The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this comes from a line bundle condition of descent.
Jacobi theta function
There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them.
One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula
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published: 04 Dec 2021
Modular forms: Theta functions
This lecture is part of an online graduate course on modular forms.
We show that the theta function of a 1-dimensional lattice is a modular form using the Poisson summation formula, and use this to prove the functional equation of the Riemann zeta function.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
published: 27 Mar 2021
Counting points on the E8 lattice with modular forms (theta functions) | #SoME2
In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of lattices and to modular forms for those with some knowledge of vector spaces and series.
--------------
References:
(Paper on MIT) Poisson summation on lattices - https://math.mit.edu/~brubaker/Math784/thetafunctions.pdf
Duals of lattices - https://cseweb.ucsd.edu/classes/wi12/cse206A-a/LecDual.pdf
Modular forms - https://wstein.org/books/modform/modform/
Modular form graph was made using Sage with code adapted from this StackExchange answer:
https://math.stackexchange.com/a/4309925
--------------
This is my submission for 3blue1brown's Summer of Math Exposition 2 (SoME2) and is also my first math YouTube video. I'm excited to ...
published: 16 Aug 2022
Modular forms: Theta functions in higher dimensions
This lecture is part of an online graduate course on modular forms.
We study theta functions of even unimodular lattices, such as the root lattice of the E8 exceptional Lie algebra. As examples we show that one cannot "her the shape of a drum", and calculate the number of minimal vectors in the Leech lattice, and show that any even unimodular lattice has to have dimension divisible by 8.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
published: 28 Mar 2021
Number Theory 2.1: A Brief Introduction to the Theta Function
published: 05 May 2020
Ramanujan tau function and the Sato--Tate conjecture
published: 27 Jul 2016
Riemann’s theta function
Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and cryptography. One of my long term projects is to develop a program to study and connect the various aspects - geometric, computational, tropical and applied - of the theta function. In this talk, I present some of the results obtained in my group about theta functions of algebraic curves, and especially how they connect to the physics of water waves through the KP equation.
published: 30 Oct 2022
mock theta function
short introduction for mock theta function
#mathematics
#mock theta function
#ramanujan
If you would like to know more about it , I recommend following books
https://amzn.to/43gb4MY
https://amzn.to/43hMwDr
https://amzn.to/3rlZxOU
published: 07 Jul 2023
How Ramanujan May Have Discovered of the Mock Theta Functions by George Andrews
The mock theta functions made their first appearance in
Ramanujan’s last letter to Hardy. Ramanujan explains that he is trying
to find functions apart from theta functions that behave like theta
functions near the unit circle. Where did he ever get the idea that
such functions might exist? Why in the world would he consider the
special q-series that he lists in his last letter? The object of this talk
is to provide a plausible explanation for the discovery of mock theta
functions.
published: 26 Mar 2021
Mock Modular Forms are Everywhere - Miranda Cheng
14.03.2019
Science Colloquium at the Institute for the Theoretical Studies, ETH Zürich
Source: https://video.ethz.ch/speakers/its/2019/spring/colloquium/9a64c3f5-be20-4b24-9966-59b16f082cb9.html
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Gallian: https://amzn.to/2zg4YEo
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This lecture is part of an online graduate course on modular forms.
We show that the theta function of a 1-dimensional lattice is a modular form using the Poi...
This lecture is part of an online graduate course on modular forms.
We show that the theta function of a 1-dimensional lattice is a modular form using the Poisson summation formula, and use this to prove the functional equation of the Riemann zeta function.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
This lecture is part of an online graduate course on modular forms.
We show that the theta function of a 1-dimensional lattice is a modular form using the Poisson summation formula, and use this to prove the functional equation of the Riemann zeta function.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of la...
In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of lattices and to modular forms for those with some knowledge of vector spaces and series.
--------------
References:
(Paper on MIT) Poisson summation on lattices - https://math.mit.edu/~brubaker/Math784/thetafunctions.pdf
Duals of lattices - https://cseweb.ucsd.edu/classes/wi12/cse206A-a/LecDual.pdf
Modular forms - https://wstein.org/books/modform/modform/
Modular form graph was made using Sage with code adapted from this StackExchange answer:
https://math.stackexchange.com/a/4309925
--------------
This is my submission for 3blue1brown's Summer of Math Exposition 2 (SoME2) and is also my first math YouTube video. I'm excited to share this with you! Let me know if there are any mistakes I made in the video, and I will make note of them in this description.
--------------
Chapters:
0:00 - Introduction
0:34 - Lattices
5:23 - Theta Functions
8:45 - Modular Forms
12:12 - Showing Modularity
15:12 - Spaces of Modular Forms
19:35 - Coefficients and Counting
21:25 - Conclusion
In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of lattices and to modular forms for those with some knowledge of vector spaces and series.
