A real arithmetic function $f$ is \emph{multiplicatively monotonous} if $f(mn)-f(m)$ has constant sign for $m,n$ positive integers. Properties and examples of such functions are discussed, with applications to positive hermitian Toeplitz-multiplicative determinants.

In this paper, we study quadric surfaces in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We show that helicoids and spheres are the only quadric surfaces of finite $III$-type.

There are three known ways to construct the minimal dilation of the discrete semigroup generated by a normal unital completely positive map on a von Neumann algebra, which are given by Arveson, Bhat-Skeide and Muhly-Solel. In this paper, we describe the relation between Bhat-Skeide's and Muhly-Solel's constructions, which is different from the one described by Skeide's commutant duality.

In this paper, a novel high-order numerical method is formulated to solve fractional differential equations. The fractional derivative is described in the Caputo sense due to its applicability to real-world phenomena. First, the fractional differential equation is reduced into a Volterra-type integral equation by applying the Laplace and inverse Laplace transform. Then, the piecewise Lagrange interpolation polynomial of degree five is utilized to approximate unknown function. The truncation error estimates for the novel schemes is derived, and it is theoretically established that the order of convergence of the numerical method is $O(h^6)$. The stability analysis of the novel method is also carefully investigated. Numerical examples are given to show the accuracy, applicability and the effectiveness of the novel method.

We introduce the concept of Meir-Keeler condensing operator in a Banach space via an arbitrary measure of weak noncompactness. We prove some generalizations of Darbo's fixed point theorem by considering a measure of weak noncompactness which not necessary has the maximum property. We prove some coupled fixed point theorems and we apply them in order to establish the existence of weak solutions for a system of functional integral equations of Volterra type.

In this note we establish the compatibility between the topological index in the differential K-theory of Freed-Lott and the analytic push-forward construction in the differential K-theory of Bunke-Schick by a direct computation.

In this paper, we introduce two new concepts, namely, a subset being $d_\delta$-connected relative to a topological space, and a subset being $D_\delta$-closed relative to the space. The former is a generalization of the concept of a subset being $\theta$-connected relative to a space, and the latter is analogous to the $H(i)$ space.

We prove an analogue of the Centralizer Theorem in the context of Artin-Tits groups.

In this paper, we give characterizations for a polynomial stability in Banach spaces. This is done by using evolution cocycles and techniques of Banach function spaces. Our characterizations are new versions of the theorems of Datko type.