The present study aimed to investigate the similarity monadic $BL$-algebras and prove the completeness of the similarity monadic basic logic. To this end, the similarity monadic $BL$-algebras were defined as: a variety of monadic $BL$-algebras equipped with unary operators $\forall$ and $\exists$, and the vital properties of the algebras were provided. In addition, the S-monadic filters in the similarity monadic $BL$-algebras were introduced and investigated. Correspondingly, the representable similarity monadic $BL$-algebras were characterized, and the notion of the logic of the similarity monadic $BL$-algebras were introduced to prove their soundness and completeness.

A phantom category is an admissible subcategory with vanishing Grothendieck group of the bounded derived category of coherent sheaves on a smooth projective variety. The goal of this paper is to study the abstract situation when such a category appears and establish some results which provide evidence for the idea that these categories are invisible on the level of Chow motives.

The diametral dimension, $\Delta(E)$, and the approximate diametral dimension, $\delta (E)$, of a nuclear Fréchet space $E$ which satisfies $\underline{DN}$ and $\Omega$, are related to corresponding invariant of power series spaces $\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}\left(\varepsilon\right)$ for some exponent sequence $\varepsilon$. In this article, we examine a question of whether $\delta (E)$ must coincide with that of a power series space if $\Delta(E)$ does the same, and vice versa. In this regard, we first show that this question has an affirmative answer in the infinite type case by showing that $\Delta (E)=\Delta\left(\Lambda_{\infty} (\varepsilon)\right)$ if and only if $\delta (E)= \delta (\Lambda_{\infty}(\varepsilon))$. Then we consider the question in the finite type case and, among other things, we prove that $\delta (E)=\delta\left(\Lambda_{1} (\varepsilon)\right)$ if and only if $\Delta (E)= \Delta (\Lambda_{1}(\varepsilon))$ and $E$ has a prominent bounded subset.

This paper is a further contribution to the classification of point-primitive finite linear spaces. Let $ p,q$ be two primes. We prove that if $\mathcal{S}$ is a non-trivial finite linear space with $2pq$ points, and $G\leq Aut(\mathcal{S})$ is point-primitive, then $G $ is line-transitive and $\mathcal{S}$ is the Ree unital $U_R(3), $ or the Hermitian unital $U_H(s). $

Let \(X\), \(Y\) be sets and let \(\Phi\), \(\Psi\) be mappings with domains \(X^{2}\) and \(Y^{2}\), respectively. We say that \(\Phi\) and \(\Psi\) are \emph{combinatorially similar} if there are bijections \(f \colon \Phi(X^2) \to \Psi(Y^{2})\) and \(g \colon Y \to X\) such that \(\Psi(x, y) = f(\Phi(g(x), g(y)))\) for all \(x\), \(y \in Y\). Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.

Let $P_1$ and $P_2$ be potential properties of a metric for which each of $P_1 \wedge P_2, P_1 \wedge \neg P_2, \neg P_1 \wedge P_2$ and $\neg P_1 \wedge \neg P_2$ is possible. We call such a pair \textit{strongly independent} if whenever a metrizable space admits a metric for which either $P_1$ or $\neg P_1$ holds and separately a metric for which either $P_2$ or $\neg P_2$ holds, then there must exist a compatible metric for which both conditions at once hold. We show that boundedly compactness and UC-ness form a strongly independent pair and so do boundedness and UC-ness. Metrizable spaces that admit a boundedly compact metric when equipped with a non-UC metric have been recently studied with respect to the lineability of the non-uniformly continuous real-valued functions defined on them within $C(X)$ [4].

Let $\mathfrak{g}$ be the 2$d$ supersymmetric Galilean conformal algebra. In this paper, we determine each super-skewsymmetric super-biderivation of $\mathfrak{g}$. In particular, we find that there exist non-inner super-biderivations of $\mathfrak{g}$

The purpose of this article is to investigate the growth of meromorphic solutions for the system of complex functional equations with a more general form. Some estimates of the growth order of its meromorphic solutions in all possible cases are obtained, and several examples show our estimates are sharp to some certain.

P-frames in Banach spaces are straightforward generalization of frames in Hilbert spaces. In this paper, motivated the concept of p-Bessel sequences, the concept of p-controlled Bessel sequences is introduced and showed that under some warily conditions, $1<p\leq 2$, they can use instead of each other. Also a close relationship between inherent properties of p-frame mappings and p-pseudo frame mappings with controlled p-frames and p-frames is found. In other words the cases that these natural mappings can be accretive, Lipschitz continuous, coercive and monotone are investigated.