--------------
References:
(Paper on MIT) Poisson summation on lattices - https://math.mit.edu/~brubaker/Math784/thetafunctions.pdf
Duals of lattices - https://cseweb.ucsd.edu/classes/wi12/cse206A-a/LecDual.pdf
Modular forms - https://wstein.org/books/modform/modform/
Modular form graph was made using Sage with code adapted from this StackExchange answer:
https://math.stackexchange.com/a/4309925
--------------
This is my submission for 3blue1brown's Summer of Math Exposition 2 (SoME2) and is also my first math YouTube video. I'm excited to share this with you! Let me know if there are any mistakes I made in the video, and I will make note of them in this description.
--------------
Chapters:
0:00 - Introduction
0:34 - Lattices
5:23 - Theta Functions
8:45 - Modular Forms
12:12 - Showing Modularity
15:12 - Spaces of Modular Forms
19:35 - Coefficients and Counting
21:25 - Conclusion
This lecture is part of an online graduate course on modular forms.
We study theta functions of even unimodular lattices, such as the root lattice of the E8 ex...
This lecture is part of an online graduate course on modular forms.
We study theta functions of even unimodular lattices, such as the root lattice of the E8 exceptional Lie algebra. As examples we show that one cannot "her the shape of a drum", and calculate the number of minimal vectors in the Leech lattice, and show that any even unimodular lattice has to have dimension divisible by 8.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
This lecture is part of an online graduate course on modular forms.
We study theta functions of even unimodular lattices, such as the root lattice of the E8 exceptional Lie algebra. As examples we show that one cannot "her the shape of a drum", and calculate the number of minimal vectors in the Leech lattice, and show that any even unimodular lattice has to have dimension divisible by 8.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and c...
Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and cryptography. One of my long term projects is to develop a program to study and connect the various aspects - geometric, computational, tropical and applied - of the theta function. In this talk, I present some of the results obtained in my group about theta functions of algebraic curves, and especially how they connect to the physics of water waves through the KP equation.
Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and cryptography. One of my long term projects is to develop a program to study and connect the various aspects - geometric, computational, tropical and applied - of the theta function. In this talk, I present some of the results obtained in my group about theta functions of algebraic curves, and especially how they connect to the physics of water waves through the KP equation.
short introduction for mock theta function
#mathematics
#mock theta function
#ramanujan
If you would like to know more about it , I recommend following book...
short introduction for mock theta function
#mathematics
#mock theta function
#ramanujan
If you would like to know more about it , I recommend following books
https://amzn.to/43gb4MY
https://amzn.to/43hMwDr
https://amzn.to/3rlZxOU
short introduction for mock theta function
#mathematics
#mock theta function
#ramanujan
If you would like to know more about it , I recommend following books
https://amzn.to/43gb4MY
https://amzn.to/43hMwDr
https://amzn.to/3rlZxOU
The mock theta functions made their first appearance in
Ramanujan’s last letter to Hardy. Ramanujan explains that he is trying
to find functions apart from thet...
The mock theta functions made their first appearance in
Ramanujan’s last letter to Hardy. Ramanujan explains that he is trying
to find functions apart from theta functions that behave like theta
functions near the unit circle. Where did he ever get the idea that
such functions might exist? Why in the world would he consider the
special q-series that he lists in his last letter? The object of this talk
is to provide a plausible explanation for the discovery of mock theta
functions.
The mock theta functions made their first appearance in
Ramanujan’s last letter to Hardy. Ramanujan explains that he is trying
to find functions apart from theta functions that behave like theta
functions near the unit circle. Where did he ever get the idea that
such functions might exist? Why in the world would he consider the
special q-series that he lists in his last letter? The object of this talk
is to provide a plausible explanation for the discovery of mock theta
functions.
14.03.2019
Science Colloquium at the Institute for the Theoretical Studies, ETH Zürich
Source: https://video.ethz.ch/speakers/its/2019/spring/colloquium/9a64c...
14.03.2019
Science Colloquium at the Institute for the Theoretical Studies, ETH Zürich
Source: https://video.ethz.ch/speakers/its/2019/spring/colloquium/9a64c3f5-be20-4b24-9966-59b16f082cb9.html
14.03.2019
Science Colloquium at the Institute for the Theoretical Studies, ETH Zürich
Source: https://video.ethz.ch/speakers/its/2019/spring/colloquium/9a64c3f5-be20-4b24-9966-59b16f082cb9.html
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5
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https://brave.com/sdp793
Buy textbooks here and help me out: https://amzn.to/31Bj9ye
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Books I like:
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
How to think about Analysis: https://amzn.to/2AIhwVm
Calculus:
OpenStax(online): https://openstax.org/subjects/math
OpenStax Vol 1: https://amzn.to/2zlreN8
OpenStax Vol 2: https://amzn.to/2TtwoxH
OpenStax Vol 3: https://amzn.to/3bPJ3Bn
My Filming Equipment:
Camera: https://amzn.to/3kx2JzE
Lense: https://amzn.to/2PFxPXA
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Lights: https://amzn.to/2XHxRT0
White Chalk: https://amzn.to/3ipu3Oh
Color Chalk: https://amzn.to/2XL6eIJ
This lecture is part of an online graduate course on modular forms.
We show that the theta function of a 1-dimensional lattice is a modular form using the Poisson summation formula, and use this to prove the functional equation of the Riemann zeta function.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of lattices and to modular forms for those with some knowledge of vector spaces and series.
