Arnold Mathematical Journal最新文献

We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.

We prove an ({{mathbb {F}}}_p)-Selberg integral formula, in which the ({{mathbb {F}}}_p)-Selberg integral is an element of the finite field ({{mathbb {F}}}_p) with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p.

When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in a neighborhood of a singular point? The present paper, of a survey nature, presents a research program around this question. A way to answer is to use normal forms. However, there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper, we discuss the case of singularities for which the normalizing transformation is k-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic k-parameter families unfolding such singularities.

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.