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Solutions to some typical exam questions. See my other videos https://www.youtube.com/channel/UCmtelDcX6c-xSTyX6btx0Cw/.
Views: 37820 Randell Heyman

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If you find our videos helpful you can support us by buying something from amazon. https://www.amazon.com/?tag=wiki-audio-20 Elliptic curve cryptography Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.ECC requires smaller keys compared to non-ECC cryptography (based on plain Galois fields) to provide equivalent security. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=UTJ2jxuyL7g
Views: 522 WikiAudio

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This is episode one of the Math Behind Bitcoin. In an effort to understand the math behind bitcoin, I try to explain it to you guys. If there are any mistakes or suggestions, please put it in the comment section below. Thanks! Resources - https://www.coindesk.com/math-behind-bitcoin/ - https://eng.paxos.com/blockchain-101-foundational-math - Mastering Bitcoin by Andreas Antonopoulos - https://www.cryptocoinsnews.com/explaining-the-math-behind-bitcoin/ - https://en.wikipedia.org/wiki/Finite_field
Views: 1134 Kevin Su

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Vídeo original: https://youtu.be/iB3HcPgm_FI Welcome to part four in our series on Elliptic Curve Cryptography. I this episode we dive into the development of the public key. In just 44 lines of code, with no special functions or imports, we produce the elliptic curve public key for use in Bitcoin. Better still, we walk you through it line by line, constant by constant. Nothing makes the process clearer and easier to understand than seeing it in straight forward code. If you've been wondering about the secp256k1 (arguably the most important piece of code in Bitcoin), well then this is the video for you. This is part 4 of our upcoming series on Elliptic Curves. Because of such strong requests, even though this is part 4, it is the first one we are releasing. In the next few weeks we will release the rest of the series. Enjoy. Here's the link to our Python code (Python 2.7.6): https://github.com/wobine/blackboard1... Here's the private key and the link to the public address that we use. Do you know why it is famous? Private Key : A0DC65FFCA799873CBEA0AC274015B9526505DAAAED385155425F7337704883E Public Address on Blockchain.info https://blockchain.info/address/1JryT... Here's the private key we use at the end: 42F615A574E9CEB29E1D5BD0FDE55553775A6AF0663D569D0A2E45902E4339DB Public Address on Blockchain.info https://blockchain.info/address/16iTd... Welcome to WBN's Bitcoin 101 Blackboard Series -- a full beginner to expert course in bitcoin. Please like, subscribe, comment or even drop a little jangly in our bitcoin tip jar 1javsf8GNsudLaDue3dXkKzjtGM8NagQe. Thanks, WBN
Views: 6053 Fabio Carpi

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Animation of elliptic curve y² = x³ + ax + b, varying Parameter a from -2..1 and b from -1..2. see also http://en.wikipedia.org/wiki/Image:EllipticCurveCatalog.svg
Views: 6366 fuckyoubugger

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From http://1anonymous.org Want to see how the NSA HACKED ECC CRYPTO? Want to see what the SNOWDEN DOCS and WIKI LEAKS really says about the NSA and the NSA HACK of ECC CRYPTO and NSA BITCOIN. The special relationship between primes and numbers on the 8 PRIMES SPIRALS is how the NSA has been able to create backdoors and cook ECC SEED KEYS. The NSA is hiding any info on the 8 PRIME SPIRALS and WIKI is now under NSA CONTROL. SEE THE PROOF DO NOT USE ANY ECC CRYPTO IT IS ALL BACK DOORED WE ARE ANONYMOUS WE ARE EVERYWHERE WE ARE LEGION WE ARE WATCHING 1 ANONYMOUS ORG
Views: 1429 1 Anonymous Org 7

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Animation of elliptic curve y² = x³ + ax + b, varying Parameter a from -2..1 and b from -1..2. see also http://en.wikipedia.org/wiki/Image:EllipticCurveCatalog.svg
Views: 1758 fuckyoubugger

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In this video I'm explaining what is that Galois Counter Mode that provides Authenticated Encryption with Associated Data (AEAD). You must have heard it combined with AES, and maybe used in TLS, ... This is just a small explanation, you can get more on the NIST specs. Errata (thanks to Casper Kejlberg-Rasmussen in the comments) error at 11:21, the last M_H that is applied before going into the TAG should not be there if you compare your drawing to the diagram on https://en.wikipedia.org/wiki/Galois/Counter_Mode. Be sure to follow me on twitter :) https://twitter.com/lyon01_david and to subscribe to my blog! http://www.cryptologie.net Cheers!
Views: 19373 David Wong

