We investigate Lagrangian submanifolds in the nearly Kähler manifold $G \times G$. First we review the construction of a nearly Kähler structure on the Lie group product $G \times G$, where $G$ is a Lie group with a bi-invariant metric. This construction was proposed by K. Sekigawa to E. Abbena and S. Garbiero. An example of this construction is the homogeneous nearly Kähler manifold $\mathbb{S}^{3}\times \mathbb{S}^{3}$, where $G=\mathbb{S}^{3}$ with its standard metric. It is known that this construction on $G \times G$ gives a nearly Kähler structure. To get our main result, we extend the notion of angle functions of a Lagrangian submanifold proposed by B. Dioos, L. Vrancken and X. Wang in the case of $\mathbb{S}^{3}\times \mathbb{S}^{3}$. These angle functions are useful to characterize minimal Lagrangian submanifolds in the NK manifold $G \times G$. We prove our main result: a Lagrangian submanifold is minimal if and only if the sum of its angle functions is constant. We give five examples of Lagrangian submanifolds: three canonical examples and other two examples using an element in the center of $G$.

We present a self-contained proof of the Khinchin inequality for multiple sums, which avoids advanced results from Probability theory. Not only our new proof is more accessible, but also it sheds lights on some properties of the inequality which may yield further generalizations.

The paper contains the complete classification of nil rings with decomposable torsion-free additive group of rank two and description of nil rings with indecomposable torsion-free additive group of rank two. Moreover, it is shown that an indecomposable non-nil torsion-free abelian group $A$ of rank two supports only nilpotent rings exactly if $A$ is the additive group of a nil ring. Some decomposition criteria for torsion-free abelian groups of rank two are also included.

A $P$-space is a topological space whose every $G_{\delta}$-set is open. In this article, basic properties of $P$-spaces are investigated in the absence of the Axiom of Choice. New weaker forms of the Axiom of Choice, all relevant to $P$-spaces or to countable intersections of $G_{\delta}$-sets, are introduced for applications. Special subrings of rings of continuous real functions are applied. New notions of a quasi Baire space and a strongly (quasi) Baire space are introduced. Several independence results are obtained. For instance, it is shown in $\mathbf{ZF}$ that if $G_{\delta}$-modifications of Tychonoff spaces are $P$-spaces, then every denumerable family of denumerable sets has a multiple choice function. In $\mathbf{ZF}$, a zero-dimensional subspace of $\mathbb{R}$ may fail to be strongly zero-dimensional, and countable intersections of $G_{\delta}$-sets of $\mathbb{R}$ may fail to be $G_{\delta}$-sets. New open problems are posed. Partial answers to some of them are given.

We want to bound the symbol length of classes in ${_{2^{m-1}}Br}(F)$ which are represented by tensor products of 5 or 6 cyclic algebras of degree $2^m$. The main ingredients are the chain lemma for quadratic forms, a form of a generalized Clifford invariant and Pfister's and Rost's descriptions of 12- and 14-dimensional forms in $I^3 F$.

In this paper, we establish a sufficient and necessary condition for a function involving the inverse hyperbolic tangent function as a best possible upper bound for the complete elliptic integral of the first kind. Equivalently, we obtain a lower bound involving the arithmetic mean and logarithmic mean for the Gauss arithmetic-geometric mean. This provides a positive answer to a conjecture proposed by Yang, Song and Chu in 2014.

Jan Okniński raised the question whether the primes dividing the size $n$ of a finite indecomposable set-theoretic solution to the Yang-Baxter equation are related to the primes dividing the order of the associated permutation group. With Cedó he proved that both prime sets are equal if $n$ is square-free. We characterize equality and prove that surjective morphisms of solutions admit a canonical factorization into a covering and a morphism given by a brace ideal. The existence of solutions with non-equality of the prime sets is reduced to irretractable solutions. It is proved that non-equality is possible, and a minimal example is constructed..