Abstract:
We prove the local in time existence and a blow up criterion of solution in the H\"{o}lder spaces for the inviscid Boussinesq system in $R^{N},N\geq2$, under the assumptions that the initial values $\theta_{0},u_{0}\in C^{r}$, with $r>1$.

Abstract:
A simplicial complex $X$ is said to be tight with respect to a field $\mathbb{F}$ if $X$ is connected and, for every induced subcomplex $Y$ of $X$, the linear map $H_\ast (Y; \mathbb{F}) \rightarrow H_\ast (X; \mathbb{F})$ (induced by the inclusion map) is injective. This notion was introduced by K\"{u}hnel in [10]. In this paper we prove the following two combinatorial criteria for tightness. (a) Any $(k+1)$-neighbourly $k$-stacked $\mathbb{F}$-homology manifold with boundary is $\mathbb{F}$-tight. Also, (b) any $\mathbb{F}$-orientable $(k+1)$-neighbourly $k$-stacked $\mathbb{F}$-homology manifold without boundary is $\mathbb{F}$-tight, at least if its dimension is not equal to $2k+1$. The result (a) appears to be the first criterion to be found for tightness of (homology) manifolds with boundary. Since every $(k+1)$-neighbourly $k$-stacked manifold without boundary is, by definition, the boundary of a $(k+1)$-neighbourly $k$-stacked manifold with boundary - and since we now know several examples (including two infinite families) of triangulations from the former class - theorem (a) provides us with many examples of tight triangulated manifolds with boundary. The second result (b) generalizes a similar result from [2] which was proved for a class of combinatorial manifolds without boundary. We believe that theorem (b) is valid for dimension $2k+1$ as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.

Abstract:
We prove estimates in H\"{o}lder spaces for some Cauchy-type integral operators representing holomorphic functions in Cartesian and symmetric products of planar domains. As a consequence, we obtain information on the boundary regularity in H\"{o}lder spaces of proper holomorphic maps between symmetric products of planar domains.

Abstract:
We prove Shao and Yu's tightness criterion for the generalized empirical process in the space with topology. Covariance inequalities are used in applying the criterion to particular types of the empirical processes. We weaken the assumptions imposed on the covariance structure as well as the properties of the underlying sequence of r.v.'s, under which presented processes converge weakly. 1. Introduction Let be a sequence of absolutely continuous identically distributed (i.d.) random variables (r.v.’s) with an unknown distribution function (d.f.) and probability density function (p.d.f.) . The empirical distribution function, based on the first r.v.’s, is defined by . It is well known, however, that this estimate does not make use of the smoothness of , that is, the existence of the p.d.f. . Therefore, the kernel estimate has been proposed, where the kernel function is a known d.f. and is a sequence of positive constants descending at an appropriate rate. Such estimator has been deeply studied in the last two decades mainly by Cai and Roussas in [1–4], Li and Yang in [5] and others. Asymptotic normality, Berry-Essen bounds for smooth estimator are only examples of their fruitful results. Recently, Li et al. proposed in [6] the so-called recursive kernel estimator of the d.f. as follows: The seemingly tiny modification they introduced to the formula of the typical kernel estimator has an important advantage. Namely, in the case of a large size of a sample, can be easily updated with each new observation since it is computable recursively by where . The authors discussed the asymptotic bias and quadratic-mean convergence and established the pointwise asymptotic normality of under relevant assumptions. In this paper, however, we will focus on the empirical process built on an estimator of the d.f. rather than itself. Let us recall that the following process: is called the empirical process built on an estimator . Yu [7] studied the case when is a standard empirical d.f. and showed weak convergence of to the Gaussian process assuming stationarity and association of the underlying r.v.’s. Cai and Roussas [1] obtained a similar result in the case when is the kernel estimator of the d.f. built on a stationary sequence of negatively associated r.v.’s. In this paper, we shall study the empirical process generated by the generalized kernel estimator of the d.f. given by the formula A1: is a sequence of absolutely continuous i.d. r.v.’s taking values in and having twice differentiable d.f. with first and second derivative bounded;A2: is a kernel function such that

Abstract:
This paper sheds new light on regularity of multifunctions through various characterizations of directional H\"older /Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional H\"older /Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional H\"older /Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

