arXiv subjects
Publications and source records attributed to Achintya Raya Polavarapu.
Fix $b\in(0,1)$, let $1\leq k\leq n$, and let $A=(A_{ij})$ be an $n\times n$ random matrix with independent real entries satisfying $$ \sup_{x\in\mathbb{R}}\mathbb{P}\{A_{ij}=x\}\leq b<1 \qquad (1\leq i,j\leq n). $$ We show that there exists $c>0$ such that $$ \mathbb{P}\{\operatorname{rank} A\leq n-k\}\leq \exp(-cnk), \qquad 1\leq k\leq n. $$
Let $A=(ξ_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb Eξ_{ij}=0,\qquad \mathbb Eξ_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|ξ_{ij}-z| 0$ and $b\in(0,1)$. We prove that, for every $δ\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,δ$, such that \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] for every $t\ge1$ and every $1\le l\le(1-δ)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le(1-δ)N$.
Let $A=(a_{ij})$ be an $n\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $κ\in (0, 1)$ such that the entries of $A$ satisfy $$ \max_{i,j}\sup_{u \in \mathbb{R}} \mathbb{P}(\lvert a_{ij} - u\rvert < 1) \le κ. $$ We prove that there are constants $c,C>0$, depending only on $K$ and $κ$, such that for every fixed invertible $n\times n$ matrix $M$ and every $\varepsilon\ge0$, $$ \mathbb{P}!\left(s_{\min}(MA) \le \frac{\varepsilon}{\lVert M^{-1}\rVert_{\mathrm{HS}}}\right) \le C\varepsilon + e^{-cn}. $$ In the Gaussian case, we also show that the above estimate is sharp in the sense that $\mathbb{E}[s_{\min}(MA)]\asymp \lVert M^{-1}\rVert_{\mathrm{HS}}^{-1}.$
For integers $k\ge 1$ and $m\ge 2$, let $g(k,m)$ be the least integer $n\ge 1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. We prove that \[ g(k,m)\le \max(m+2k-2,\,3k+1) \] for all $k\ge 1$ and $m\ge 2$, establishing the 1983 conjecture of Thomassen that $g(k,k+1)\le 3k+1$. The key new ingredient is a Hall-feasibility argument replacing the final numerical step in the proof of Nguyen.
A functional calculus for an order complete vector lattice $\mathcal{E}$ was developed by Grobler in 2014 using the Daniell integral. We show that if one represents the universal completion of $\mathcal{E}$ as $C^\infty(K)$, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in $C^\infty(K)$. This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in $C^\infty(K)$. We obtain a representation that is analogous to what is expected in probability theory.
It was showed by Donner in 1982 that every order complete vector lattice $X$ may be embedded into a cone $X^s$, called the sup-completion of $X$. We show that if one represents the universal completion of $X$ as $C^\infty(K)$, then $X^s$ is the set of all continuous functions from $K$ to $[-\infty,\infty]$ that dominate some element of $X$. This provides a functional representation of $X^s$, as well as an easy alternative proof of its existence.