arXiv subjects
Publications and source records attributed to Khazhgali Kozhasov.
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries are very well-conditioned on the average.
We investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a random point of view. More precisely, we say that smooth convex hypersurfaces $X_1, \ldots, X_{d_{k,n}}\subset \mathbb{R}\textrm{P}^n$, where $d_{k,n}=(k+1)(n-k)$, are in random position if each one of them is randomly translated by elements $g_1, \ldots, g_{d_{k,n}}$ sampled independently and uniformly from the Orthogonal group; we denote by $τ_k(X_1, \ldots, X_{d_{k,n}})$ the average number of $k$-dimensional projective subspaces (flats) which are simultaneously tangent to all the hypersurfaces. We prove that $$τ_k(X_1, \ldots, X_{d_{k,n}})=δ_{k,n}\cdot\prod_{i=1}^{d_{k,n}}\frac{|Ω_k(X_i)|}{|\textrm{Sch}(k,n)|},$$ where $δ_{k,n}$ is the expected degree (the average number of $k$-flats incident to $d_{k,n}$ many random $(n-k-1)$-flats), $|\textrm{Sch}(k,n)|$ is the volume of the Special Schubert variety of $k$-flats meeting a $(n-k-1)$-flat and $|Ω_k(X)|$ is the volume of the manifold of all $k$-flats tangent to $X$. We give a formula for the evaluation of $|Ω_k(X)|$ in term of some curvature integral of the embedding $X\hookrightarrow \mathbb{R}\textrm{P}^n$ and we relate it with the notion of intrinsic volumes of a convex set: $$\frac{|Ω_{k}(\partial C)|}{|\textrm{Sch}(k, n)|}=4|V_{n-k-1}(C)|,\quad k=0,\ldots,n-1.$$ We prove the upper bound: $$τ_k(X_1,\ldots,X_{d_{k,n}})\leqδ_{k, n}\cdot 4^{d_{k,n}}.$$ In the case $k=1,n=3$ for every $m>0$ we provide examples of smooth convex hypersurfaces $X_1,\ldots,X_4$ such that the intersection $Ω_1(X_1)\cap\cdots\capΩ_1(X_4)\subset\mathbb{G}(1,3)$ is transverse and consists of at least $m$ lines. We also present analogous results for semialgebraic hypersurfaces satisfying some nondegeneracy assumption.
We introduce a first order flow of $G_{2}$-structures and construct its explicit solution in case of a cone over $S^3\!\times\! S^3$. Also we prove for this situation that starting from certain initial datum the flow deforms corresponding to $G_2$-structure metric to a conic metric up to homotheties.