In complex analysis, an elliptic function is a meromorphicfunction that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a boundedentire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles (counting multiplicity) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of any simple poles must cancel.
Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms.
Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem
Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.
This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)!
QUESTIONS:
1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?
2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it th...
published: 26 Jul 2020
Elliptic Integrals and Elliptic Functions, a brief history
published: 15 May 2021
Intro to Jacobi Elliptic Functions
In this video I introduce Jacobi Elliptic Functions.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
published: 03 Aug 2020
Elliptic functions 1. Weierstrass function.
This lecture is part of a series of lectures on Elliptic functions
This lecture covers the basic properties of the Weierstrass P function
The pictures of elliptic functions in the video come from the book by Jahnke and Emde,
which can be found here: https://archive.org/details/in.ernet.dli.2015.212842
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj50t6hrPaJ0GruNrN-xPcFTI
published: 18 Feb 2024
Elliptic Curves - Computerphile
Just what are elliptic curves and why use a graph shape in cryptography? Dr Mike Pound explains.
Mike's myriad Diffie-Hellman videos: https://www.youtube.com/playlist?list=PLzH6n4zXuckpoaxDKOOV26yhgoY2S-xYg
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
published: 16 Jan 2018
Jacobi Elliptic Function Intuition
In this video I look at the period of Jacobi Elliptic Functions, and some more of the analogies between elliptic functions and sine and cosine.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
published: 04 Aug 2020
Hypergeometric functions and Elliptic Integrals -- Part 1
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published: 16 Dec 2021
Elliptic function Meaning
Video shows what elliptic function means. Any function of a complex variable which is periodic in two directions. Elliptic function Meaning. How to pronounce, definition audio dictionary. How to say elliptic function. Powered by MaryTTS, Wiktionary
published: 27 Apr 2015
Javier Fresán - On G-functions of differential order 2
This talk was part of the Workshop on "Algebraicity and Transcendence for Singular Differential Equations" held at the ESI October 7 -- 19, 2024.
published: 04 Nov 2024
The Pendulum and Weierstrass Elliptic Functions
This is a presentation of something recent
https://arxiv.org/abs/2306.10653
which is kind of cute as it involves very classical topics: the Weierstrass elliptic function and the pendulum, both topics which occur in undergraduate mathematics. Traditionally the pendulum is solved using Jacobi elliptic functions. The use of the Weierstrass elliptic function is cute too.
Since I forgot my camera at home this morning, I had to record with my laptop and phone. The video quality of the camera showing the board is therefore lower than usual (1960 px) rather than the usual 4K.
Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.
This is by far the ...
Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.
This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)!
QUESTIONS:
1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?
2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory?
3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it?
4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)?
5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this?
6. Does anyone understand Andrew Wiles’ paper? LOL
SOURCES I USED TO STUDY:
Keith Conrad’s Lectures on Modular Forms (8 part video series):
https://www.youtube.com/watch?v=LolxzYwN1TQ
Keith Conrad’s Notes on Modular Forms:
https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNTmodularforms.pdf
“Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo.
(The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.)
“The Arithmetic of Elliptic Curves” by Joseph Silverman
HOMEWORK IDEA CREDIT goes to Looking Glass Universe!
SAGE RESOURCES:
“Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : http://www.gregorybard.com/Sage.html
Download SAGE: https://www.sagemath.org/download.html
Proof of the Hasse-Weil Bound on Terry Tao’s Blog: https://terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-bound/
OTHER VIDEOS ON THESE TOPICS:
Numberphile Playlist: https://www.youtube.com/playlist?list=PLt5AfwLFPxWLD3KG-XZQFTDFhnZ3GHMlW
Elliptic Curves and Modular Forms: https://www.youtube.com/watch?v=A8fsU97g3tg
SOFTWARE USED TO MAKE THIS VIDEO:
SAGE for the code and the graphs
https://github.com/hernanat/dcolor for domain coloring
Adobe Premiere Elements For Video Editing
MUSIC:
Music Info: Documentary - AShamaluevMusic.
