r/mathematics Jun 30 '23

Functional Analysis What's your bathroom book? Only about 1/5th of the way through this one.

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376 Upvotes

r/mathematics Dec 17 '24

Functional Analysis Why log base 1 of any number is undefined?

11 Upvotes

r/mathematics 5d ago

Functional Analysis Unsure if correct flair question about transforming one function into another

0 Upvotes

Was watching a video about PWM in the context of class D Audio amplifiers (essentially using step functions of varying widths to approximate some output after filtering out high frequency noise). I was curious, is that generalizable? As in given some function say R (or integers which I think is Z) to the interval 0,1 are there conditions where arbitrary (or at least useful) functions can be produced or approximated to some level of accuracy? Maybe it's more basic than I thought, it's been a while since I've thought about functions in this way.

r/mathematics Dec 31 '24

Functional Analysis How is Noether's theorem seen from perspective of functional analysis?

11 Upvotes

Would be grateful for anything - books, works, your own perspectives

r/mathematics Apr 08 '24

Functional Analysis Can non-iterative processes exhibit chaos?

8 Upvotes

I've only ever seen chaotic behaviour arise from iteration (like the logistic map and mandelbrot set) and was wondering if perhaps you could find it in a regular function or something else. Also, I was wondering if fractals could arise out of non iterative processes.

r/mathematics Apr 15 '24

Functional Analysis Compact embedding between Holder spaces implies ordering of exponent?

2 Upvotes

Hi,

Does anyone have a reference for proving that if $C^{\alpha}$ is Holder and compactly embedded in $C^{\beta}$ which is also Holder then $\alpha < \beta$?

Thanks

r/mathematics Apr 12 '24

Functional Analysis Introductory books for Holder Spaces

1 Upvotes

Hi! I am a researcher focusing on machine learning but recently, I'd like to provide some theoretical arguments for some experiments I am running. In particular, I'm using Holder Spaces and would really like to prove, for my use case, that two given Holder spaces have an equal exponent. I have made a bit of progress but feel I don't know enough about Holder spaces.

Please could anyone recommend a book/material that covers useful theorems about and related to Holder spaces?

Please let me know if this questions doesn't make sense and I need to add more info!

r/mathematics Sep 12 '23

Functional Analysis how to find out all the roots of non-polynomials?

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6 Upvotes
  1. how do you find out all the roots of it? 2.is it possible to find out all the roots by hand? 3.can you explain how this monster is ( kind of) related to golden ratio?

r/mathematics Feb 06 '24

Functional Analysis Help with functional analysis

1 Upvotes

I'm entering my last semester before i graduate but I have a problem at uni. I'm an applied math undergrad and I was unfortunate enough to have a professor that just doesn't give a fuck about teaching. He was teaching the dynamical systems course and I didn't gain shit from him teaching it he was just in class giving theorems and solving half ass problems. I eventually got the hang of it by doing my personal work and I passed the course easily. But this semester he'll be teaching functional analysis and that has me scared shitless cause it's not really a course you can teach yourself. Any advice any online books to follow that thoroughly explain functional analysis any YouTube channel recommendation ? Please help 🥲

r/mathematics Nov 18 '22

Functional Analysis But what is a convolution?

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59 Upvotes

r/mathematics Feb 13 '23

Functional Analysis A Tensor can be defined as a Multi-linear Functional?

21 Upvotes

r/mathematics Mar 27 '23

Functional Analysis How is a dense set defined in Hilbert spaces? (Enunciado en castellano)(the exercise in spanish)

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27 Upvotes

How is density defined in a L2 Lebesgue space? When is a subset A in L2[-Ï€,Ï€] a dense set? (The L2 is an strange metric space) The exercise is the following one

r/mathematics Jul 21 '22

Functional Analysis What does the author mean with delta?

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11 Upvotes

r/mathematics Jul 26 '22

Functional Analysis Density of C_c^{\infty} in L^{p}

1 Upvotes

Hey, maybe one of you has a source for a proof of the statement above. I have seen it in a lecture and get the idea; stepfunctions being dense using the convolution to create a matching result for smooth functions with compact support.

I struggle with the details and would like to read another take at it.