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If you find our videos helpful you can support us by buying something from amazon. https://www.amazon.com/?tag=wiki-audio-20 Modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=S0BTCAta6gw
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A course on how bitcoin works and how to program bitcoin stuff with the javascript bitcoin library Yours Bitcoin. Taught by Ryan X. Charles, Cofounder & CEO of Yours, and former cryptocurrency engineer of reddit. The third lecture covers elliptic curves and in particular secp256k1, the curve used by bitcoin. This curve is used for public keys and ECDSA, the digital signature algorithm of bitcoin. https://github.com/yoursnetwork/yours-bitcoin https://github.com/yoursnetwork/yours-bitcoin-examples https://www.yours.network https://www.ryanxcharles.com/ https://twitter.com/ryanxcharles
Views: 779 Yours

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This video is a demonstration of some infamous Elliptic curves. http://en.wikipedia.org/wiki/Elliptic_curve The parameter "a" running from -3 to 3

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In this video I demonstrate getting the ECDSA Z value from a bitcoin transaction with only one input. I also show the R and S values. The ECDSA R, S and Z values are used throughout the many layers of bitcoin to validate a transaction, The Z value is also sometimes referred to as the signed message. Transactions that don't contain valid inputs can be safely ignored, and the Z value is one of the properties that is used to check validity. This video shows me dissecting a very basic transaction with only 1 input and 1 output. The urls I show in this video are https://2coin.org/index.html?txid=bf474b96908ba7769120b2e8f2bfcbd2deca80c99b576b4b63bf18fb69e3d242 https://en.bitcoin.it/wiki/Protocol_documentation#tx https://2coin.org/doublesha256.html
Views: 3512 seanwasere ytbe

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For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

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GCM does AES-256 encryption and, simutaneously, performs message authentication. View this video to understand how it works.
Views: 6132 Vidder, Inc.

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What is CRYPTOGRAPHIC SPLITTING? What does CRYPTOGRAPHIC SPLITTING mean? CRYPTOGRAPHIC SPLITTING definition - CRYPTOGRAPHIC SPLITTING explanation. Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license. SUBSCRIBE to our Google Earth flights channel - https://www.youtube.com/channel/UC6UuCPh7GrXznZi0Hz2YQnQ Cryptographic splitting, also known as cryptographic bit splitting or cryptographic data splitting, is a technique for securing data over a computer network. The technique involves encrypting data, splitting the encrypted data into smaller data units, distributing those smaller units to different storage locations, and then further encrypting the data at its new location. With this process, the data is protected from security breaches, because even if an intruder is able to retrieve and decrypt one data unit, the information would be useless unless it can be combined with decrypted data units from the other locations. The technology was filed for patent consideration in June 2003, and the patent was granted in June 2008. Cryptographic splitting utilizes a combination of different algorithms to provide the data protection. A block of data is first encrypted using the AES-256 government encryption standard. The encrypted bits are then split into different shares and then each share is hashed using the National Security Agency's SHA-256 algorithm. One application of cryptographic splitting is to provide security for cloud computing. The encrypted data subsets can be stored on different clouds, with the information required to restore the data being held on a private cloud for additional security. Security vendor Security First Corp uses this technology for its Secure Parser Extended (SPx) product line. In 2009, technology services company Unisys gave a presentation about using cryptographic splitting with storage area networks. By splitting the data into different parts of the storage area network, this technique provided data redundancy in addition to security. Computer giant IBM has written about using the technology as part of its Cloud Data Encryption Services (ICDES). The technology has also been written about in the context of more effectively using sensitive corporate information, by entrusting different individuals within a company (trustees) with different parts of the information.
Views: 216 The Audiopedia

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This is my video on the modularity theorem for the #breakthroughjuniorchallenge
Views: 2082 Chris Williams

