Extending shelling orders and a hierarchy of functions of unimodal simple polytopes

Let P be a simplicial d-polytope with n facets ((d − 1)-dimensional faces) in R^d. A shelling of P is an ordering of the facets of P such that the intersection of each facet F with the union of all facets that precede it the ordering is a nonempty union of (d − 2)-faces of F. The following open question was raised by Tverberg and is recorded in [4]. Suppose for some k < n, there is an ordering of k of the facets of P so that the intersection of each of these facets with the union of all of the facets that precede it in the ordering is a nonempty union of (d − 2)-faces. Can this initial “segment” be extended to a shelling of all the facets? This question is open even in the case that P is the dual of the d-dimensional hypercube. The question in this case has resurfaced several times since G. Danaraj and V. Klee (1978) in a variety of forms. It is related to the hierarchies of completely unimodal pseudo-Boolean functions studied in P.L. Hammer et al. (1988), the author (1988) and D. Wiedemann (1986). (A pseudo-Boolean function is a function mapping the vertices of the d-dimensional hypercube into the reals). In this paper, the hierarchies are compared and combined. This hierarchy is then extended to general simple polytopes, and the relationship to the above open question is explained.     

On the use of augmenting chains in chain packings

In a graph G = (X, E), we assign to each node υ a positive integer b(υ)≤d_G(υ), where d_G(υ) is the degree of υ in G. Let P be a collection of edge-disjoint chains such that no two chains in P have a common endpoint and such that in the partial graph H = (X, E(P)) formed by the edge set E(P) of P we have d_H(υ)≤b(υ) for each node υ. P is called a chain packing.

We extend the augmenting chain theorem of matchings to chain packings and we find an analogue of matching matroids. We also study chain packings by short chains, i.e., chains of lengths one or two. We show that we may restrict ourselves to packings by short chains when we want to find a packing containing a maximum number of chains. We show that the use of augmenting chains fails in general to produce a new short chain packing from an old one, even for bipartite graphs, but that it does do so for the special case of trees. For the case of trees, we also find a min-max result for packings by short chains.     

Planar and axially symmetric configurations which are circumvented with the maximum critical mach number

The structure of planar and axially symmetric configurations which, by satisfying a number of geometrical constraints, are circumvented in a boundless space or in a cylindrical channel by an ideal (non-viscous and non-thermally conducting) gas with a maximal critical Mach number M* is found. The analysis is carried out using the “rectilinearity property” of a sonic line in “subsonic” flows (SF), the “principle of a maximum” for an SF and “comparison theorems” which are either taken from /1/ or serve as a generalization of the corresponding assertions from /1/. Following /1/, configurations are considered which have a plane or axis of symmetry parallel to the velocity V_∞ of the approach stream, while flows in which (including the boundary) the Mach number M 1 are said to be “subsonic”. As usual, by M* we mean a value of M_∞ such that the inequality M1, which is satisfied in the whole stream when M_∞ M*, is violated when M_∞>M*.

The configurations investigated include closed bodies and the leading (trailing) parts of a semi-infinite plate or a circular cylinder in an unbounded flow and in a channel as well as lattices of symmetric profiles. Both in /1/, where the structure of closed planar and axially symmetric bodies was found, as well as in /2/, where such bodies were constructed numerically, the generatrices of all the configurations investigated contain the end planes or the segments replacing them of the maximum permissible slope (in modulus) and the “free” streamlines with M 1. Now, however, unlike in /1, 2/, segments of the horizontals are added to it in the general case. Furthermore, in the case of flows in channels and lattices, the configurations which have been found can be circumvented with the development of finite domains of advancing sonic flow.     

“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.     

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 extremal problem on potentially Kr,s-graphic sequences

We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ(K_r,s,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(K_r,s,n) is potentially K_r,s-graphic, where K_r,s is a r×s complete bipartite graph, i.e., π has a realization G containing K_r,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially K_r,s-graphic, and then we determine σ(K_r,r,n) for r=3,4.     

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.     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.     

Harmonic trees

A graph G is defined to be harmonic if there is a constant λ (necessarily a natural number) such that, for every vertex v, the sum of the degrees of the neighbors of v equals λd_G (v) where d_G (v) is the degree of v. We show that there is exactly one finite harmonic tree for every λ ε , and we give a recursive construction for all infinite harmonic trees.     

Recognizing polygonal parts from width measurements

Automatic recognition of parts is an important problem in many industrial applications. One model of the problem is: given a finite set of polygonal parts, use a set of “width” measurements taken by a parallel-jaw gripper to determine which part is present. We study the problem of computing efficient strategies (“grasp plans”), with the goal to minimize the number of measurements necessary in the worst case. We show that finding a minimum length grasp plan is -hard, and give a polynomial time approximation algorithm that is simple and produces a solution that is within a log factor from optimal.     