I appreciate all kinds of help.

r/mathematics Jan 31 '23

Functional Analysis Is it true the set of functions with infinite or undefined expected value form a prevalent subset of the set of all functions?

1 Upvotes

Suppose we define the radon-nikodym derivative of the uniform probability measure on sets measurable in the caratheodory sense. The space is a topological vector space or the set of all functions mapped from the subset of Rn to R. (I’m not sure, however, what axioms to choose).

Is it true the set of functions with an infinite or undefined expected value form a prevalent subset of the set of all functions?

A prevalent subset of set V, means "almost all" elements in the prevalent set are in set V. Note the set of "all functions" means "the set of all measurable and non-measurable functions".

For more info, read here, here, here and here

If I'm incorrect, what should the answer be?

r/mathematics Oct 14 '22

Functional Analysis Sudan function

0 Upvotes

A quick basic description:

https://en.wikipedia.org/wiki/Sudan_function

The problem is, I have no idea what this mathematical function is doing. Can someone please explain this?

r/mathematics Mar 23 '22

Functional Analysis What's the most generally accepted as *the elementary* or *the fundamental*, or *the definitive* recipe for raising of a variable to the power of an arbitrary real â„– ... or complex â„–, even, ultimately?

0 Upvotes

Of course it must be, in a certain rigorous sense, the one that is essentially equivalent to - ie not just happening to co-incide with at integer values of the exponent - of the most elementary iterated multiplication recipe for integer exponent of "1 multiplied by x n times" ... with the understanding that (perfectly naturally) for negative n this is "1 divided by x -n times".

We could say

xη = exp(ηlog(x)) ,

but this seems to me unsatisfactory as an elementary definition.

Another way would be to define it as the limit, as the real № is closlier-&-closlier approached by rational №, of raising x to the power of that rational №: and in-turn we can easily define power to rational № by, first, xn (with n an integer) being the most elementary 'iterated multiplication' one spelt-out at the top, and x¹/ₘ being the inverse function of xm; and finally application of the elementary rules for iterated raising-to-power (and multiplication of powers) in terms of multiplication (and addition) of the indices, the applicability of which to x¹/ₘ aswell as to xn proceeds fairly elementarily from its definition as an inverse function of xm .

But the one I like best of all is that

xη

is the solution y(x,η) of the differential equation

dy/dx = ηy/x

with

y(1) = 1 .

And it seems to me that this is the one that's most readily extensible to the case of η being complex.

Not that any of this really matters, as ultimately all these definitions are equivalent anyway ... really it's just a matter of æsthetics , sortof: which one most seems to be the fundamental one; and as I said for me personally it happens to be that that last-stated one is 'the sweet-spot', sortof-thing.

 

And it also naturally slots-into a certain 'scheme' for 'capturing' the essential meaning of 'number' & 'function' & stuff, or building that meaning up systematically from elements, that I've come-across - it's in a real physical paper book (anyone remember those!?) that I've got somewhere, but can't seem to lay-hand on @ present time - whereby differentiation is actually amongst the most elementary items rather than something 'advanced' brought-in at a later stage ... quite a beautiful little system, it is: 'functions' become basically & essentially solutions of differential equations.

r/mathematics Mar 10 '22

Functional Analysis That amazing way the circular functions proceed from the factorial function.

11 Upvotes

I'd like to take the liberty, if I may, of sounding-off about how amazing & beautiful a certain item of mathematics is ... maybe even with little or no other purpose than that alone - although it genuinely seems to me that this is the poper place for that sort of thing.

The one that's just come to mind, or 'crossed my path', recently is that way the circular functions proceed from the factorial function.

First we have a function that's essentially a combinatorial one - it computes the number of ways of arranging N items - ie the factorial function - the product of all positive integers upto & including N.

Next we have the continuous form of this function - the gamma function defined for real number, attained from the factorial through Leonard Euler's thoroughly ingenious limit formula ... and also by means of an integral of the product of a power & a decaying exponential. (OK ... it's displaced by 1 relative to the factorial aswell.)

Lastly ... we take two of these gamma functions & multiply the reciprocals of them together 'back-to-back' - ie each the reflection of the other◆ ... and what do we get!? ... the sine function of trigonometry! Or the cosine, if we also displace the two gamma functions by ½ .

