In this paper, we study the ranges of (absolute value) cosine trans- forms for which we give a proof for an extended surjectivity t heorem by mak- ing applications of the Fredholm’s theorem in inte…

Source: On the Range of Cosine Transform of Distributions for Torus-invariant Complex Minkowski Spaces

In this paper, we study the q-singular values of random matrices with pre-Gaussian entries defined in terms of the q-quasinorm with 0 < q ≤ 1. In this paper, we mainly consider the decay of the …


Posted by: Louis Y. Liu | March 1, 2011

On the Infinity Norm

The infinity norm is often used as a model in analysis and geometry, not for continuous scenarios, but for the discrete or singular scenarios.

We know that ||(x,y)||_{\infty}:=max(|x|,|y|), for any (x,y)\in \mathbb{R}^{2}, can be expressed as an integral on S^{1} by spreading a measure on it, which is

||(x,y)||_{\infty}=c\int_{S^{1}}|x\cos\theta+y\sin\theta|(\delta_{\frac{\pi}{4}}(\theta)+\delta_{\frac{3\pi}{4}}(\theta))d\theta (1)

for some constant c because max(|x|,|y|)=\frac{1}{2}(|x+y|+|x-y|), where

$latex \delta_{\frac{\pi}{4}}(\theta)=\begin{cases}
+\infty, & \theta=\frac{\pi}{4}\\
0, & \theta\ne\frac{\pi}{4}.\end{cases}$ (2)

which satisfies \int_{-\infty}^{\infty}\delta_{\frac{\pi}{4}}(\theta)\, dx=1 and

$latex \delta_{\frac{3\pi}{4}}(\theta)=\begin{cases}
+\infty, & \theta=\frac{3\pi}{4}\\
0, & \theta\ne\frac{3\pi}{4}.\end{cases}$ (3)

satisfying \int_{-\infty}^{\infty}\delta_{\frac{3\pi}{4}}(\theta)\, dx=1 are modified Dirac delta functions.

One might also be able to put a measure on S^{3} for the complex norm ||(z,w)||_{\infty}=max(|z|,|w|), (z,w)\in \mathbb{C}^{2}, which can be written in real norm as ||(x,y,u,v)||:=max(\sqrt{x^{2}+y^{2}},\sqrt{u^{2}+v^{2}}). A problem is what function f would satisfy

||(x,y)||_{\infty}=\int_{S^{3}}|x\cos\xi_{1}\cos\eta+y\cos\xi_{2}\sin\eta|f(\xi_{1},\xi_{2},\eta)d\xi_{1}d\xi_{2}d\eta (4)

if one parametrizes S^{3} by the Hopf coordinates

z=e^{i\xi_{1}}\sin\eta,\, w=e^{i\xi_{2}}\cos\eta. (5)

One can show that the function f is actually some function independent of \xi_{1} and \xi_{2} by the invariance of the complex norm under U(1)\times U(1) action, so it can be denoted as f(\eta). But it is unknown yet what exactly the function f(\eta) is, because the invariance yields that f(\eta)=\delta_{\frac{\pi}{4}}(\eta)+\delta_{\frac{3\pi}{4}}(\eta), but this doesn’t produce an appropriate function in the general expression for the complex infinity norm. If one looks back the case of real infinity norm, the way of expressing the infinity norm is by making use of the absolute values on sum and difference smartly to sift out the maximum from two number. But in the complex case the double integral on the torus

\left\{ (\frac{\sqrt{2}}{2}e^{i\xi_{1}},\frac{\sqrt{2}}{2}e^{i\xi_{2}}):\xi_{1},\xi_{2}\in[0,2\pi]\right\} (6)

eliminates the feasibility of sifting out the maximal modulus.

Posted by: Louis Y. Liu | October 5, 2008

On Stokes’ Theorem

Stokes’ theorem plays an important role in computing integrals of differential forms on manifolds, and its consequence, the Cauchy integral formula, is quite fundamental in complex analysis in one variable as well.

Suppose M is a compact 2-dimensional submanifold with smooth boundary in the plane mathbb{R}^{2}. From symplectic geometry, the natural symplectic form on T^{*}mathbb{R}^{2} can be written as omega=dalpha, where alpha=xi dx+eta dy. If one computes the integral,

int_{D^{*}(partial M)}alphawedgeomega=int_{D^{*}(partial M)}xi dxwedge dywedge deta+int_{D^{*}(partial M)}eta dywedge dxwedge dxi(1)

where D^{*}(partial M) denotes the codisk bundle of partial M, since partial M is an one dimensional curve in mathbb{R}^{2}, it makes the integral containing dxwedge dy measuring the perturbations of base points of cotangent vectors in D^{*}(partial M) be zero. Hence each term of the right hand side of (1) is zero. So one has that int_{D^{*}(partial M)}alphawedgeomega=0, which is also true in general when M is a compact hypersurface contained in some affine n-1-plane in mathbb{R}^{n}.

By the way, Gowers wrote a post, one way of looking at Cauchy theorem, which is very interesting and tells us that the Cauchy’s theorem is the natural generalization in 2-dimension of the statement f'=0Longrightarrow f is a constant for f:mathbb{Rrightarrow R}, that, we know, is a direct consequence of the second fundamnetal theorem of calculus which is the special case of the Stokes’ theorem in one dimension. So in this sense, Stokes’ theorem is the “generator” of all the theorems and statements of this kind.