L-functions and Arithmetic

Views: 17
Mathnet Korea

Trace functions are arithmetic functions defined modulo $q$ (some prime number) obtained as Frobenius trace function of $\ell$-adic sheaves. The basic example is that of a Dirichlet character of modulus $q$ but there are many other examples of interest for instance (hyper)-Kloosterman sums. In this series of lectures we will explain how they arise in classical problems of analytic number theory and how (basi) methods from $\ell$-adic cohomology allow to extract a lot out of them. Most of these lectures are based on works of E. Fouvry, E. Kowalski, myself and W. Sawin.

Views: 305
Preserve Knowledge

In this video we look at the explicit problem of finding a factorization of the large number z= 10 tri 10 + 23 into prime factors. We claim that, contrary to the Fundamental theorem of Arithmetic, also called the Unique Factorization theorem, this is impossible. Ultimately issues of complexity overwhelm us, and we want to see explicitly how this comes about.
After some perhaps amusing discussion of "dark numbers", we turn to explicit examples of factorizations of large numbers, along the lines of the video FMP1 to get some intuition about what happens when we try to factor larger and larger numbers.
This video is really acknowledging a major fork in the road for pure mathematics: continue with idealism in the direction of what we would like to be true, or moving towards a constructive view of the mathematical world which is in line with what our computers can tell us.
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 MathFoundations184 to 212 can be found at my WildEgg website here: http://www.wildegg.com/store/p105/product-Math-Foundations-C2-screenshots-pdf

Views: 4186
Insights into Mathematics

SPEAKER: Francois Rodier TITLE: Asymptotic nonlinearity of Boolean functions ABSTRACT: The nonlinearity of Boolean functions on the space Fm2 is important in cryptography. It is used to measure the strength of cryptosystems when facing linear attacks. In the case low degree of approximation attacks, we examine the nonlinearity of order r of a Boolean function, which equals the number of necessary substitutions in its truth table needed to change it into a function of degree at most r. Studies aimed at the distribution of Boolean functions according to the r-th order nonlinearity. Asymptotically, a lower bound is established in the higher order cases for almost all Boolean functions, whereas a concentration point is shown in the first and second order nonlinearity case. In the case of vectorial Boolean functions, a concentration point is shown in the first order nonlinearity case. SPEAKER: Sorina Ionica TITLE: Pairing-based methods for genus 2 curve jacobians with maximal endomorphism ring ABSTRACT: Algorithms for constructing jacobians of genus 2 curves with nice cryptographic properties involve the computation of Igusa class polynomials for CM quartic fields. The CRT method used to compute these polynomials needs to find first a jacobian with maximal endomorphism ring over a finite field, and then enumerates all others jacobians having maximal endomorphism ring using horizontal isogenies. For $\ell 2$, we use Galois cohomology and the Tate pairing to compute the action of the Frobenius on the $\ell$-torsion. In view of application to Igusa class polynomials computation, we deduce an algorithm to verify whether the jacobian of a genus 2 curve has locally maximal endomorphism ring at $\ell$. Moreover, we derive a method to construct horizontal isogenies starting from a jacobian with maximal endomorphism ring.

Views: 70
Microsoft Research

Hunter speaks on Deuring's Theorem that L-functions of CM elliptic curves are L-functions of Hecke characters.

Views: 427
Johnson Jia

Hey everybody!
This video shows future incoming guest students of Montanuniversitaet Leoben (MUL) how to pre-register online.
The pre-registration is the first and essential step in completing the registration process at the admission's office (also called Registrar's Office; in German 'Studien und Lehrgänge').
The link to the pre-registration is: https://starter.unileoben.ac.at/en/3435/ ;
you can also search for 'Montanuniversitaet pre registration' through Google or another search engine.
We hope this video makes the registration process easier :-) See you soon in Leoben!
Best regards,
the MIRO -team
Website: international.unileoben.ac.at
Email: [email protected]
Facbeook: https://www.facebook.com/MULmiro/
Instagram: https://www.instagram.com/miro_montanuni/?hl=de

Views: 53
MIRO Montanuni International Office

Joint IAS/Princeton University Number Theory Seminar
Topic: On small sums of roots of unity
Speaker: Philipp Habegger
Affiliation: University of Basel
Date: March 9, 2017

Views: 557
Institute for Advanced Study

A very brief overview of expander graphs, which were the topic of my honours thesis.

Views: 1939
gilesgardam

Averages of p-torsion in class groups over function fields---good and bad primes - Melanie Wood (University of Wisconsin-Madison)
When we consider the average size of the p-torsion in class groups of quadratic fields, the behavior for p=2 is controlled by genus theory and is different from the conjectured behavior for odd primes, which is uniform in a certain sense over all odd primes. In this talk, we will consider the question when quadratic fields are replaced by fields of higher degree--for which primes p are the class group averages "bad" versus "good"? We will explain some theorems on class group averages for extensions of function fields that show some subtleties in the classification of good and bad primes
MAY 01, 2017
MONDAY

Views: 448
Graduate Mathematics

Talk at pkc 2010. Authors: Robert Granger, Michael Scott. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=23419

Views: 120
TheIACR

Ugolini, S.*
*Dipartimento di Matematica,
Università degli studi di Trento,
Povo (Trento), ITALY
Email: [email protected]
http://www.sciencedirect.com/science/article/pii/S0022314X15000414
Manuscript Number: JNT-D-14-00274R1

