Cyclic Difference Packing and Covering Arrays

Let n and k be positive integers. Let C_q be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (a_ij) with entries a_ij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from C_q such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of C_q occurs in the difference list at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.     

d- Dimensional Dual Hyperovals in PG(d + n, 2) for d + 1 ≤ n ≤ 3d − 7

Assume that d ≥  4. Then there exists a d -dimensional dual hyperoval in PG(d +  n, 2) for d +  1  ≤  n ≤  3 d −  7.     

On Score Sequences ofk-Hypertournaments

Given two nonnegative integers n and k withn ≥ k > 1, a k -hypertournament on n vertices is a pair (V, A), where V is a set of vertices with | V | = n and A is a set of k -tuples of vertices, called arcs, such that for any k -subset S ofV , A contains exactly one of the k!k -tuples whose entries belong to S. We show that a nondecreasing sequence (r₁, r₂, , r_n) of nonnegative integers is a losing score sequence of a k -hypertournament if and only if for each j(1 ≤ j ≤ n),with equality holding whenj = n. We also show that a nondecreasing sequence (s₁,s₂ , , s_n) of nonnegative integers is a score sequence of somek -hypertournament if and only if for each j(1 ≤ j ≤ n),with equality holding whenj = n. Furthermore, we obtain a necessary and sufficient condition for a score sequence of a strong k -hypertournament. The above results generalize the corresponding theorems on tournaments.     

Greedily Finding a Dense Subgraph

Given an n-vertex graph with nonnegative edge weights and a positive integer k ≤ n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2 + n/2k)² − O(n ^− 1/3) ≤ R ≤ (1/2 + n/2k)² + O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k²) for k < n/3. For k = n/2, for example, these bounds are 9/4 ± O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming.     

On the spectral function of the Poisson–Voronoi cells

Denote by (t)=∑_n1e^−λ_nt, t>0, the spectral function related to the Dirichlet Laplacian for the typical cell of a standard Poisson–Voronoi tessellation in . We show that the expectation E(t), t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E(t), when t→0⁺,+∞. In particular, we prove that the law of the first eigenvalue λ₁ of satisfies the asymptotic relation lnP{λ₁t}−2^dω_dj_(d−2)/2^d·t^−d/2 when t→0⁺, where ω_d and j_(d−2)/2 are respectively the Lebesgue measure of the unit ball in and the first zero of the Bessel function J_(d−2)/2.     

Distributions of Points in <Emphasis Type="Italic">d</Emphasis> Dimensions and Large <Emphasis Type="Italic">k</Emphasis>-Point Simplices

We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1] ^d , such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ _k,d (n), the supremum of this minimum volume over all distributions of n points in [0,1] ^d , we show that c _k,d ⋅(log n)^1/(d−k+1)/n ^k/(d−k+1)≤Δ _k,d (n)≤c _k,d ′/n ^k/d for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ _k,d (n)≤c _k,d ″/n ^{k/d+(k−1)/(2d(d−1))}, where c _k,d ,c _k,d ′,c _k,d ″>0 are constants. A preliminary version of this paper appeared in COCOON ’05.     

Selection in Monotone Matrices and Computingkth Nearest Neighbors

Anm×nmatrix =(a_{i, j}), 1≤i≤mand 1≤j≤n, is called atotally monotonematrix if for alli₁, i₂, j₁, j₂, satisfying 1≤i₁<i₂≤m, 1≤j₁<j₂≤n.[formula]We present an[formula]time algorithm to select thekth smallest item from anm×ntotally monotone matrix for anyk≤mn. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm of the same time complexity for ageneralized row-selection problemin monotone matrices. Given a setS={p₁,…, p_n} ofnpoints in convex position and a vectork={k₁,…, k_n}, we also present anO(n^4/3log^c n) algorithm to compute thek_ith nearest neighbor ofp_ifor everyi≤n; herecis an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points ofSare arbitrary, then thek_ith nearest neighbor ofp_i, for alli≤n, can be computed in timeO(n^7/5 log^c n), which also improves upon the previously best-known result.     

Prophet inequalities for cost of observation stopping problems

Comparisons are made between the expected gain of a prophet (an observer with complete foresight) and the maximal expected gain of a gambler (using only non-anticipating stopping times) observing a sequence of independent, uniformly bounded random variables where a non-negative fixed cost is charged for each observation. Sharp universal bounds are obtained under various restrictions on the cost and the length of the sequence. For example, it is shown for X₁, X₂, … independent, [0, 1]-valued random variables that for all c ≥ 0 and all n ≥ 1 that E(max_{1 ≤ j ≤ n}(X_j − jc)) − sup_{t Tn} E(X_t − tc) ≤ 1/e, where T_n is the collection of all stopping times t which are less than or equal to n almost surely.     