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Many years ago I came across a clickable flash animation that explained how the Rijndael cipher works. And even though Rijndael is pure, complex math, the animated visualizations made the whole process so crystal clear that I had to bend down to the floor afterwards to pick up my dropped jaw. Since then I know how powerful animated visualizations can be, even (or rather especially) for abstract and/or complex topics. When I started my Go blog, I knew I had to use animations because they are worth a thousand words. I did the same in my Go videos that you can find over here in my channel, and also in my Go course. This video is a recoding of the flash animation while I click through it. The flash animation is still available at formaestudio.com (link below), but no sane browser would agree to play any flash content anymore, so a video capture is the best we can get. I hope the pace of clicking through the steps is just right for you. NOTE: The video has no audio part. This is not a bug, the Flash animation simply had no sounds. The Rijndael Animation (and another Flash program called Rijndael Inspector): http://www.formaestudio.com/rijndaelinspector/ (c) Enrique Zabala. License terms: "Both these programs are free of use." (I guess that publishing a video of the animation is covered by these terms.) My blog: https://appliedgo.net My course: https://appliedgo.com/p/mastergo
Views: 28320 AppliedGo

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Between Logjam, FREAK, POODLE, and Heartbleed, TLS hasn't had a good year. TLS is the most commonly deployed cryptographic protocol, but is notoriously difficult to both implement and deploy, resulting in widespread security issues for many of the top services on the Internet. For the past three years, we've been working to improve the global state of TLS deployment through measurement-based approaches, including tracking the impact of Heartbleed and other vulnerabilities. Based on measurement data, we conducted one of the largest-ever mass vulnerability notification campaigns, discovered failures in how Diffie-Hellman has been deployed in practice, and uncovered the Logjam attack against TLS. In this talk, we'll briefly examine what TLS is and how it fails, and present the Logjam attack. We'll also discuss ZMap, the Internet-wide network scanner we use for our research, and show how ZMap helped lead to the discovery of Logjam.
Views: 1573 Duo Security

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Cryptography is the practice and study of techniques for secure communication in the presence of third parties . More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and which are related to various aspects in information security such as data confidentiality, data integrity, authentication, and non-repudiation. Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce. This video targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
Views: 243 encyclopediacc

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For slides, a problem set and more on learning cryptography, visit www.crypto-textbook.com

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A classical approach of constructing elliptic curves that can be used for cryptographic purposes relies on the theory of complex multiplication. A key ingredient in the algorithm is to compute the Hilbert class polynomial P_D for a suitable discriminant D. The polynomial P_D has integer coefficients, and is the minimal polynomial of the modular j-value j(O_D) for the imaginary quadratic order O_D of discriminant D. The polynomial P_D can be computed using complex analytic techniques. In this talk we present a new p-adic algorithm to compute P_D. One of the advantages of working over a p-adic field is that we do not have to worry about rounding errors, and the p-adic algorithm is the first algorithm with a rigorous run time analysis. When implemented carefully, the p-adic algorithm is very fast in practice and easily competes with the complex analytic approach. Many examples will be given.
Views: 76 Microsoft Research

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Eleventh IACR Theory of Cryptography Conference TCC 2014 February 24-26, 2014 Amos Beimel and Aner M. Ben-Efraim and Carles Padró and Ilya Tomkin
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Cryptography is a systems problem (or) 'Should we deploy TLS' Given by Matthew Green, Johns Hopkins University
Views: 5734 Dartmouth

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Steven J Miller and M. Ram Murty Williams College Department of Mathematics and Statistics Bronfman Science Center, Rm. 202 Williamstown, MA 01267 Email: [email protected] Manuscript Number: JNT-D-10-00120 R3
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Views: 4892 Dr. Julian Hosp

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Views: 2436743 3Blue1Brown

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If you find our videos helpful you can support us by buying something from amazon. https://www.amazon.com/?tag=wiki-audio-20 Discrete logarithm In mathematics, a discrete logarithm is an integer k solving the equation bk = g, where b and g are elements of a finite group.Discrete logarithms are thus the finite-group-theoretic analogue of ordinary logarithms, which solve the same equation for real numbers b and g, where b is the base of the logarithm and g is the value whose logarithm is being taken. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=2SVqP_0RCHc
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14th ALGA meeting - Commutative Algebra and Algebraic Geometry Fabien Pazuki (University of Copenhagen) Bad reduction of curves with CM jacobianss Página do Programa: http://www.impa.br/opencms/en/eventos/store_2017/evento_1704 Download dos Vídeos: http://video.impa.br/index.php?page=14th-alga-meeting For twenty years, the ALGA meetings have been bringing together the Brazilian community of Commutative Algebra and Algebraic Geometry, and its foreign collaborators. They have been fundamental for the consolidation and strengthening of the research group. The 14th edition of ALGA celebrates its 20th anniversary. The program includes invited lectures and sessions of "Presentations by Young Researchers". Young researchers and Ph.D. students interested in making a presentation can submit a proposal through the registration form below. IMPA - Instituto de Matemática Pura e Aplicada © http://www.impa.br | http://video.impa.br

