Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiagonal sum of infinite image partition regular matrices
61
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition regular matrices, an interesting subclass of centrally image partition regular matrices. Also we shall see that many of known centrally image partition regular matrices are C-image partition regular.
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 matri- ces that are known to be infinite image partition regular near zero. In this present work we have produced several new examples of such classes.
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., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries.
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 subset of $M_2(mathbb F_q)$ defined by $ D_i := {xin M_2(mathbb F_q): det(x)=i}.$ Then our results can be stated as follows. First of all, we show that when $E$ and $F$ are subsets of $D_i$ and $D_j$ for some $i, jin mathbb{F}_q^*$, respectively, we have $$det(E+F)=mathbb F_q,$$ whenever $|E||F|ge {15}^2q^4$, and then provide a concrete construction to show that our result is sharp. Next, as an application of the first result, we investigate a distribution of the determinants generated by the sum set $(Ecap D_i) + (Fcap D_j),$ when $E, F$ are subsets of the product type, i.e., $U_1times U_2subseteq mathbb F_q^2times mathbb F_q^2$ under the identification $ M_2(mathbb F_q)=mathbb F_q^2times mathbb F_q^2$. Lastly, as an extended version of the first result, we prove that if $E$ is a set in $D_i$ for $i e 0$ and $k$ is large enough, then we have [det(2kE):=det(underbrace{E + dots + E}_{2k~terms})supseteq mathbb{F}_q^*,] whenever the size of $E$ is close to $q^{frac{3}{2}}$. Moreover, we show that, in general, the threshold $q^{frac{3}{2}}$ is best possible. Our main method is based on the discrete Fourier analysis.
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 $n$ algorithm deciding whether $R(M) leq k$ (whose complexity may depend on $k$). We also give a polynomial in $n$ algorithm computing a number $m$ such that $m/2 leq R(M) leq m$. These results have applications to graph drawings on non-orientable surfaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
