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Register a new userNew Classes of Infinite Image Partition Regular Matrices Near Zero
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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.
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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 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.
In cite{dehind1}, the concept of image partition regularity near zero was first instigated. In contrast to the finite case , infinite image partition regular matrices near zero are very fascinating to analyze. In this regard the abstraction of Centrally image partition regular matrices near zero was introduced in cite{biswaspaul}. In this paper we propose the notion of matrices that are C-image partition regular near zero for dense subsemigropus of $((0,infty),+)$.
Every Hadamard matrix $H$ of order $n > 1$ induces a graph with $4n$ vertices, called the Hadamard graph $Gamma(H)$ of $H$. Since $Gamma(H)$ is a distance-regular graph with diameter $4$, it induces a $4$-class association scheme $(Omega, S)$ of order $4n$. In this article we deal with fission schemes of $(Omega, S)$ under certain conditions, and for such a fission scheme we estimate the number of isomorphism classes with the same intersection numbers as the fission scheme.
In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $vin Phi_1cup Phi_2 cup Phi_3 cup Phi_4$; and for $uin {3,5}$ and $vin Phi_1cup Phi_2 cup Phi_3$. Here, $Phi_1={q^2:qequiv 1pmod{4}mbox{ is a prime power}}$, $Phi_2={n^4in mathbb{N}:nequiv 1pmod{2}} cup {9n^4in mathbb{N}:nequiv 1pmod{2}}$, $Phi_3={5}$ and $Phi_4={13,37}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $uin Phi_1cup Phi_2$ and $vin Phi_1cup Phi_2 cup Phi_3$.
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