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RoyK
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19 May 2012, 5:13 pm

Does anyone know how to find a zero of the Riemann zeta function?



ruveyn
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21 May 2012, 12:34 am

RoyK wrote:
Does anyone know how to find a zero of the Riemann zeta function?


See http://en.wikipedia.org/wiki/Riemann_ze ... hypothesis


ruveyn



AspieRogue

21 May 2012, 1:53 pm

marshall wrote:
Fatal-Noogie wrote:
I'm not bored, but I need a break...

What do you think is the most inherently frightening branch of mathematics? :chin:

For me, I think maybe systems of non-linear equations,
followed by Laplace and Fourier transformations.
(That's as far as I got in my mechanical engineering training.)


Differential equations is one of the more ugly branches of mathematics. At least on a rigorous theoretical level. I mean, it's fun looking at numerical solutions to non-linear systems on a computer but actually proving things about non-linear equations is next to impossible.

As for the most frightening, I'd have to vote for tensor calculus. The traditional definitions and notation are abhorrent. If you've had to take a fluid dynamics course you'll understand the horror of being introduced to tensor notation. Tensors simply do not make sense the way fluid dynamics people talk about them in their wretched introductory texts.



I tend to think of tensors as transformation matrices that allow you to determine the components of a vector in multiple coordinate systems. They make a lot of sense once you've studied matrix algebra and Vector spaces. Because a Tensor is a basically a cartesian product of vector spaces mapped into yet another vector space.


As for differential equations, it turns out that the most powerful technique for solving them is to use Lie Groups(specifically linear transformation groups). Lie Groups of course require an understand of differential geometry and general topology. I heart Lie Groups.



AspieRogue

21 May 2012, 1:58 pm

anxiouspoet wrote:
marshall wrote:
I guess despite the supposed stereotype there aren't a lot of aspies who love math. Right now I'm trying to self-teach myself axiomatic set theory.



By "learning axiomatic set theory" do you mean learning ZFC (zermelo-fraenkel set theory with axiom of choice) and learning how to construct basic mathematical objects? That's fascinating stuff the first time you see it. It really helps build philosophical skill and faith in math to judge itself critically.

It can also be mind-bending to really examine those different levels of infinity. Not to mention when you get into constructability of universes, inner model theory, and forcing etc.etc.


OF COURSE he means ZFC set theory! :lol:

The other 3 set theories are based on ZFC+AC set theory so that is truly the foundation for axiomatic set theory. Model theory, which includes the study of constructable universes and Boolean algebras and introduces the forcing technique, is firmly rooted in first order logic. Most predicate logic classes are put under the philosophy department for some reason.



marshall
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27 May 2012, 11:59 pm

AspieRogue wrote:
anxiouspoet wrote:
marshall wrote:
I guess despite the supposed stereotype there aren't a lot of aspies who love math. Right now I'm trying to self-teach myself axiomatic set theory.



By "learning axiomatic set theory" do you mean learning ZFC (zermelo-fraenkel set theory with axiom of choice) and learning how to construct basic mathematical objects? That's fascinating stuff the first time you see it. It really helps build philosophical skill and faith in math to judge itself critically.

It can also be mind-bending to really examine those different levels of infinity. Not to mention when you get into constructability of universes, inner model theory, and forcing etc.etc.


OF COURSE he means ZFC set theory! :lol:

The other 3 set theories are based on ZFC+AC set theory so that is truly the foundation for axiomatic set theory. Model theory, which includes the study of constructable universes and Boolean algebras and introduces the forcing technique, is firmly rooted in first order logic. Most predicate logic classes are put under the philosophy department for some reason.


I like the NBG formulation of set theory as it can be generated from a finite number of axioms while being logically identical to ZFC for statements not involving proper classes.



marshall
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28 May 2012, 10:30 am

AspieRogue wrote:
Model theory, which includes the study of constructable universes and Boolean algebras and introduces the forcing technique, is firmly rooted in first order logic. Most predicate logic classes are put under the philosophy department for some reason.

