On extremal sort sequences

A sort sequence S_n is a sequence of all unordered pairs of indices inI _n = {1, 2, ..., n}. With a sort sequenceS_n = (s₁ ,s₂ ,...,s_{( 2ⁿ )} )S_n = (s_1 ,s_2 ,...,s_{\left( {_2^n } \right)} ), one can associate a predictive sorting algorithm A(S_n). An execution of the algorithm performs pairwise comparisons of elements in the input setX in the order defined by the sort sequence S_n except that the comparisons whose outcomes can be inferred from the results of the preceding comparisons are not performed. A sort sequence is said to be extremal if it maximizes a given objective function. First we consider the extremal sort sequences with respect to the objective function ω(S_n) — the expected number of active predictions inS _n. We study ω-extremal sort sequences in terms of their prediction vectors. Then we consider the objective function Ω(S_n) — the minimum number of active predictions in S_n over all input orderings.     

A note on a special case of the Frobenius problem

For a set of positive and relative prime integers A = {a ₁…,a _k }, let Γ(A) denote the set of integers of the form a ₁ x ₁+…+a _k x _k with each x _j ≥ 0. Let g(A) (respectively, n(A) and s(A)) denote the largest integer (respectively, the number of integers and sum of integers) not in Γ(A). Let S*(A) denote the set of all positive integers n not in Γ(A) such that n + Γ(A) \ {0} ? Γ((A)\{0}. We determine g(A), n(A), s(A), and S*(A) when A = {a, b, c} with a | (b + c).     

On a problem of rota

Let S(n, k) denote Stirling numbers of the second kind; for each n, let K_n be such that S(n, K_n) ? S(n, k) for all k. Also, let P(n) denote the lattice of partitions of an n-element set. Say that a collection of partitions from P(n) is incomparable if no two are related by refinement. Rota has asked if for all n, the largest possible incomparable collection in P(n) contains S(n, K_n) partitions. In this paper, we construct for all n sufficiently large an incomparable collection in P(n) containing strictly more than S(n, K_n) partitions. We also estimate how large n must be for this construction to work.     

Bounds on the location of the maximum Stirling numbers of the second kind

Let S(n,k) denote the Stirling number of the second kind, and let K_n be such that

S(n,K_n−1)<S(n,K_n)≥S(n,K_n+1).     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

(m)-covering of a triangulation

Let G be a graph triangularly imbedded into a surface S, G_(m) is the graph constructed from G by replacing each vertex x by m vertices (xx,0), (x, 1), ..., (x, m − 1) and joining two vertices (x, i) and (y, j) by an edge if and only if x and y are joined in G. The main result is that the construction of G_(m) is possible whenever n is an odd prime and a well separating cycle (mod m) can be determined.     

An analysis of (h,k, 1)-Shellsort

One classical sorting algorithm, whose performance in many cases remains unanalyzed, is Shellsort. Let h be a t-component vector of positive integers. An h-Shellsort will sort any given n elements in t passes, by means of comparisons and exchanges of elements. Let S>_j(h; n) denote the average number of element exchanges in the jth pass, assuming that all the n! initial orderings are equally likely. In this paper we derive asymptotic formulas of S_j(h; n) for any fixed h = (h, k, l), making use of a new combinatorial interpretation of S₃. For the special case h = (3, 2, 1), the analysis is further sharpened to yield exact expressions.     

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x₁,…,x_n} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [x_i,x_j] of x_i and x_j. The set S is said to be gcd-closed if for any x_i,x_j∈S,(x_i,x_j)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r?3, there exists a gcd-closed set S satisfying max_x∈S{ω(x)}=r, such that the LCM matrix [S] is singular.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

More on the generalized macaulay theorem — II

Let k₁ ? k₂? ? ? k_n be given positive integers and let S denote the set of vectors x = (x₁, x₂, … ,x_n) with integer components satisfying 0 ? x₁ ? k_ni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l.     

On the set of natural numbers which only yield orders of Abelian groups

Let A denote the set of all natural numbers n such that every group of order n is Abelian. Let C denote the set of all natural numbers n such that every group of order n is cyclic. We prove that Σ_{n ≤ x,n?A?C}1 has roughly the order of magnitude x(log log x)^?1.     

