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Register a new userSign patterns that require $mathbb{H}_n$ exist for each $ngeq 4$
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The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary eigenvalues of $A$. For $n geq 3$, the set of refined inertias $mathbb{H}_n={(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an $ntimes n$ sign pattern ${cal A}$ requires $mathbb{H}_n$ if $mathbb{H}_n={text{ri}(B) | B in Q({cal A})}$. Bodine et al. conjectured that no $ntimes n$ irreducible sign pattern that requires $mathbb{H}_n$ exists for $n$ sufficiently large, possibly $nge 8$. However, for each $n geq 4$, we identify three $ntimes n$ irreducible sign patterns that require $mathbb{H}_n$, which resolves this conjecture.
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A {it sign pattern matrix} is a matrix whose entries are from the set ${+,-, 0}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this paper that for any $m times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer $ngeq 9$, there exists a nonnegative integer $m$ such that there exists an $ntimes m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $mtimes n$ sign patterns $A$ with minimum rank $n-1$ is given (which solves an open problem in Brualdi et al. cite{Bru10}), along with a more general description of sign patterns with minimum rank $r$, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of $k$-dimensional subspaces of $mathbb R^n$ are obtained. In particular, it is shown that the maximum number of sign vectors of $2$-dimensional subspaces of $mathbb R^n$ is $4n+1$. Several related open problems are stated along the way.
Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve $C$ in $mathbb{A}^3$ has Markov complexity $m(C)$ two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no $din mathbb{N}$ such that $m(C)leq d$ for all monomial curves $C$ in $mathbb{A}^4$. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in $mathbb{A}^n, ngeq 4$.
A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the identity matrix, the all-one matrix; it is closed with respect to correction{matrix transposition}, Schur-Hadamard (entrywise) multiplication and the Jordan product $A*B=frac 12 (AB+BA)$, where $AB$ is the usual matrix product. The suggested axiomatics (with some natural additional requirements) implies an equivalent reformulation in terms of symmetric binary relations on a vertex set of cardinality $n$. The appearing graph-theoretical structure is called a Jordan scheme of order $n$ and rank $r$. A significant source of Jordan schemes stems from the symmetrization of association schemes. Each such structure is called a non-proper Jordan scheme. The question about the existence of proper Jordan schemes was posed a few times by Peter J. Cameron. In the current text an affirmative answer to this question is given. The first small examples presented here have orders $n=15,24,40$. Infinite classes of proper Jordan schemes of rank 5 and larger are introduced. A prolific construction for schemes of rank 5 and order $n=binom{3^d+1}{2}$, $din {mathbb N}$, is outlined. The text is written in the style of an essay. The long exposition relies on initial computer experiments, a large amount of diagrams, and finally is supported by a number of patterns of general theoretical reasonings. The essay contains also a historical survey and an extensive bibliography.
A emph{sign pattern (matrix)} is a matrix whose entries are from the set ${+, -, 0}$. The emph{minimum rank} (respectively, emph{rational minimum rank}) of a sign pattern matrix $cal A$ is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of $cal A$. A sign pattern $cal A$ is said to be emph{condensed} if $cal A$ has no zero row or column and no two rows or columns are identical or negatives of each other. In this paper, a new direct connection between condensed $m times n $ sign patterns with minimum rank $r$ and $m$ point--$n$ hyperplane configurations in ${mathbb R}^{r-1}$ is established. In particular, condensed sign patterns with minimum rank 3 are closed related to point--line configurations on the plane. It is proved that for any sign pattern $cal A$ with minimum rank $rgeq 3$, if the number of zero entries on each column of $cal A$ is at most $r-1$, then the rational minimum rank of $cal A$ is also $r$. Furthermore, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3.
Providing architectural support is crucial for newly arising applications to achieve high performance and high system efficiency. Currently there is a trend in designing accelerators for special applications, while arguably a debate is sparked whether we should customize architecture for each application. In this study, we introduce what we refer to as Gene-Patterns, which are the base patterns of diverse applications. We present a Recursive Reduce methodology to identify the hotspots, and a HOtspot Trace Suite (HOTS) is provided for the research community. We first extract the hotspot patterns, and then, remove the redundancy to obtain the base patterns. We find that although the number of applications is huge and ever-increasing, the amount of base patterns is relatively small, due to the similarity among the patterns of diverse applications. The similarity stems not only from the algorithms but also from the data structures. We build the Periodic Table of Memory Access Patterns (PT-MAP), where the indifference curves are analogous to the energy levels in physics, and memory performance optimization is essentially an energy level transition. We find that inefficiency results from the mismatch between some of the base patterns and the micro-architecture of modern processors. We have identified the key micro-architecture demands of the base patterns. The Gene-Pattern concept, methodology, and toolkit will facilitate the design of both hardware and software for the matching between architectures and applications.
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