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We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite…
A generalization of Wilsonian lattice gauge theory may be obtained by considering the possible self-adjoint extensions of the electric field operator in the Hamiltonian formalism. In the special case of 3D $\mathrm{U}(1)$ gauge theory these…
In this paper we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of…
Hibi showed that the polynomial in the numerator of the Ehrhart series of a reflexive polytope is palindromic. We proved that those in the numerator of the Ehrhart series of every graph polytope (defined later) of the bipartite graph is…
The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the…
We study solutions of three-term recurrence relations whose $N$-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more.…
We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
In this paper, we construct new, uniformly-rotating solutions of the vortex sheet equation bifurcating from circles with constant vorticity amplitude. The proof is accomplished via a Lyapunov-Schmidt reduction and a second order expansion…
Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $\omega_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems…
We prove a splitting theorem for globally hyperbolic, weighted spacetimes with metrics and weights of regularity $C^1$ by combining elliptic techniques for the negative homogeneity $p$-d'Alembert operator from our recent work in the smooth…
We study the global structure of Lorentzian manifolds with partial sectional curvature bounds. In particular, we prove completeness theorems for homogeneous and isotropic cosmologies as well as static spherically symmetric spacetimes. The…
We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil-McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of…
Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
We give an explicit proof of a Bogomolov-type inequality for $c_3$ of reflexive sheaves on $\mathbb{P}^3$. Then, using resolutions of rank-two reflexive sheaves on $\mathbb{P}^3$, we prove that some strata of the moduli of rank-two…
We discuss the statistics of tunnelling rates in the presence of chaotic classical dynamics. This applies to resonance widths in chaotic metastable wells and to tunnelling splittings in chaotic symmetric double wells. The theory is based on…
We define the periodic Full Kostant-Toda on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from…
We give a proof of the hard Lefschetz theorem for orbifolds that does not involve intersection homology. This answers a question of Fulton. We use a foliated version of the hard Lefschetz theorem due to El Kacimi.
We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation,…