Design and analysis of predictive sorting algorithms

The focus of this research is the class of sequential algorithms, called predictive sorting algorithms, for sorting a given set ofn elements using pairwise comparisons. The order in which these pairwise comparisons are made is defined by a fixed sequence of all unordered pairs of distinct integers {1, 2, ...,n} called a sort sequence. A predictive sorting algorithm associated with a sort sequence specifies pairwise comparisons of elements in the input set in the order defined by the sort sequence, except that the comparisons whose outcomes can be inferred from the preceding pairs of comparisons are not performed. In this paper predictive sorting algorithms are obtained, based on known sorting algorithms, and are shown to be required on the averageO(n logn) comparisons.     

On P.I. ring and standard identities

An example is given of a ringR (with 1) satisfying the standard identityS ₆[x ₁, ...,x ₆] butM ₂(R), the 2 × 2 matrix ring overR, does not satisfyS ₁₂[x ₁, ...,x ₁₂]. This is in contrast to the caseR=M _n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS _2n [x ₁, ...,x _2n] andM ₂(R) satisfiesS _4n [x ₁, ...,x _n]. Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.     

Inequalities concerning numbers of subsets of a finite set

Let S(n) denote the set of subsets of an n-element set. For an element x of S(n), let Γx and Px denote, respectively, all (|x| ?1)-element subsets of x and all (|x| + 1)-element supersets of x in S(n). Several inequalities involving Γ and P are given. As an application, an algorithm for finding an x-element antichain $\text{X}{}^1$ in S(n) satisfying $\text{| YX}{}^1\text{| ? | YX |}$ for all x-element antichains X in S(n) is developed, where YX is the set of all elements of S(n) contained in an element of X. This extends a result of Kleitman [9] who solved the problem in case x is a binomial coefficient.     

The existence of non-degenerate functions on a compact differentiablem-manifoldM

Summary Let M be a compact differentiable m-manifold of class C^m in E_n, n=2m+1. Let x=(x₁, ..., x_n) represent a point in E_n. The union of the direction c on the direction sphere S_n−1 in E_n such that the scalar product c · x defines a non-degenerate fonction on M is an open subset of S_n−1 whose complement θ has a Lebesgue measure zero on S_n−1. When M is non-compact θ can be everywhere dense on S_n−1, but still has Lebesgue measure zero. To Giovanni Sansone on his 70^th birth day.     

A characteristic property of spheres

Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k₁ ≥ ... ≥ k_n denote its principle curvatures. Let φ(ξ₁, ..., ξ_n) be such that . Then if φ(k₁, ..., k_n)=const on S and S is subject to some additional general conditions (those(II ₀) or(II) n^o 1), S is a sphere. To Enrico Bompiani on his scientific Jubilee     

On the complexity of testing membership in the core of min-cost spanning tree games

LetN = {1,...,n} be a finite set of players andK _N the complete graph on the node setN∪{0}. Assume that the edges ofK _N have nonnegative weights and associate with each coalitionS∪N of players as costc(S) the weight of a minimal spanning tree on the node setS∪{0}. Using transformation from EXACT COVER BY 3-SETS, we exhibit the following problem to beNP-complete. Given the vectorxε?^itN withx(N) =c(N). decide whether there exists a coalitionS such thatx(S) >c(S).     

Linear maps on matrices: the invariance of symmetric polynomials

Let L be a linear map on the space M_n of all n by n complex matrices. Let h(x₁,…,x_n) be a symmetric polynomial. If X is a matrix in M_n with eigenvalues λ₁,…,λ_n, denote h(λ₁,…,λ_n) by h(X). For a large class of polynomials h, we determine the structure of the linear maps L for which h(X)=h(L(X)), for all X in M_n.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Многомерные вариаци и множеств и их контин генции

LetE be a compact set inR ⁿ (n≧2), and denote byV ₀(E) the number of the components ofE. Letp=1,2, ...,n?1;k=0,1, ...,p, and $$V_k (E;n,p) = \int\limits_{\Omega _k^n } {V_0 (E \cap \tau )^{{{(n - p)} \mathord{\left/ {\vphantom {{(n - p)} {(n - k)}}} \right. \kern-\nulldelimiterspace} {(n - k)}}} d\mu _\tau ,}$$ whereΩ _k ⁿ is the set of all (n-k)-dimensional hyperplanesτ?R ⁿ anddμ _τ is the Haar measure on the spaceΩ _k ⁿ ; furthermore, let $$V_n (E;n,n - 1) = mes_n E.$$ . Theorem. Let E?Rⁿ, p=1, 2 ..., n?1, V_p+1(E;n,p)=0, and V_k(E; n, p)<∞ for k= =0,1, ..., p. Then the contingency of the set E at a point x∈E coincides with a certain p-dimensional hyperplane for almost all points x∈E in the sense of Hausdorff p-measure.