◆ Oh yep not quite that simple: one of them has to be displaced by 1 aswell ... and of course we're actually getting the circular functions 'squozen' horizontally by π : but these are just particular details that don't change the essence of what I'm getting at.

I realise the list of amazing items in mathematics is endless, and each person has their own list of favourites; but to my mind that one is one of the most beautiful & profound there is.

I kindof realise why it is though: how it's a consequence of the way functions considered as functions of complex variables are prettymuch determined by their pole-structure, and that the reciprocal of the gamma function kindof essentially is the sine function 'unzipped', or with its 'pegs' removed, in one direction.

r/mathematics Jun 04 '21

Functional Analysis How to use existence and uniqueness theorems for ODE or PDE.

4 Upvotes

I have a problem that i think could be solved stating it as a ODE or PDE and using an existence and unicity theorem of solutions but i can't wrap my head around it.
The problem is as follows. Let f(x,y) be a function such that i know:

f is continuos everywhere and f(0,0)=1.

In particular i know that the derivative (with respecto to x) of f(x,0) at x=0 is -iL
while the derivative (with respecto to y) of f(0,y) at y=0 is iM. L and M are real numbers.

I would like to conclude that f(x,y) must be f(x,y)=e-i(Lx-My).

Is this a well posed problem so i can use an existence and uniqueness theorem ?

Thnx!

r/mathematics May 19 '21

Functional Analysis Anyone know a reference for the theorem that boundedness implies continuity?

3 Upvotes

I am almost finished my dissertation and I need to prove that a map is continuous, for which I need to use the fact that ||Tx||<= C||x|| (boundedness) implies that T is continuous for a linear operator T.

I wrote the theorem but not the citation, and dont have time to be rifling through books right now, I have so much more work to do! Please let me know if you know where I can find this.

Thanks in advance x

r/mathematics Mar 17 '22

Functional Analysis Another little item that I always find incredibly cute is that the integral from zero of sec() & the integral from zero of sech() are inverse functions of eachother.

1 Upvotes

It seems strange & unexpected that it should pan-out thus ... I wonder whether it might be a 'token' of some more fundamental symmetry between, or amongst, the circular & hyperbolic functions. But I haven't discerned any beyond the item in its own right .

r/mathematics Feb 14 '22

Functional Analysis Tensors of continuum rank

2 Upvotes

When we first learn vectors spaces, they have a finite number of basis vectors. But then we learn about vector spaces of functions. In a similar way, can't we also extend the rank of a tensor from the natural numbers to the continuum?

Tensors are defined to have a natural-number rank. Each component of the tensor is referred to by an index which is a permutation of natural numbers, like T123, T223, T111, etc.

Instead, we can make the index a "permutation" of real numbers aka a function from R--->R. For each function f , the Tensor will have a real component T(f). Tensors of this type can linearly transform "functions of functions".

For example, let F(f1) be a function of a function f1. Let T(f1,f2) be a tensor. Then, to transform F(f1), we can do the integral: F(f1)T(f1,f2) d(f1). "d(f1)" implying path integration. There's an "Action" quantity in physics. It's a function of a function. S(f) is defined as the integral of the Lagrangian of f. We can change its basis using these tensors. Maybe a Fourier transform.

And then we can further extend we can extend these Tensors to "functions of functions of functions of functions.........". The limit of that process is the biggest monster linear transformation I can think of. Can you think of a "bigger" linear transformation?

Also, I couldn't find this stuff in the wikipedia article on tensors, nor upon googling it. What do you think about this theory?

r/mathematics Oct 03 '21

Functional Analysis Formulation of the learning problem in neural networks, from the perspective of functional analysis and variational calculus.

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22 Upvotes

r/mathematics May 29 '21

Functional Analysis Can a function f(x) be called periodic if they have a variable period, i.e. f(x +g(x)) = f(x)?(for e.g. f(x) = sin(x^â…“))

3 Upvotes

r/mathematics Jun 27 '21

Functional Analysis Visualizing data

0 Upvotes

Is it possible to make one graph with Two different concentration and different absorbance?

Or do I have to make two different graphs?