In this paper, we study the asymptotic and blow-up dynamics of the attraction Keller-Segel chemotaxis system of equations in scale of Banach spaces $E^\alpha_q = H^{2\alpha,q}(\Omega), −1 \le \alpha \le 1,1 \lt q \lt \infty$, where $\Omega \subset \mathbb{R}^N$ is a bounded spatial domain. We show that the system of equations is well-posed for a perturbed analytic semigroup, whenever $2\chi + a \lt \left( \frac{Ne\pi}{2} \right)^{\beta+\frac{\gamma}{2}-\frac{1}{2}}$, where $\chi$ is the chemical attractivity coefficient, $a$ is the rate of production of chemical, and $q, \beta, \gamma$ are of the scale spaces. Thus, as $t\nearrow\infty$, the asymptotic dynamics are captured in the limit set $\mathcal{M}\cup \{0\}$, where $\mathcal{M} = |\Omega|L^1 -$spatial average solutions. The constants for the sharp space embedding $E^\alpha_q \subset L ^\Theta(\Omega) (1\lt\Theta\le\infty)$ indicate that for either the application of Banach fixed point theorem, or the global existence of solutions, no need of either the time for a contraction mapping, nor the initial data of the system of equations, to be small, respectively. In blow-up dynamics, we prove that the solutions blow-up at the borderline scale spaces $E^\alpha_q, \alpha = \frac{N}{2q}$, independent of time $t > 0$, if the chemo-attractivity coefficient dominates the Moser-Trudinger threshold value. An analysis of the finite time bounds for blow-up of solutions in norm of $L^{2p}(\Omega),1 \le p \le 6$ and $\Omega \subset \mathbb{R}^N,N = 2, 3$, is also furnished.

In this article, we study a coupled Cahn-Hilliard-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with a Cahn-Hilliard model for the order (phase) parameter. We prove the existence of an attractor using the theory of pullback attractors.

In this note, we extend the idea of startpoint to a quasi-uniform space. We present two main results, first for single-valued maps and second for multi-valued maps.

A locally asymptotically optimal test is constructed for log-return processes. The behavior of the test statistic is studied under the null and under a sequence of local alternatives. A local asymptotic normality (LAN) result is previously established. Applying the test to log-return data, one rejects the hypothesis that they are independent and identically distributed (iid).

Let $C$ be a semidualizing module over any commutative ring $R$. We investigate the semidualizing module $C$ with finite injective dimension. In particular, we obtain some equivalent characterizations of $C$ under the trivial extension of $R$ by $C$. Moreover, we get that the supremum of the $C$-Gorenstein projective dimensions of all $R$-modules and the supremum of the $C$-Gorenstein injective dimensions of all $R$-modules are equal. Hence the $C$-Gorenstein global dimension of the ring $R$ is definable. At last, we consider the weak $C$-Gorenstein global dimension.

The well-known Hilali conjecture stated in [9] is one claiming that if $X$ is a simply connected elliptic space, then $ \dim \pi_*(X)\otimes {\mathbb Q} \leq \dim H_*(X; {\mathbb Q})$. In this paper we propose that if $f:X \to Y$ is a continuous map of simply connected elliptic spaces, then $\dim {\rm Ker} \ \pi_*(f)_{\mathbb Q}\leq \dim {\rm Ker}\ H_*(f; {\mathbb Q})+1$, and we prove this for certain reasonable cases. Our proposal is a relative version of the Hilali conjecture and it includes the Hilali conjecture as a special case.

Symmetric kernel estimators of an unknown density function on a partial or totally bounded support suffer from edge effects and several authors considered specific asymmetric kernels, belonging in the large class of continuous associated kernels. Asymptotic properties of the corresponding estimators have been examined on a case-by-case basis. In this paper, it is proposed general asymptotic results for continuous associated kernel estimators; in particular, weak and strong global convergences are shown with respect to both uniform and $L^1$ norms. Three lognormal kernel estimators have used for illustrations and discussions. Finally, some concluding remarks are made.