Music Link: https://www.ashamaluevmusic.com
Follow me!
Twitter: https://twitter.com/00aleph00
Instagram: https://www.instagram.com/00aleph00
Intro: (0:00)
Elliptic Curves: (0:58)
Modular Forms: (3:26)
Taniyama Shimura Conjecture: (7:26)
Fermat's Last Theorem: (8:02)
Questions for you!: (8:51)
Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.
This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)!
QUESTIONS:
1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?
2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory?
3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it?
4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)?
5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this?
6. Does anyone understand Andrew Wiles’ paper? LOL
SOURCES I USED TO STUDY:
Keith Conrad’s Lectures on Modular Forms (8 part video series):
https://www.youtube.com/watch?v=LolxzYwN1TQ
Keith Conrad’s Notes on Modular Forms:
https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNTmodularforms.pdf
“Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo.
(The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.)
“The Arithmetic of Elliptic Curves” by Joseph Silverman
HOMEWORK IDEA CREDIT goes to Looking Glass Universe!
SAGE RESOURCES:
“Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : http://www.gregorybard.com/Sage.html
Download SAGE: https://www.sagemath.org/download.html
Proof of the Hasse-Weil Bound on Terry Tao’s Blog: https://terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-bound/
OTHER VIDEOS ON THESE TOPICS:
Numberphile Playlist: https://www.youtube.com/playlist?list=PLt5AfwLFPxWLD3KG-XZQFTDFhnZ3GHMlW
Elliptic Curves and Modular Forms: https://www.youtube.com/watch?v=A8fsU97g3tg
SOFTWARE USED TO MAKE THIS VIDEO:
SAGE for the code and the graphs
https://github.com/hernanat/dcolor for domain coloring
Adobe Premiere Elements For Video Editing
MUSIC:
Music Info: Documentary - AShamaluevMusic.
Music Link: https://www.ashamaluevmusic.com
Follow me!
Twitter: https://twitter.com/00aleph00
Instagram: https://www.instagram.com/00aleph00
Intro: (0:00)
Elliptic Curves: (0:58)
Modular Forms: (3:26)
Taniyama Shimura Conjecture: (7:26)
Fermat's Last Theorem: (8:02)
Questions for you!: (8:51)
In this video I introduce Jacobi Elliptic Functions.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9...
In this video I introduce Jacobi Elliptic Functions.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
In this video I introduce Jacobi Elliptic Functions.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
This lecture is part of a series of lectures on Elliptic functions
This lecture covers the basic properties of the Weierstrass P function
The pictures of ellip...
This lecture is part of a series of lectures on Elliptic functions
This lecture covers the basic properties of the Weierstrass P function
The pictures of elliptic functions in the video come from the book by Jahnke and Emde,
which can be found here: https://archive.org/details/in.ernet.dli.2015.212842
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj50t6hrPaJ0GruNrN-xPcFTI
This lecture is part of a series of lectures on Elliptic functions
This lecture covers the basic properties of the Weierstrass P function
The pictures of elliptic functions in the video come from the book by Jahnke and Emde,
which can be found here: https://archive.org/details/in.ernet.dli.2015.212842
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj50t6hrPaJ0GruNrN-xPcFTI
Just what are elliptic curves and why use a graph shape in cryptography? Dr Mike Pound explains.
Mike's myriad Diffie-Hellman videos: https://www.youtube.com/p...
Just what are elliptic curves and why use a graph shape in cryptography? Dr Mike Pound explains.
Mike's myriad Diffie-Hellman videos: https://www.youtube.com/playlist?list=PLzH6n4zXuckpoaxDKOOV26yhgoY2S-xYg
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
Just what are elliptic curves and why use a graph shape in cryptography? Dr Mike Pound explains.