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If you find our videos helpful you can support us by buying something from amazon. https://www.amazon.com/?tag=wiki-audio-20 Classical cipher A cipher is a means of concealing a message, where letters of the message are substituted or transposed for other letters, letter pairs, and sometimes for many letters.In cryptography, a classical cipher is a type of cipher that was used historically but now has fallen, for the most part, into disuse. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=rMJMymx6s08
Views: 270 WikiAudio

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This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/List_of_important_publications_in_mathematics 00:00:49 1 Algebra 00:00:59 1.1 Theory of equations 00:01:08 1.1.1 iBaudhayana Sulba Sutra/i 00:01:57 1.1.2 iThe Nine Chapters on the Mathematical Art/i 00:02:17 1.1.3 iHaidao Suanjing/i 00:02:46 1.1.4 iSunzi Suanjing/i 00:03:07 1.1.5 iAryabhatiya/i 00:03:23 1.1.6 iJigu Suanjing/i 00:04:15 1.1.7 iBrāhmasphuṭasiddhānta/i 00:04:38 1.1.8 iAl-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala/i 00:05:07 1.2 iLīlāvatī/i, iSiddhānta Shiromani/i and iBijaganita/i 00:05:49 1.2.1 iYigu yanduan/i 00:06:12 1.2.2 iMathematical Treatise in Nine Sections/i 00:06:30 1.2.3 iCeyuan haijing/i 00:07:05 1.2.4 iJade Mirror of the Four Unknowns/i 00:07:25 1.2.5 iArs Magna/i 00:07:46 1.2.6 iVollständige Anleitung zur Algebra/i 00:08:18 1.2.7 iDemonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse/i 00:08:55 1.3 Abstract algebra 00:09:05 1.3.1 Group theory 00:09:27 1.3.1.1 iRéflexions sur la résolution algébrique des équations/i 00:09:43 1.3.2 iArticles Publiés par Galois dans les Annales de Mathématiques/i 00:09:53 1.3.3 iTraité des substitutions et des équations algébriques/i 00:10:08 1.3.4 iTheorie der Transformationsgruppen/i 00:10:26 1.3.5 iSolvability of groups of odd order/i 00:11:10 1.3.6 Homological algebra 00:11:24 1.3.7 iHomological Algebra/i 00:11:53 1.3.8 "Sur Quelques Points d'Algèbre Homologique" 00:12:47 2 Algebraic geometry 00:13:28 2.1 "Theorie der Abelschen Functionen" 00:14:04 2.2 iFaisceaux Algébriques Cohérents/i 00:14:13 2.3 iGéométrie Algébrique et Géométrie Analytique/i 00:14:44 2.4 "Le théorème de Riemann–Roch, d'après A. Grothendieck" 00:15:11 2.5 iÉléments de géométrie algébrique/i 00:15:21 2.6 iSéminaire de géométrie algébrique/i 00:16:33 3 Number theory 00:18:03 3.1 iBrāhmasphuṭasiddhānta/i 00:19:33 3.2 iDe fractionibus continuis dissertatio/i 00:20:25 3.3 iRecherches d'Arithmétique/i 00:21:05 3.4 iDisquisitiones Arithmeticae/i 00:22:26 3.5 "Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält" 00:22:35 3.6 "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" 00:23:21 3.7 iVorlesungen über Zahlentheorie/i 00:23:50 3.8 iZahlbericht/i 00:24:50 3.9 iFourier Analysis in Number Fields and Hecke's Zeta-Functions/i 00:26:34 3.10 "Automorphic Forms on GL(2)" 00:27:25 3.11 "La conjecture de Weil. I." 00:28:39 3.12 "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" 00:29:30 3.13 "Modular Elliptic Curves and Fermat's Last Theorem" 00:30:11 3.14 iThe geometry and cohomology of some simple Shimura varieties/i 00:30:54 3.15 "Le lemme fondamental pour les algèbres de Lie" 00:31:24 4 Analysis 00:31:48 4.1 iIntroductio in analysin infinitorum/i 00:32:38 4.2 Calculus 00:33:25 4.2.1 iYuktibhāṣā/i 00:34:01 4.2.2 iNova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus/i 00:34:26 4.2.3 iPhilosophiae Naturalis Principia Mathematica/i 00:34:36 4.2.4 iInstitutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum/i 00:36:17 4.2.5 iÜber die Darstellbarkeit einer Function durch eine trigonometrische Reihe/i 00:36:26 4.2.6 iIntégrale, longueur, aire/i 00:37:30 4.3 Complex analysis 00:38:09 4.3.1 iGrundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse/i 00:39:30 4.4 Functional analysis 00:40:41 4.4.1 iThéorie des opérations linéaires/i 00:41:30 4.5 Fourier analysis 00:41:54 4.5.1 iMémoire sur la propagation de la chaleur dans les corps solides/i 