User yaoxiao - MathOverflow

Spectrum theorem for p-adic matrix analysis

2013-03-04T14:05:20Z

Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\

2: whether invertible p-adic matrix is dense in $$C_{p}^{n\times n}$$

3:whether diagonalizable p-adic matrix is dense in $$C_{p}^{n \times n}$$

Any reference and comments will be appreciated.

Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems

2011-12-21T13:48:49Z

Recently I have much interest in algebraic topology and algebraic geometry, I am a student of field of complex dynamic systems. According to my knowledge, my friends told me that there are many applications of algebraic geometry and algebraic topology in dimension of reduction in statistics and some other fields. I want to konw whether there exists some interesting applications of algebraic geometry and albebraic topology in dynamics system. Any comments and advice will be appreciated. Thanks.

p-adic Lie theory

2013-03-06T08:30:57Z

It is well known that exponential map in $C^{n\times n}$ will cover all non-sigular matrix $GL(n,C)$, which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases.

Recent I may need facts in classical matrix analysis to be generalized in p-aidc matrix cases. something are really trivial, however some others may have essential difficulties at least to me. it is quite striking fact that entire p-adic map doesn't omit value in $C_{p}$, and the p-adic exponential map can not defined in whole $C_{p}$

Any references and comments will be appreciated.

the dual space of C(X) (X is noncompact metric space)

2012-02-23T11:50:19Z

It is well known that when X is compact space (or locally compact space), $C(X)={f:f X\rightarrow X \text{is continous,and bounded} }$ ,the dual space of $C(X) C(X)^{*} $is correspond to $M(X)$ space of Radon measure with bounded variation.

However according to my knowledge. there are few books discuss case that when X is noncompact, for example complete vsperable metric space.

even for the simplest example, when taking X is R, means real line, what does $(C(X))^{*}$ mean.

Any advice and reference will be appreaciated.

Normal Family for several complex variable (from $C^{n} $to$ C^{n}\U$)

2012-10-12T06:44:38Z

Recently I face up with a problem, which I realized that have close connection with the following problem. ${ f_{n} }_{n=1}^{\infty}$ is analytic map from $C^{n}$ to $C^{n} $\ $U$ where U is open neighborhood of 0, whether f is a normal family.

I know when n=1, this is really Montel Normal family criterion, However I did know whether it is true for high dimension. also I heard that the for any two topological equivalent simple connected domain in$ C^{n}$ $(n\geq 2)$, the probability of holomorphic equivalent for this two domain is 0. I want to know what is the precise statement for this theorem.

Any advice and comments will be appreciated.

If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded

2012-09-08T16:25:33Z

Recently when I read a paper about Fatou component, I met the following theorem which cited in Professor Eremenko's paper "on the iteration of entire functions"

If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded.

it is really known theorem proved by I.N Baker in his paper in "The domains of normality of an entire function". However I really did know how to find this paper. any comments and materials about this theorem will be very appreciated. thanks.

a problem about All roots in the left half plane

2012-03-26T09:43:22Z

Suppose a polynomial$ u(x)=ax^{2}+bx+c$，with a b c positive,$g(x)=d*xe^{-\lambda_1 x}+fe^{-\lambda_2 x} $with d,f positve, let $ H(x)=u(x)-g(x)$, when we reserach some bahavior of a dynamic system, we want to give some condtion such that all root of H(x) will locate in the half-left complex, however I did not how to deal with, any reference and advice will be appreatiated.

triangle equality in manifold

2012-02-22T15:44:57Z

For a generilized triangle on a manifold, (distance can be regarded as geodesic length)it is well known that for Eucilidean Geometry，the following is true:

Consider a triangle $ABC$, $D$ is the midpoint of $BC$, then $AD\leq \frac{1}{2}(AB+AC)$

what about some other cases in manifolds. according to my knowledge, it is also true in spheres with dimension n. However I did not konw general cases.

Any advice will be appreaciated.

Answer by yaoxiao for Weak L_1-convergence of squares

2012-02-22T16:06:21Z

It is obvious true. $f_{k}\rightarrow f$ in$ L^{1}$ strongly,(for $ \forall p <2$) $f_{k}^{2}\rightarrow f_{2}$ strongly, of course weakly, then $F=f^{2}$

finite groups with faithful real two dimensional representation

2011-10-06T03:50:25Z

Recently I have met an interesting problem $\rho$: $G \rightarrow SL(2,R)$ be a faithful representions of a finite group by real 2\times 2 matrices of determinant 1, then we can get this group is cylic.

what is more, how can we determine all finite groups wich have a faithful real two dimensional representation? I feel it have some connections with finite subgroup in $SL(2,R),SU_{2},U_{2}$

more generally, determine all finite groups which have a faithful real n dimensional representation?

