Irrationally elliptic closed characteristics on symmetric compact star-shaped hypersurfaces in $$\mathbf{R}^\mathbf{4}$$

In this paper, we prove that if a symmetric compact star-shaped hypersuface in \(\mathbf{R}^4\) carries exactly two geometrically distinct closed characteristics, then both of them must be irrationally elliptic.     

Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ A X + B Recursion with Triangular Matrices

Journal of Theoretical Probability - We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the...     

Quaternary universal forms over \mathbf{Q}(\sqrt{13})

Let be a real quadratic field over Q with m a square-free positive rational integer and be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x ₁,…,x _n )=∑_1≤i,j≤n α _ij x _i x _j ( ) is called universal if f represents all totally positive integers in . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. In this paper we prove that there are only two quaternary universal forms (up to equivalence) over . For the proof of universality we apply the theory of quadratic lattices.      

Simply transitive geodesic ball packings to \mathbf{S}^2\!\times \!\mathbf{R} space groups generated by glide reflections

The \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry can be derived by the direct product of the spherical plane \(\mathbf{S}^2\) and the real line \(\mathbf{R}\) . In (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 42:235–250, 2001), Farkas has classified and given the complete list of the space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . The \(\mathbf{S}^2\!\times \!\mathbf{R}\) manifolds were classified by Molnár and Farkas in [2] by similarity and diffeomorphism. In Szirmai (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 52(2):413–430, 2011), we have studied the geodesic balls and their volumes in \(\mathbf{S}^2\!\times \!\mathbf{R}\) space; moreover, we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . In this paper, we study the locally optimal ball packings to the \(\mathbf{S}^2\!\times \!\mathbf{R}\) space groups having Coxeter point groups, and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The density of the densest packing is \(\approx 0.80407553\) , may be surprising enough in comparison with the Euclidean result \(\frac{\pi }{\sqrt{18}}\approx 0.74048\) . Molnár has shown in (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 38(2):261–288, 1997) that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere \(\mathcal PS ^3(\mathbf{V}^4,\varvec{V}_4,\mathbb R )\) . In our work, we shall use this projective model of \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry.     

The Cocompleteness of the Category $$\mathbf{Tych}^G$$

Mathematical Notes - The cocompleteness of the category $$\mathbf{Tych}^G$$ is proved.     

Actions of $$\mathbf {SAut}\varvec{(F_n)}$$

We study actions of SAut\((F_n)\), the unique subgroup of index two in the automorphism group of a free group of rank n, and obtain rigidity results for its representations. In particular, we show that every smooth action of SAut\((F_n)\) on a low dimensional torus is trivial.     

2-Dimensional complete self-shrinkers in $$\mathbf {R}^3$$

It is our purpose to study complete self-shrinkers in Euclidean space. By making use of the generalized maximum principle for \(\mathcal {L}\)-operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in \(\mathbb R^3\). Ding and Xin (Trans Am Math Soc 366:5067–5085, 2014) have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

The Algebra of {{\mathbf{SL}}_3}\left( \mathbb{C} \right) Conformal Blocks

We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL₃( $ \mathbb{C} $ ) bundles on a smooth, marked curve (C, $ \vec{p} $ ): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0; 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli $ {{\mathcal{M}}_{{C,\vec{p}}}}\left( {\mathrm{S}{{\mathrm{L}}_3}\left( \mathbb{C} \right)} \right) $ of quasi-parabolic principal SL₃( $ \mathbb{C} $ ) bundles on (C, $ \vec{p} $ ). Along the way we recover positive polyhedral rules for counting conformal blocks.     

Alternating groups as a quotient of $${{\varvec{PSL}}}\left( \mathbf{2}, \pmb {\mathbb {Z}} \left[ {{\varvec{i}}}\right] \right) $$

In this study, we developed an algorithm to find the homomorphisms of the Picard group \(\textit{PSL}(2,Z[i])\) into a finite group G. This algorithm is helpful to find a homomorphism (if it is possible) of the Picard group to any finite group of order less than 15! because of the limitations of the GAP and computer memory. Therefore, we obtain only five alternating groups \( A_{n}\), where \(n=5,6,9,13\) and 14 are quotients of the Picard group. In order to extend the degree of the alternating groups, we use coset diagrams as a tool. In the end, we prove our main result with the help of three diagrams which are used as building blocks and prove that, for \(n\equiv 1,5,6(\mathrm { mod}\, 8)\), all but finitely many alternating groups \(A_{n}\) can be obtained as quotients of the Picard group \(\textit{PSL}(2,Z[i])\). A code in Groups Algorithms Programming (GAP) is developed to perform the calculation.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

Ergodicity of non-homogeneous $$\mathbf {p}$$ p -majorizing quadratic stochastic operators

Positivity - In this paper, we study the strong ergodicity of non-homogeneous $$\mathbf {p}$$ -majorizing quadratic stochastic operators acting on the finite dimension simplex. We first provide a...     

Local linear estimation for stochastic processes driven by \alpha -stable L\acute{\mathbf{e}}vy motion

The $\alpha $ α -stable L $\acute{\mathrm{e}}$ e ´ vy motion together with the Poisson process and Brownian motion are the most important examples of L $\acute{\mathrm{e}}$ e ´ vy processes, which form the first class of stochastic processes being studied in the modern spirit. In this paper, the stochastic processes driven by $\alpha $ α -stable L $\acute{\mathrm{e}}$ e ´ vy motion are considered, local linear estimator of the drift function for these processes is discussed. Under mild conditions, we derive consistency of the local linear estimator of the drift function. The performance of the proposed estimator is assessed by simulation study.     

Restriction map in a regular reduction of $\mathbf{SU}(n)^{2g}$

The quasi-Hamiltonian reduction of at a regular value, in the centre of SU(n), of the moment map is isomorphic to a moduli-space of semi-stable vector bundles over a Riemann surface. We describe the restriction map from the equivariant cohomology of to the cohomology of the moduli space in terms of natural multiplicative generators of these cohomologies. Received: May 28, 2002