I'd say Model theory is the application of algebraic and set-theoretic concepts to first-order logic itself. It's a weird concept to wrap my head around as my mind is trained to think in terms of logic without necessarily analyzing it from a mathematical point of view.



ruveyn
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28 May 2012, 11:08 am

[quote="AspieRogue"


I tend to think of tensors as transformation matrices that allow you to determine the components of a vector in multiple coordinate systems. They make a lot of sense once you've studied matrix algebra and Vector spaces. Because a Tensor is a basically a cartesian product of vector spaces mapped into yet another vector space.


[/quote]

An F valued tensor is a multi-linear mapping from a cartesian product of vectors spaces and their dual spaces into the field F.

F could be the real numbers or F could be the complex numbers. The classical definition of tensors in terms of their components is an equivalence relation between components under a linear mapping.

ruveyn



AspieRogue

28 May 2012, 12:04 pm

ruveyn wrote:
AspieRogue wrote:


I tend to think of tensors as transformation matrices that allow you to determine the components of a vector in multiple coordinate systems. They make a lot of sense once you've studied matrix algebra and Vector spaces. Because a Tensor is a basically a cartesian product of vector spaces mapped into yet another vector space.




An F valued tensor is a multi-linear mapping from a cartesian product of vectors spaces and their dual spaces into the field F.

F could be the real numbers or F could be the complex numbers. The classical definition of tensors in terms of their components is an equivalence relation between components under a linear mapping.

ruveyn


How does what I posted about tensors conflict with the definition you presented? Short answer: It doesn't.



ruveyn
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28 May 2012, 1:24 pm

AspieRogue wrote:
ruveyn wrote:
AspieRogue wrote:


I tend to think of tensors as transformation matrices that allow you to determine the components of a vector in multiple coordinate systems. They make a lot of sense once you've studied matrix algebra and Vector spaces. Because a Tensor is a basically a cartesian product of vector spaces mapped into yet another vector space.




An F valued tensor is a multi-linear mapping from a cartesian product of vectors spaces and their dual spaces into the field F.

F could be the real numbers or F could be the complex numbers. The classical definition of tensors in terms of their components is an equivalence relation between components under a linear mapping.

ruveyn


How does what I posted about tensors conflict with the definition you presented? Short answer: It doesn't.


You said a tensor is a cartesian product. Nay, nay. It is a function. The cartesian product is the domain of the function. You also forgot to mention the dual vector spaces.

ruveyn



AspieRogue

28 May 2012, 4:32 pm

ruveyn wrote:
AspieRogue wrote:
ruveyn wrote:
AspieRogue wrote:


I tend to think of tensors as transformation matrices that allow you to determine the components of a vector in multiple coordinate systems. They make a lot of sense once you've studied matrix algebra and Vector spaces. Because a Tensor is a basically a cartesian product of vector spaces mapped into yet another vector space.




An F valued tensor is a multi-linear mapping from a cartesian product of vectors spaces and their dual spaces into the field F.

F could be the real numbers or F could be the complex numbers. The classical definition of tensors in terms of their components is an equivalence relation between components under a linear mapping.

ruveyn


How does what I posted about tensors conflict with the definition you presented? Short answer: It doesn't.


You said a tensor is a cartesian product. Nay, nay. It is a function. The cartesian product is the domain of the function. You also forgot to mention the dual vector spaces.

ruveyn



It is both. I was talking about the Tensor Product.



marshall
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30 May 2012, 12:49 pm

AspieRogue wrote:
ruveyn wrote:
AspieRogue wrote:
ruveyn wrote:
AspieRogue wrote:


I tend to think of tensors as transformation matrices that allow you to determine the components of a vector in multiple coordinate systems. They make a lot of sense once you've studied matrix algebra and Vector spaces. Because a Tensor is a basically a cartesian product of vector spaces mapped into yet another vector space.




An F valued tensor is a multi-linear mapping from a cartesian product of vectors spaces and their dual spaces into the field F.

F could be the real numbers or F could be the complex numbers. The classical definition of tensors in terms of their components is an equivalence relation between components under a linear mapping.

ruveyn


How does what I posted about tensors conflict with the definition you presented? Short answer: It doesn't.


You said a tensor is a cartesian product. Nay, nay. It is a function. The cartesian product is the domain of the function. You also forgot to mention the dual vector spaces.

ruveyn



It is both. I was talking about the Tensor Product.


I'm trying to understand the connection between the mathematical definition of tensor and the definition physicists and fluid dynamicists use. I don't think I'm quite there yet. I usually prefer the definitions mathematicians devise in terms of precision and clarity, but they can be lacking in intuitive motivation.

