537 and Peter Phelan 20046800
Views: 921 AidanMcCarthy91
Type Theory Foundations, Lecture 1
 
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Robert Harper - Type Theory Foundations, Lecture 1, Oregon Programming Languages Summer School 2012, University of Oregon For more info about the summer school please visit http://www.cs.uoregon.edu/research/summerschool/summer12/
Views: 25499 p473r
Ravi Kannan -- Foundations of Data Science
 
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On 25th February 2014, Prof. Ravi Kannan delivered the inaugural lecture on the "Foundations of Data Science". Speaker Bio: Ravi Kannan is a Principal Researcher at Microsoft Research India, where he leads the algorithms research group, and is an Adjunct Professor in the Department of Computer Science and Automation at the Indian Institute of Science. Before joining Microsoft, Prof. Kannan was the William K. Lanman Jr. Professor of Computer Science and Applied Mathematics at Yale University. He has also taught at MIT and CMU. He was awarded the Knuth Prize in 2011 for developing influential algorithmic techniques aimed at solving long-standing computational problems, the Fulkerson Prize in 1991 for his work on estimating the volume of convex sets, and the Distinguished Alumnus award of the Indian Institute of Technology, Bombay in 1999.
Encyclopedia of Applied and Computational Mathematics
 
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Learn more at: http://www.springer.com/978-3-540-70528-4. Leads the field of applied mathematics with comprehensive coverage. Emphasizes the strong links of applied mathematics with major areas of science. Provides a working tool for all types of mathematicians, scientists and engineers.
Views: 110 SpringerVideos
Lecture 1 - Propositional Logic
 
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Discrete Mathematical Structures
Views: 945750 nptelhrd
Logical difficulties with the modern theory of limits (I)|Real numbers + limits Math Foundations 109
 
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This is the first of two videos that will look at the official formal definition of a limit of a sequence, as initiated by Bolzano, Cauchy and Weierstrass. Although commonly regarded as a pillar of modern analysis, in fact this definition has serious logical problems. We state what these problems are, and then start to try to explain them. This lecture will involve some more complicated notions, often only introduced in introductory analysis courses at the 2nd or 3rd year undergraduate level. However by looking carefully at some examples, I hope to show you what the definition is trying to say, and then ultimately why it doesn't work. 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
Lenore Blum - Alan Turing and the other theory of computing and can a machine be conscious?
 
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Abstract Most logicians and theoretical computer scientists are familiar with Alan Turing’s 1936 seminal paper setting the stage for the foundational (discrete) theory of computation. Most however remain unaware of Turing’s 1948 seminal paper which introduces the notion of condition, setting the stage for a natural theory of complexity for the “other theory of computation.” Computational mathematics, the “other theory of computation,” emanates from the classical tradition of numerical analysis, equation solving and the continuous mathematics of calculus. This talk will recognize Alan Turing’s work in the foundations of numerical computation (in particular, his 1948 paper “Rounding-Off Errors in Matrix Processes”), its influence in complexity theory today, and how it provides a unifying concept for the two major traditions of the Theory of Computation. It is based on a plenary talk given on the eve of Turing’s 100th birthday in June 2012 at the Turing Centenary Conference at the University of Cambridge. Biography Lenore Blum (PhD, MIT) is distinguished career professor of Computer Science at Carnegie Mellon University and Founding Director of Project Olympus, an innovation center bridging the gap between cutting-edge university research/innovation and economy-promoting commercialization. Project Olympus has been catalytic in the Pittsburgh renaissance and is a good example of Blum’s determination to make a real difference in the academic community and the world beyond. Lenore is internationally recognized for her work in increasing the participation of girls and women in Science, Technology, Engineering, and Math (STEM) fields. She was a founder of the Association for Women in Mathematics and recipient of the US Presidential Award for Excellence in Science, Mathematics, and Engineering Mentoring. At Carnegie Mellon, Lenore founded the [email protected] program, where women comprise almost half of new majors in computer science. Lenore’s research, from her early work in model theory and differential fields (logic and algebra) to her more recent work in developing a theory of computation and complexity over the real numbers (mathematics and computer science), has focused on merging seemingly unrelated areas. The latter work, founding a theory of computation and complexity over continuous domains, forms a theoretical basis for scientific computation. On the eve of Alan Turing’s 100th birthday in June 2012, she was plenary speaker at the Turing Centenary Celebration at the University of Cambridge, England, demonstrating how a lesser known Turing paper is fundamental to this theory. Lenore is a Fellow of the American Association for the Advancement of Science and an Inaugural Fellow of the American Mathematical Society. #TuringLectures
James Worrell: Computational Learning Theory I
 
