# YANG LIU'S MATH BLOG

## The Probabilistic Estimates of the Largest Strictly Convex p-singular Value of Pregaussian Random Matrices

In this paper, the  p -singular values of random matrices with  Gaussian entries defined in terms of the  l p -p -norm for  p> 1, as is studied.  Mainly, using analytical techniques, we show the probabilist ic estimate,  precisely, the decay, on the upper tail probability of the larges t strictly  convex singular values, when the number of rows of the mat rices becomes  very large and the lower tail probability of theirs as well.  These results  provide probabilistic description or picture on the behaviors  of the largest  p -singular values of random matrices in probability for  p> 1. Also, we show  some numerical experiential results, which verify the theoretical  results.

## On the Decay of the Smallest Singular Value of Submatrices of Rectangular Matrices

In this paper, we study the decay of the smallest singular value of submatrices that consist of bounded column vectors. We find that that the smallest singular value of submatrices is related to the minimal distance of points to the lines connecting other two points in a bounded point set. Using a technique from integral geometry and from the perspective of combinatorial geometry, we show the decay rate of the minimal distance for the sets of points if the number of the points that are on the boundary of the convex hull of any subset is not too large, relative to the cardinality of the set. In the numeral or computational aspect, we conduct some numerical experiments for many sets of points and analyze the smallest distance for some extremal configurations.

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

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 integral equa tions, and show a Hermitian characterization theorem for complex Minkowsk i metrics on $C^n$ . Moreover, we parametrize the Grassmannian in an elementary linear algebra approach, and give a characterization on the image of the (ab solute value) co- sine transform on the space of distributions on the Grassman nian $Gr_2 ( C^ 2 )$, by computing the coefficients in the Legendre series expansion o f distributions.

## THE PROBABILISTIC ESTIMATES ON THE LARGEST AND SMALLEST q-SINGULAR VALUES OF RANDOM MATRICES

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 lower and upper tail probabilities of the largest q-singular value $s ^{(q)}_ 1$ , when the number of rows of the matrices becomes very large. Based on the results in probabilistic estimates on the largest q-singular value, we also give probabilistic estimates on the smallest q-singular value for pre-Gaussian random matrices.

## 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

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

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

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