Views: 183
JournalNumberTheory

Introduction to Galois Fields Part 3 for the Paar Lectures on Cryptography

Views: 1097
Project Rhea

Topic: Lifting Galois representations
Speaker: Daniel Le, Member, School of Mathematics
Time/Room: 2:30pm - 2:45pm/S-101
More videos on http://video.ias.edu

Views: 271
Institute for Advanced Study

Treviño, Enrique*
*Department of Mathematics and Computer Science
Lake Forest College
Lake Forest, Illinois 60045, USA
Email: [email protected]
http://www.journals.elsevier.com/journal-of-number-theory/
Manuscript Number: JNT-D-11-00259R1

Views: 267
JournalNumberTheory

2012 Shannon Lecture
Networks
Abbas El Gamal
Stanford University
Biography: Abbas El Gamal is the Hitachi America Professor in the School of Engineering and Professor of Electrical Engineering at Stanford University. He received the B.Sc. (honors) degree in electrical engineering from Cairo University in 1972 and the M.S. degree in statistics and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1977 and 1978, respectively. His research interest and contributions are in the areas of network information theory, wireless communications, digital imaging, and integrated circuit design. He has authored or coauthored over 200 papers and 30 patents in these areas. He is coauthor of the book Network Information Theory (Cambridge Press 2011). He has won several honors and awards, including the 2004 Infocom best paper award, the 2009 Padovani lecture, and the 2012 Shannon Award. He has been serving on the Board of Governors of the IT Society since 2009 and is currently the Second Vice President.

Views: 272
IEEE Information Theory Society

Paper presented at Eurocrypt 2017, by Jean-François Biasse and Thomas Espitau and Pierre-Alain Fouque and Alexandre Gélin and Paul Kirchner
See https://iacr.org/cryptodb/data/paper.php?pubkey=28037

Views: 192
TheIACR

Jared Weinstein
Institute for Advanced Study
October 27, 2010
The usual Katz-Mazur model for the modular curve X(pn)X(pn) has horribly singular reduction. For large n there isn't any model of X(pn)X(pn) which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for GL(2)GL(2) .

Views: 133
Institute for Advanced Study

The values of the elliptic modular function $j$ at imaginary quadratic numbers $\tau$ are called singular moduli. They are of fundamental importance in the study of elliptic curves and in algebraic number theory, including the study of elliptic curves over finite fields. The theorem of Gross and Zagier has provided striking congruences satisfied by these numbers. Various aspects of this theorem were generalized in recent years by Jan Bruinier and Tonghai Yang and by Kristin Lauter and the speaker. These may be viewed as concerning the theory of curves of genus 2 and their singular moduli that are obtained by evaluating the Igusa invariants at certain 2x2 complex matrices $\tau$. I will describe some of the recent ideas introduced in this area and, time allowing, describe some current projects.

Views: 55
Microsoft Research

Nalli, Ayse* and Özyılmaz, Çağla
*Department of Mathematics
Karabuk University
Karabuk, TURKEY
Email: [email protected]
Manuscript Number: JNT-D-14-00030R2

Views: 457
JournalNumberTheory

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Field_(mathematics)
00:02:28 1 Definition
00:03:27 1.1 Classic definition
00:03:32 1.2 Alternative definition
00:04:39 2 Examples
00:05:00 2.1 Rational numbers
00:05:24 2.2 Real and complex numbers
00:06:14 2.3 Constructible numbers
00:06:44 2.4 A field with four elements
00:06:55 3 Elementary notions
00:08:09 3.1 Consequences of the definition
00:08:14 3.2 The additive and the multiplicative group of a field
00:08:24 3.3 Characteristic
00:13:25 3.4 Subfields and prime fields
00:14:21 4 Finite fields
00:15:41 5 History
00:17:55 6 Constructing fields
00:18:49 6.1 Constructing fields from rings
00:19:03 6.1.1 Field of fractions
00:20:03 6.1.2 Residue fields
00:20:22 6.2 Constructing fields within a bigger field
00:20:34 6.3 Field extensions
00:21:06 6.3.1 Algebraic extensions
00:21:19 6.3.2 Transcendence bases
00:21:36 6.4 Closure operations
00:23:31 7 Fields with additional structure
00:25:49 7.1 Ordered fields
00:26:43 7.2 Topological fields
00:27:49 7.2.1 Local fields
00:28:49 7.3 Differential fields
00:29:33 8 Galois theory
00:30:27 9 Invariants of fields
00:30:58 9.1 Model theory of fields
00:31:19 9.2 The absolute Galois group
00:32:16 9.3 K-theory
00:33:53 10 Applications
00:38:01 10.1 Linear algebra and commutative algebra
00:38:12 10.2 Finite fields: cryptography and coding theory
00:39:50 10.3 Geometry: field of functions
00:42:45 10.4 Number theory: global fields
00:43:59 11 Related notions
00:45:03 11.1 Division rings
00:46:20 12 Notes
00:47:03 13 References
00:47:24 Algebraic extensions
00:50:40 Transcendence bases
00:52:23 Closure operations
00:55:18 Fields with additional structure
00:55:42 Ordered fields
00:57:04 x2
NaN:NaN:NaN xn
NaN:NaN:NaN Topological fields
NaN:NaN:NaN Local fields
NaN:NaN:NaN Differential fields
NaN:NaN:NaN Galois theory
NaN:NaN:NaN X5 − 4X + 2 (and E
NaN:NaN:NaN Invariants of fields
NaN:NaN:NaN Model theory of fields
NaN:NaN:NaN The absolute Galois group
NaN:NaN:NaN H2(F, Gm).
NaN:NaN:NaN K-theory
NaN:NaN:NaN F×. Matsumoto's theorem shows that K2(F) agrees with K2M(F). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.
NaN:NaN:NaN Applications
NaN:NaN:NaN === Linear algebra and commutative algebra
NaN:NaN:NaN bhas a unique solution x in F, namely x
NaN:NaN:NaN Finite fields: cryptography and coding theory
NaN:NaN:NaN x3 + ax + b.Finite fields are also used in coding theory and combinatorics.
NaN:NaN:NaN Geometry: field of functions
NaN:NaN:NaN Number theory: global fields
NaN:NaN:NaN Related notions
NaN:NaN:NaN Division rings
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.780616326590952
Voice name: en-US-Wavenet-B
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.
A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle can not be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable.
Fields serve as foundational notions ...