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Lecture 1, Sunday 1 July 2018, part of the FoPSS Logic and Learning School at FLoC 2018 - see http://fopss18.mimuw.edu.pl/ and www.floc2018.org for further information.
Sir Roger Penrose - "Consciousness and the foundations of physics”
 
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Sir Roger Penrose - “Consciousness and the Foundations of Physics”, delivered at the Ian Ramsey Centre - Humane Philosophy Project 2014-2015 Seminar. Chaired by Ralph Weir, Alister McGrath and Mikolaj Slawkowski-Rode. The introduction of quantum mechanics in the early 20th Century led many physicists to question the “Newtonian” type of picture of an objective deterministic physical reality that had been previously regarded as an essential background to a fully scientific picture of the world. Quantum measurement, as described in standard theory however, requires a fundamental indeterminism, and issues such as Bell non-locality cause basic difficulties with a picture of objective reality that is consistent with the principles of relativity. Accordingly, many philosophers of science have felt driven to viewpoints according to which “reality” itself takes on subjective qualities, seemingly dependent upon the experiences of conscious beings. My own position is an essentially opposite one, and I argue that conscious experience itself arises from a particular objective feature of physical law. This, however, must go beyond our current understanding of the laws of quantum processes and their relation to macroscopic phenomena. I argue that this objective feature has to do with implications of Einstein’s general theory of relativity and, moreover, must lie beyond the scope of a fully computational universe. SIR ROGER PENROSE OM FRS is a renowned mathematical physicist, mathematician and philosopher of science. He is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford, as well as an Emeritus Fellow of Wadham College. He is known for his work in mathematical physics, especially his contributions to general relativity and cosmology. He has received numerous prizes and awards, including the 1988 Wolf Prize for physics, which he shared with Stephen Hawking for their contribution to our understanding of the Universe. In 1972 he was elected a Fellow of the Royal Society of London in 1972. He was knighted for services to science in 1994 and appointed to the Order of Merit in 2000. He also holds honorary doctorate degrees from many distinguished universities including Warsaw, Leuven, York and Bath.
Views: 73148 IanRamseyCentre
4. Comparative Genomic Analysis of Gene Regulation
 
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MIT 7.91J Foundations of Computational and Systems Biology, Spring 2014 View the complete course: http://ocw.mit.edu/7-91JS14 Instructor: Christopher Burge Prof. Burge discusses comparative genomics. He begins with a review of global alignment of protein sequences, then talks about Markov models, the Jukes-Cantor model, and Kimura models. He discusses types of selection: natural, negative, and positive. License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
Views: 12004 MIT OpenCourseWare
Mathematical space and a basic duality in geometry | Rational Geometry Math Foundations 122
 
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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
17. Logic Modeling of Cell Signaling Networks
 
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MIT 7.91J Foundations of Computational and Systems Biology, Spring 2014 View the complete course: http://ocw.mit.edu/7-91JS14 Instructor: Doug Lauffenburger Prof. Doug Lauffenburger delivers a guest lecture on the topic of logic modeling of cell signaling networks. He begins by giving a conceptual background of the subject, and then discusses an example involving hepatocyes (liver cells). License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
Views: 7532 MIT OpenCourseWare

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