Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

Foundations of Computational Mathematics - We show that there exist real parameters $$c\in (-2,0)$$ for which the Julia set $$J_{c}$$ of the quadratic map $$z^{2} + c$$ has arbitrarily high...     

Improved convergence estimates for the Schröder–Siegel problem

Annali di Matematica Pura ed Applicata (1923 -) - We reconsider the Schröder–Siegel problem of conjugating an analytic map in $$\mathbb {C}$$ in the neighborhood of a fixed point to its...     

Existence,uniqueness and regularity of the projection onto differentiable manifolds

Annals of Global Analysis and Geometry - We investigate the maximal open domain $${\mathscr {E}}(M)$$ on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ can be...     

Exponent of a finite group of odd order with an involutory automorphism

Let $$w = w(x_1, \ldots , x_n)$$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $$\widetilde{w}: G^n\rightarrow G$$ where $$\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$$ for every sequence $$(g_1, \ldots , g_n)$$ . Let $$\mathcal G$$ be a simple and simply connected group which is defined and split over an infinite field K and let $$G =\mathcal G(K)$$ . For the case when $$w = w_1w_2 w_3 w_4$$ and $$w_1, w_2, w_3, w_4$$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $$G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$$ where Z(G) is the center of the group G and $${{\text { Im}}}\, {\widetilde{w}}$$ is the image of the word map $$\widetilde{w}$$ . For the case when $$G = {{\text {SL}}}_n(K)$$ and $$n \ge 3$$ , in the same paper of Hui et al. (2015) it was shown that the inclusion $$G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$$ holds for a product $$w = w_1w_2 w_3$$ of any three non-trivial words $$ w_1, w_2, w_3$$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $$B_2, G_2$$ .     

The reflexivity of a Segre product of projective varieties

We study the reflexivity of a Segre product of a projective space and a projective variety Y, and give a criterion for to be reflexive in terms of m, the dimension of Y, the rank of the general Hessian of Y and the characteristic of the ground field. Our study is closely related to a question raised by Kleiman and Piene on the relationship between the conormal map and the Gauss map.     

Modules for map Virasoro conformal algebras

Archiv der Mathematik - Any nontrivial finite irreducible module for the map Lie conformal algebra $$\hbox {Vir}\bigotimes {\mathcal {A}}$$ , where $$\hbox {Vir}$$ is the Virasoro conformal algebra...     

Common periodic trajectories of interval maps

We prove that for any continuous piecewise monotone or smooth interval map f and any subset of the set of periods of periodic trajectories of f, there is another map such that the set of periods of periodic trajectories common for f and , which is denoted by , coincides with . At the same time, for each integer , there exists a continuous map f such that for any map if is an infinite set. Dedicated to Vladimir Igorevich Arnold     

Desargues Configurations via Polar Conics of Plane Cubics

Using polar conics of plane cubics we define a rational map from the moduli space of stable binary sextics into the moduli space of Desargues configurations. We show that this map is the inverse of a birational map defined via the von Staudt conic. In particular Ψ _m is a birational map.     

Regularity for weakly Dirac-harmonic maps to hypersurfaces

We prove that a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface is smooth. Supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.     

Partitions of the complete hypergraph $$K_6^3$$ and a determinant-like function

Journal of Algebraic Combinatorics - In this paper, we introduce a determinant-like map $$\mathrm{det}^{S^3}$$ and study some of its properties. For this, we define a graded vector space $$\Lambda...     

Near coverings and cosystolic expansion

Archiv der Mathematik - Let X, Y be simplicial complexes and let $$f:Y \rightarrow X$$ be a simplicial surjective map. We introduce a notion of deficiency of f, denoted by $$m_f(Y)$$ ,...     

Continuous maps with the disjoint support property

Periodica Mathematica Hungarica - It is well known that for each $$n\ge 0$$ , there is a continuous map $$ f :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}$$ with the disjoint support property....     

Word maps in finite simple groups

Elements of the free group define interesting maps, known as word maps, on groups. It was previously observed by Lubotzky that every subset of a finite simple group that is closed under endomorphisms occurs as the image of some word map. We improve upon this result by showing that the word in question can be chosen to be in $$v(\mathbf F _n),$$ the verbal subgroup of the free group generated by the word v, provided that v is not a law on the finite simple group in question. In addition, we provide an example of a word w that witnesses the chirality of the Mathieu group $$M_{11}$$. The paper concludes by demonstrating that not every subset of a group closed under endomorphisms occurs as the image of a word map.     

Braids,mapping class groups,and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology. The first author was supported by Inha University research grant.     

Spectra of elements in operator space tensor products of $$\hbox {C}^*$$ C ∗ -algebras

Positivity - Given two operator spaces E and F, injectivity of the canonical map from the $$\lambda $$ -tensor product $$E\bigotimes ^\lambda F$$ into the operator space injective tensor product...     

Quantum channels that preserve entanglement

Let M and N be full matrix algebras. A unital completely positive (UCP) map is said to preserve entanglement if its inflation has the following property: for every maximally entangled pure state ρ of , is an entangled state of . We show that there is a dichotomy in that every UCP map that is not entanglement breaking in the sense of Horodecki–Shor–Ruskai must preserve entanglement, and that entanglement preserving maps of every possible rank exist in abundance. We also show that with probability 1, all UCP maps of relatively small rank preserve entanglement, but that this is not so for UCP maps of maximum rank.     

Continuity of $$L_{p}$$ Balls and an Application to Input-Output Systems

Mathematical Notes - In this paper, the continuity of the set-valued map $$p\rightarrow B_{\Omega,\mathcal{X},p}(r)$$ , $$p\in (1,+\infty)$$ , is proved where $$B_{\Omega,\mathcal{X},p}(r)$$ is the...     

The coarse moduli space of a flat analytic groupoid

Annali di Matematica Pura ed Applicata (1923 -) - Let $$\mathcal X $$ be a flat analytic groupoid $$R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$$ such that the holomorphic map...     

Moduli spaces of quadratic complexes and their singular surfaces

We construct the coarse moduli space of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism . Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.