Rates of A-statistical convergence of positive linear operators

In this paper we study the rates of $A$-statistical convergence of sequences of positive linear operators mapping the weighted space $C_{{\rho }_1}$ into the weighted space $B_{{\rho }_2}$.     

Γ-flatness and Bishop–Phelps–Bollobás type theorems for operators

The Bishop–Phelps–Bollobás property deals with simultaneous approximation of an operator T and a vector x at which T nearly attains its norm by an operator $T_0$ and a vector $x_0$, respectively, such that $T_0$ attains its norm at $x_0$. In this note we extend the already known results about the Bishop–Phelps–Bollobás property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: Γ-flat operators and Banach spaces with ${\mathrm{ACK}}_{\rho }$ structure. In particular, we prove a general BPB-type theorem for Γ-flat operators acting to a space with ${\mathrm{ACK}}_{\rho }$ structure and show that uniform algebras and spaces with the property β have ${\mathrm{ACK}}_{\rho }$ structure. We also study the stability of the ${\mathrm{ACK}}_{\rho }$ structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces Y such that the Bishop–Phelps–Bollobás property for Asplund operators is valid for all pairs of the form ($X,Y$).     

Metabelian Q1-groups

A finite group G is called a ${\mathbb{Q}}_1$-group if all of its non-linear irreducible characters are rational valued. In this paper, we will find the general structure of a metabelian ${\mathbb{Q}}_1$-group.     

Embeddings of rank-2 tori in algebraic groups

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type $A_2$, $G_2$ and $F_4$ defined over k, via their rank-2 k-tori. Simple, simply connected groups of type $A_2$ play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of $A_n$ type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type $A_2$, $G_2$ and $F_4$ in terms of the mod-2 Galois cohomological invariants attached with these groups. The results in this paper and our earlier paper ([6]) show that the mod-2 invariants of groups of type $G_2,F_4$ and $A_2$ are controlled by their k-subgroups of type $A_1$ and $A_2$ as well as the unitary k-tori embedded in them.     

Singularities of the representation variety of the braid group on 3 strands

The singularities of ${\mathtt{rep}}_n{B}_3$ are studied, where $B_3$ is the knot group on 3 strands. Specifically, we determine which semisimple representations are smooth points of ${\mathtt{rep}}_n{B}_3$.     

A class of minimal cyclic codes over finite fields

Let ${\mathbb{F}}_q$ be a finite field of odd order $q$ and $n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where $a,a_1,a_2$ are positive integers, $p_1,p_2$ are distinct odd primes and $4p_1{p}_2|q?1$. In this paper, we study the irreducible factorization of $x^n?1$ over ${\mathbb{F}}_q$ and all primitive idempotents in the ring ${\mathbb{F}}_q\left[x\right]∕〈x^n?1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length $n$ over ${\mathbb{F}}_q$.     

On the compositum of integral closures of valuation rings

It is well known that if $K_1,K_2$ are algebraic number fields with coprime discriminants, then the composite ring $A_{K_1}{A}_{K_2}$ is integrally closed and $K_1,K_2$ are linearly disjoint over the field of rationals, $A_{K_i}$ being the ring of algebraic integers of $K_i$. In an attempt to prove the converse of the above result, in this paper we prove that if $K_1,K_2$ are finite separable extensions of a valued field $(K,v)$ of arbitrary rank which are linearly disjoint over $K=K_1\cap K_2$ and if the integral closure $S_i$ of the valuation ring $R_v$ of v in $K_i$ is a free $R_v$-module for $i=1,2$ with $S_1{S}_2$ integrally closed, then the discriminant of either $S_1/R_v$ or of $S_2/R_v$ is the unit ideal. We quickly deduce from this result that for algebraic number fields $K_1,K_2$ linearly disjoint over $K=K_1\cap K_2$ for which $A_{K_1}{A}_{K_2}$ is integrally closed, the relative discriminants of $K_1/K$ and $K_2/K$ must be coprime.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.