Views: 3
wikipedia tts

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Field_(mathematics)
00:02:01 1 Definition
00:02:50 1.1 Classic definition
00:02:54 1.2 Alternative definition
00:03:50 2 Examples
00:04:09 2.1 Rational numbers
00:04:29 2.2 Real and complex numbers
00:05:12 2.3 Constructible numbers
00:05:37 2.4 A field with four elements
00:05:47 3 Elementary notions
00:06:48 3.1 Consequences of the definition
00:06:52 3.2 The additive and the multiplicative group of a field
00:07:01 3.3 Characteristic
00:11:22 3.4 Subfields and prime fields
00:12:09 4 Finite fields
00:13:16 5 History
00:15:07 6 Constructing fields
00:15:52 6.1 Constructing fields from rings
00:16:05 6.1.1 Field of fractions
00:16:56 6.1.2 Residue fields
00:17:13 6.2 Constructing fields within a bigger field
00:17:24 6.3 Field extensions
00:17:51 6.3.1 Algebraic extensions
00:18:02 6.3.2 Transcendence bases
00:18:17 6.4 Closure operations
00:19:52 7 Fields with additional structure
00:21:44 7.1 Ordered fields
00:22:30 7.2 Topological fields
00:23:25 7.2.1 Local fields
00:24:13 7.3 Differential fields
00:24:50 8 Galois theory
00:25:35 9 Invariants of fields
00:26:00 9.1 Model theory of fields
00:26:17 9.2 The absolute Galois group
00:27:04 9.3 K-theory
00:28:27 10 Applications
00:31:49 10.1 Linear algebra and commutative algebra
00:31:59 10.2 Finite fields: cryptography and coding theory
00:33:20 10.3 Geometry: field of functions
00:35:47 10.4 Number theory: global fields
00:36:50 11 Related notions
00:37:44 11.1 Division rings
00:38:48 12 Notes
00:39:25 13 References
00:39:42 Algebraic extensions
00:42:28 Transcendence bases
00:43:55 Closure operations
00:46:22 Fields with additional structure
00:46:43 Ordered fields
00:47:53 x2
NaN:NaN:NaN xn
NaN:NaN:NaN Topological fields
NaN:NaN:NaN Local fields
NaN:NaN:NaN Differential fields
NaN:NaN:NaN Galois theory
NaN:NaN:NaN X5 − 4X + 2 (and E
NaN:NaN:NaN Invariants of fields
NaN:NaN:NaN Model theory of fields
NaN:NaN:NaN The absolute Galois group
NaN:NaN:NaN H2(F, Gm).
NaN:NaN:NaN K-theory
NaN:NaN:NaN F×. Matsumoto's theorem shows that K2(F) agrees with K2M(F). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.
NaN:NaN:NaN Applications
NaN:NaN:NaN === Linear algebra and commutative algebra
NaN:NaN:NaN bhas a unique solution x in F, namely x
NaN:NaN:NaN Finite fields: cryptography and coding theory
NaN:NaN:NaN x3 + ax + b.Finite fields are also used in coding theory and combinatorics.
NaN:NaN:NaN Geometry: field of functions
NaN:NaN:NaN Number theory: global fields
NaN:NaN:NaN Related notions
NaN:NaN:NaN Division rings
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8818678708892833
Voice name: en-US-Wavenet-D
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.
A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle can not be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable.
Fields serve as foundational notion ...