In physics tensors are linear relations between vector quantities that preserve their geometric meaning under different choices of coordinate bases. Physical vectors have the same meaning as first order tensors and higher order tensors are linear geometrically preserving relations between two lower order tensors. My problem is the notation physicists use leads one to confuse the multi-dimensional arrays of numbers that represent tensors with tensors themselves. Tensors are really something more than just indexed multi-dimensional arrays.



AspieRogue

01 Jun 2012, 4:36 pm

marshall wrote:


I'm trying to understand the connection between the mathematical definition of tensor and the definition physicists and fluid dynamicists use. I don't think I'm quite there yet. I usually prefer the definitions mathematicians devise in terms of precision and clarity, but they can be lacking in intuitive motivation.

In physics tensors are linear relations between vector quantities that preserve their geometric meaning under different choices of coordinate bases. Physical vectors have the same meaning as first order tensors and higher order tensors are linear geometrically preserving relations between two lower order tensors. My problem is the notation physicists use leads one to confuse the multi-dimensional arrays of numbers that represent tensors with tensors themselves. Tensors are really something more than just indexed multi-dimensional arrays.


You know about linear transformations, right? A linear transformation can qualify as a rank-2 tensor provided that it obeys the requisite transformation rules. Physicists generally work with tensors of rank 2. Tensors of higher rank are generally encountered more frequently in (pure)differential geometry and general relativity.



Declension
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01 Jun 2012, 4:50 pm

modustollens wrote:
I am not as advanced as set theory yet.


Congratulations! In nine words, you have just revealed why reductionist theories of mathematics are silly. Although they can be satisfying when you are feeling obsessive.

I'm doing a Master's in mathematics; my topic is do with topology. I like topology because when I'm with topologists I can often just draw a picture to convince people of something, instead of having to jump through formal hoops. The formal hoops are important, of course, because they allow us to have a shared background theory within which we can interpret the pictures correctly.



marshall
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01 Jun 2012, 8:32 pm

AspieRogue wrote:
marshall wrote:


I'm trying to understand the connection between the mathematical definition of tensor and the definition physicists and fluid dynamicists use. I don't think I'm quite there yet. I usually prefer the definitions mathematicians devise in terms of precision and clarity, but they can be lacking in intuitive motivation.

In physics tensors are linear relations between vector quantities that preserve their geometric meaning under different choices of coordinate bases. Physical vectors have the same meaning as first order tensors and higher order tensors are linear geometrically preserving relations between two lower order tensors. My problem is the notation physicists use leads one to confuse the multi-dimensional arrays of numbers that represent tensors with tensors themselves. Tensors are really something more than just indexed multi-dimensional arrays.


You know about linear transformations, right? A linear transformation can qualify as a rank-2 tensor provided that it obeys the requisite transformation rules. Physicists generally work with tensors of rank 2. Tensors of higher rank are generally encountered more frequently in (pure)differential geometry and general relativity.


I guess I'd like to understand precisely how the definition Ruveyn gave is equivalent to the definition used for continuum/fluid mechanics. I'm not sure how stress tensor and strain tensor are linear transformations



slave
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02 Jun 2012, 12:08 am

anxiouspoet wrote:
marshall wrote:
I guess despite the supposed stereotype there aren't a lot of aspies who love math. Right now I'm trying to self-teach myself axiomatic set theory.


I'm a math senior who just got accepted to a PhD school in pure math. And I guess there are times when me and math are on good terms when we're not having lovers' quarrels.
By "learning axiomatic set theory" do you mean learning ZFC (zermelo-fraenkel set theory with axiom of choice) and learning how to construct basic mathematical objects? That's fascinating stuff the first time you see it. It really helps build philosophical skill and faith in math to judge itself critically.

It can also be mind-bending to really examine those different levels of infinity. Not to mention when you get into constructability of universes, inner model theory, and forcing etc.etc.


Congratulations on entering a doctoral program!
I hope that you love the remainder of your education and your career in Mathematics!

slave :D



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02 Jun 2012, 1:59 pm

Me :)

BSc Pure Maths
MSc Mathematics and Statistics
PhD Geospatial Statistical Modelling

LOVE my statistics :)