The maximum saving partition problem

The input to the MAXIMUM SAVING PARTITION PROBLEM consists of a set V={1,…,n}, weights w_i, a function f, and a family S of feasible subsets of V. The output is a partition (S₁,…,S_l) such that S_i∈S, and is maximized. We present a general -approximation algorithm, and improved algorithms for special cases of the function f.     

Interval partitions and Stanley depth

In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1,2,…,n} into intervals. Specifically, given a positive integer n, they asked whether there exists a partition P(n) of the non-empty subsets of {1,2,…,n} into intervals, so that |B|?n/2 for each interval [A,B] in P(n). We answer this question in the affirmative by first embedding it in a stronger result. We then provide two alternative proofs of this second result. The two proofs use entirely different methods and yield non-isomorphic partitions. As a consequence, we establish that the Stanley depth of the ideal (x₁,…,xn)⊆K[x₁,…,xn] (K a field) is ⌈n/2⌉.     

Path spectra for trees

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive integers is spectral if it is the path spectrum of a tree. We characterize the spectral sets containing at most two odd integers (and arbitrarily many even ones) and obtain several necessary conditions for a set to be spectral. We show that for each even integer s≥2 at least 1/4 of all subsets of the set {2,3,…,s} are spectral and conjecture that all the subsets with at least 3s/4 integers are spectral.     

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set {1,2,…,n}, the local direction of each edge (ij) is from i to j if i<j. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ=e₁¹e₂²… of a tree on the vertex set {1,2,…,n} is a partition of n−1. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.     

Sets characterized by missing sums and differences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We show that the probability that a uniform random subset of {0,1,…,n} is an MSTD set approaches some limit ρ>4.28×¹⁰−4. This improves the previous result of Martin and O?Bryant that there is a lower limit of at least 2×¹⁰−7. Monte Carlo experiments suggest that ρ≈4.5×¹⁰−4. We present a deterministic algorithm that can compute ρ up to arbitrary precision. We also describe the structure of a random MSTD set S⊆{0,1,…,n}. We formalize the intuition that fringe elements are most significant, while middle elements are nearly unrestricted. For instance, the probability that any “middle” element is in S approaches 1/2 as n→∞, confirming a conjecture of Miller, Orosz, and Scheinerman. In general, our results work for any specification on the number of missing sums and the number of missing differences of S, with MSTD sets being a special case.     

Walking in circles

Consider the unit circle S¹ with distance function d measured along the circle. We show that for every selection of 2n points x₁,…,x_n,y₁,…,y_n∈S¹ there exists i∈{1,…,n} such that . We also discuss a game theoretic interpretation of this result.     

A random intersection digraph: Indegree and outdegree distributions

Let S(1),…,S(n),T(1),…,T(n) be random subsets of the set [m]={1,…,m}. We consider the random digraph D on the vertex set [n] defined as follows: the arc i→j is present in D whenever S(i)∩T(j)≠0?. Assuming that the pairs of sets (S(i),T(i)), 1≤i≤n, are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n,m→∞.     

Nevanlinna-Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z₁,…,z_k in the open unit disk D and a set of complex numbers {f_i,0,f_i,1,…,f_i,ni−1} assigned to each z_i, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at z_i has f_i,0,f_i,1,…,f_i,ni−1 as its first n_i coefficients.     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

Constructing MSTD sets using bidirectional ballot sequences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We construct a new dense family of MSTD subsets of {0,1,2,…,n−1}. Our construction gives Θ(n²/n) MSTD sets, improving the previous best construction with Ω(n²/n⁴) MSTD sets by Miller, Orosz, and Scheinerman.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.     

On the Bateman-Horn Conjecture

Let r be a positive integer and f₁,…,f_r be distinct polynomials in Z[X]. If f₁(n),…,f_r(n) are all prime for infinitely many n, then it is necessary that the polynomials f_i are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f₁(n)···f_r(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363-367) proposed a conjectural asymptotic estimate on the number of positive integers n?x such that f₁(n),…,f_r(n) are all primes. In the present paper, we apply the Hardy-Littlewood circle method to study the Bateman-Horn conjecture when r?2. We consider the Bateman-Horn conjecture for the polynomials in any partition {f₁,…,f_s}, {f_s+1,…,f_r} with a linear change of variables. Our main result is as follows: If the Bateman-Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman-Horn conjecture for f₁,…,f_r is equivalent to a plausible error term conjecture for the minor arcs in the circle method.     

