{\mathcal{F}}_\phi and {\mathcal{F}}_g Classes

In this paper we give a generalization of the Zhao F(p, q, s)-spaces by using operators instead of functions. In this way we unify and simplify several important results about the classic spaces D_p, Q_p{\mathcal{Q}}_{p} ,B^α, etc.     

A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }

In this paper, we give a construction of partial difference sets in _p ² x _p ² x ... x _p ²using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.     

Monotone hulls for {\mathcal {N}}\cap {\mathcal {M}}

Using the method of decisive creatures [see Kellner and Shelah (J Symb Log 74:73–104, 2009)] we show the consistency of “there is no increasing \(\omega _2\) –chain of Borel sets and \(\mathrm{non}({\mathcal N})= \mathrm{non}({\mathcal M})=\mathrm{non}({\mathcal N}\cap {\mathcal M})=\omega _2=2^\omega \) ”. Hence, consistently, there are no monotone Borel hulls for the ideal \({\mathcal M}\cap {\mathcal N}\) . This answers Balcerzak and Filipczak (Math Log Q 57:186–193, 2011 [Questions 23, 24]). Next we use finite support iteration of ccc forcing notions to show that there may be monotone Borel hulls for the ideals \({\mathcal M},{\mathcal N}\) even if they are not generated by towers.     

The rational cohomology of $${\overline{\mathcal{M}_4}}$$

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space of stable complex curves of genus 4, with its Hodge structure.     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

The Gorenstein conjecture fails for the tautological ring of \mathcal{\overline{M}}_{2,n}

We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring $R^{\bullet}(\overline {\mathcal {M}}_{2,N})$ is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on $\overline {\mathcal {M}}_{2,N}$ . By work of Graber and Pandharipande, such a class exists on $\overline {\mathcal {M}}_{2,20}$ , and we present some evidence indicating that N is in fact 20.     

Moduli Spaces of Stable Polygons and Symplectic Structures on\overline {\mathcal{M}} _{0,n}

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to , brings up a new tool to study the Kähler topology of . A wild but precise conjecture on the shape of the Kähler cone of is given in the end.     

Extremal effective divisors on \overline{\mathcal M}_{1,n}

For every \(n\ge 3\) , we exhibit infinitely many extremal effective divisors on \(\overline{\mathcal M}_{1,n}\) , the Deligne-Mumford moduli space parameterizing stable genus one curves with \(n\) ordered marked points.     

Note on $$ \mathcal{I}R $$-closed and $$ A_{\mathcal{I}R} $$-sets

We characterize and discuss the properties of $ \mathcal{I}R $ \mathcal{I}R -closed sets and $ A_{\mathcal{I}R} $ A_{\mathcal{I}R} -sets. Also, we give characterizations of weakly $ \mathcal{I} $ \mathcal{I} -locally closed sets and $ \mathcal{I} $ \mathcal{I} -submaximal spaces. A characterization of codense ideals in terms of $ \mathcal{I}R $ \mathcal{I}R -closed sets is also given.     

Scaling of conformal blocks and generalized theta functions over $$\overline{\mathcal {M}}_{g,n}$$

By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Notes on the partition property of {\mathcal{P}_\kappa\lambda}

We investigate the partition property of ${\mathcal{P}_{\kappa}\lambda}$ . Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that ${\mathcal{P}_{\kappa}\lambda}$ carries a (λ ^κ , 2)-distributive normal ideal without the partition property, then λ is ${\Pi^1_n}$ -indescribable for all n?<?ω but not ${\Pi^2_1}$ -indescribable. (2) If cf(λ) ≥?κ, then every ineffable subset of ${\mathcal{P}_{\kappa}\lambda}$ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over ${\mathcal{P}_{\kappa}\lambda}$ has the partition property.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

The universal difference variety over \overline{\mathcal{M }}_g

We determine the Kodaira dimension of the Deligne–Mumford compactification \(\overline{\mathfrak{Diff }}_g\) of the universal difference variety over the moduli space of curves.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

\boldsymbol{\mathcal{Q}}-*-Categories

We consider the theory of categories enriched in an involutive quantaloid : the -*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in to define -*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.      

The exact order of\sum\nolimits_{k = 1}^{\text{n}} {|{\text{x - x}}_{\text{k}} } |^8 |{\text{l}}_{\text{k}} ({\text{x)|}}^{\text{t}}

In this paper we give the exact order of \(\sum\nolimits_{k = 1}^{\text{n}} {|{\text{x - x}}_{\text{k}} } |^5 .\) for any fixed nonnegative integers s and t, which is n^?s, n^?s lnn and n^1?t for s≤t?2, s=t?1 and s≥t, respectively.