Authors:William Floyd; Gregory Kelsey; Sarah Koch; Russell Lodge; Walter Parry; Kevin M. Pilgrim; Edgar Saenz Pages: 365 - 395 Abstract: We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena. PubDate: 2017-09-01 DOI: 10.1007/s40598-017-0071-0 Issue No:Vol. 3, No. 3 (2017)

Authors:Nikolai Nadirashvili; Serge Vlăduţ Pages: 397 - 421 Abstract: We develop an integral geometry of stationary Euler equations defining some function w on the Grassmannian of affine lines in \(\mathbb {R}^3\) depending on a putative compactly supported solution (v; p) of the system and deduce some linear differential equations for w. We conjecture that \(w=0\) everywhere and prove that this conjecture implies that \(v=0.\) PubDate: 2017-09-01 DOI: 10.1007/s40598-017-0072-z Issue No:Vol. 3, No. 3 (2017)

Authors:Cordian Riener; Nicolai Vorobjov Pages: 423 - 443 Abstract: Fix any real algebraic extension \(\mathbb K\) of the field \(\mathbb Q\) of rationals. Polynomials with coefficients from \(\mathbb K\) in n variables and in n exponential functions are called exponential polynomials over \({\mathbb K}\) . We study zero sets in \({\mathbb R}^n\) of exponential polynomials over \(\mathbb K\) , which we call exponential-algebraic sets. Complements of all exponential-algebraic sets in \({\mathbb R}^n\) form a Zariski-type topology on \({\mathbb R}^n\) . Let \(P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]\) be a polynomial and denote $$\begin{aligned} V:=\{ (x_1, \ldots , x_n) \in {\mathbb R}^n \> P(x_1, \ldots ,x_n,, e^{x_1}, \ldots ,e^{x_n})=0 \}. \end{aligned}$$ The main result of this paper states that, if the real zero set of a polynomial P is irreducible over \(\mathbb K\) and the exponential-algebraic set V has codimension 1, then, under Schanuel’s conjecture over the reals, either V is irreducible (with respect to the Zariski topology) or each of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of P. In the case of a single exponential (i.e., when P is independent of \(U_2, \ldots , U_n\) ) stronger statements are shown which are independent of Schanuel’s conjecture. PubDate: 2017-09-01 DOI: 10.1007/s40598-017-0073-y Issue No:Vol. 3, No. 3 (2017)

Authors:Pearce Washabaugh; Stephen C. Preston Pages: 175 - 185 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: 2017-06-01 DOI: 10.1007/s40598-016-0058-2 Issue No:Vol. 3, No. 2 (2017)

Authors:Gabriel Katz Pages: 281 - 317 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: 2017-06-01 DOI: 10.1007/s40598-016-0059-1 Issue No:Vol. 3, No. 2 (2017)

Authors:Nikita Kalinin; Mikhail Shkolnikov Abstract: The sum (resp. the sum of squares) of the defects in the triangle inequalities for the area one lattice parallelograms in the first quadrant has a surprisingly simple expression. Namely, let \(f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}\) . Then, where the sum runs by all \(a,b,c,d\in {\mathbb {Z}}_{\ge 0}\) such that \(ad-bc=1\) . We present a proof of these formulae and list several directions for the future studies. PubDate: 2017-11-02 DOI: 10.1007/s40598-017-0075-9

Authors:Alexey Basalaev; Atsushi Takahashi; Elisabeth Werner Abstract: This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund–Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold. PubDate: 2017-11-02 DOI: 10.1007/s40598-017-0076-8

Authors:Dmitry Fuchs; Alexandre Kirillov; Sophie Morier-Genoud; Valentin Ovsienko Abstract: We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton’s essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix. PubDate: 2017-08-22 DOI: 10.1007/s40598-017-0074-x

Authors:Pinaki Mondal Abstract: We study two variants of the following question: “Given two finitely generated \(\mathbb {C}\) -subalgebras \(R_1, R_2\) of \(\mathbb {C}[x_1, \ldots , x_n]\) , is their intersection also finitely generated?” We show that the smallest value of n for which there is a counterexample is 2 in the general case, and 3 in the case that \(R_1\) and \(R_2\) are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of \(\mathbb {C}^n\) and to the moment problem on semialgebraic subsets of \(\mathbb {R}^n\) . The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of \(\mathbb {C}^2\) via key forms of valuations centered at infinity. PubDate: 2017-06-06 DOI: 10.1007/s40598-017-0068-8

Authors:Gaiane Panina Abstract: We describe and study an explicit structure of a regular cell complex \(\mathcal {K}(L)\) on the moduli space M(L) of a planar polygonal linkage L. The combinatorics is very much related (but not equal) to the combinatorics of the permutohedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space M is a sphere, the complex \(\mathcal {K}\) is dual to the boundary complex of the permutohedron.The dual complex \(\mathcal {K}^*\) is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron. PubDate: 2017-05-29 DOI: 10.1007/s40598-017-0070-1

Authors:Yutaka Ishii 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: 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: 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: 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: 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: 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: 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: 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: 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

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