The Primitive Permutation Groups of Degree Less Than 4096

In this article we use the Classification of the Finite Simple Groups, the O'Nan–Scott Theorem, and Aschbacher's theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and MAGMA.     

Completing Hermann Schubert's Work on the Enumerative Geometry of Cuspidal Cubics in ℙ3

Abstract

We outline the results of our revisiting Hermann Schubert's work on the enumerative geometry of cuspidal cubics in ?³(Sec. 23 of his Kalkül der abzählenden Geometrie, Teubner, [1879] Schubert, H. 1879. Kalkül der abzählenden Geometrie Teubner. Rep. in 1979 by Springer-Verlag [Google Scholar]. Rep. in 1979 by Springer-Verlag). There are three main aspects that we would like to point to. First, we describe the spaces parameterizing cuspidal cubics in ?³, as well as several different degenerations, using modern algebraic geometry language and techniques. Then we get formulas, by means of today's intersection theory, for the relevant relations among conditions and degenerations, and for allthe intersection numbers in which Schubert was in principle interested. And finally there is the computational aspect, which has been an adventure on its own: the computations have been performed by means of the mathematical computation system OmegaMath, together with the WITmodule. They are discussed briefly in the final Section, with references to detailed information, and here we would just like to say that one of our motivations has been to test that system in what has turned out to be an interesting computational project. Our final table for the cuspidal cubics, which has the 19778 nonzero numbers involving the nine first-order conditions considered by Schubert, fully confirms the fraction of numbers computed by Schubert, as listed on pages 140–143 of the Kalkül. (For those interested in getting our table, please see the indications in the last section of the full paper.) For an assessment of whether or not the numbers computed by Schubert are fully representative of the problems involved in computing all of them, see the Remark at the end of Sec. 3.     

Explicit conformally constrained Willmore minimizers in arbitrary codimension

In our previous work (Ndiaye and Schätzle, 2014), we proved that the flat constant mean curvature tori $$\begin{aligned} T_r := r S^1 \times \sqrt{1 - r^2} S^1 \subseteq S^3 \quad \hbox {for } 0 < r \le 1/\sqrt{2} \end{aligned}$$ minimize the Willmore energy in their conformal class in codimension one when \(r \approx 1 / \sqrt{2}\) , that is \(T_r\) is close to the Clifford torus \(T_{Cliff} = T_{1/\sqrt{2}}\) . In this article, we extend this to arbitrary codimension. Moreover we prove that the Clifford torus minimizes the Willmore energy in an open neighbourhood of its conformal class, again in arbitrary codimension, but the neighbourhood may depend on the codimension.     

Theta divisors with curve summands and the Schottky problem

We prove the following converse of Riemann’s Theorem: let \((A,\Theta )\) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \(\Theta =C+Y\). Then C is smooth, A is the Jacobian of C, and Y is a translate of \(W_{g-2}(C)\). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.     

The Asymptotic Stability of x(n + k) + ax(n) + bx(n-l) = 0

We established necessary and sufficient conditions for the asymptotic stability of the difference equation where the coefficients a and b are real numbers and k and l are nonnegative integers.     

Multifractal analysis of the divergence of Fourier series: The extreme cases

We study the size, in terms of the Hausdorff dimension, of the subsets of T such that the Fourier series of a generic function in L ¹(T), L ^p (T), or C(T) may behave badly. Genericity is related to the Baire Category Theorem or the notion of prevalence. This paper is a continuation of [3].     

A note on the Petri loci

Let ${\mathcal {M}_g}$ be the coarse moduli space of complex projective nonsingular curves of genus g. We prove that when the Brill?CNoether number ??(g, r, n) is non-negative every component of the Petri locus ${P^r_{g,n} \subset \mathcal {M}_g}$ whose general member is a curve C such that ${W^{r+1}_n(C) = \emptyset}$ , has codimension one in ${\mathcal {M}_g}$ .     

Some ergodic and rigidity properties of discrete Heisenberg group actions

The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group \(\mathcal{H}\). We establish the decomposition of the tangent space of any C^∞ compact Riemannian manifold M for Lyapunov exponents, and show that all Lyapunov exponents for the central elements are zero. We obtain that if an \(\mathcal{H}\) action contains an Anosov element, then under certain conditions on the eigenvalues of this element, the action of each central element is of finite order. In particular, there is no faithful codimension one Anosov Heisenberg group action on any compact manifold, and there is no faithful codimension two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic \(\mathcal{H}\) actions by toral automorphisms, using a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann–Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5 Carrillo , J. A. , Desvillettes , L. , Fellner , K. ( 2008 ). Fast-reaction limit for the inhomogeneous aizenman-bak model . Kinetic and Related Models 1 : 127 – 137 . [Google Scholar]], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.     

A Lower Bound for the Determinantal Complexity of a Hypersurface

We prove that the determinantal complexity of a hypersurface of degree \(d > 2\) is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the \(3 \times 3\) permanent is 7. We also prove that for \(n> 3\), there is no nonsingular hypersurface in \({\mathbb {P}}^n\) of degree d that has an expression as a determinant of a \(d \times d\) matrix of linear forms, while on the other hand for \(n \le 3\), a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.     

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$

Robust heteroclinic cycles in equivariant dynamical systems in \({\mathbb R}^4\) have been a subject of intense scientific investigation because, unlike heteroclinic cycles in \({\mathbb R}^3\), they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyze the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in \({\mathbb R}^4\).     

Reflexive and Spanned Sheaves on {\mathbb{P}^3}

We investigate the reflexive sheaves on ${\mathbb{P}^3}$ spanned in codimension 2 with very low first Chern class c ₁. We also give the sufficient and necessary conditions on numeric data of such sheaves for indecomposabiity. As a by-product we obtain that every reflexive sheaf on ${\mathbb{P}^3}$ spanned in codimension 2 with c ₁ = 2 is spanned.     

The Inversion Results for the Limit <Emphasis Type="Italic">q</Emphasis>-Bernstein Operator

The limit q-Bernstein operator \(B_q\) appears as a limit for a sequence of the q-Bernstein or for a sequence of the q-Meyer-König and Zeller operators in the case \(0<q<1.\) Lately, various features of this operator have been investigated from several angles. It has been proved that the smoothness of \(f\in C[0,1]\) affects the possibility for an analytic continuation of its image \(B_qf.\) This work aims to investigate the reciprocal: to what extent the smoothness of f can be retrieved from the analytical properties of \(B_qf\).     

Self-shrinkers for the mean curvature flow in arbitrary codimension

In this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere ${bf S}^{n}(\sqrt{2n})$ is the only complete embedded connected $F$ -stable self-shrinker in $\mathbf{R}^{n+k}$ with $\mathbf{H}\ne 0$ , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in $\mathbf{R}^4$ with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen–Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not $F$ -stable.     

On the Upper Bound of Some Geometric Constants of Absolute Normalized Norms on $${\mathbb{R}^2}$$

This is a continuation of the paper (Mizuguchi and Saito, Ann Funct Anal 2:22–33, 2011). We consider the Banach space \({X=(\mathbb{R}^2, \|\cdot\|)}\) with a normalized, absolute norm. We treat three geometric constants; the von Neumann–Jordan constant C _NJ(X), the modified von Neumann–Jordan constant \({C^{\prime}_{\rm NJ}(X)}\) and the Zb?ganu constant C _Z (X). We consider the conditions in which these constants coincide with their upper bound.