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.     

The realization graph of a degree sequence with majorization gap 1 is Hamiltonian

It is known that the degree sequences of threshold graphs are characterized by the property that they are not majorized strictly by any degree sequence. Consequently every degree sequence d can be transformed into a threshold sequence by repeated operations consisting of subtracting I from a degree and adding 1 to a larger or equal degree. The minimum number of these operations required to transform d into a threshold sequence is called the majorization gap of d. A realization of a degree sequence d of length n is a graph on the vertices 1, …, n, where the degree of vertex i is d_i. The realization graph %plane1D;4A2;(d) of a degree sequence d has as vertices the realizations of d, and two realizations are neighbors in %plane1D;4A2;(d) if one can be obtained from the other by deleting two existing edges [a, b], [c, d] and adding two new edges [a, d]; [b, c] for some distinct vertices a, b, c, d. It is known that %plane1D;4A2;(d) is connected. We show that if d has a majorization gap of 1, then %plane1D;4A2;(d) is Hamiltonian.     

A note on a property of a weak self-similar perfect set

Let S be a compact, weak self-similar perfect set based on a system of weak contractions f_j, j=1,…,m each of which is characterized by a variable contraction coefficient _j(l) as d(f_j(x),f_j(y)) _j(l)d(x,y), d(x,y)<l, l>0. If the relation ∑^m_j=1j(l₀)<1 holds at at least one point l₀, then every nonempty compact metric space is a continuous image of the set S.     

Choosability of planar graphs

A graph G = G(V, E) with lists L(v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L(v). We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with L(v) = k v V(G).

Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.     

On uniform k-partition problems

We study various uniform k-partition problems which consist in partitioning m sets, each of cardinality k, into k sets of cardinality m such that each of these sets contains exactly one element from every original set. The problems differ according to the particular measure of “set uniformity” to be optimized. Most problems are polynomial and corresponding solution algorithms are provided. A few of them are proved to be NP-hard. Examples of applications to scheduling and routing problems are also discussed.     

Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

A number of results in hamiltonian graph theory are of the form “ implies ”, where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. An example of such a theorem is the well-known Chvátal–Erd s Theorem, which states that every graph G with κ is hamiltonian. Here κ is the vertex connectivity of G and is the cardinality of a largest set of independent vertices of G. In another paper Chvátal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than κ independent vertices. In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of a well-known theorem of Jung (Ann. Discrete Math. 3 (1978) 129) for graphs on 16 or more vertices.     

Strong approximation for cross-covariances of linear variables with long-range dependence

Suppose {_k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {c_j}^∞_j=0, {d_j}^∞_j=0 are sequences of real numbers such that Σ_jc²_j < ∞, Σ_jd²_j < ∞. Then, under appropriate moment conditions on {_k, −∞ < k < ∞}, y_k Σ^∞_j=0c_jk-j, z_k Σ^∞_j=0d_jk-j exist almost surely and in ⁴ and the question of Gaussian approximation to S_[t]Σ^[t]_k=1 (y_k z_k − E{y_k z_k}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S_[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on _k for “time” k ≤ 0, weaken the stationarity assumptions on {_k, −∞ < k < ∞}, and improve the summability conditions on {c_j}^∞_j=0, {d_j}^∞_j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let d_j = c_j-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.     

Multiple bifurcation from “simple” eigenvalues

Consider an operator equation G(u,λ) = 0 where λ is a real parameter. Suppose 0 is a “simple” eigenvalue of the Fréchet derivative G_u at (u₀, λ₀). We give a hierarchy of conditions which completely determines the solution structure of the operator equation. It will be shown that multiple bifurcation as well as simple bifurcation can occur. This extends the standard bifurcation theory from a simple eigenvalue in which only one branch bifurcates. We also discuss limit point bifurcations. Applications to semilinear elliptic equations and the homotopy method for the matrix eigenvalue problem are also given.     

On some integrals arising in mathematical physics

This paper presents in the first section the exact evaluation of three single integrals relating to the dielectric behavior of two-dimensional electron plasmas. In the second section we present a procedure for reducing 3d-dimensional integrals of the form: ∫∫∫dqdpdkD(q)(p+k+q)ƒ(p)[1−ƒ(p+q)]ƒ(k)[1−ƒ(k+q)], where the vectors lie in d-dimensional space and ƒ denotes the Fermi function, to tractable form. The second-order exchange integral for a d-dimensional electron gas is taken as an example and is evaluated in closed form as a function of d.