Mike's myriad Diffie-Hellman videos: https://www.youtube.com/playlist?list=PLzH6n4zXuckpoaxDKOOV26yhgoY2S-xYg
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
In this video I look at the period of Jacobi Elliptic Functions, and some more of the analogies between elliptic functions and sine and cosine.
For more videos...
In this video I look at the period of Jacobi Elliptic Functions, and some more of the analogies between elliptic functions and sine and cosine.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
In this video I look at the period of Jacobi Elliptic Functions, and some more of the analogies between elliptic functions and sine and cosine.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5
Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1
Patreon: https://www.patre...
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5
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Buy textbooks here and help me out: https://amzn.to/31Bj9ye
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Sacred Mathematics: Japanese Temple Geometry: https://amzn.to/2ZIadH9
Electricity and Magnetism for Mathematicians: https://amzn.to/2H8ePzL
Abstract Algebra:
Judson(online): http://abstract.ups.edu/
Judson(print): https://amzn.to/2Xg92wD
Dummit and Foote: https://amzn.to/2zYOrok
Gallian: https://amzn.to/2zg4YEo
Artin: https://amzn.to/2LQ8l7C
Differential Forms:
Bachman: https://amzn.to/2z9wljH
Number Theory:
Crisman(online): http://math.gordon.edu/ntic/
Strayer: https://amzn.to/3bXwLah
Andrews: https://amzn.to/2zWlOZ0
Analysis:
Abbot: https://amzn.to/3cwYtuF
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Video shows what elliptic function means. Any function of a complex variable which is periodic in two directions. Elliptic function Meaning. How to pronounce, ...
Video shows what elliptic function means. Any function of a complex variable which is periodic in two directions. Elliptic function Meaning. How to pronounce, definition audio dictionary. How to say elliptic function. Powered by MaryTTS, Wiktionary
Video shows what elliptic function means. Any function of a complex variable which is periodic in two directions. Elliptic function Meaning. How to pronounce, definition audio dictionary. How to say elliptic function. Powered by MaryTTS, Wiktionary
This is a presentation of something recent
https://arxiv.org/abs/2306.10653
which is kind of cute as it involves very classical topics: the Weierstrass elliptic...
This is a presentation of something recent
https://arxiv.org/abs/2306.10653
which is kind of cute as it involves very classical topics: the Weierstrass elliptic function and the pendulum, both topics which occur in undergraduate mathematics. Traditionally the pendulum is solved using Jacobi elliptic functions. The use of the Weierstrass elliptic function is cute too.
Since I forgot my camera at home this morning, I had to record with my laptop and phone. The video quality of the camera showing the board is therefore lower than usual (1960 px) rather than the usual 4K.
This is a presentation of something recent
https://arxiv.org/abs/2306.10653
which is kind of cute as it involves very classical topics: the Weierstrass elliptic function and the pendulum, both topics which occur in undergraduate mathematics. Traditionally the pendulum is solved using Jacobi elliptic functions. The use of the Weierstrass elliptic function is cute too.
Since I forgot my camera at home this morning, I had to record with my laptop and phone. The video quality of the camera showing the board is therefore lower than usual (1960 px) rather than the usual 4K.
Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.
This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)!
QUESTIONS:
1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?
2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory?
3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it?
4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)?
5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this?
6. Does anyone understand Andrew Wiles’ paper? LOL
SOURCES I USED TO STUDY:
Keith Conrad’s Lectures on Modular Forms (8 part video series):
https://www.youtube.com/watch?v=LolxzYwN1TQ
Keith Conrad’s Notes on Modular Forms:
https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNTmodularforms.pdf
“Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo.
(The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.)
“The Arithmetic of Elliptic Curves” by Joseph Silverman
HOMEWORK IDEA CREDIT goes to Looking Glass Universe!
