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Let $a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $a$ is gaussian. Our investigation reveals an interesting fact that the core matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $a$ is gaussian. Consequently, our general estimate involves the norm $|M|$. In the special case when $|M|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.
We show that for an $ntimes n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $xi$ with mean $0$ and variance $1$, [mathbb{P}[s_n(A_n) le epsilon/sqrt{n}] le O_{xi}(epsi
The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh - suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation be
This chapter describes gene expression analysis by Singular Value Decomposition (SVD), emphasizing initial characterization of the data. We describe SVD methods for visualization of gene expression data, representation of the data using a smaller num
We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.
In this work we continue our research on nonharmonic analysis of boundary value problems as initiated in our recent paper (IMRN 2016). There, we assumed that the eigenfunctions of the model operator on which the construction is based do not have zero