I do not konw how I takle with these problems

Answer by yaoxiao for Elementary+Short+Useful

2011-04-10T08:56:59Z

Every matrix can be represented by the linear combinations of four orthogonal matrix

Is there any conclusions generalized Singular Value Decomposition into Hilbert Space

2011-04-09T04:02:47Z

Spectrum decomposition can be regarded as the generalizations of the following fact that: Every Hermitian matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix

Singular vector decomposition can be expressed as Every Matrix $A_{mn}$ can be decomposed in to $A=U\Lambda V^{*}$, where $U$,and $V$ are unitary matrices. Does it can be extended in to decompostion of linear operators on Hilbert Space. ?

I searched the internet and several traditional books about the topic "Singular Vector Decomposition into Hilbert space", However, to my disappointment, I find no similar conclusion. Thanks for your help.

Whether the system of matrix equations is always solvable

2011-01-03T10:38:20Z

In recent days, I learned a linear algebra problem from one of my friends. It can be stated as follows.

Given four matrices $A,B,C,D$, find three matrices $E,G,F$, not simultaneously zero, such that the following conditions (1), (2), (3) are satisfied: $$ \begin{align*} (1) &\quad AE=EA, \cr (2) &\quad BG=GB, \cr (3) &\quad AF-FB=ED-CG. \end{align*} $$ The question is whether such $E,G,F$ always exist.

Also it is obvious that we can obtain $E,G$ by (1) and (2) easily. However the hard die is to satisfy condition (3). I just know when $A$ and $B$ have different spectra, we can obtain $F$ in a unique way.

How to prove a unit norm matrix is the average of two unitary matrix

2011-01-02T17:35:37Z

thanks for your answers

Comment by yaoxiao

2013-03-04T15:03:15Z

@Will Sawin thanks for your comments.

Comment by yaoxiao

2012-10-15T15:22:33Z

@Yemon Choi, sorry for my poor english, which I means is for every matrix $Y \in C^{n\times n}$, whether sin(X)=Y, has a solution.

Comment by yaoxiao

2012-10-14T13:49:26Z

Thank you for your answer and statement. I know the proof of unit ball and polydisk are not holomorphically equivalent.

Comment by yaoxiao

2012-09-14T02:29:25Z

Thank you professor Eremenko. Your survey article is really great help, thanks.

Comment by yaoxiao

2012-03-26T13:33:45Z

@Willie Wong， I am sorry，It is just means times

Comment by yaoxiao

2012-02-22T23:34:33Z

@Joseph O'Rourke. Oh, thanks. Sorry for my poor statement, here I means the shortest geodesic

Comment by yaoxiao

2012-02-22T23:32:42Z

@Anton Petrunin Sorry for my statement for this problem, I means the shortest geodesic,your counter example is just$ AB=BC=AC=\frac{\pi}{3}$

Comment by yaoxiao

2011-12-22T01:16:06Z

Thanks for your advice, last year a professor in Chern's institue have given a report about algebraic geometry's applications in dimension reuduction's about the accuracy test of Fisher's table test, which give me deep and powerful impression.

Comment by yaoxiao

2011-10-06T05:26:08Z

thanks for your answers， can you give me some materials about n=3，thanks

Comment by yaoxiao

2011-04-12T06:26:04Z

This is my first time to know about the MRB constant，does it play a very important part in number theory?

Comment by yaoxiao

2011-04-11T02:56:33Z

According to history of mathematics, at the beginning of Bernoulli's original idea, he did not begin defined as the Taylor coefficients of the function x/(e^x-1) at 0.

Comment by yaoxiao

2011-04-10T09:02:39Z

As far as I see， the theorem related to number “4” will be the best： First： four color theorem Second：Every positive integer can be represented by squares of four integers. Third: Every matrix can be represented by linear combinations of four matrix and so on

Comment by yaoxiao

2011-04-10T08:44:35Z

Without special functions, I am afraid we can not give a direct result on this problem

Comment by yaoxiao

2011-04-09T12:33:21Z

Oh，thank you. may be this theorem "Every bounded operator acting on a Hilbert space has a decomposition , where and is a partial isometry. This decomposition is called polar decomposition. If is invertible, then can be chosen to be unitary." given by mathworld is related to some deep operator theory.

Comment by yaoxiao

2011-04-09T10:27:42Z

Thanks for Yemon Choi and Dennis Serre's comment.