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Register a new userDiagonal Form of the Varchenko Matrices
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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.
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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.
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.
We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A = I-E$ is an $n times n$ matrix and the elements of $E$ are bounded in absolute value by $varepsilon le 1/n$, then a lower bound of Ostrowski (1938) is $det(A) ge 1-nvarepsilon$. We show that if, in addition, the diagonal elements of $E$ are zero, then a best-possible lower bound is [det(A) ge (1-(n-1)varepsilon),(1+varepsilon)^{n-1}.] Corresponding upper bounds are respectively [det(A) le (1 + 2varepsilon + nvarepsilon^2)^{n/2}] and [det(A) le (1 + (n-1)varepsilon^2)^{n/2}.] The first upper bound is stronger than Ostrowskis bound (for $varepsilon < 1/n$) $det(A) le (1 - nvarepsilon)^{-1}$. The second upper bound generalises Hadamards inequality, which is the case $varepsilon = 1$. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order $n$ and all positive $varepsilon$ is the existence of a skew-Hadamard matrix of order $n$.
The author presents a computer implementation, calculating the terms of the Saneblidze-Umble diagonals on the permutahedron and the associahedron. The code is analyzed for correctness and presented in the paper, the source code of which simultaneously represents both the paper and the program.
For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices ${1,dots,n}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/log n)$, whereas the best-known lower bound is $Omega((n/log n)^{4/3})$, and Conlon et al. hypothesize that $r_<(M, K_3) = O(n^{2-epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random matchings with interval chromatic number $2$.
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