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A finite or infinite matrix $A$ is image partition regular provided that whenever $mathbb N$ is finitely colored, there must be some $vec{x}$ with entries from $mathbb N$ such that all entries of $Avec{x}$ are in some color class. In [6], it was proved that the diagonal sum of a finite and an infinite image partition regular matrix is also image partition regular. It was also shown there that centrally image partition regular matrices are closed under diagonal sum. Using Theorem 3.3 of [2], one can conclude that diagonal sum of two infinite image partition regular matrices may not be image partition regular. In this paper we shall study the image partition regularity of diagonal sum of some infinite image partition regular matrices. In many cases it will produce more infinite image partition regular matrices.
A finite or infinite matrix $A$ is image partition regular provided that whenever $mathbb{N}$ is finitely colored, there must be some $overset{rightarrow}{x}$ with entries from $mathbb{N}$ such that all entries of $A overset{rightarrow}{x}$ are in th
Image partition regular matrices near zero generalizes many classical results of Ram- sey Theory. There are several characterizations of finite image partition regular matrices near zero. Contrast to the finite cases there are only few classes of mat
Varchenko defined the Varchenko matrix associated to any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e.
Let $mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $det S$ for certain types of subsets $S$ in the ring $M_2(mathbb F_q)$ of $2times 2$ matrices with entries in $mathbb F_q$. For $iin mathbb{F}_q$, let $D_i$ be the
For an $n times n$ matrix $M$ with entries in $mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a polynomial in