Arnold Mathematical Journal最新文献

This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021.

We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or (tau )-model), or as three permutations on the set of half-edges (rotation system or (sigma )-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

We consider an embedded n-dimensional compact complex manifold in (n+d) dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of (C_n) in (M_{n+d}) is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in (M_{n+d}) having (C_n) as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations.

Let ({mathcal {M}}) denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote (k^{times }), and a uniformizer we denote (pi ). In this paper, we consider the map (T_{v}: {mathcal {M}} rightarrow {mathcal {M}}) defined by

where b(x) denotes the equivalence class to which (frac{pi ^{v(x)}}{x}) belongs in (k^{times }). We show that (T_v) preserves Haar measure (mu ) on the compact abelian topological group ({mathcal {M}}). Let ({mathcal {B}}) denote the Haar (sigma )-algebra on ({mathcal {M}}). We show the natural extension of the dynamical system (({mathcal {M}}, {mathcal {B}}, mu , T_v)) is Bernoulli and has entropy (frac{#( k)}{#( k^{times })}log (#( k))). The first of these two properties is used to study the average behaviour of the convergents arising from (T_v). Here for a finite set A its cardinality has been denoted by (# (A)). In the case (K = {mathbb {Q}}_p), i.e. the field of p-adic numbers, the map (T_v) reduces to the well-studied continued fraction map due to Schneider.

A (d^{{n}})-cage (mathsf K) is the union of n groups of hyperplanes in (mathbb P^n), each group containing d members. The hyperplanes from the distinct groups are in general position, thus producing (d^n) points where hyperplanes from all groups intersect. These points are called the nodes of (mathsf K). We study the combinatorics of nodes that impose independent conditions on the varieties (X subset mathbb P^n) containing them. We prove that if X, given by homogeneous polynomials of degrees (le d), contains the points from such a special set (mathsf A) of nodes, then it contains all the nodes of (mathsf K). Such a variety X is very special: in particular, X is a complete intersection.

Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that the value of the weight system constructed from the Lie algebra (mathfrak {sl}_2) on a chord diagram depends on the intersection graph of the diagram, rather than the diagram itself. This inspired a question due to Lando about whether it is possible to extend the weight system (mathfrak {sl}_2) to a graph invariant satisfying the four term relations for graphs. We show that for all graphs with up to 8 vertices such an extension exists and is unique, thus answering in affirmative to Lando’s question for small graphs.

We consider a pair of games where two players alternately select previously unselected elements of (mathbb {Z}_n) given a particular starting element. On each turn, the player either adds or multiplies the element they selected to the result of the previous turn. In one game, the first player wins if the final result is 0; in the other, the second player wins if the final result is 0. We determine which player has the winning strategy for both games except for the latter game with nonzero starting element when (n in {2p,4p}) for some odd prime p.

The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville’s theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family’s limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the (N=4) case).