“Drawings since hit tables in lotteries” and a new multivariate geometric distribution

In number lotteries people choose r numbers out of s. Weekly published “drawings since hit tables” indicate how many drawings have taken place since each of the s numbers was last selected as a winning number. Among many lotto players, they enhance the widespread belief that numbers should be “due” if they have not come up for a long time. Under the assumptions of independence of the drawings and equiprobability of all possible combinations, the random s-vectors Y_n, n 1, of entries in a drawings since hit table after n drawings form a Markov chain. The limit distribution of Y_n as n → ∞ is a new multivariate generalization of the geometric distribution. The determination of the distribution of the maximum entry in a drawings since hit table within the first n draws of a lottery seems to be an open problem.     

Landmarks in graphs

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. On a graph, however, there is neither the concept of direction nor that of visibility. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks.

Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. In this paper we present some results about this problem. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(log n), and some properties of graphs with metric dimension two.     

3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting—preprocess a set of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure—preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching— preprocess a set C of n convex fat polygons, so that the k objects intersecting a “not-too-large” query polygon can be reported efficiently, and (iv) bounded-size segment shooting—preprocess a set C as in (iii), so that the first object (if exists) hit by a “not-too-long” oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(λ_s(n)log³n), where s is the maximum number of intersections between the boundaries of the (xy-projections) of any pair of objects, and λ_s(n) is the maximum length of (n, s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(λ_s(n)log²n). The query time in the first problem is O(log⁴n), the query time in the second and third problems is O(log³n + klog²n), and the query time in the fourth problem is O(log³n).

We also present a simple algorithm for computing a depth order for a set as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for , if exists, is a linear order of , such that, if K₁, K₂ and K₁ lies vertically above K₂, then K₁ precedes K₂.) Unlike the algorithm of Agarwal et al. (1995) that might output a false order when a depth order does not exist, the new algorithm is able to determine whether such an order exists, and it is often more efficient in practical situations than the former algorithm.     

Interval orders based on weak orders

One definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “ ”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “ ”, a six-element crown or a six-element fence.     

Eigenvalue interlacing and weight parameters of graphs

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some “weights” (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights “regularize” the graph, and hence allow us to define a kind of regular partition, called “pseudo-regular,” intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lovász, it is shown that the weight Shannon capacity Θ^* of a connected graph Γ, with n vertices and (adjacency matrix) eigenvalues λ₁ > λ₂ … λ_n, satisfies

where Θ is the (standard) Shannon capacity and v is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived.     

Combinatorial aspects of geometric graphs

As a special case of our main result, we show that for all L> 0, each k-nearest neighbor graph in d dimensions excludes K_h as a depth L minor if h = Ω(L^d). More generally, we prove that the overlap graphs defined by Miller, Teng, Thurston and Vavasis (1993) have this combinatorial property. By a construction of Plotkin, Rao and Smith (1994), our result implies that overlap graphs have “good” cut-covers, answering an open question of Kaklamanis, Krizanc and Rao (1993). Consequently, overlap graphs can be emulated on hypercube graphs with a constant factor of slow-down and on butterfly graphs with a factor of O(log^* n) slow-down. Therefore, computations on overlap graphs, such as finite element and finite difference methods on “well-conditioned” meshes and image processing on k-nearest neighbor graphs, can be performed on hypercubic parallel machines with a linear speed-up. Our result, in conjunction with a result of Plotkin, Rao and Smith, also yields a combinatorial proof that overlap graphs have separators of sublinear size. We also show that with high probability, the Delaunay diagram, the relative neighborhood graph, and the k-nearest neighbor graph of a random point set exclude K_h as a depth L minor if h = Ω(L^d/2 log n).     

An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS(v)”. Our main idea is to use transitive resolvable idempotent symmetric quasigroups instead of TKTS. As an application, we can combine the known results on LKTS and LR-designs to obtain the existence of an LKTS(3ⁿm(2·13^n₁+1)(2·13^n_t+1)) for n1, m{1,5,11,17,25,35,43,67,91,123}{2^2r+125^s+1 : r0,s0}, t0 and n_i1 (i=1,…,t).     

On some conjectures on the spectra of τ-matrices

We will consider three conjectures of Schneider and Varga concerning the location of eigenvalues of ω- and τ-matices in the complex plane, and extend the known results to n ≤ 4. We will further show that in the neighborhood of certain cyclic M-matices the conjectures hold for all n.     

