Locally anisotropic toposes

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.     

Topology and measure in logics for region-based theories of space

Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. On an alternative, region-based approach to space, extended regions together with the relations of ‘parthood’ and ‘contact’ are taken as primitive; points are represented as mathematical abstractions from regions.Region-based theories of space have been traditionally modeled in regular closed (or regular open) algebras, in work that goes back to [5] and [21]. Recently, logics for region-based theories of space were developed in [3] and [19]. It was shown that these logics have both a nice topological and relational semantics, and that the minimal logic for contact algebras, ${\mathbb{L}}_{min}^{cont}$ (defined below), is complete for both.The present paper explores the question of completeness of ${\mathbb{L}}_{min}^{cont}$ and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that ${\mathbb{L}}_{min}^{cont}$ is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension ${\mathbb{L}}_{min}^{cont}+(Con)$ is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic ${\mathbb{L}}_{min}^{cont}+(Univ)$ is weakly complete for the infinite binary tree.     

Quasi-Baer module hulls and applications

Let V be a module with $S=\mathrm{End}(V)$. V is called a quasi-Baer module if for each ideal J of S, $r_V(J)=eV$ for some $e^2=e\in S$. On the other hand, V is called a Rickart module if for each $?\in S$, $\mathrm{Ker}(?)=eV$ for some $e^2=e\in S$. For a module N, the quasi-Baer module hull $\mathbf{qB}(N)$ (resp., the Rickart module hull $\mathbf{Ric}(N)$) of N, if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull $E(N)$ of N. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R-module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let $\{K_i|i\in \mathrm{\Lambda }\}$ be any set of R-submodules of $F_R$. For an R-module $M_R$ with ${\mathrm{Ann}}_R(M)\ne 0$, we show that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has a quasi-Baer module hull if and only if $M_R$ is semisimple. This quasi-Baer hull is explicitly described. An example such that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which $N/t(N)$ is projective and ${\mathrm{Ann}}_R(t(N))\ne 0$, where $t(N)$ is the torsion submodule of N, we show that the quasi-Baer hull $\mathbf{qB}(N)$ of N exists if and only if $t(N)$ is semisimple. We prove that the Rickart module hull also exists for such modules N. Furthermore, we provide explicit constructions of $\mathbf{qB}(N)$ and $\mathbf{Ric}(N)$ and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum $N_R$ of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.     

Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let 25584X:media:mana201600285:mana201600285-math-0001" class="section_image" src="/cms/asset/f8c1d6cc-d982-4389-95db-6fb2603c04ef/mana201600285-math-0001.png"> be a compact gradient shrinking Ricci soliton satisfying 25584X:media:mana201600285:mana201600285-math-0002" class="section_image" src="/cms/asset/975dee4e-3f59-43eb-9e16-3beaa8dfe7a6/mana201600285-math-0002.png"> with 25584X:media:mana201600285:mana201600285-math-0003" class="section_image" src="/cms/asset/87eb8449-5452-46e8-9851-d50a38723e88/mana201600285-math-0003.png"> constant. We show that if 25584X:media:mana201600285:mana201600285-math-0004" class="section_image" src="/cms/asset/5c997f4c-1f21-4f6d-ba95-1d5f570b0b6f/mana201600285-math-0004.png"> satisfies 25584X:media:mana201600285:mana201600285-math-0005" class="section_image" src="/cms/asset/a844a38c-276f-4d33-b9fa-d2e1dad5b907/mana201600285-math-0005.png">, then 25584X:media:mana201600285:mana201600285-math-0006" class="section_image" src="/cms/asset/b2cb54b8-01bc-4049-a3ea-d10c6f9589ed/mana201600285-math-0006.png"> is Einstein. Here 25584X:media:mana201600285:mana201600285-math-0007" class="section_image" src="/cms/asset/e3de4ee0-26a1-4eb5-b9fa-42f8c5e4c068/mana201600285-math-0007.png"> denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in 10 , 14 and 19 . Finally, we prove that if 25584X:media:mana201600285:mana201600285-math-0008" class="section_image" src="/cms/asset/714b1406-4e99-403b-a8d0-77cf958eb51c/mana201600285-math-0008.png"> is a complete noncompact gradient steady Ricci soliton satisfying 25584X:media:mana201600285:mana201600285-math-0009" class="section_image" src="/cms/asset/7eca335c-234b-41fc-9aff-e5d058f88105/mana201600285-math-0009.png">, and if the scalar curvature attains its maximum at some point in the interior of M, then either 25584X:media:mana201600285:mana201600285-math-0010" class="section_image" src="/cms/asset/49990ec9-ef01-4af1-b132-abd540301d01/mana201600285-math-0010.png"> is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in 3 .     

On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process ${(X_t)}_{t\ge 0}$,

$\underset{t\to 0}{\lim }?\frac{1}{t}({\mathbb{E}}^{x}f(X_t)?f(x)),x\in {\mathbb{R}}^d,$

for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x\in {\mathbb{R}}^d$ if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of ${(X_t)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.     

Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function 25584X:media:mana201700376:mana201700376-math-0001" class="section_image" src="/cms/asset/96f31c91-6ef2-4487-a60e-3e958a47a478/mana201700376-math-0001.png"> on the upper half‐spaces. Using the decomposition, we study the boundedness of 25584X:media:mana201700376:mana201700376-math-0002" class="section_image" src="/cms/asset/943d9cea-1e55-4c0d-a66d-edbbe6510b3b/mana201700376-math-0002.png">. We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

1,3,4-(口恶)二唑硫醚酰胺为核心骨架的新型多杂环分子的合成及对Cdc25B与PTP1B的抑制活性

首先利用含有三嗪的芳香酰肼(3)构筑了1,3,4-噁二唑衍生物(5), 然后将化合物5与含有1,3,4-噻二唑的衍生物(6)拼合合成了18个目标分子. 利用红外光谱(IR)、 核磁共振波谱(NMR)和高分辨质谱(HRMS)等技术对其结构进行了表征. 考察了目标分子对细胞分裂周期25磷酸酯酶B(Cdc25B)和蛋白酪氨酸磷酸酯酶1B(PTP1B)的抑制活性. 结果表明, 有8个目标分子的抑制活性优于其阳性对照物, 有望成为潜在的Cdc25B抑制剂; 有12个目标分子的抑制活性优于其对照物, 有望成为潜在的PTP1B抑制剂.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).