Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Constructible \nabla -modules on curves

Let $\mathcal{V }$ be a complete discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal{V }$ . We introduce the notion of constructible convergent $\nabla $ -module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fiber of $X$ . A constructible module is an $\mathcal{O }_{X_{K}^{\mathrm{an}}}$ -module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$ . The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla $ -modules to the category of $\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$ -modules. We show that specialization induces an equivalence between constructible $F$ - $\nabla $ -modules and perverse holonomic $F$ - $\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$ -modules.     

The distribution of the zeros of the Goss zeta-function for A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)

Let $F$ be a global function field over a finite constant field and $\infty $ a place of $F$ . The ring $A$ of functions regular away from $\infty $ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta $ -function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty $ of $F$ at $\infty $ . He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$ . In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in  $F_\infty $ . In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$ —this $A$ has class number $1$ , it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty $ is $\mathbb{F }\,\!{}_2$ -rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty $ is a rational place of $F$ , we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus  $2$ .     

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.     

The differentiability of Banach space-valued functions of bounded variation

In this paper we consider Banach space-valued functions with the compact range. It is shown that if a Banach space-valued function $F:[0,1] \rightarrow X$ is of bounded variation with respect to the Minkowski functional $||.||_{F}$ associated to the closed absolutely convex hull $C_{F}$ of $F([0,1])$ , then $F$ is differentiable almost everywhere on $[0,1]$ .     

On the third gap for proper holomorphic maps between balls

Let $F$ be a proper rational map from the complex ball $\mathbb B ^n$ into $\mathbb B ^N$ with $n>7$ and $3n+1 \le N\le 4n-7$ . Then $F$ is equivalent to a map $(G, 0, \dots , 0)$ where $G$ is a proper holomorphic map from $\mathbb B ^n$ into $\mathbb B ^{3n}$ .     

Metrics on tiling spaces, local isomorphism and an application of Brown’s lemma

We give an application of a topological dynamics version of multidimensional Brown’s lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$ , one can find a patch such that, for each scale factor $\lambda $ , there is a vector $\vec {t}_\lambda $ so that copies of this patch appear in the tilling “nearly” centered on $\lambda F+\vec {t}_\lambda $ once we allow “bounded perturbations” in the structure of the homothetic copies of $F$ . Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group “actions” on patches and we discuss the local isomorphism property of tilings within this setting.     

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

On a convolution series attached to a Siegel Hecke cusp form of degree 2

We prove that the “naive” convolution Dirichlet series $D_{2}(s)$ attached to a degree 2 Siegel Hecke cusp form $F$ , has a pole at $s=1$ . As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor-zeta function attached to $F$ is $s = 1$ .     

The equidistribution of small points for strongly regular pairs of polynomial maps

In this paper, we prove the equidistribution of periodic points of a regular polynomial automorphism $f : \mathbb{A }^n \rightarrow \mathbb{A }^n$ defined over a number field $K$ : let $f$ be a regular polynomial automorphism defined over a number field $K$ and let $v\in M_K$ . Then there exists an $f$ -invariant probability measure $\mu _{f,v}$ on $\mathrm{Berk }\bigl ( \mathbb{P }^n_\mathbb{C _v} \bigr )$ such that the set of periodic points of $f$ is equidistributed with respect to $\mu _{f,v}$ .     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume

We deal with a compact hypersurface $M$ without boundary immersed in Euclidean space $R^{n+1}$ with the quotient of anisotropic mean curvatures $\frac{(r+1)C_{n}^{r+1}H^F_{r+1}}{a(k+1)C_{n}^{k+1}H^F_{k+1}-b}=constant$ , for real numbers $a$ and $b$ . Such a hypersurface is a critical point for the variational problem preserving a linear combination (with coefficientes $a$ and $b$ ) of the $(k,F)$ -area and the $(n + 1)$ -volume enclosed by $M$ . We show that $M$ is $(r,k,a,b)$ -stable if and only if, up to translations and homotheties, it is the Wulff shape of $F$ , under some assumptions on $a$ and $b$ proved to be sharp. For $a=0$ and $b=1$ , this gives the known $r$ -stability of the $r$ -area for volume preserving variations; if also $F\equiv 1$ it yields the stability studied by Alencar-do Carmo-Rosenberg and Barbosa-Colares. For $b=0$ we also prove a characterization of the Wulff shape as a critical point of the $(r,F)$ -area for variations preserving the $(k, F)$ -area, $0\le k<r<n$ , without the $r$ -stability hypothesis.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Spectral self-affine measures with prime determinant

The self-affine measure $\mu _{M,D}$ relating to an expanding matrix $M\in M_{n}(\mathbb Z )$ and a finite digit set $D\subset \mathbb Z ^n$ is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of $\mu _{M,D}$ in the case when $|\det (M)|=p$ is a prime. The main result shows that under certain mild conditions, if there are two points $s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}$ such that the exponential functions $e_{s_{1}}(x), e_{s_{2}}(x)$ are orthogonal in $L^{2}(\mu _{M,D})$ , then the self-affine measure $\mu _{M,D}$ is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

Regularity of the Monge–Ampère equation in Besov’s spaces

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi $ be the optimal transportation mapping pushing forward $\mu $ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$ . In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}$ in the Besov spaces $W^{\gamma ,1}_{loc}$ . We prove that $D^2 \Phi \in W^{\gamma ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$ . Our proof does not rely on the previously known regularity results.