Authors:Yutaka Ishii Abstract: Abstract The Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer. This complicated object is the Julia set of a complex Hénon map \(f_{c, b}(x, y)=(x^2+c-by, x)\) defined on \(\mathbb {C}^2\) together with its stable and unstable manifolds, hence it is a fractal set in the real 4-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of \(\mathbb {C}^2\) including complex Hénon maps with an emphasis on the combinatorial and topological aspects of their Julia sets. PubDate: 2017-04-25 DOI: 10.1007/s40598-017-0066-x

Authors:Laura Brillon; Revaz Ramazashvili; Vadim Schechtman; Alexander Varchenko Abstract: Abstract Using the ideas coming from the singularity theory, we study the eigenvectors of the Cartan matrices of finite root systems, and of q-deformations of these matrices PubDate: 2017-04-13 DOI: 10.1007/s40598-017-0065-y

Authors:Pavel Etingof Abstract: Abstract We explain a proof of the Broué–Malle–Rouquier conjecture on Hecke algebras of complex reflection groups, stating that the Hecke algebra of a finite complex reflection group W is free of rank W over the algebra of parameters, over a field of characteristic zero. This is based on previous work of Losev, Marin– Pfeiffer, and Rains and the author. PubDate: 2017-04-12 DOI: 10.1007/s40598-017-0069-7

Authors:Dami Lee Abstract: Abstract In this paper, we will construct an example of a closed Riemann surface X that can be realized as a quotient of a triply periodic polyhedral surface \(\Pi \subset \mathbb {R}^3\) where the Weierstrass points of X coincide with the vertices of \(\Pi .\) First we construct \(\Pi \) by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface. The symmetries of X allow us to construct hyperbolic structures and various translation structures on X that are compatible with its conformal type. The translation structures are the geometric representations of the holomorphic 1-forms of X. Via the basis of 1-forms we find an explicit algebraic description of the surface that suggests the Fermat’s quartic. Moreover the 1-forms allow us to identify the Weierstrass points. PubDate: 2017-04-12 DOI: 10.1007/s40598-017-0067-9

Authors:Trevor Hyde; Jeffrey C. Lagarias Abstract: Abstract We study for each n a one-parameter family of complex-valued measures on the symmetric group \(S_n\) , which interpolate the probability of a monic, degree n, square-free polynomial in \(\mathbb {F}_q[x]\) having a given factorization type. For a fixed factorization type, indexed by a partition \(\lambda \) of n, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to \(S_n\) -subrepresentations of the cohomology of the pure braid group \(H^{\bullet }(P_n, \mathbb {Q})\) . We deduce that the splitting measures for all parameter values \(z= -\frac{1}{m}\) (resp. \(z= \frac{1}{m}\) ), after rescaling, are characters of \(S_n\) -representations (resp. virtual \(S_n\) -representations). PubDate: 2017-03-06 DOI: 10.1007/s40598-017-0064-z

Authors:J. Gordon; F. Petrov Abstract: Abstract Let \(\rho \) be a metric on the set \(X=\{1,2,\dots ,n+1\}\) . Consider the n-dimensional polytope of functions \(f:X\rightarrow \mathbb {R}\) , which satisfy the conditions \(f(n+1)=0\) , \( f(x)-f(y) \leqslant \rho (x,y)\) . The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by (Vershik, Arnold Math J 1(1):75–81, 2015). We prove that for any “generic” metric the number of \((n-m)\) -dimensional faces, \(0\leqslant m\leqslant n\) , equals \(\left( {\begin{array}{c}n+m\\ m,m,n-m\end{array}}\right) =(n+m)!/m!m!(n-m)!\) . This fact is intimately related to regular triangulations of the root polytope (convex hull of the roots of \(A_n\) root system). Also we get two-sided estimates for the logarithm of the number of Vershik classes of metrics: \(n^3\log n\) from above and \(n^2\) from below. PubDate: 2017-02-09 DOI: 10.1007/s40598-017-0063-0

Authors:Laura DeMarco; Kathryn Lindsey Abstract: Abstract Any planar shape \(P\subset {\mathbb {C}}\) can be embedded isometrically as part of the boundary surface S of a convex subset of \(\mathbb {R}^3\) such that \(\partial P\) supports the positive curvature of S. The complement \(Q = S {\setminus } P\) is the associated cap. We study the cap construction when the curvature is harmonic measure on the boundary of \(({\hat{{\mathbb {C}}}}{\setminus } P, \infty )\) . Of particular interest is the case when P is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy. PubDate: 2017-01-31 DOI: 10.1007/s40598-016-0061-7

Authors:Viktor L. Ginzburg; Dmitrii V. Pasechnik Abstract: Abstract We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology. PubDate: 2017-01-25 DOI: 10.1007/s40598-016-0062-6

