Related papers: Probabilistic Schubert Calculus: asymptotics

The Waldschmidt constant $\alphahat(I)$ of a radical ideal $I$ in the coordinate ring of $\PP^N$ measures (asymptotically) the degree of a hypersurface passing through the set defined by $I$ in $\PP^N$. Nagata's approach to the 14th Hilbert…

Let us consider i.i.d. random variables $\{a_k,b_k\}_{k \geq 1}$ defined on a common probability space $(\Omega, \mathcal F, \mathbb P)$, following a symmetric Rademacher distribution and the associated random trigonometric polynomials…

The main result of this paper is that for any $1/2 \leq s < 2 - \sqrt{2} \approx 0.5858$, there is a number $\sigma = \sigma(s) < s$ with the following property. Let $\delta > 0$ be small, assume that $A \subset [0,1]$ is a…

We find equivalent hypergeometric- and difference-equation-based formulas, $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,\ldots,9$, for that (rational-valued) portion of the total separability probability for generalized…

We investigate Bertrand's probabilistic paradox through the lens of discrete geometry and old-fashioned but reliable discrete probability. We approximate the plane unit circle with $1/n$ times $1/n$ boxes and count the pairs of boxes…

We consider parabolic systems in divergence form with piecewise $C^{(s+\delta)/2,s+\delta}$ coefficients and data in a bounded domain consisting of a finite number of cylindrical subdomains with interfacial boundaries in $C^{s+1+\mu}$,…

We introduce a variety $Y_{n,k}$, which we call the \textit{affine $\Delta$-Springer fiber}, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an $S_n$ action and a bigrading that corresponds to the…

The degree of the Grassmannian with respect to the Pl\"ucker embedding is well-known. However, the Pl\"ucker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the…

We study two kinds of push-forwards of $\delta$-forms and define the pull-backs of $\delta$-forms. As a generalization of Gubler-K\"unnemann, we prove the projection formula and the tropical Poincar\'e-Lelong formula. As an application, we…

The Szemer\'edi-Trotter theorem gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that $n$ lines and $n$ points determine $O(n^{4/3})$ incidences. Let us suppose that an…

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…

We consider the Schur-Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is…

Denote by $u(n)$ the largest principal specialization of the Schubert polynomial: $ u(n) := \max_{w \in S_n} \mathfrak{S}_w(1,\ldots,1) $ Stanley conjectured in [arXiv:1704.00851] that there is a limit $\lim_{n\to \infty} \, \frac{1}{n^2}…

The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear…

We revisit the noisy binary search model of Karp and Kleinberg, in which we have $n$ coins with unknown probabilities $p_i$ that we can flip. The coins are sorted by increasing $p_i$, and we would like to find where the probability crosses…

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a…

We establish some $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $\Sigma$ as a…

Let $L_{q,\mu}$, $1\leq q\leq\infty$, denotes the weighted $L_q$ space of functions on the unit ball $\Bbb B^d$ with respect to weight $(1-\|x\|_2^2)^{\mu-\frac12},\,\mu\ge 0$, and let $W_{2,\mu}^r$ be the weighted Sobolev space on $\Bbb…

We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second…

We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability…