There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened — these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category ∆ is replaced by a certain ‘fat’ delta of ‘coloured ordinals’, where the d...

Rota conjectured that, given n disjoint bases of a rank-n matroid M , there are n disjoint transversals of these bases that are all bases of M . We prove a stronger statement for the class of paving matroids.

In 1998, Reed conjectured that for any graph G, χ(G) ≤ ⌈ 2 ⌉, where χ(G), ω(G), and ∆(G) respectively denote the chromatic number, the clique number and the maximum degree of G. In this paper, we study this conjecture for some expansions of graphs, that is graphs obtained with the well known operation composition of graphs. We prove that Reed’s Conjecture holds for expansions of bipartite graph...

In this article the elementary mathematical methods are used to prove Beal’s Conjecture, Fermat’s Collatz Conjecture and Goldbach Conjecture.

in this paper, we investigate the zassenhaus conjecture for $psl(4,3)$ and $psl(5,2)$. consequently, we prove that the prime graph question is true for both groups.

Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...

In this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. We obtain one results, which can give a negative answer to an uniqueness question relate to the Bruck conjecture dealt by Liu and Yang. Meanwhile, we also establish a difference analogue of the Bruck conjecture for entire functions of order less than 2, ...

Let $\pi$ be a proper subset of the set all primes. Denote by $r$ smallest prime which does not belong to and $m = r$ if $r 2$ or $3$ r-1$ \geqslant 5$. We study following conjecture: conjugacy class $D$ finite group $G$ is contained in $\pi$-radical $\mathrm{O}_\pi(G)$ only every $m$ elements generate $\pi$-subgroup. confirm this conjecture for each whose nonabelian composition factors are iso...

ardy and littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. they believed that the number $mathcal{r}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{r}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...