00:42:04 4.5.2 iSur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données/i 00:42:38 4.5.3 iOn convergence and growth of partial sums of Fourier series/i 00:42:47 5 Geometry 00:43:44 5.1 iBaudhayana Sulba Sutra/i 00:43:53 5.2 iEuclid's/i iElements/i 00:45:06 5.3 iThe Nine Chapters on the Mathematical Art/i 00:46:10 5.4 iThe Conics/i 00:46:41 5.5 iSurya Siddhanta/i 00:46:51 5.6 iAryabhatiya/i 00:47:56 5.7 iLa Géométrie/i 00:48:16 5.8 iGrundlagen der Geometrie/i 00:49:15 5.9 iRegular Polytopes/i 00:50:13 5.10 Differential geometry 00:50:51 5.10.1 iRecherches sur la courbure des surfaces/i 00:51:47 5.10.2 iDisquisitiones generales circa superficies curvas/i 00:52:28 5.10.3 iÜber die Hypothesen, welche der Geometrie zu Grunde Liegen/i 00:53:00 5.10.4 iLeçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal/i 00:53:47 6 Topology 00:54:32 6.1 iAnalysis situs/i 00:54:42 6.2 iL'anneau d'homologie d'une représentation/i, iStructure de l'anneau d'homologie d'une r ...
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This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Matrix_(mathematics) 00:04:39 1 Definition 00:05:49 1.1 Size 00:06:48 2 Notation 00:11:06 3 Basic operations 00:11:48 3.1 Addition, scalar multiplication and transposition 00:12:07 3.2 Matrix multiplication 00:12:34 3.3 Row operations 00:12:47 3.4 Submatrix 00:13:50 4 Linear equations 00:14:15 5 Linear transformations 00:14:49 6 Square matrix 00:15:04 6.1 Main types 00:15:07 6.1.1 Diagonal and triangular matrix 00:18:33 6.1.2 Identity matrix 00:20:36 6.1.3 Symmetric or skew-symmetric matrix 00:21:11 6.1.4 Invertible matrix and its inverse 00:23:14 6.1.5 Definite matrix 00:26:00 6.1.6 Orthogonal matrix 00:28:37 6.2 Main operations 00:28:49 6.2.1 Trace 00:29:32 6.2.2 Determinant 00:30:09 6.2.3 Eigenvalues and eigenvectors 00:30:19 7 Computational aspects 00:30:50 8 Decomposition 00:32:33 9 Abstract algebraic aspects and generalizations 00:33:03 9.1 Matrices with more general entries 00:33:19 9.2 Relationship to linear maps 00:34:21 9.3 Matrix groups 00:34:40 9.4 Infinite matrices 00:35:04 9.5 Empty matrices 00:36:24 10 Applications 00:36:28 10.1 Graph theory 00:37:53 10.2 Analysis and geometry 00:38:56 10.3 Probability theory and statistics 00:39:05 10.4 Symmetries and transformations in physics 00:40:53 10.5 Linear combinations of quantum states 00:40:57 10.6 Normal modes 00:42:02 10.7 Geometrical optics 00:43:15 10.8 Electronics 00:44:21 11 History 00:44:55 11.1 Other historical usages of the word "matrix" in mathematics 00:47:16 12 See also 00:48:01 13 Notes 00:50:24 14 References 00:50:42 14.1 Physics references 00:51:17 14.2 Historical references 00:51:31 15 External links 00:52:29 Matrices with more general entries 00:54:34 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible. Matrices over superrings are called supermatrices.Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring; but their sizes must fulfil certain compatibility conditions. 00:55:24 Relationship to linear maps 00:55:35 Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A 00:58:02 m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn. 00:58:15 Matrix groups 00:59:55 Infinite matrices 01:04:24 Empty matrices 01:05:34 Applications 01:07:44 Graph theory 01:08:44 Analysis and geometry 01:11:51 Probability theory and statistics 01:13:16 1, …, Nwhich can be formulated in terms of matrices, related to the singular value decomposition of matrices.Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics. 01:13:42 Symmetries and transformations in physics 01:13:54 Linear transformations and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors. For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses. 01:15:08 Linear combinations of quantum states 01:15:19 The first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states. This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.Another matrix serves as a k ...