Views: 6
wikipedia tts

This is a report on joint work with Kristin Lauter and Peter Stevenhagen. Broker and Stevenhagen have shown that in practice it is not hard to produce an elliptic curve (over some finite field) with a given number N of points, provided that the factorization of N is known. In his talk this week, Stevenhagen will show that the natural generalization of this method to produce genus-2 curves with a given number of points on their Jacobian is an exponential algorithm. I will consider the related problem of constructing a genus-2 curve over some finite field such that the curve itself has a given number N of points. The idea of explicit

Views: 131
Microsoft Research

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Field_(mathematics)
00:01:55 1 Definition
00:02:42 1.1 Classic definition
00:02:46 1.2 Alternative definition
00:03:39 2 Examples
00:03:56 2.1 Rational numbers
00:04:15 2.2 Real and complex numbers
00:04:56 2.3 Constructible numbers
00:05:20 2.4 A field with four elements
00:05:30 3 Elementary notions
00:06:27 3.1 Consequences of the definition
00:06:31 3.2 The additive and the multiplicative group of a field
00:06:41 3.3 Characteristic
00:10:35 3.4 Subfields and prime fields
00:11:19 4 Finite fields
00:12:23 5 History
00:14:09 6 Constructing fields
00:14:52 6.1 Constructing fields from rings
00:15:04 6.1.1 Field of fractions
00:15:52 6.1.2 Residue fields
00:16:08 6.2 Constructing fields within a bigger field
00:16:19 6.3 Field extensions
00:16:45 6.3.1 Algebraic extensions
00:16:56 6.3.2 Transcendence bases
00:17:11 6.4 Closure operations
00:18:41 7 Fields with additional structure
00:20:27 7.1 Ordered fields
00:21:11 7.2 Topological fields
00:22:04 7.2.1 Local fields
00:22:50 7.3 Differential fields
00:23:26 8 Galois theory
00:24:10 9 Invariants of fields
00:24:34 9.1 Model theory of fields
00:24:50 9.2 The absolute Galois group
00:25:37 9.3 K-theory
00:26:54 10 Applications
00:30:06 10.1 Linear algebra and commutative algebra
00:30:16 10.2 Finite fields: cryptography and coding theory
00:31:33 10.3 Geometry: field of functions
00:33:50 10.4 Number theory: global fields
00:34:50 11 Related notions
00:35:41 11.1 Division rings
00:36:42 12 Notes
00:37:17 13 References
00:37:34 Algebraic extensions
00:40:09 Transcendence bases
00:41:31 Closure operations
00:43:51 Fields with additional structure
00:44:11 Ordered fields
00:45:17 x2
NaN:NaN:NaN xn
NaN:NaN:NaN Topological fields
NaN:NaN:NaN Local fields
NaN:NaN:NaN Differential fields
NaN:NaN:NaN Galois theory
NaN:NaN:NaN X5 − 4X + 2 (and E
NaN:NaN:NaN Invariants of fields
NaN:NaN:NaN Model theory of fields
NaN:NaN:NaN The absolute Galois group
NaN:NaN:NaN H2(F, Gm).
NaN:NaN:NaN K-theory
NaN:NaN:NaN F×. Matsumoto's theorem shows that K2(F) agrees with K2M(F). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.
NaN:NaN:NaN Applications
NaN:NaN:NaN === Linear algebra and commutative algebra
NaN:NaN:NaN bhas a unique solution x in F, namely x
NaN:NaN:NaN Finite fields: cryptography and coding theory
NaN:NaN:NaN x3 + ax + b.Finite fields are also used in coding theory and combinatorics.
NaN:NaN:NaN Geometry: field of functions
NaN:NaN:NaN Number theory: global fields
NaN:NaN:NaN Related notions
NaN:NaN:NaN Division rings
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9588288041795844
Voice name: en-US-Wavenet-E
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.
A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle can not be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable.
Fields serve as foundational notion ...

Views: 5
wikipedia tts

Yelton, Jeff*
Mathematics Department
The Pennsylvania State University
University Park, State College, PA 16802
Email: [email protected]
http://www.journals.elsevier.com/journal-of-number-theory/
Manuscript Number: JNT-D-13-00616R3

Views: 306
JournalNumberTheory

The lecture was held within the framework of the Hausdorff Trimester Program: Symplectic Geometry and Representation Theory.
Abstract:
The cyclotomic KLR algebras are certain quotients of the quiver Hecke algebras, or Khovanov–Lauda–Rouquier algebras. These algebras are important because they categorify the highest weight representations of the corresponding quantum group. We will start by discussing these algebras in arbitrary type, where surprisingly little is known. We then focus on type A where Brundan and Kleshchev’s graded isomorphism theorem tells us that these algebras are isomorphic to the cyclotomic Hecke algebras of type A, which are a family of deformation algebras that include as special cases the group algebras of the symmetric groups and their Iwahori–Hecke algebras. The main aim of my lectures is to understand the Ariki–Brundan–Kleshchev categorification theorem in terms of the representation of the cyclotomic KLR algebras of type A.

Views: 163
Hausdorff Center for Mathematics

Support the label and artist, buy it here:
https://ekman.bandcamp.com/album/primus-motor-shiplp08
https://www.deejay.de/Ekman_Primus_Motor_SHIPLP08_Vinyl__300709
https://www.juno.co.uk/products/ekman-primus-motor/671941-01/
https://www.triplevision.nl/release/SHIPLP08/
Artist: Ekman
Label: Shipwrec
Genre: Electronic
Release date: 25 Jan 2018
Tracklist:
Holomorpic Functions
Glitch Primes
Polymath8
Goldbach Number
Mills' Constant
e To The Pi i
Riemann Zeta Function
Euler's Lucky Numbers
"Primus Motor" sees Ekman return to many of the sounds that gained him a reputation as an artist on labels like Bunker and Berceuse Heroique. Although harsh bucking acid lines, pockmarked notes, brutal beats and eerie echo are all present, a new tone has been added to the caustic palette. A psychological nuance, an undercurrent of the inchoate has been investigated and exploited to chilling effect. Crippling psychoses. Debilitating neuroses. Physical pain. Suffering stalks this first full vinyl album. Sinister sounds encircle the eight offerings, hovering like vultures over inhospitable plains of ashen grey, jagged peaks and dark sweeping swamps. Hope is all but drawn away in this bleached audio landscape, drained by sheer synthlines and reduced renderings. Amidst the sorrow and strain there are pin pricks of joy, albeit bleak and pallid ones. An album that unnerves as it engages, a collection of icy radiance and industrial indifference from a dutch master.