The Dempster–Shafer calculus for statisticians

The Dempster–Shafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p, q, r) where p is the probability “for” the assertion, q is the probability “against” the assertion, and r is the probability of “don’t know”. Arguments are presented for the necessity of “don’t know”. Elements of the calculus are sketched, including the extension of a DS model from a margin to a full state space, and DS combination of independent DS uncertainty assessments on the full space. The methodology is applied to inference and prediction from Poisson counts, including an introduction to the use of join-tree model structure to simplify and shorten computation. The relation of DS theory to statistical significance testing is elaborated, introducing along the way the new concept of “dull” null hypothesis.     

Krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function ψ: C → C whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations uinj + 1 = ψ(A)u_j, j = 0, 1,…, where u₀ is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector u_n for some sufficiently large n. We can also apply the simultaneous iteration method to the subspace span{x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_k} with some sufficiently large n, where x^(j+1)_m = ψ(A)x^(j)_m, j = 0, 1,…, m = 1,…, k. In all cases the resulting Ritz pairs are approximations to the eigenpairs of A with the special property. We provide a rather thorough convergence analysis of the approach involving all three methods as n → ∞ for the case in which A is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature.     

Quadratic expansions of spectral functions

A function, F, on the space of n×n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A)=F(UAU^T) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on : those that are invariant under arbitrary swapping of their arguments. In this paper we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessian' of the spectral function. In the case of convex functions we show that a positive definite ‘Hessian' of f implies positive definiteness of the ‘Hessian' of F.     

All trees are 1-embeddable and all except stars are 2-embeddable

A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))=. It is known that all trees T with n (≥2) vertices and T K_1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e};and that it is 2-embeddable if,for every pair e₁, e₂ of edges, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e₁, e₂}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K_1,n−1 are 2-embeddable. In a certain sense, this result is sharp.     

Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces

We present an optimal Θ(n)-time algorithm for the selection of a subset of the vertices of an n-vertex plane graph G so that each of the faces of G is covered by (i.e., incident with) one or more of the selected vertices. At most n/2 vertices are selected, matching the worst-case requirement. Analogous results for edge-covers are developed for two different notions of “coverage”. In particular, our linear-time algorithm selects at most n−2 edges to strongly cover G, at most n/3 diagonals to cover G, and in the case where G has no quadrilateral faces, at most n/3 edges to cover G. All these bounds are optimal in the worst-case. Most of our results flow from the study of a relaxation of the familiar notion of a 2-coloring of a plane graph which we call a face-respecting 2-coloring that permits monochromatic edges as long as there are no monochromatic faces. Our algorithms apply directly to the location of guards, utilities or illumination sources on the vertices or edges of polyhedral terrains, polyhedral surfaces, or planar subdivisions.     

The complexity of irredundant sets parameterized by size

An irredundant set of vertices V′V in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size n−k is fixed-parameter tractable.     

One-dimensional, two-fluid modelling of turbulent premixed flames

The “two-fluid” mathematical model for turbulent combustion is applied to a one-dimensional, premixed, stabilized ducted flame. The flame is assumed to consist of two interspersed fluids (“reactants” and “products”), each characterized by its own properties and interacting through the exchange of mass, heat, and momentum. The distributions of pressure, densities, velocities, and volume fractions across the duct were successfully simulated. From a parametric study on the effects of the empirical constants involved in the interfluid relations, the significant dependence of the system on the parameters that characterize the mass transfer rate and the relative effect of mass transfer to momentum transfer was confirmed. The application of the model to transient states proved its ability to predict system oscillations.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ' of σ which consists of real numbers, we prove that if σ' satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

A new approach to solving three combinatorial enumeration problems on planar graphs

The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising “spin glass” model of statistical mechanics. These alternative algorithms require O(n²) arithmetic operations on an n-vertex planar grapha, and are relatively easy to implement.     

The stability of nonlinear age-dependent population equation

A kind of size-dependent age-structured single species population equation with a random gestation period is discussed. A generalized population size E, called “the newborn equivalent quantity” is defined. The stability of a positive equilibrium is studied when the control function is chosen to be E. It is proved that if E is unfavorable to both survival and reproduction, the unique positive equilibrium is globally asymptotically stable.     

Reporting curve segment intersections using restricted predicates

We investigate how to report all k intersecting pairs among a collection of n x-monotone curve segments in the plane, using only predicates of the following forms: is an endpoint to the left of another? is an endpoint above a segment? do two segments intersect? By studying the intersection problem in an abstract setting that assumes the availability of certain “detection oracles”, we obtain a near-optimal randomized algorithm that runs in expected time. In the bichromatic case (where segments are colored red or blue with no red/red or blue/blue intersections), we find a better algorithm that runs in O((n+k)log_2+k/nn) worst-case time, by modifying a known segment-tree method. Two questions of Boissonnat and Snoeyink are thus answered to within logarithmic factors.