Abstract:
In [Isett,13], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with H\"{o}lder regularity not exceeding $1/3$. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space $L_t^\infty B_{3,\infty}^{1/3}$ due to low regularity of the energy profile. In this paper, we establish two theorems that may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with H\"{o}lder exponent less than $1/5$. Our first result shows that any non-negative function with compact support and H\"older regularity $1/2$ can be prescribed as the energy profile of an Euler flow in the class $C^{1/5-\epsilon}_{t,x}$. The exponent $1/2$ is sharp in view of a regularity result of [Isett,13]. The proof employs an improved greedy algorithm scheme that builds upon that in [Buckmaster-De Lellis-Sz\'ekelyhidi, 13]. Our second result shows that any given smooth Euler flow can be perturbed in $C^{1/5-\epsilon}_{t,x}$ on any pre-compact subset of ${\mathbb R}\times {\mathbb R}^3$ to violate energy conservation. In particular, there exist nonzero $C^{1/5-\epsilon}_{t,x}$ solutions to Euler with compact space-time support, generalizing previous work of the first author [Isett,12] to the nonperiodic setting.

Abstract:
We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of ${\cal W}_k(d)$ for $d\geq 2k$. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of ${\cal W}^{\ast}_k(d)$ for $d \geq 2k+2$. As another application, we prove that, when $d \neq 2k+1$, all members of ${\cal W}^{\ast}_k(d)$ are tight. We also characterize the tight members of ${\cal W}^{\ast}_k(2k + 1)$ in terms of their $k^{\rm th}$ Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for triangulated manifolds in which the members of ${\cal W}_1(d)$ provide the equality case. This generalises a result (the $d=4$ case) due to Walkup and Kuehnel. As a consequence, it is shown that every tight member of ${\cal W}_1(d)$ is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuehnel and Lutz asserting that tight triangulated manifolds should be strongly minimal.

Abstract:
This study is devoted to investigating the regularity criterion of weak solutions of the micropolar fluid equations in . The weak solution of micropolar fluid equations is proved to be smooth on when the pressure satisfies the following growth condition in the multiplier spaces , . The previous results on Lorentz spaces and Morrey spaces are obviously improved. 1. Introduction Consider the Cauchy problem of the three-dimensional (3D) micropolar fluid equations with unit viscosities associated with the initial condition: where , and are the unknown velocity vector field and the microrotation vector field. is the unknown scalar pressure field. and represent the prescribed initial data for the velocity and microrotation fields. Micropolar fluid equations introduced by Eringen [1] are a special model of the non-Newtonian fluids (see [2–6]) which is coupled with the viscous incompressible Navier-Stokes model, microrotational effects, and microrotational inertia. When the microrotation effects are neglected or , the micropolar fluid equations (1.1) reduce to the incompressible Navier-Stokes flows (see, e.g., [7, 8]): That is to say, Navier-Stokes equations are viewed as a subclass of the micropolar fluid equations. Mathematically, there is a large literature on the existence, uniqueness and large time behaviors of solutions of micropolar fluid equations (see [9–15] and references therein); however, the global regularity of the weak solution in the three-dimensional case is still a big open problem. Therefore it is interesting and important to consider the regularity criterion of the weak solutions under some assumptions of certain growth conditions on the velocity or on the pressure. On one hand, as for the velocity regularity criteria, by means of the Littlewood-Paley decomposition methods, Dong and Chen [16] proved the regularity of weak solutions under the velocity condition: with Moreover, the result is further improved by Dong and Zhang [17] in the margin case: On the other hand, as for the pressure regularity criteria, Yuan [18] investigated the regularity criterion of weak solutions of the micropolar fluid equations in Lebesgue spaces and Lorentz spaces: where is the Lorents space (see the definitions in the next section). Recently, Dong et al. [19] improved the pressure regularity of the micropolar fluid equations in Morrey spaces: where Furthermore, Jia et al. [20] refined the regularity from Morrey spaces to Besov spaces: with One may also refer to some interesting results on the regularity criteria of Newtonian and non-Newtonian fluid equations

Abstract:
We investigate the sample paths regularity of operator scaling alpha-stable random fields. Such fields were introduced as anisotropic generalizations of self-similar fields and satisfy a scaling property for a real matrix E. In the case of harmonizable operator scaling random fields, the sample paths are locally H\"{o}lderian and their H\"{o}lder regularity is characterized by the eigen decomposition with respect to E. In particular, the directional H\"{o}lder regularity may vary and is given by the eigenvalues of E. In the case of moving average operator scaling random alpha-stable random fields, with 0