We give a proof of Grilliot's unpublished theorem that all sets definable by a $\Sigma^1_1$ induction on $\mathbb{N}$ are also definable by a positive $\Sigma^1_1$ induction on $\mathbb{N}$. The proof is recursion-theoretic and likely very similar to Grilliot's original proof.

We define the concept of small subsets of asymptotic resemblance spaces and obtain some of their properties. In addition, we show that if the ideal of all small subsets of a metric space $X$ coincides with the ideal of all bounded subsets of $X$, then the asymptotic dimension of $X$ is equal to zero. We also show that a subset $A$ of a proper metric space $X$ is small, if and only if, the interior of the boundary of $A$ in the Higson corona of $X$ is empty.

We construct two infinite families of regular $3$-polytopes of type $\{3, 8m\}$ with $192m^2$ and $384m^2$ automorphisms for every positive integer $m$, respectively. The automorphism groups of these polytopes are solvable groups, and when $m$ is a power of $2$, they provide examples with automorphism groups of order $3\cdot2^n$ where $n$ can be any integer greater than $5$. In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss.

We study the notions of action and semidirect product for digroups and left skew braces.

We study the existence of the positive mild solutions of evolution systems with delay in an ordered Banach space. Firstly, the existence of mild solution is obtained by means of the characteristics of $C_0$-semigroup $T(t)(t\geq 0)$ and monotone iterative technique. Secondly, without assuming the existence of upper and lower solutions, we deal with the existence of positive mild solutions of an evolution equation with delay on infinite intervals by introducing the generalized Arzel\`{a}-Ascoli theorem and Kuratowski measure of noncompactness. The results in this paper are generalization and continuation of the recent results on this issue. Finally, as the application of abstract results, an example is given to illustrate our main results.

We use the idea of normal family to investigate the uniqueness problems for entire functions: $f$ shares one finite non-zero value with its first derivative $f^{(1)}$ together with its certain type of linear differential polynomial $L(f)$ of constant coefficients. The results here improve and generalize results obtained by Yang, Li and Yang, and Li

Let $R$ be a commutative Noetherian ring, ${\mathfrak a}$ be an ideal of $R$, $n$ be a non-negative integer, $X$ be an arbitrary $R$-module and $L$ be a finitely generated $R$-module. We characterize when $H^i_{\mathfrak a}(X)$ and $H^i_{\mathfrak a}(L,X)$ are $(FD_{\prec n}, {\mathfrak a})$-cofinite for all $i$, whenever one of the following statements holds: (a) ${\rm ara}({\mathfrak a})\leq 1$, (b) ${\rm dim} R/{\mathfrak a} \leq n+1$, (c) ${\rm dim} R\leq n+2$ or (d) $X$ is an $FD_{\prec n+2}~R$-module. As a consequence, we show that ${\rm Ext}^i_R(L,X)$ is $FD_{\prec n}$ for all $i$ and any ${\frak a}$-torsion finitely generated $R$-module $L$ with ${\rm dim} L\leq n+1$ if one of the above statements holds. Moreover, we obtain that, if $R$ is a semi-local ring with ${\rm dim} R/{\mathfrak a}\leq 2$ and $X$ is an $R$-module such that ${\rm Ext}^j_R(R/{\mathfrak a}, X)$ is $FD_{\prec 1}$ for all $j\leq {\rm dim} X$, then ${\rm Ass}_R(H^i_{\mathfrak a}(L, X))$ is finite for all $i$.

Let $\mathbb{K}$ be a non-archimedean complete valued field and let $(E,\|.\| )$ be a non-archimedean normed space over $\mathbb{K}$. Let $\left\vert \mathbb{K}^{\times }\right\vert :=\left\{ \left\vert \lambda \right\vert :\lambda \in \mathbb{K}\backslash \left\{ 0\right\} \right\} $ and $\left\Vert E^{\times}\right\Vert :=\left\{ \left\Vert x\right\Vert :x\in E\backslash \left\{0\right\} \right\}$. By Serre's Renorming Problem we mean the following question: can one for every non-archimedean normed space $E$ introduce a norm $\|.\| _{\bullet }$ on $E$ that is equivalent to the given norm in the sense that it determines the same topology and has the property $\|E^{\times }\|_{\bullet }=\left\vert \mathbb{K}^{\times }\right\vert $? An affirmative solution of this problem is presented.

By means of the contour integration method, two variants of the Dirichlet beta function are investigated. Two general summation formulae are established. Several interesting particular infinite series evaluations are given as consequences.

Let $f$ and $g$ be functions in Fock-Sobolev spaces $F_{-\alpha}^2$ of negative orders. In this paper we give a complete characterization of the boundedness and compactness of the mixed product $H_{\overline{f}}T_{\overline{g}}$ from $F_{-\alpha}^2$ to $L^2(\mathbb{C},d\widehat{\mu}_{-\alpha})$.