Arnold Mathematical Journal最新文献

The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let A(t) be an (n times n) matrix whose entries are Laurent series in t. We show that, as (t rightarrow 0), the logarithms of singular values of A(t) approach the invariant factors of A(t). This leads us to suggest logarithms of singular values of an (n times n) complex matrix as an analog of the logarithm map on ((mathbb {C}^*)^n) for the matrix group ({text {GL}}(n, mathbb {C})).

The Kepler orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits (mathrm { PSL}_2(mathbb {R})) as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre (y^{prime prime }= omega (x, y, y^{prime })), vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new.

Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in ({mathbb {C}}^2), and we give an asymptotic counting result using lattice point counting techniques.

We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, such as non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics.

We describe a topological relationship between slices of the parameter space of cubic maps. In the paper [9], Milnor defined the curves (mathcal {S}_{p}) as the set of all cubic polynomials with a marked critical point of period p. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves (mathcal {S}_{1}) and (mathcal {S}_{2}).

We study the categorical entropy and counterexamples to Gromov–Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan–Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston’s classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.

We show that there are infinitely many nonisomorphic quandle structures on any topogical space X of positive dimension. In particular, we disprove Conjecture 5.2 in Cheng et al. (Topology Appl 248:64–74, 2018), asserting that there are no nontrivial quandle structures on the closed unit interval [0, 1].

We show how the formalism of 2-traces can be applied in the setting of derived algebraic geometry to obtain a generalization of the holomorphic Atiyah–Bott formula to the case when an endomorphism is replaced by a correspondence.