Authors:Aleksandr V. Pukhlikov Abstract: Abstract The famous \(4n^2\) -inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is greater than \(4n^2\mu \) , where \(\mu \) is the multiplicity of the singular point. PubDate: 2016-11-30 DOI: 10.1007/s40598-016-0060-8

Authors:Gabriel Katz Abstract: Abstract This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on n-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage. PubDate: 2016-11-23 DOI: 10.1007/s40598-016-0059-1

Authors:Pearce Washabaugh; Stephen C. Preston Abstract: Abstract The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold M can give information about the stability of inviscid, incompressible fluid flows on M. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by \(\mathcal {D}_{\mu ,E}(M)\) , has positive sectional curvature in every section containing the field \(X = u(r)\partial _\theta \) iff \(\partial _r(ru^2)>0\) . This is in sharp contrast to the situation on \(\mathcal {D}_{\mu }(M)\) , where only Killing fields X have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on \(\mathcal {D}_{\mu ,E}(M)\) along the geodesic defined by X. PubDate: 2016-10-27 DOI: 10.1007/s40598-016-0058-2

Authors:Eric Bucher Abstract: Abstract Given a marked surface (S, M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work along with previous work by others seeks to establish a base case in answering the question of whether a given finite mutation type cluster algebra exhibits a maximal green sequence. PubDate: 2016-09-29 DOI: 10.1007/s40598-016-0057-3

Abstract: Abstract We consider parameters \(\lambda \) for which 0 is preperiodic under the map \(z\mapsto \lambda e^z\) . Given k and l, let n(r) be the number of \(\lambda \) satisfying \(0< \lambda \le r\) such that 0 is mapped after k iterations to a periodic point of period l. We determine the asymptotic behavior of n(r) as r tends to \(\infty \) . PubDate: 2016-09-19 DOI: 10.1007/s40598-016-0056-4

Abstract: Abstract Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations. PubDate: 2016-09-19 DOI: 10.1007/s40598-016-0054-6

Authors:Alexandre Eremenko; Andrei Gabrielov Abstract: Abstract We study spherical quadrilaterals whose angles are odd multiples of \(\pi /2\) , and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, but not both, and the numbers of families of these two types are equal. The results can be translated to classification of Heun’s equations with real parameters, whose exponent differences are odd multiples of 1 / 2, with unitary monodromy. PubDate: 2016-09-09 DOI: 10.1007/s40598-016-0055-5

Authors:Józef H. Przytycki Abstract: Abstract We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers of Arnold Journal of Mathematics is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology. PubDate: 2016-08-19 DOI: 10.1007/s40598-016-0053-7

Authors:Victor Katsnelson Abstract: Abstract Let \(\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)\) be the roots of the equation \(R(z)=t\) , where R(z) is a rational function of the form $$\begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned}$$ \(\mu _k\) are pairwise distinct real numbers, \(\alpha _k>0,\,1\le {}k\le {}n\) . Then for each real \(\xi \) , the function \(e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}\) is exponentially convex on the interval \(-\infty<t<\infty \) . PubDate: 2016-08-02 DOI: 10.1007/s40598-016-0051-9

Authors:Faustin Adiceam; David Damanik; Franz Gähler; Uwe Grimm; Alan Haynes; Antoine Julien; Andrés Navas; Lorenzo Sadun; Barak Weiss Abstract: Abstract This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field. PubDate: 2016-07-11 DOI: 10.1007/s40598-016-0046-6

Authors:Guy Valette Abstract: Abstract We consider generic smooth closed curves on the sphere \(S^2\) . These curves (oriented or not) are classified relatively to the group \(\text{ Diff }(S^2)\) or its subgroup \(\text{ Diff }^+(S^2)\) ), with the Gauss diagrams as main tool. V. I. Arnold determined the numbers of orbits of curves with n double points when \(n<6\) . This paper explains how a preliminary classification of the Gauss diagrams of order 5, 6 and 7 allows to draw up the list of the realizable chord diagrams of these orders. For each such diagram \(\Gamma \) and for each Arnold symmetry type T, we determine the number of orbits of spherical curves of type T realizing \(\Gamma \) . As a consequence, we obtain the total numbers of curves (oriented or not) with 6 or 7 double points on the sphere (oriented or not) and also the number of curves with special properties (e.g. having no simple loop). PubDate: 2016-07-11 DOI: 10.1007/s40598-016-0049-3

Authors:Oleg R. Musin Abstract: Abstract Tucker and Ky Fan’s lemma are combinatorial analogs of the Borsuk–Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan’s lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT–manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT–manifolds. PubDate: 2016-06-16 DOI: 10.1007/s40598-016-0045-7