research – Page 2 – yuliy baryshnikov

analytic combinatorics in several variables (aimath, san jose 4-9.4.2022)

\(\def\Comp{\mathbb{C}}\def\Proj{\mathbb{P}}\def\Nat{\mathbb{N}}\) This workshop is about complex analytic techniques usable in applications from classical combinatorial problems to asymptotic representation theory and cluster algebras. The scope is approximately what is covered by Pemantle-Wilson(-Melczer) book. I plan to report on what is going on here (reporting is sporadic and idiosyncratic). Lectures are streamed. Day One (all times PDT): […]

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derivatives and random matrices

research / ymb

In connection with our IGL project, I remembered a result that might be useful, and is not widely enough known. Let \(z_1\leq \ldots\leq z_N\) be a sequence of real numbers forming the spectrum of a Hermitian operator \(A\) acting in \(N\)-dimensional Hermitian space \(U\):\[P_A(z):=\det(zE-A)=\prod_k (z-z_k).\] Proposition: Let \(V\) be a codimension \(1\) hyperplane in \(U\),

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averaging inflections in plane curves

problems, research / ymb / inflections, Rice-Kac

We all know that a generic complex plane curve of degree \(d\) has \(3d(d-2)\) inflections. Since Klein it is also known what the (strict) upper bound on the number of real inflections in real plane curves of degree \(d\) is: \(d(d-2)\), one third of the number of complex inflections. But what about the average real

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martingales and isochrones

problems, research / ymb

Limit Cycles and Isochrones Consider smooth vector field \(v\) on a manifold with an exponentially stable limit cycle \(\gamma\): in other words, any trajectory starting close enough to \(\gamma\), converges to it exponentially fast. In this situation Guckenheimer proved the existence of isochrons, – a foliation of an open vicinity \(U\supset \gamma\) of the limit

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gallery of quantum random walks, scarlet edition

research, visuals / ymb / quantum random walks

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singularities, biparametric persistence and cubical complexes

problems, research / ymb / persistence, singularities

\(\def\Real{\mathbb{R}}\def\phd{\mathbf{P}H}\def\CAT{\mathtt{CAT}}\) The goal of this note is to define the biparametric persistence diagrams for smooth generic mappings \(h=(f,g):M\to\Real^2\) for smooth compact manifold (M). Existing approaches to multivariate persistence are mostly centered on the workaround of absence of reasonable algebraic theories for quiver representations for lattices of rank 2 or higher, or similar artificial obstacles. Singularities

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