Arnold Mathematical Journal最新文献

We propose a set of questions on the dynamics of Hénon maps from the real, complex, algebraic and arithmetic points of view.

The aim of this note is to draw attention to recent results about the so called Jordan property of groups. (The name was motivated by a classical theorem of Jordan about finite subgroups of matrix groups). We explore interrelations between geometric properties of complex projective varieties and compact Kähler manifolds and the Jordan property (or the lack of it) of their automorphism groups of birational and biregular selfmaps, and of bimeromorphic and biholomorphic maps, respectively.

We investigate differential geometric properties of a parabolic point of a surface in the Euclidean three space. We introduce the contact cylindrical surface which is a cylindrical surface having a degenerate contact type with the original surface at a parabolic point. Furthermore, we show that such a contact property gives a characterization to the (mathcal {A})-singularity of the orthogonal projection of a surface from the asymptotic direction.

Linear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence ({F}_{n}={F}_{n-1}+{F}_{n-2}), have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of (zeta left(3right)), which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as (pi) and (e). More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as (pi), (e), (zeta left(3right)), and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., (zeta left(5right))).

Let A denote the cylinder ({mathbb {R}} times S^1) or the band ({mathbb {R}} times I), where I stands for the closed interval. We consider 2-moderate immersions of closed curves (“doodles”) and compact surfaces (“blobs”) in A, up to cobordisms that also are 2-moderate immersions in (A times [0, 1]) of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order (ge 3) to the fibers of the obvious projections (A rightarrow S^1),  (A times [0, 1] rightarrow S^1 times [0, 1]) or (A rightarrow I),  (A times [0, 1] rightarrow I times [0, 1]). These bordisms come in different flavors: in particular, we consider one flavor based on regular embeddings of doodles and blobs in A. We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on (A= {mathbb {R}} times I), our computations of 2-moderate immersion bordisms (textbf{OC}^{textsf{imm}}_{mathsf {moderate le 2}}(A)) are near complete: we show that they can be described by an exact sequence of abelian groups

where (textbf{OC}^{textsf{emb}}_{mathsf {moderate le 2}}(A) approx {mathbb {Z}} times {mathbb {Z}}), the epimorphism ({mathcal {I}} rho ) counts different types of crossings of immersed doodles, and the kernel ({textbf{K}}) contains the group (({mathbb {Z}})^infty ) whose generators are described explicitly.

We study the existence of points on a compact oriented surface at which a symmetric bilinear two-tensor field is conformal to a Riemannian metric. We give applications to the existence of conformal points of surface diffeomorphisms and vector fields.

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra (mathfrak g_2). In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

We prove that the Euler characteristic of a collapsing Alexandrov space (in particular, a Riemannian manifold) is equal to the sum of the products of the Euler characteristics with compact support of the strata of the limit space and the Euler characteristics of the fibers over the strata. This was conjectured by Semyon Alesker.

In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, (X=T^{*}{text {Gr}}(k,n)). This integral representation can be used to compute the (hbar rightarrow infty ) limit of the vertex function, where (hbar ) denotes the equivariant parameter of a torus acting on X by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the A-series of the Grassmannian ({text {Gr}}(k,n)) with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.

We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension (> 1,) which were previously unknown for dimensions equal to (4n+1.) The argument does not involve understanding factorizations in the symplectic mapping class group.