Let $f$ be a centralizing automorphism of a semiprime ring $R$. Then for all $x \in R (f(x) - x) \in Z(R)$; that is the mapping $f -1$ maps $R$ into its centre.

The main purpose of this paper is using the important works of Heath-Brown, Yoichi Motohashi, Masanori Katsurada and Kohji Matsumoto, and the properties of Dirichlet $L$-functions to study the number of squarefree primitive roots, squarefull primitive roots and squarefree quadratic residues modulo a prime $p$, and give three much sharper asymptotic formulae.

We obtain equations of geodesic lines in the Lie group Sol and prove that the ideal boundary of the Sol is a set $\mathcal{R} = \{(x, y, z)| xy = 0,\text{ and } x^{2} +y^{2}+z^{2} = 1\}$ with a degenerate Tits metric, i.e., the distance between different points equals $\infty$.

Spectral and scattering properties of positive self-adjoint operators that correspond to singular perturbations of the Schrödinger operator homogeneous with respect to the scaling transformations are investigated by a method based on the Lax-Phillips ideas in scattering theory.

Cohen-Macaulay dimension for modules over a commutative ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.

We give a sufficient condition to construct non-trivial $\mu$-symmetric diffusion processes on a locally compact separable metric measure space $(M,\rho , \mu )$. These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of $\Gamma$-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing diffusions can be applied to these fractals.

In this paper we consider the initial value problems for the second order mixed monotone type of impulsive differential inclusions. By introducing a special partial order, we present the existence of maximal and minimal fixed points for mixed monotone multivalued operators in Banach spaces. Applying the results, we establish the existence and uniqueness of above impulsive differential inclusions.

We consider the Cauchy problem for a system of semilinear wave equations with multiple propagation speeds in three space dimensions. We assume that quadratic terms in its nonlinearity consist of products of unknowns whose speeds are different from each other. It is known that some classical solution to the above type of system may blow up in finite time. In this article, we get the sharp lower bound for the lifespan of classical solutions to this kind of system.

We construct a complex manifold $X$, $dimX \geq 3$, which is an increasing union of (1, 1) convex-concave open subsets having the same fixed convex boundary, and a holomorphic line bundle $L$ on $X$, such that the cohomology group $H^{1}(X,L)$ is not separated.The manifold $X$ is constructed as a proper modification of the (1, 1) convex-concave manifold $\mathbb{C}^{k} \backslash \{0\}$ at a discrete subset. It is also remarked that an increasing union of 1-concave manifolds has always separated cohomology (for locally free sheaves).

We prove a boundary version of the Unique Continuation Principle for pseudo-holomorphic curves. It is a consequence of the boundary regularity of pseudo-holomorphic curves. which can be achieved by a bootstrap method.