Views: 280
Electronic Love Collective

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Number_theory
00:01:39 1 History
00:01:48 1.1 Origins
00:01:56 1.1.1 Dawn of arithmetic
00:06:49 1.1.2 Classical Greece and the early Hellenistic period
00:10:09 1.1.3 Diophantus
00:14:18 1.1.4 Āryabhaṭa, Brahmagupta, Bhāskara
00:16:20 1.1.5 Arithmetic in the Islamic golden age
00:17:07 1.1.6 Western Europe in the Middle Ages
00:17:50 1.2 Early modern number theory
00:18:00 1.2.1 Fermat
00:22:36 1.2.2 Euler
00:25:47 1.2.3 Lagrange, Legendre, and Gauss
00:28:37 1.3 Maturity and division into subfields
00:30:30 2 Main subdivisions
00:30:39 2.1 Elementary tools
00:31:42 2.2 Analytic number theory
00:33:38 2.3 Algebraic number theory
00:38:56 2.4 Diophantine geometry
00:45:03 3 Recent approaches and subfields
00:45:38 3.1 Probabilistic number theory
00:47:16 3.2 Arithmetic combinatorics
00:49:50 3.3 Computations in number theory
00:52:57 4 Applications
00:53:55 5 Prizes
00:54:16 6 See also
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
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Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
"There is only one good, knowledge, and one evil, ignorance."
- Socrates
SUMMARY
=======
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).
The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.

Views: 13
wikipedia tts

Get Yelawolf's "Love Story" - http://smarturl.it/YelaLoveStory
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Music video by Yelawolf performing Johnny Cash. (C) 2015 Interscope Records
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#Yelawolf #JohnnyCash #Vevo #HipHop #OfficialMusicVideo

Views: 11555776
YelawolfVEVO

The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics.
Abstract:
Picard modular surfaces X, which are smooth compactifications of the quotients Y of the complex ball by a discrete subgroups Γ of SU(2,1), have been studied from various points of view. They are often defined over an imaginary quadratic field M, and we are interested in the rational points of X over finite extensions k of M. In a joint work with M. Dimitrov we deduced some finiteness results for the points on the open surfaces Y of congruence type with a bit of level (using theorems of Faltings, Rogawski and Nadel), which we will first recall.A recent result we have is an analogue, for polarized abelian threefolds with multiplication by OM and without CM factors, an analogue of the classical theorem of Manin asserting that for p prime, there is a universal r=r(p,k) such that for any non-CM elliptic curve E, the pr-division subgroup of E has no k-rational line. If time permits, we will explain the final objective of our ongoing program.

Views: 122
Hausdorff Center for Mathematics

Charlie Wilson Chief Revenue Officer Green Bits, the leading retail management and automatic compliance platform for the legal cannabis industry where he leads business development, sales, marketing, and customer success. Charlie has served in leadership roles at numerous leading technology, payments and commerce companies for more than 20 years. He spent nearly half that time at Visa, the world's largest payment network, where he led the company's ecommerce initiatives, including Visa's $2 billion acquisition of merchant services provider CyberSource in 2010. Before joining Green Bits, Charlie was founder and CEO of Commerce Sync, a venture-backed provider of accounting automation services for small businesses. He has also held executive roles at Green Dot, a publicly traded provider of general purpose reloadable prepaid cards, and served as President and CEO of IP Commerce, a venture-backed provider of merchant payments services. Charlie earned a B.S.E in Industrial Engineering at Arizona State University and an M.S. in Management Science and Engineering at Stanford University.
Mark Denzin President Crypto Value Management System. Mark and his team have created a propriety blockchain utility token called ICCE which stands for Instate Crypto Commodity Exchange. This was created for the Cannabis Business by offering banking and B2B transactions that is FinCEN and AML compliant. Crypto Cowboy is a fully turnkey vertical solutions provider to make your coin or token business come alive. From Pre-design, Business Plan & Requirements, Development, Deployment, Support, Legal Guidance and Marketing, our team takes full ownership of your offering and ensures it is managed, maintained, and adheres to all regulatory requirements to ensure your success. Mark has a long history in merger and acquisitions in technology over his 20 years building and operating companies such as Godaddy.com, Endurance International and Newtek Business Services who experienced tremendous success by rapidly implementing marketing, sales, fundraising and acquisition strategies for product and service firms. He has helped companies win with customer excellence and execution on product business strategies to drive success.
Stacey Jackson General Counsel Golden Bear Insurance Company and M.J. Hall & Company, Inc. She began her legal career as a prosecutor for the San Joaquin County District Attorney’s office before becoming a civil litigator. She was partner in the civil defense firm of Thayer, Harvey, Gregerson, Hedberg and Jackson and represented insureds throughout California before coming to “the Bear.” Stacey represents Golden Bear as an active member of PADIC- Pacific Association of Domestic Insurance Companies. Stacey also holds a Master’s Degree in English and a teaching credential from the University of the Pacific. On point with the coming presentation, Stacey was instrumental in drafting coverage for the Cannabis package policy offered by Golden Bear. It is the only admitted cannabis product in the State of California.
Joe Barton, Associate Sheppard Mullin in Los Angeles. Mr. Barton advises companies and individuals in the cannabis space on compliance with federal, state, and local government regulations and defends them in government enforcement actions. Mr. Barton also defends companies and individuals generally in federal criminal and civil False Claims Act cases as well as trade secret cases. Mr. Barton actively performs pro bono work. Mr. Barton’s most recent pro bono projects involved composing a manual of civil tort remedies for domestic violence victims and litigating disabled access claims against the New York Transit Authority.
Moderator: Mark Crager, Founder/Managing Partner Next Stage Partners, a firm that works with early-stage Fintech companies and Financial Institutions to develop strategy and implement new products and services in the payments space and other key growth verticals. Prior to NSP, he held a number of executive positions at Visa Inc. resulting in the launch of multiple products and significant development of key industries. He has a proven track record of moving products from concept to full commercialization in consumer and B2B environments. Prior to his career at Visa, Mark launched new divisions for Sony, J&J and Nestle. Additionally, Mark is a Mentor at the Silicon Valley incubator Plug&Play and also maintains a Member Board Advisor position with StartX, the Stanford University incubator for fintech startups and other growth industries. For further info check out http://www.fintechsv.com