Nonexistence of [n, 5, d]q Codes Attaining the Griesmer Bound for q4 ?2q2 ?2q+1 ≤ d≤ q4?2q2 ?q

We prove that there does not exist a [q⁴+q³−q²−3q−1, 5, q⁴−2q²−2q+1]_q code over the finite field for q≥ 5. Using this, we prove that there does not exist a [g_q(5, d), 5, d]_q code with q⁴ −2q² −2q +1 ≤ d ≤ q⁴ −2q² −q for q≥ 5, where g_q(k,d) denotes the Griesmer bound.MSC 2000: 94B65, 94B05, 51E20, 05B25     

The Fine Structure of the n-Widths of H-Spaces in Lq(−1, 1)

The n-widths of the unit ball A_p of the Hardy space H^p in L_q( −1, 1) are determined asymptotically. It is shown that for 1 ≤ q < p ≤∞ there exist constants k₁ and k₂ such that [formula]≤ d_n(A_p, L_q(−1, 1)),dⁿ(A_p, L_q(−1, 1)), δ_n(A_p, L_q(−1, 1))[formula]where d_n, dⁿ, and δ_n denote the Kolmogorov, Gel′fand and linear n-widths, respectively. This result is an improvement of estimates previously obtained by Burchard and Höllig and by the author.     

Best Approximation with Linear Constraints: A Model Example

We consider best approximation in L_p( ), 1 ≤ p ≤ ∞, by means of entire functions y of exponential type subject to additional constraints Γ_j(y) = 0, j = 1, ..., K. Here Γ_j are (unbounded) linear functionals of the form Γ_j(y) = Dⁿy(s_j) − ∑ a_kD^ky(s_j) where s_j are fixed points.     

Prophet regions and sharp inequalities for pth absolute moments of martingales

Exact comparisons are made relating E|Y₀|^p, E|Y_n−1|^p, and E(max_j≤n−1 |Y_j|^p), valid for all martingales Y₀,…,Y_n−1, for each p ≥ 1. Specifically, for p > 1, the set of ordered triples {(x, y, z) : X = E|Y₀|^p, Y = E |Y_n−1|^p, and Z = E(max_j≤n−1 |Y_j|^p) for some martingale Y₀,…,Y_n−1} is precisely the set {(x, y, z) : 0≤x≤y≤z≤Ψ_n,p(x, y)}, where Ψ_n,p(x, y) = xψ_n,p(y/x) if x > 0, and = a_n−1,py if x = 0; here ψ_n,p is a specific recursively defined function. The result yields families of sharp inequalities, such as E(max_j≤n−1 |Y_j|^p) + ψ_n,p^*(a) E |Y₀|^p ≤ aE |Y_n−1|^p, valid for all martingales Y₀,…,Y_n−1, where ψ_n,p^* is the concave conjugate function of ψ_n,p. Both the finite sequence and infinite sequence cases are developed. Proofs utilize moment theory, induction, conjugate function theory, and functional equation analysis.     

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

Extremal Regular Graphs with Prescribed Odd Girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let x_e denote the smallest even integer no less than x, δ(g) = (−1)^{g − 1}/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20.     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

Uniform Estimates for a Class of Evolution Equations

L^∞ estimates are derived for the oscillatory integral ∫⁺₀^∞e^{−i(xλ + (1/m) tλm)}a(λ) dλ, where 2 ≤ m and (x, t) × ₊. The amplitude a(λ) can be oscillatory, e.g., a(λ) = e^{it (λ)} with (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)^k with 0 ≤ k ≤ (m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.     

A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of |V| and the growth behaviors of the constants {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

Some Properties of Nonstar Steps in Addition Chains and New Cases Where the Scholz Conjecture Is True

Let ℓ(n) be the smallest possible length of addition chains for a positive integer n. Then Scholz conjectured that ℓ(2ⁿ − 1) ≤ n + ℓ(n) − 1, which still remains open. It is known that the Scholz conjecture is true when ν(n) ≤ 4, where ν(n) is the number of 1's in the binary representation of n. In this paper, we give some properties of nonstar steps in addition chains and prove that the Scholz conjecture is true for infinitely many new integers including the case where ν(n) = 5.     

An optimal convex hull algorithm in any fixed dimension

We present a deterministic algorithm for computing the convex hull ofn points inE ^d in optimalO(n logn+n ^⌞d/2⌟ ) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n ^⌜d/2⌝ ) time. This research was supported in part by the National Science Foundation under Grant CCR-9002352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. A preliminary version of this paper has appeared in “An optimal convex hull algorithm and new results on cuttings”,Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science, October 1991, pp. 29–38. The convex hull algorithm given in the present paper, although similar in spirit, is considerably simpler than the one given in the proceedings.