## Recursos de colección

1. #### Lower bounds for numbers of real solutions in problems of Schubert calculus

Mukhin, Evgeny; Tarasov, Vitaly
We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian ${\mathop{\rm Gr}(n,d)}$ related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to ${\mathop{\rm gl}_n}$ . The Gaudin Hamiltonians are self-adjoint with respect to a non-degenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form.

2. #### Universality in several-matrix models via approximate transport maps

Figalli, Alessio; Guionnet, Alice
We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy–Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m-points correlation functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity.

3. #### Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

Bernard, Patrick; Kaloshin, Vadim; Zhang, Ke
We prove a form of Arnold diffusion in the a-priori stable case. Let $H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$ be a nearly integrable system of arbitrary degrees of freedom ${n \geqslant 2}$ with a strictly convex H0. We show that for a “generic” ${\epsilon H_1}$ , there exists an orbit ${(\theta,p)}$ satisfying $\|p(t)-p(0)\| > l(H_{1}) > 0,$ where ${l(H_1)}$ is independent of ${\epsilon}$ . The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances. ¶ For the proof we use a combination of geometric and variational methods,...

Appel, P.

6. #### On the theory of infinite systems of differential equations and their application to the theory of stochastic processes and the perturbation theory of quantum mechanics

Arley, Niels; Borchsenius, Vibeke

7. #### Über die absoluten Beträge der Wurzeln algebraischer Gleichungen

Batschelet, Eduard

Wiman, A.

9. #### Über analytische Funktionen auf transzendenten zweiblättrigen Riemannschen Flächen mit reellen Verzweigungspunkten

Myrberg, P. J.

10. #### Zur numerischen Lösung von Randwertaufgaben bei gewöhnlichen Differentialgleichungen

Nyström, E. J.

11. #### Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires

Riesz, Marcel

12. #### On the representation of numbers in the form ax2+by2+cz2+dt2

Kloosterman, H. D.

Roux, J.

Ore, Öystein

15. #### Über Die Arithmetische Reduktion Der Formenscharen

Myrberg, P. J.

Hahn, Hans

Stekloff, W.

18. #### Sur la recherche des fonctions primitives

Lebesque, Henri

19. #### Le principe de huyghens dans le cas de quatre variables indépendantes

For any natural number m(>1) let P(m) denote the greatest prime divisor of m. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $\frac{P(2^n-1)}{n} \to \infty \quad {\rm as}\, n \to \infty$ (see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem,...