In this paper, we introduce a general Ricci flow system, which is closely linked with the Ricci flow and the renormalization group flow, etc. We prove the short-time existence, the entropy functionals, the higher derivatives estimates and the compactness theorem for this general Ricci flow system on closed Riemannian manifolds. These basic results are useful tools to understand the singularities of this system.

We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). We give some examples of maps on infra-solvmanifolds of dimension three for which the corresponding density is positive.

The paper studies some new ${\mathbb{C}}^n$-generalizations of Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We find the values of parameter ${\beta}$ for which these operators are bounded on mixed norm spaces L(p, q, ${\beta}$) over the unit ball in ${\mathbb{C}}^n$. Moreover, these operators are bounded projections as well, and the images of L(p, q, ${\beta}$) under the projections are found.

In this paper, a homotopy continuation method for obtaining the unique Hermitian positive definite solution of the nonlinear matrix equation $X-{\sum_{i=1}^{m}}A^{\ast}_iX^{-p_i}A_i=I$ with $p_i$ > 1 is proposed, which does not depend on a good initial approximation to the solution of matrix equation.

Let K be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$. Let E be an elliptic curve with complex multiplication by ${\mathcal{O}}_K$, and let $h_E$ be the Weber function on E. Let $N{\in}\{2,3,4,6\}$. We show that $h_E$ alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo $N{\mathcal{O}}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.

In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.

In this paper we introduce a new tensor named semi-projective curvature tensor which generalizes the projective curvature tensor. First we deduce some basic geometric properties of semi-projective curvature tensor. Then we study pseudo semi-projective symmetric manifolds $(PSPS)_n$ which recover some known results of Chaki [5]. We provide several interesting results. Among others we prove that in a $(PSPS)_n$ if the associated vector field ${\rho}$ is a unit parallel vector field, then either the manifold reduces to a pseudosymmetric manifold or pseudo projective symmetric manifold. Moreover we deal with semi-projectively flat perfect fluid and dust fluid spacetimes respectively. As a consequence we obtain some important theorems. Next we consider the decomposability of $(PSPS)_n$. Finally, we construct a non-trivial Lorentzian metric of $(PSPS)_4$.

General (${\alpha},\;{\beta}$)-metrics form a rich and important class of Finsler metrics. In this paper, we obtain a differential equation which characterizes a general (${\alpha},\;{\beta}$)-metric with isotropic E-curvature, under a certain condition. We also solve the equation in a particular case.

A set {$a_1,\;a_2,{\ldots},\;a_m$} of positive integers is called a Diophantine m-tuple if $a_ia_j+1$ is a perfect square for all $1{\leq}i$ < $j{\leq}m$. In this paper, we show that the form of third element in Diophantine pairs and develop some results which are needed to prove the extendibility of the Diophantine pair {a, b} with some conditions. By using this result, we prove the extendibility of Diophantine pairs {$F_{k-2}F_{k+1},\;F_{k-1}F_{k+2}$} and {$F_{k-2}F_{k-1},\;F_{k+1}F_{k+2}$}, where $F_n$ is the n-th Fibonacci number.

Types (over parameters) in the theory of atomless random variable structures correspond precisely to (conditional) distributions in probability theory. Moreover, the logic (resp. metric) topology on the type space corresponds to the topology of weak (resp. strong) convergence of distributions. In this paper, we study metrics between types. We show that type spaces under $d^{\ast}-metric$ are isometric to Wasserstein spaces. Using optimal transport theory, two formulas for the metrics between types are given. Then, we give a new proof of an integral formula for the Wasserstein distance, and generalize some results in optimal transport theory.

In this article, some projection methods (or fractional-step methods) are proposed and analyzed for the micropolar Navier-Stokes equations (MNSE). These methods allow us to decouple the MNSE system into two sub-problems at each timestep, one is the linear and angular velocities system, the other is the pressure system. Both first-order and second-order projection methods are considered. For the classical first-order projection scheme, the stability and error estimates for the linear and angular velocities and the pressure are established rigorously. In addition, a modified first-order projection scheme which leads to some improved error estimates is also proposed and analyzed. We also present the second-order projection method which is unconditionally stable. Ample numerical experiments are performed to confirm the theoretical predictions and demonstrate the efficiency of the methods.