Views: 71
FinTech Silicon Valley

At http://homepages.math.uic.edu/~rtakloo/atkin2012.html
http://wstein.org/talks/2012-04-28-talk/

Views: 192
William Stein

The complex multiplication method (CM method) builds an algebraic curve over a given finite field GF(q) and having an easily computable cardinality. Used at first for elliptic curves, this method is one of the building blocks of the ECPP algorithm that proves the primality of large integers, and it appeared interesting for other applications, the most recent of which being the construction of pairing friendly curves. The aim of the talk is to recall the method, give some applications, and survey recent advances on several parts of the method, due to various authors, concentrating on elliptic curves. This includes class invariant computations, and the potential use of the Montgomery/Edwards parametrization of elliptic curves.

Views: 99
Microsoft Research

I talk with Paladin, a member of the white hats and a forensic financial accountant with a background in law enforcement who authors a guest blog on my site from time to time.
PALADIN is a member of a group of the 'good guys' who work behind the scenes in intel agencies, military, ex-military and others who have revealed the financial backing of the deep state and the hypocrisy that fuels it.
Go here for more info: http://whitehatsreport.com/
KERRY CASSIDY
PROJECT CAMELOT
http://projectcamelot.tv
http://projectcamelotportal.com

Views: 45994
Project Camelot

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Group_theory
00:01:10 1 Main classes of groups
00:01:34 1.1 Permutation groups
00:02:42 1.2 Matrix groups
00:03:20 1.3 Transformation groups
00:04:13 1.4 Abstract groups
00:06:05 1.5 Groups with additional structure
00:08:21 2 Branches of group theory
00:08:30 2.1 Finite group theory
00:09:52 2.2 Representation of groups
00:11:55 2.3 Lie theory
00:13:00 2.4 Combinatorial and geometric group theory
00:16:37 3 Connection of groups and symmetry
00:19:10 4 Applications of group theory
00:19:42 4.1 Galois theory
00:20:35 4.2 Algebraic topology
00:21:42 4.3 Algebraic geometry and cryptography
00:22:58 4.4 Algebraic number theory
00:23:40 4.5 Harmonic analysis
00:24:04 4.6 Combinatorics
00:24:24 4.7 Music
00:24:41 4.8 Physics
00:25:21 4.9 Chemistry and materials science
00:27:30 4.10 Statistical mechanics
00:27:43 5 History
00:27:53 6 See also
00:28:08 7 Notes
00:30:20 8 References
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
"There is only one good, knowledge, and one evil, ignorance."
- Socrates
SUMMARY
=======
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

Views: 1
wikipedia tts

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/History_of_mathematics
00:04:37 1 Prehistoric
00:07:45 2 Babylonian
00:13:09 3 Egyptian
00:15:38 4 Greek
00:30:28 5 Roman
00:35:28 6 Chinese
00:44:11 7 Indian
00:51:23 8 Islamic empire
00:58:41 9 Maya
00:59:50 10 Medieval European
01:05:46 11 Renaissance
01:10:29 12 Mathematics during the Scientific Revolution
01:10:41 12.1 17th century
01:13:34 12.2 18th century
01:15:08 13 Modern
01:15:17 13.1 19th century
01:21:05 13.2 20th century
01:31:21 13.3 21st century
01:32:11 14 Future
01:32:48 15 See also
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.75313900926134
Voice name: en-US-Wavenet-B
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.
The most ancient mathematical texts available are from Mesopotamia and Egypt - Plimpton 322 (Babylonian c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 2000–1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples and so, by inference, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a "demonstrative discipline" 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. Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century. At the end of the 19th century the International Congress of Mathematicians was founded a ...