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This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Matrix_(mathematics) 00:04:07 1 Definition 00:05:09 1.1 Size 00:06:01 2 Notation 00:09:53 3 Basic operations 00:10:30 3.1 Addition, scalar multiplication and transposition 00:10:48 3.2 Matrix multiplication 00:11:12 3.3 Row operations 00:11:24 3.4 Submatrix 00:12:20 4 Linear equations 00:12:42 5 Linear transformations 00:13:13 6 Square matrix 00:13:27 6.1 Main types 00:13:30 6.1.1 Diagonal and triangular matrix 00:16:29 6.1.2 Identity matrix 00:18:19 6.1.3 Symmetric or skew-symmetric matrix 00:18:51 6.1.4 Invertible matrix and its inverse 00:20:38 6.1.5 Definite matrix 00:23:04 6.1.6 Orthogonal matrix 00:25:23 6.2 Main operations 00:25:34 6.2.1 Trace 00:26:13 6.2.2 Determinant 00:26:46 6.2.3 Eigenvalues and eigenvectors 00:26:55 7 Computational aspects 00:27:23 8 Decomposition 00:28:55 9 Abstract algebraic aspects and generalizations 00:29:23 9.1 Matrices with more general entries 00:29:37 9.2 Relationship to linear maps 00:30:33 9.3 Matrix groups 00:30:49 9.4 Infinite matrices 00:31:12 9.5 Empty matrices 00:32:22 10 Applications 00:32:26 10.1 Graph theory 00:33:41 10.2 Analysis and geometry 00:34:38 10.3 Probability theory and statistics 00:34:46 10.4 Symmetries and transformations in physics 00:36:25 10.5 Linear combinations of quantum states 00:36:28 10.6 Normal modes 00:37:26 10.7 Geometrical optics 00:38:30 10.8 Electronics 00:39:29 11 History 00:39:59 11.1 Other historical usages of the word "matrix" in mathematics 00:42:04 12 See also 00:42:44 13 Notes 00:44:49 14 References 00:45:06 14.1 Physics references 00:45:37 14.2 Historical references 00:45:49 15 External links 00:46:40 Matrices with more general entries 00:48:31 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible. Matrices over superrings are called supermatrices.Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring; but their sizes must fulfil certain compatibility conditions. 00:49:15 Relationship to linear maps 00:49:25 Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A 00:51:37 m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn. 00:51:49 Matrix groups 00:53:18 Infinite matrices 00:57:16 Empty matrices 00:58:17 Applications 01:00:13 Graph theory 01:01:06 Analysis and geometry 01:03:52 Probability theory and statistics 01:05:08 1, …, Nwhich can be formulated in terms of matrices, related to the singular value decomposition of matrices.Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics. 01:05:31 Symmetries and transformations in physics 01:05:42 Linear transformations and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors. For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses. 01:06:47 Linear combinations of quantum states 01:06:58 The first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states. This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.Another matrix serves as a k ...
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic modulo 12. 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 is congruent to 0 modulo 12. This video is targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
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Factoring a number using pollard rho http://www.awright2009.com/factor.c Edit: Just realized people might want a windows version. Had it in my email a long time ago (Professor didnt seem to believe in me or Linux) I believe this is compiled for conroe based intel machine's, but better than nothing. http://www.awright2009.com/factor.zip I'll put the source and binary on my http://github.com/akw0088/factor in a bit (had to repull the windows binary from email as the university servers went down)
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Please Subscribe our goal is 200 subscriber for this month :) Please give us a THUMBS UP if you like our videos!!! Source:http://en.wikipedia.org/wiki/History_of_mathematics The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Chinese mathematics made early contributions, including a place value system. The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
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