--------------
References:
(Paper on MIT) Poisson summation on lattices - https://math.mit.edu/~brubaker/Math784/thetafunctions.pdf
Duals of lattices - https://cseweb.ucsd.edu/classes/wi12/cse206A-a/LecDual.pdf
Modular forms - https://wstein.org/books/modform/modform/
Modular form graph was made using Sage with code adapted from this StackExchange answer:
https://math.stackexchange.com/a/4309925
--------------
This is my submission for 3blue1brown's Summer of Math Exposition 2 (SoME2) and is also my first math YouTube video. I'm excited to share this with you! Let me know if there are any mistakes I made in the video, and I will make note of them in this description.
--------------
Chapters:
0:00 - Introduction
0:34 - Lattices
5:23 - Theta Functions
8:45 - Modular Forms
12:12 - Showing Modularity
15:12 - Spaces of Modular Forms
19:35 - Coefficients and Counting
21:25 - Conclusion
This lecture is part of an online graduate course on modular forms.
We study theta functions of even unimodular lattices, such as the root lattice of the E8 exceptional Lie algebra. As examples we show that one cannot "her the shape of a drum", and calculate the number of minimal vectors in the Leech lattice, and show that any even unimodular lattice has to have dimension divisible by 8.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F
Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and cryptography. One of my long term projects is to develop a program to study and connect the various aspects - geometric, computational, tropical and applied - of the theta function. In this talk, I present some of the results obtained in my group about theta functions of algebraic curves, and especially how they connect to the physics of water waves through the KP equation.
short introduction for mock theta function
#mathematics
#mock theta function
#ramanujan
If you would like to know more about it , I recommend following books
https://amzn.to/43gb4MY
https://amzn.to/43hMwDr
https://amzn.to/3rlZxOU
The mock theta functions made their first appearance in
Ramanujan’s last letter to Hardy. Ramanujan explains that he is trying
to find functions apart from theta functions that behave like theta
functions near the unit circle. Where did he ever get the idea that
such functions might exist? Why in the world would he consider the
special q-series that he lists in his last letter? The object of this talk
is to provide a plausible explanation for the discovery of mock theta
functions.
14.03.2019
Science Colloquium at the Institute for the Theoretical Studies, ETH Zürich
Source: https://video.ethz.ch/speakers/its/2019/spring/colloquium/9a64c3f5-be20-4b24-9966-59b16f082cb9.html
The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this comes from a line bundle condition of descent.
Jacobi theta function
There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them.
One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula
“However, each of these brain waves have been associated with specific neural processes – theta brain waves have been associated with the direction of attention and selection of a specific thing ...
These reports include allegations of at least one incident where students were drugged or otherwise unknowingly ingested substances that caused some level of incapacitation while attending a social function at the Theta Chi chapter house.”.
The MoneyWave is a digital program that is designed to help improve cognitive function and maintain a healthy brain ... The Money Wave audio file activates the theta waves in the brain that help to bring a positive impact on brain functions.
Scientists categorise brainwaves based on their frequency ranges (measured in Hertz), which are associated with different functions. The brainwaves researchers mainly explore in putting are the theta band (4 -7 Hz.
The GeniusWave utilizes advanced soundwave technology to stimulate your brain’s Theta waves, which are crucial for creativity, learning, and overall cognitive function...Once activated, the Theta ...
Scientists categorise brainwaves based on their frequency ranges (measured in Hertz), which are associated with different functions . The brainwaves researchers mainly explore in putting are the theta band (4 -7 Hz.
Scientists categorise brainwaves based on their frequency ranges (measured in Hertz), which are associated with different functions . The brainwaves researchers mainly explore in putting are the theta band (4 -7 Hz.
ThetaNetwork (THETA) ... Key Features of Theta Network ... Theta’s key features are outlined as follows ... Theta supports the creation of various Web3 applications, including NFTs and DAOs, enhancing the platform’s versatility and functionality.
ThetaNetwork (THETA). Theta Network focuses on decentralizing streaming services through its modified Proof of Stake (PoS) protocol ... THETA and TFUEL tokens are fundamental to its ecosystem, where THETA ...
It explains the science behind Theta waves, their benefits, and how they can be harnessed to improve cognitive function... It explains the science behind Theta waves, their benefits, and how they can be harnessed to improve cognitive function.
These improvements were linked to specific changes in brain activity, especially in certain types of brain waves called alpha and theta waves ...Higher theta power suggests greater encoding capacity, crucial for retaining information in working memory.
With a steady rise in price since its partnership with EdgeCloud, THETA showcases significant growth potential ... THETA’s blockchain integration of AI enhances DeFi functionality, paving the way for enhanced user experiences.
The Theta engineering team’s creation of generative AI model pipelines resulted in the launch of EdgeCloud ... Theta recently released an upgrade to the EdgeCloud AI Showcase ... The Theta EdgeCloud is ...
Rolling theta ... This is because an option sheds value from time decay (theta). This time decay comes from the time value of an option and is a function of an option’s time to expiry and implied volatility. You can gain from time decay in two ways ... ....