Views: 42
wikipedia tts

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are
B0 = 1, B1 = ±1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30.
This video is targeted to blind users.
Attribution:
Article text available under CC-BY-SA
Creative Commons image source in video

Views: 942
Audiopedia

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
This video is targeted to blind users.
Attribution:
Article text available under CC-BY-SA
Creative Commons image source in video

Views: 1366
Audiopedia

In mathematics, the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power 3 is 1000: 1000 = 10 × 10 × 10 = 103. More generally, for any two real numbers b and x where b is positive and b ≠ 1,
The logarithm to base 10 (b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the irrational (transcendental) number e (≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base 2 (b = 2) and is prominent in computer science.
This video is targeted to blind users.
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Article text available under CC-BY-SA
Creative Commons image source in video

Views: 293
Audiopedia

Please Subscribe our goal is 200 subscriber for this month :)
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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.

Views: 6945
Wikivoicemedia

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics
00:00:09 A
00:10:55 B
00:11:43 C
00:23:36 D
00:26:35 E
00:30:01 F
00:32:12 G
00:36:57 H
00:38:40 I
00:40:29 J
00:40:37 K
00:42:10 L
00:44:09 M
00:46:45 N
00:48:45 O
00:50:02 P
00:52:20 Q
00:52:59 R
00:56:17 S
01:01:29 T
01:03:46 U
01:04:34 V
01:05:36 W
01:05:53 X
01:06:02 Y
01:06:10 Z
01:06:19 See also
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.7678575093241932
Voice name: en-AU-Wavenet-A
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
This is a glossary of terms that are or have been considered areas of study in mathematics.

Views: 5
wikipedia tts

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Group_(mathematics)
00:02:56 1 Definition and illustration
00:03:06 1.1 First example: the integers
00:04:32 1.2 Definition
00:04:59 1.3 Second example: a symmetry group
00:05:36 2 History
00:06:36 3 Elementary consequences of the group axioms
00:07:10 3.1 Uniqueness of identity element and inverses
00:07:41 3.2 Division
00:10:57 4 Basic concepts
00:14:22 4.1 Group homomorphisms
00:14:45 4.2 Subgroups
00:16:01 4.3 Cosets
00:19:23 4.4 Quotient groups
00:19:58 5 Examples and applications
00:20:39 5.1 Numbers
00:21:52 5.1.1 Integers
00:22:05 5.1.2 Rationals
00:22:36 5.2 Modular arithmetic
00:22:48 5.3 Cyclic groups
00:23:24 5.4 Symmetry groups
00:24:48 5.5 General linear group and representation theory
00:26:14 5.6 Galois groups
00:26:40 6 Finite groups
00:27:01 6.1 Classification of finite simple groups
00:27:36 7 Groups with additional structure
00:27:56 7.1 Topological groups
00:28:08 7.2 Lie groups
00:28:52 8 Generalizations
00:30:44 9 See also
00:31:45 10 Notes
00:32:19 11 Citations
00:33:15 12 References
00:33:34 12.1 General references
00:33:57 12.2 Special references
00:34:07 12.3 Historical references
00:34:12 Quotient groups
00:34:45 {gN, g ∈ G}, "G modulo N".This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN)
00:35:39 N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1
00:36:08 fvR. The group operation on the quotient is shown at the right. For example, U • U
00:36:19 (fv • fv)R
00:37:19 r1, the right rotation and f
00:37:48 f 2
00:39:05 Examples and applications
00:42:05 Numbers
00:42:51 Integers
00:43:48 2 is an integer, but the only solution to the equation a · b
00:44:22 Rationals
00:47:30 Modular arithmetic
00:49:27 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4
00:50:57 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2
00:52:04 Cyclic groups
00:52:31 e, a, a2, a3, ...,where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1
00:53:49 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n
00:54:31 5, 3 is a generator since 31
00:55:47 Symmetry groups
00:59:43 General linear group and representation theory
01:02:00 Galois groups
01:04:17 Finite groups
01:07:12 Classification of finite simple groups
01:08:01 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.
01:09:16 Groups with additional structure
01:09:27 Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.
01:10:05 Topological groups
01:12:39 Lie groups
01:15:10 Generalizations
01:16:56 See also
01:17:09 Notes
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.7336106663255026
Voice name: en-AU-Wavenet-A
"I ...