The base sets of quasi-primitive zero-symmetric sign pattern matrices with zero trace

Cheng and Liu [Bo Cheng, Bolian Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731] showed that the base set of quasi-primitive zero-symmetric (generalized) sign pattern matrices is {1,2,…,2n}. The matrices with zero trace play a prominent role in matrix theory. In this paper, we investigate the bases of quasi-primitive zero-symmetric (generalized) sign pattern matrices with zero trace and prove that the base set of such matrices is {2,3,…,2n-1}.     

Multiple cross-intersecting families of signed sets

A k-signed r-set on[n]={1,…,n} is an ordered pair (A,f), where A is an r-subset of [n] and f is a function from A to [k]. Families A₁,…,Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.     

Frame cellular automata: Configurations, generating sets and related matroids

We introduce a frame cellular automaton as a broad generalization of an earlier study on quasigroup-defined cellular automata. It consists of a triple (F,R,E_F) where, for a given finite set X of cells, the frame F is a family of subsets of X (called elementary frames, denoted S_i, i=1,…,n), which is a cover of X. A matching configuration is a map which attributes to each cell a state in a finite set G under restriction of a set of local rules R={R_i∣i=1,…n}, where R_i holds in the elementary frame S_i and is determined by an (|S_i|-1)-ary quasigroup over G. The frame associated map E_F models how a matching configuration can be grown iteratively from a certain initial cell-set. General properties of frames and related matroids are investigated. A generating set S⊂X is a set of cells such that there is a bijection between the collection of matching configurations and G^S. It is shown that for certain frames, the algebraically defined generating sets are bases of a related geometric-combinatorially defined matroid.     

On a problem of Katona on minimal separating systems

Let S be an n-element set. In this paper, we determine the smallest number f(n) for which there exists a family of subsets of S{A₁,A₂,…,A_f(n)} with the following property: Given any two elements x, y ∈ S (x ≠ y), there exist k, l such that A_k ∩ A_l= ?, and x ∈ A_k, y ∈ A_l. In particular it is shown that f(n)= 3 log₃n when n is a power of 3.     

Kernel estimation of density level sets

Let f be a multivariate density and f_n be a kernel estimate of f drawn from the n-sample X₁,…,X_n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f?t} and its plug-in estimator {f_n?t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.     

The chromatic number of 5-valent circulants

A circulant C(n;S) with connection set S={a₁,a₂,…,a_m} is the graph with vertex set Z_n, the cyclic group of order n, and edge set E={{i,j}:|i−j|∈S}. The chromatic number of connected circulants of degree at most four has been previously determined completely by Heuberger [C. Heuberger, On planarity and colorability of circulant graphs, Discrete Math. 268 (2003) 153-169]. In this paper, we determine completely the chromatic number of connected circulants C(n;a,b,n/2) of degree 5. The methods used are essentially extensions of Heuberger’s method but the formulae developed are much more complex.     

Super edge-magic strength of fire crackers, banana trees and unicyclic graphs

A graph G(V,E) is called super edge-magic if there exists a bijection f from V∪E to {1,2,3,…,|V|+|E|} such that f(u)+f(v)+f(uv)=c(f) is constant for any uv∈E and f(V)={1,2,3,…,|V|}. Such a bijection is called a super edge-magic labeling of G. The super edge-magic strength of a graph G is defined as the minimum of all c(f) where the minimum runs over all super edge-magic labelings of G and is denoted by sm(G). The super edge-magic strength of some families of graphs are obtained in this paper.     

New classes of spectrally arbitrary ray patterns

An n×n ray pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the ray pattern class of S such that its characteristic polynomial is f(λ). In this article we give new classes of spectrally arbitrary ray pattern matrices.