A quadratic form on ${\mathrm{ℝ}}^n$ is a function $Q$ defined on ${\mathrm{ℝ}}^n$ whose value at a vector $x$ in ${\mathrm{ℝ}}^n$ is an expression of the form $Q\left(x\right)=x^{T}Ax$, where $A$ is $n\times n$ symmetric matrix. In this note we discuss how to prove two theorems, Theorem A and Theorem B, for finding the maxima of two quadratic forms under some constraints. Of course, the proofs of these theorems are well known. But in this note we give an improved proof of Theorem A and a simpler proof of Theorem B.

In [18], Tallafha and Khalil introduced a new type of uniform space which is a mixture of topology and analysis, namely semi-linear uniform spaces. In this paper, we continue and extend this study by presenting a new generalization of a semi-linear uniform space, namely weak semi-linear uniform spaces. Then we present the concept of proximinal sets without any tools that the metric structure usually offers. We show that every finite set and every convergent sequence with its limit are proximinal sets. Furthermore, we give a characterization of metrizable spaces and we study some of the topological properties of weak semi-linear uniform spaces.

In this paper, we study the properties of $\left(a\right)$-semi-compact, $\left(a\right)$-semi-Lindelöf and $\left(a\right)$-semi-Menger in $\left(a\right)$topological spaces by using $\left(a\right)$-semi-open sets. We investigate the relationships of these properties with the other known covering properties. Further, we construct several examples to discuss the relationships between various covering properties.

The Seidel Laplacian matrix of the Unitary Cayley graph $X_n$ is $S_L\left(X_n\right)=D_S\left(X_n\right)-S\left(X_n\right)$. In this paper, we obtain the eigenvalues of the Seidel Laplacian matrix and the Seidel Laplacian Energy of Unitary Cayley Graphs $X_n$.

The aim of this article is to introduce the concept of graded $2$-absorbing coprimary submodules as a generalization of graded strongly $2$-absorbing second submodules, and explore some properties of this class. A nonzero graded $R$-submodule $N$ of a graded $R$-module $M$ is called a graded $2$-absorbing coprimary $R$-submodule if whenever $x,y$ are homogeneous elements of $R$ and $K$ is a graded $R$-submodule of $M$ such that $xyN\subseteq K$, then either $x$ or $y$ is in the graded radical of $\left(K\hspace{0.17em}{:}_R\hspace{0.17em}N\right)$ or $xy\in An{n}_R\left(N\right)$. Several results have been achieved.

The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. In this paper, we compute a closed formula for the $a$-number of $X^{q+1}+Y^{q+1}+Z^{q+1}=0$ with $q=p^s$ over the finite field ${\mathbb{F}}_{q^2}$, using the action of the Cartier operator on $H^0(\mathcal{X},{\mathrm{\Omega }}^1)$.

We define an abelian loop on the set $S$ consisting of $1$ and all odd prime numbers with an operation $•$, where for $a,b\in S,a•b$ is the smallest element of $S$ strictly larger than $|a-b|$. We use theorems from number theory to prove several properties of the loop and we use conjectures from number theory to state analogous conjectures about the loop.

A complex fuzzy $\mathrm{\Gamma }$-ring is a fuzzy $\mathrm{\Gamma }$-ring whose membership function takes values in the closure of the unit disk around the origin (or closed unit disk around the origin) in the complex plane. In this paper, we define the complex fuzzy sub-$\mathrm{\Gamma }$-ring and complex fuzzy ideals of $\mathrm{\Gamma }$-rings. We establish an analogue of the isomorphism theorems of $\mathrm{\Gamma }$-rings out of complex fuzzy $\mathrm{\Gamma }$-rings.

The concept of intuitionistic hesitant fuzzy subsemigroups was introduced by Siripitukdet and Thongban in 2020 who extended the concept of hesitant fuzzy semigroups by establishing the concept of intuitionistic hesitant fuzzy semigroups. In this paper, we study necessary and sufficient conditions of coincidences of types of intuitionistic hesitant fuzzy ideals in types of semigroups.

Let $f:X⟶X$ be a mapping. Consider $$\mathcal{P}\left(f\right)=\{O\subseteq X:f^{-1}(O)\subseteq O\}.$$ Then $\mathcal{P}\left(f\right)$ is an Alexandroff topology. A topological space $X$ is called a primal space if its topology coincides with a $\mathcal{P}\left(f\right)$ for some mapping $f:X⟶X$. Let $A$ be an alphabet and $A^{*}$ be the set of all finite words over $A$. A word is called primitive if it is not empty and not a proper power of another word. Let $u$ be a nonempty word; then there exists a unique primitive word $z$ and a unique integer $k\ge 1$ such that $u=z^k; z$ is called the primitive root of $u$; we denote by $z=p_A\left(u\right)$. It is convenient to set $p_A\left({\epsilon }_A\right)={\epsilon }_A$, where ${\epsilon }_A$ is the empty word over $A$. By a primitive primal space we mean a space $X$ that is homeomorphic to the subspace $A^{+}$ of $A^{*}$ equipped with the topology $\mathcal{P}\left(p_A\right)$ for some alphabet $A$. Our main result provides a structure theorem of primitive primal spaces.

In this note, we present a sufficient condition for the automorphism group of the direct product of groups of finite exponent to be isomorphic to the direct product of the automorphism groups of the factor groups, and give a complete description for the automorphism group of the direct product of two simple groups. Finally, we prove that the automorphism group of the direct product of any finitely many non-isomorphic simple groups is isomorphic to the direct product of the automorphism group of the groups.

In 1900, mathematics in America was just beginning to mature. In Missouri colleges, as in the rest of the country, there was, except in the largest institutions, only one “Professor of Mathematics” who handled all the mathematics instruction and for whom research was not only not expected but also in some cases discouraged. This article is a survey of the mathematical community in Missouri around that time.