SAGE RESOURCES:
“Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : http://www.gregorybard.com/Sage.html
Download SAGE: https://www.sagemath.org/download.html
Proof of the Hasse-Weil Bound on Terry Tao’s Blog: https://terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-bound/
OTHER VIDEOS ON THESE TOPICS:
Numberphile Playlist: https://www.youtube.com/playlist?list=PLt5AfwLFPxWLD3KG-XZQFTDFhnZ3GHMlW
Elliptic Curves and Modular Forms: https://www.youtube.com/watch?v=A8fsU97g3tg
SOFTWARE USED TO MAKE THIS VIDEO:
SAGE for the code and the graphs
https://github.com/hernanat/dcolor for domain coloring
Adobe Premiere Elements For Video Editing
MUSIC:
Music Info: Documentary - AShamaluevMusic.
Music Link: https://www.ashamaluevmusic.com
Follow me!
Twitter: https://twitter.com/00aleph00
Instagram: https://www.instagram.com/00aleph00
Intro: (0:00)
Elliptic Curves: (0:58)
Modular Forms: (3:26)
Taniyama Shimura Conjecture: (7:26)
Fermat's Last Theorem: (8:02)
Questions for you!: (8:51)
In this video I introduce Jacobi Elliptic Functions.
For more videos in this series, visit:
https://www.youtube.com/playlist?list=PL2uXHjNuf12a0PebiLp91iOQZ4-9OisCy
This lecture is part of a series of lectures on Elliptic functions
This lecture covers the basic properties of the Weierstrass P function
The pictures of elliptic functions in the video come from the book by Jahnke and Emde,
which can be found here: https://archive.org/details/in.ernet.dli.2015.212842
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj50t6hrPaJ0GruNrN-xPcFTI
Just what are elliptic curves and why use a graph shape in cryptography? Dr Mike Pound explains.
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This video was filmed and edited by Sean Riley.
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In this video I look at the period of Jacobi Elliptic Functions, and some more of the analogies between elliptic functions and sine and cosine.
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Video shows what elliptic function means. Any function of a complex variable which is periodic in two directions. Elliptic function Meaning. How to pronounce, definition audio dictionary. How to say elliptic function. Powered by MaryTTS, Wiktionary
This is a presentation of something recent
https://arxiv.org/abs/2306.10653
which is kind of cute as it involves very classical topics: the Weierstrass elliptic function and the pendulum, both topics which occur in undergraduate mathematics. Traditionally the pendulum is solved using Jacobi elliptic functions. The use of the Weierstrass elliptic function is cute too.
Since I forgot my camera at home this morning, I had to record with my laptop and phone. The video quality of the camera showing the board is therefore lower than usual (1960 px) rather than the usual 4K.
In complex analysis, an elliptic function is a meromorphicfunction that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a boundedentire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles (counting multiplicity) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of any simple poles must cancel.
Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms.
NationalMathematics Day ... Also read ... Also read ... Also read ... He also made notable contributions like the hypergeometric series, the Riemann series, the elliptic integrals, the theory of divergent series, and the functional equations of the zeta function.
Initiated in 2012 by then-Prime MinisterManmohan Singh, the day honours Ramanujan's remarkable contributions to mathematics ... The next year, he was elected to the prestigious Royal Society for his work on Elliptic Functions and number theory ... See more.
Satoshi wrote, “If SHA-256 became completely broken, I think we could come to some agreement about what the honest blockchain was before the trouble started, lock that in, and continue from there with a new hash function.”.
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Specific Next-Generation Technology... Verkle Trees. The Solution ... Various cryptographic techniques, such as elliptic curve pairings and hash functions, have been employed by the designers so that Verkle trees can support efficient zero-knowledge proofs.
These wallets are derived from private keys using elliptic curve multiplication. Quantum computers can reverse this process, making elliptic curve cryptography vulnerable. However, hash functions like ...
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The American College of Sports Medicine suggests doing both types of exercise to take advantage of their unique benefits for improving health and daily functioning and reducing chronic disease risk.
The American College of Sports Medicine suggests doing both types of exercise to take advantage of their unique benefits for improving health and daily functioning and reducing chronic disease risk.
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