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**dannyv**- Replies: 0

Hello,

I have the following problem. Let U be an unitary matrix (i.e., assume that it is a rotation in some Hilbert space H), and I want to decompose U as a product of two reflections through two subspaces A,B in H, i.e.,

where

is a projection onto subspace X, and I is the identity.

what are the basis vectors for subspaces A and B?

- Mathematics : suugaku （数学）

- Algebra : daisuu（代数）

- Linear Algebra : senkeidaisuu （線形代数）

- Integration : sekibun （積分）

- Derivation : bibun （微分）

- Logarithm : taisuu （対数）

- Probability : kakuritsu （確率）

- Probability Distribution : kakuritsubunpuu （確率分布）

- Function : kansuu （関数）

- Continuous Function : renzokukansuu （連続関数）

- Complex Number : musuu （無数）

- Composite Number : gouseisuu （合成数）

- Factor : insuu （因数）

- Transformation : henkan （変換）

- Set : shuugou （集合）

- Complement : houshuugou （補集合）

- Union Set : washuugou （和集合）

- Intersection Set : sekishuugou （積集合）

the problem is between line4 and 5. You cannot distribute the square root, i mean,

To my knowledge there is no way to decompose a square root

listening wrote:

Good idea! Can you build other words from the 1-words?

This is an interesting way to state that symbols and letters are the same thing. I wonder if I could prove that statement false...

I'm not very comfortable with proving.

You don't need to prove that, the difference here is that Sigma is an alphabet and S is a language. Just remember that Strings/Word of lenght 1 are not symbols, and symbols are not strings of lenght 1.

mathsyperson wrote:

I read the question and thought no, before scolling down and reading pretty much the reasoning I had just gone through.

Words are made by stringing letters together. You can't combine words together to make new words. Therefore, letters are not automatically length-1 words.

Actually, languages have operations defined on it, like concatenation, union, transitive closure, and others, but those are the basics. Alphabets, these operators and some rules makes a really nice algebra.

hard right? I just asked my supervisor and he told me that he will look for the answer. Let's see.

**dannyv**- Replies: 1

I've been fighting with these over the last weeks and cannot solve it yet Can somebody please give a hand? This is the problem, I have these equations

where t>0 is the time,

n is in Z (the integers),

and

then apply to (4) and (5) the inverse quantum fourier transform defined by

(note that there is a minus sign in the exponential function of the quantum inverse transform)

and obtain equation (6) and (7)

My problem is that I cannot write equation (4,5) in the form of (6,7). I've sent an email to the author and he replied

"The equations (6,7) follow from (4,5) by making the substitution k' = \pi - k in the second term, and noticing that the integrand is periodic with period 2\pi"

I trying this but its not working, can someone give me a hand with this??

Thanks!!!

Well, I will keep looking for a method, and If I find one I'll post it here.

Thanks for the link. First time I hear about Haar's method.

I like De Bruijn's book, but I like more Wong's, it has everything on asymptotics, applied math in a nutshell :-) But what I like of De Bruijn's book is that you have an explanation in words before going deep into the math. The chapter on the steepest descent method and the solution is very clear using the analogy of a guy going through a valley with hills.

Going back to posted problem, what I have found is (I'm not sure) that you cannot obtain a general solution for general f because the expansion is directly related to the shape of the landscape near the saddle points of f. In these books, the authors explain the case for the natural exponential because we know very well the shape of the landscape, and functions of that kind appear very often.

It seems that this is an open problem.

**dannyv**- Replies: 4

I have the following problem for which I cannot find an answer. I posted this same problem in "physicsforums" and "mathhelpforum" and still nobody was able to give me an anwser. I looked everywhere in the internet (journal, books, etc) and also found nothing. Maybe some of you guys can give me some hints.

Given the following contour integral

where C is a contour, f and g are analytic functions defined over C. What is the asymptotic approximation of the closed-form solution when t->infinity?

In general you cannot use the classical methods from complex analysis like steepest descent or saddle point methods because these methods require

for some h analytic.

I found that there are solutions for general f when t is multiplying f, i.e.,

this is the general steepest descent. There are also solutions when

Working with f, you'll find out that you can't cheat, e.g.,

won't help you. It only works for small values of t.

So, for general f it seems that there is no general solution. What do you think?

