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Sign patterns are matrices with only the sign of each entry specified. The refined inertia of a matrix categorizes the eigenvalues as positive, negative, zero or nonzero imaginary, and the refined inertia of a sign pattern is the set of all refined inertias allowed by real matrices with that sign pattern. The complete sets of allowed refined inertias for all tree sign patterns of orders 2 and 3 (up to equivalence and negation) are determined.
We consider the triples of integer numbers that are solutions of the equation , where is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and describe all generators for each . We also show that if the group has a generator with the third coordinate being a power of 2, such generator is unique up to multiplication by .
A permutation on elements is called a -derangement () if no -element subset is mapped to itself. One can form the -derangement graph on the set of all permutations on elements by connecting two permutations and if is a -derangement. We characterize when such a graph is connected or Eulerian. For an odd prime power, we determine the independence, clique and chromatic numbers of the 2-derangement graph.
The rook polynomial of a board counts the number of ways of placing nonattacking rooks on the board. In this paper, we describe how the properties of the two-dimensional rook polynomials generalize to the rook polynomials of “boards” in three and higher dimensions. We also define families of three-dimensional boards which generalize the two-dimensional triangle boards and the boards representing the problème des rencontres. The rook coefficients of these three-dimensional boards are shown to be related to famous number sequences such as the central factorial numbers, the number of Latin rectangles and the Genocchi numbers.
This paper presents a new way to construct confidence intervals for the unknown parameter in a first-order autoregressive, or AR(1), time series. Typically, one might construct such an interval by centering it around the ordinary least-squares estimator, but this new method instead centers the interval around a linear combination of a weighted least-squares estimator and the sample autocorrelation function at lag one. When the sample size is small and the parameter has magnitude closer to zero than one, this new approach tends to result in a slightly thinner interval with at least as much coverage.
We describe which knots can be obtained as cycles in the canonical book representation of the complete graph , and we conjecture that the canonical book representation of attains the least possible number of knotted cycles for any embedding of . The canonical book representation of contains a Hamiltonian cycle that is a composite knot if and only if . When and are relatively prime, the torus knot is a Hamiltonian cycle in the canonical book representation of . For each knotted Hamiltonian cycle in the canonical book representation of , there are at least Hamiltonian cycles that are ambient isotopic to in the canonical book representation of . Finally, we list the number and type of all nontrivial knots that occur as cycles in the canonical book representation of for .
Two vertices and in a nontrivial connected graph are twins if and have the same neighbors in . If and are adjacent, they are referred to as true twins, while if and are nonadjacent, they are false twins. For a positive integer , let be a vertex coloring where adjacent vertices may be assigned the same color. The coloring induces another vertex coloring defined by for each , where is the closed neighborhood of . Then is called a closed modular -coloring if in for all pairs , of adjacent vertices that are not true twins. The minimum for which has a closed modular -coloring is the closed modular chromatic number of . A rooted tree of order at least 3 is even if every vertex of has an even number of children, while is odd if every vertex of has an odd number of children. It is shown that for each even rooted tree and if is an odd rooted tree having no vertex with exactly one child. Exact values are determined for several classes of odd rooted trees .
Given a map and an initial argument , we can iterate the map to get a finite forward orbit modulo a prime . In particular, for a quadratic map , where is constant, work by Pollard suggests that the forward orbit should have length on the order of . We give a heuristic argument that suggests that the statistical properties of this orbit might be very similar to the birthday problem random variable , for an day year, and offer considerable experimental evidence that the limiting distribution of the orbit lengths, divided by , for as , converges to the limiting distribution of , as .
We study singular discrete third-order boundary value problems with mixed boundary conditions of the form
over a finite discrete interval . We prove the existence of a positive solution by means of the lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems.