The symbol length of ${_pBr}(k(\!(\alpha_1)\!)\dots(\!(\alpha_n)\!))$ for an algebraically closed field $k$ of $\operatorname{char}(k) \neq p$ is known to be $\lfloor \frac{n}{2} \rfloor$. We prove that the symbol length for the case of $\operatorname{char}(k) = p$ is rather $n-1$. We also show that pairs of anisotropic quadratic or bilinear $n$-fold Pfister forms over this field need not share an $(n-1)$-fold factor.

We prove that any two Schauder bases $(x_n)$ and $(y_n)$ in non-normable Köthe spaces $E$ and $F$ (over a non-Archimedean field $\mathbb K$) have block basic sequences $(u_n)$ and $(v_n)$, respectively, that are equivalent. Moreover we show that any Schauder basis in a non-normable Köthe space has a block basic sequence that is equivalent to the coordinate Schauder basis in some generalized power series space of infinite type; the generalized power series spaces are the most known and important examples of nuclear Köthe spaces. It follows that any two non-normable Köthe spaces $E$ and $F$, have closed subspaces $E_0$ and $F_0$, respectively, that are isomorphic to the same generalized power series space of infinite type $D_g(a,\infty)$ for some $g\in \Phi_c$ and $a\in \Gamma$.

Let $(M,g(t))$ be a closed Riemannian manifold of dimension $n\geq 2$. In this paper we obtain the evolution formula for the lowest constant $\lambda^{b}_{a}(g)$ under the normalized and unnormalized Yamabe flow such that the equation \begin{equation*} -{\rm \Delta} f + af\log f + bRf= \lambda^{b}_{a}(g) f, \end{equation*} with $\int_M f^2\, {\rm dV}=1,$ has positive solutions, where $a$ and $b$ are two real constants. Then we construct various monotonic quantities under the normalized and unnormalized Yamabe flow. We also show that the scalar curvature of a steady Yamabe breather with nonnegative scalar curvature is identically zero.

Let $k$ be a non-zero complex number. In this paper, we consider a $k$-circulant matrix with binomial coefficients. The inverse of such invertible matrix is determined. We also obtain the Moore\,-\,Penrose inverse (and the group inverse) of such singular matrix. The obtained results are illustrated by examples at the end of this paper.

A ring in which the relation of being an ideal is transitive is called fully filial. The aim of this paper is to give the classification theorem for torsion-free fully filial rings.

Closures of $\mathcal{Q}_K(p,q)$-type spaces in the Logarithmic Bloch space are investigated in this paper. Moreover, we characterize the boundedness and compactness of composition operators from the Logarithmic Bloch space to the closure of $\mathcal{Q}_K(p,q)$ type spaces in the Logarithmic Bloch space

In this paper, we prove that if $2^\lambda < \kappa$ and $\mathcal A \subset [\kappa]^{\lambda}$ is a MADF then the generalized Mrówka space $\Psi (\mathcal A\setminus \mathcal B)$ has no $G_\lambda$-diagonal, where $\mathcal B \subset \mathcal A$ and $|\mathcal B|\le\lambda$. If a topological space $X$ has a regular $G_\lambda$-diagonal and a local decreasing base of cardinality $\chi(X)$ for each $x \in X$, then $|X| \le 2^{dc(X) \cdot \chi(X) \cdot \lambda}$, where $dc(X)$ is the discrete cellularity of $X$ and $\chi(X)$ is the character of $X$. As a corollary, we prove that if $X$ is a DCCC (and hence, $DC(\omega_1)$, weakly Lindelöf or star Lindelöf) first countable space with a regular $G_\delta$-diagonal then $|X|\le 2^\omega$, which gives a positive answer to a question in [17]. Finally, we give another counterexample to a question of Ginsburg and Woods [6], which has nicer properties than the counterexamples constructed in [15] and [16].

In this paper, we consider the linear first order singularly perturbed Fredholm integro-differential equation. For the solution of this problem, fitted difference scheme is constructed on a Shishkin mesh. The method is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. The method is proved to be second-order convergent in the discrete maximum norm. Also, numerical results are given to support theoretical analysis.

In the present paper we introduce a new class of sequences called $GMS\left(p, \beta ,r\right) ,$ which is the generalization of a class considered by Tikhonov in [9] and Szal in [10]. Moreover, we obtained in this note sufficient and necessary conditions for the uniform convergence of sine and cosine series with $\left(p,\beta ,r\right) -$ general monotone coefficients.

We show that the basis constant of every retractional Schauder basis on the Free space of a graph circle increases with the radius. As a consequence, there exists a uniformly discrete subset $M\subseteq\R^2$ such that $\F(M)$ does not have a retractional Schauder basis. Furthermore, we show that for any net $ N\subseteq\R^n$, $n\geq 2$, there is no retractional unconditional basis on the Free space $\mathcal F(N)$.

In this paper, we introduce a viscosity-type proximal point algorithm which comprises of a finite sum of resolvents of monotone operators, and a generalized asymptotically nonexpansive mapping. We prove that the algorithm converges strongly to a common zero of a finite family of monotone operators, which is also a fixed point of a generalized asymptotically nonexpansive mapping in an Hadamard space. Furthermore, we give two numerical examples of our algorithm in finite dimensional spaces of real numbers and one numerical example in a non-Hilbert space setting, in order to show the applicability of our results.

The parameters of the association scheme on the points off a quadric are computed. This corrects a mistake in the literature.