Views: 14
wikipedia tts

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Group_(mathematics)
00:02:10 1 Definition and illustration
00:02:19 1.1 First example: the integers
00:03:26 1.2 Definition
00:03:48 1.3 Second example: a symmetry group
00:04:18 2 History
00:05:06 3 Elementary consequences of the group axioms
00:05:33 3.1 Uniqueness of identity element and inverses
00:05:59 3.2 Division
00:08:27 4 Basic concepts
00:11:02 4.1 Group homomorphisms
00:11:22 4.2 Subgroups
00:12:20 4.3 Cosets
00:14:51 4.4 Quotient groups
00:15:19 5 Examples and applications
00:15:52 5.1 Numbers
00:16:48 5.1.1 Integers
00:17:00 5.1.2 Rationals
00:17:25 5.2 Modular arithmetic
00:17:36 5.3 Cyclic groups
00:18:06 5.4 Symmetry groups
00:19:10 5.5 General linear group and representation theory
00:20:15 5.6 Galois groups
00:20:37 6 Finite groups
00:20:53 6.1 Classification of finite simple groups
00:21:18 7 Groups with additional structure
00:21:33 7.1 Topological groups
00:21:43 7.2 Lie groups
00:22:18 8 Generalizations
00:23:45 9 See also
00:24:33 10 Notes
00:25:00 11 Citations
00:25:44 12 References
00:26:01 12.1 General references
00:26:20 12.2 Special references
00:26:30 12.3 Historical references
00:26:33 Quotient groups
00:27:01 {gN, g ∈ G}, "G modulo N".This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN)
00:27:43 N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1
00:28:07 fvR. The group operation on the quotient is shown at the right. For example, U • U
00:28:16 (fv • fv)R
00:29:03 r1, the right rotation and f
00:29:27 f 2
00:30:27 Examples and applications
00:32:43 Numbers
00:33:18 Integers
00:34:04 2 is an integer, but the only solution to the equation a · b
00:34:33 Rationals
00:37:00 Modular arithmetic
00:38:29 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4
00:39:39 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2
00:40:31 Cyclic groups
00:40:54 e, a, a2, a3, ...,where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1
00:41:55 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n
00:42:27 5, 3 is a generator since 31
00:43:26 Symmetry groups
00:46:21 General linear group and representation theory
00:48:05 Galois groups
00:49:51 Finite groups
00:52:03 Classification of finite simple groups
00:52:41 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.
00:53:36 Groups with additional structure
00:53:46 Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.
00:54:15 Topological groups
00:56:15 Lie groups
00:58:10 Generalizations
00:59:30 See also
00:59:41 Notes
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Group_(mathematics)
00:03:02 1 Definition and illustration
00:03:12 1.1 First example: the integers
00:04:41 1.2 Definition
00:05:08 1.3 Second example: a symmetry group
00:05:45 2 History
00:06:48 3 Elementary consequences of the group axioms
00:07:22 3.1 Uniqueness of identity element and inverses
00:07:54 3.2 Division
00:11:14 4 Basic concepts
00:14:42 4.1 Group homomorphisms
00:15:05 4.2 Subgroups
00:16:23 4.3 Cosets
00:19:50 4.4 Quotient groups
00:20:25 5 Examples and applications
00:21:07 5.1 Numbers
00:22:21 5.1.1 Integers
00:22:35 5.1.2 Rationals
00:23:07 5.2 Modular arithmetic
00:23:19 5.3 Cyclic groups
00:23:56 5.4 Symmetry groups
00:25:21 5.5 General linear group and representation theory
00:26:47 5.6 Galois groups
00:27:14 6 Finite groups
00:27:36 6.1 Classification of finite simple groups
00:28:12 7 Groups with additional structure
00:28:34 7.1 Topological groups
00:28:46 7.2 Lie groups
00:29:32 8 Generalizations
00:31:28 9 See also
00:32:32 10 Notes
00:33:06 11 Citations
00:34:05 12 References
00:34:26 12.1 General references
00:34:49 12.2 Special references
00:35:00 12.3 Historical references
00:35:05 Quotient groups
00:35:39 {gN, g ∈ G}, "G modulo N".This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN)
00:36:36 N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1
00:37:06 fvR. The group operation on the quotient is shown at the right. For example, U • U
00:37:17 (fv • fv)R
00:38:21 r1, the right rotation and f
00:38:50 f 2
00:40:09 Examples and applications
00:43:11 Numbers
00:43:57 Integers
00:44:56 2 is an integer, but the only solution to the equation a · b
00:45:31 Rationals
00:48:49 Modular arithmetic
00:50:48 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4
00:52:20 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2
00:53:29 Cyclic groups
00:53:57 e, a, a2, a3, ...,where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1
00:55:18 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n
00:55:59 5, 3 is a generator since 31
00:57:14 Symmetry groups
01:01:16 General linear group and representation theory
01:03:37 Galois groups
01:06:00 Finite groups
01:09:00 Classification of finite simple groups
01:09:50 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.
01:11:06 Groups with additional structure
01:11:17 Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.
01:11:57 Topological groups
01:14:36 Lie groups
01:17:10 Generalizations
01:18:56 See also
01:19:10 Notes
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.7181211225309111
Voice name: en-US-Wavenet-D
"I ...

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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/History_of_mathematics
00:03:27 1 Prehistoric
00:05:49 2 Babylonian
00:09:54 3 Egyptian
00:11:48 4 Greek
00:22:56 5 Roman
00:26:44 6 Chinese
00:33:15 7 Indian
00:38:41 8 Islamic empire
00:44:10 9 Maya
00:45:03 10 Medieval European
00:49:32 11 Renaissance
00:53:05 12 Mathematics during the Scientific Revolution
00:53:16 12.1 17th century
00:55:27 12.2 18th century
00:56:39 13 Modern
00:56:48 13.1 19th century
01:01:09 13.2 20th century
01:08:51 13.3 21st century
01:09:30 14 Future
01:09:59 15 See also
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9803638922155543
Voice name: en-US-Wavenet-A
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.
The most ancient mathematical texts available are from Mesopotamia and Egypt - Plimpton 322 (Babylonian c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 2000–1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples and so, by inference, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a "demonstrative discipline" 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. Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century. At the end of the 19th century the International Congress of Mathematicians was founded ...

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wikipedia tts