For a good reading on asymptotics, you can look at these books from google books

http://books.google.es/books?id=_tnwmvHmVwMC&printsec=frontcover#v=onepage&q=&f=false

http://books.google.es/books?id=xooq99A9anMC&printsec=frontcover#v=onepage&q=&f=false

http://books.google.es/books?id=KQHPHPZs8k4C&printsec=frontcover#v=onepage&q=&f=false

k takes values on the reals and maps into the complex numbers

yes, I meant maximum of the module. I the question, as I wrote above

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

thanks for the reply. I understand that solves the problem, but actually what I want to know if this:

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

**dannyv**- Replies: 8

Hi, I have the following problem

given a function f(k) defined on the reals and a complex constant z0, what is the maximum of following function?

The maximum of the module is clearly the value k such that

right? because when you take the module, the squares of the real and imaginary parts are maximum and hence the module is maximum.

But what happens when you cannot factorize the complex constants? e.g., given the following fuction

where k is only real, and z1 and z2 are complex constants. Can we still derivate g, make it equal to 0 and still say we can find a critical point? i.e., does solving

gives you a critical point?

thanks in advance for the help

what's IBP?

for a good reference on asymptotics, this book seems really nice

http://books.google.co.jp/books?id=KQHPHPZs8k4C&printsec=frontcover#v=onepage&q=&f=false

and you can read it a lot for free from google books!

Maybe Laplaces's method could work. It requires an integral with the form:

Take a maximum point

of the function , then the integral is:I think that this method only requires

and to be analytic.**dannyv**- Replies: 6

Hi, can someone give me a hand with this little integral please.

where

t is the time, which is discrete

s is between 0 and t

k has domain [-pi, pi]

n is a natural number

a, b are constants

actually this integral is the binomial expansion of this other integral

there are some other constants, which I omitted. Also, I am omitting the coefficients and summation of the binomial expansion in the first integral.

I looking for a closed form, and I tried to use asymptotic approximations but it doesn't work because in general to solve it using asymptotic techniques you need to write in this form:

and I can't do that. Maybe there are some other asymptotic techniques that I don't know, maybe you can give me a hand on this.

For some applications in computer sciences you can read some papers like:

http://portal.acm.org/citation.cfm?id=1352928.1353015

http://www.springerlink.com/content/q06j6u2578j78742/

Hi, it seems correct, the only obscure part for me was the following:

coffeeking wrote:

Since

has analytic domain

But after looking at it for a couple of minutes it became clear. If you write the complex logarithm it is easier.

And what about Automata Theory, Complexity, Formal Semantics, Type Theory, Formal Languages/Grammar. I think that we can also include here Markov Process, because now in computer science markov processes are being used for building random algorithms. Actually, right now the most efficient algorithm solving Binary Constraint programming problems is based on random walk!!

Also, one very important unsolved problem in mathematics is on complexity: P vs NP

http://mathworld.wolfram.com/UnsolvedProblems.html

**dannyv**- Replies: 7

Hi, althoug I've signed up as a user of this forum around 2 years ago, I don't normally post topics nor comments. I always prefered to just read and entertain myself with all the fun problems that other people post here. But latetly I have notice that it is really important to do networking in every sense of the word (talking with people that shares similar tastes in conferences, virtual forums, etc), and found it to be even more fun that way.

One of the reasons that kept me away from posting seems to be that my interests are in computability and automata theory. It is really hard to find a good forum on theoretical computing subjects, and the computer sciences forums you can find are mainly related to programming, databases, etc., not math.

I really like this forum because I can see that people here really enjoys doing math (as I do), and problems are so cool, and allowed me to learn a lot of stuffs that I never see before. I have a background in computer sciences and always liked mathematics, and when I discover computability, automata and complexity, I just simply felled in love of these subjects, and I knew that I had to learn more mathematics, and this site allowed me to do that. And at the same time, I learned to love mathematics in general.

The main subject of theoretical computing that I like the most is computability and complexity issues of computing paradigms (quantum computing, molecular computing, etc). All these paradigms have their own abstract model with so many questions still to answer. Right now, as a graduate student I studying these subjects, from a strictly mathematical point of view, and I now that this is what I want to do for the rest of my life.

Well, I just wanna to write this and see what other people think about theoretical computing in general. Does somebody knows about these topics? As mathematicians (future or present) what are your thoughts about the field? Is there somebody working/studying one/some/several of these topics right now?