Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

Fast canonization of circular strings

A circular string A = (a₁,…,a_n) is a string that has n equivalent linear representations A_i = a_i,…,a_n,a₁,…,a_i?1 for i = 1,…,n. The a_i's are assumed to be well ordered. We say that A_i < A_j if the word a_i…a_na₁…a_i?1 precedes the word a_j…a_na₁…a_j?1 in the lexicographic order, A_i ? A_j if either A_i < A_j or A_i = A_j. A_i0 is a minimal representation of A if A_i0 ? A_i for all 1 ≤ i ≤ n. The index i₀ is called a minimal starting point (m.s.p.). In this paper we discuss the problem of finding m.s.p. of a given circular string. Our algorithm finds, in fact, all the m.s.p.'s of a given circular string A of length n by using at most $\text{n + ?}\text{d}\text{2}\text{?}$ comparisons. The number d denotes the difference between two successive m.s.p.'s (see Lemma 1.1) and is equal to n if A has a unique m.s.p. Our algorithm improves the result of 3n comparisons given by K. S. Booth. Only constant auxiliary storage is used.     

Decomposable bilinear numerical radii

Let A be an n-square normal matrix over $\text{C}$, and Q_{m, n} be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Q_{m, n} denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i₁,…,i_k, i_k+1,…,i_m, such that α(i_j)=β(i_j), j=1,…,k, and {α(i_k+1),…,α(i_m)};∩{β(i_k+1),…,β(i_m)}=ø. Let

be the group of n-square unitary matrices. Define the nonnegative number

$\text{?}{}_{\mathrm{k}}\text{(A)=}\text{max}\text{U∈}\textcolor{red}{}\text{|}\text{det}\text{(U}{}^1\text{AU) [α|β]|}$

, where |α∩β|=k. Theorem 1 establishes a bound for ?_k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that

$\text{?}{}_{\mathrm{m}}\text{(A)??}{}_{\mathrm{m?1}}\text{(A)????}{}_0\text{(A)}$

.     

A Fisher type inequality

In this paper we study subsets of a finite set that intersect each other in at most one element. Each subset intersects most of the other subsets in exactly one element. The following theorem is one of our main conclusions. Let S₁,… S_m be m subsets of an n-set S with |S₁| ? 2 (l = 1, …,m) and |S_i ∩ S_j| ? 1 (i ≠ j; i, j = 1, …, m). Suppose further that for some fixed positive integer c each S_i has non-empty intersection with at least m ? c of the remaining subsets. Then there is a least positive integer M(c) depending only on c such that either m ? n or m ? M(c).     

On orthogonal matrices with constant diagonal

In connection with the problem of finding the best projections of k-dimensional spaces embedded in n-dimensional spaces Hermann König asked: Given m∈$\text{R}$ and n∈$\text{N}$, are there n×n matrices C=(c_ij), i, j=1,…,n, such that c_ii=m for all i, |c_ij|=1 for i≠j, and C²=(m²+n?1)I_n? König was especially interested in symmetric C, and we find some families of matrices satisfying this condition. We also find some families of matrices satisfying the less restrictive condition CC^T=(m²+n?1)I_n.     

Extensions of band matrices with band inverses

Let R_ij be a given set of μ_i× μ_j matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[F_ij], i,j=1,…, n, such that F_ij=R_ij for |i?j|?m, F admits either a left or right block triangular factorization, and (F^?1)_ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given R_ij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

Complementing and exactly covering sequences

The following is proved (in a slightly more general setting): Let α₁, …, α_m be positive real, γ₁, …, γ_m real, and suppose that the system [nα_i + γ_i], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then $\text{α}{}_{\mathrm{i}}\text{α}{}_{\mathrm{j}}$ is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all α_i but one are integers; (iii) m ? 5, two of the α_i are integers, at least one of them prime; (iv) m ? 5 and α_n ? 2ⁿ for n = 1, 2, …, m ? 4.For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m ? 1 sequences.     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.     

Crisscross Latin squares

Let A be a Latin square of order n. Then the jth right diagonal of A is the set of n cells of A: {(i,j+i):i=0,1…,n?1(modn); and the jth left diagonal of A is the set {(i,j?i):i=0,1…,n?1(modn); A diagonal is said to be complete if every element appears in it exactly once. For n = 2m even, we introduce the concept of a crisscross Latin square which is something in between a diagonal Latin square and a Knut Vik design. A crisscross Latin square is a Latin square such that all the jth right diagonals for even j and all the jth left diagonals for odd j are complete. We show that a necessary and sufficient condition for the existence of a crisscross Latin square of order 2m is that m is even.     

A stochastic permutation-flowshop scheduling problem minimizing in distribution the schedule length

Consider n jobs (J₁,J₂,…,J_n) and m machines (M₁,M₂…,M_m). Upon completion of processing of J_i, 1 ? i ? n, on M_j 1 ? j ? m ? 1, it departs with probability p_i or moves to M_j+1 with the complementary probability, 1?p_i. A job completing service on M_m departs. The processing time of j_i on M_j possesses a distribution function F_j. It is proved that sequencing the jobs in a nondecreasing order of p_i minimizes in distribution the schedule length.     

On monotone functions of tree structures

Let T be a rooted tree structure with n nodes a₁,…,a_n. A function f: {a₁,…,a_n} into {1 < ? < k} is called monotone if whenever a_i is a son of a_j, then f(a_i) ≥ f(a_j). The average number of monotone bijections is determined for several classes of tree structures. If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ?^?nn^{?$\text{3}\text{2}$} (n → ∞) are obtained for several classes of tree structures.     

Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Subsets of a finite set that intersect each other in at most one element

In this paper we study de Bruijn-Erdös type theorems that deal with the foundations of finite geometries. The following theorem is one of our main conclusions. Let S₁,…, S_n be n subsets of an n-set S. Suppose that |S_i| ? 3 (i = 1,…,n) and that |S_i ∩ S_j| ? 1 (i ≠ j;i,j = 1,…,n). Suppose further that each S_i has nonempty intersection with at least n ? 2 of the other subsets. Then the subsets S₁,…,S_n of S are one of the following configurations. (1) They are a finite projective plane. (2) They are a symmetric group divisible design and each subset has nonempty intersection with exactly n ? 2 of the other subsets. (3) We have n = 9 or n = 10 and in each case there exists a unique configuration that does not satisfy (1) or (2).     

A variable projection method for solving separable nonlinear least squares problems

Consider the separable nonlinear least squares problem of findinga εR ⁿ and α εR ^k which, for given data (y _i ,t _i ),i=1,2,...m, and functions ? _j (α,t),j=1,2,...,n(m>n), minimize the functional $$r(a,\alpha ) = \left\| {y - \Phi (\alpha )a} \right\|_2^2$$ where θ(α) _ij =? _j (α,t _i ). Golub and Pereyra have shown that this problem can be reduced to a nonlinear least squares problem involvingα only, and a linear least squares problem involvinga only. In this paper we propose a new method for determining the optimalα which computationally has proved more efficient than the Golub-Pereyra scheme.     

Langford sequences: perfect and hooked

A sequence {d, d+1,…, d+m?1} of m consecutive positive integers is said to be perfect if the integers {1, 2,…, 2m} can be arranged in disjoint pairs {(a_i, b_i): 1?i?m} so that {b_i?a_i: 1?i?m}={d,d+1,…,d+m?1}. A sequence is hooked if the set {1, 2,…, 2m?1 2m+1} can be arranged in pairs to satisfy the same condition. Well known necessary conditions for perfect sequences are herein shown to be sufficient. Similar necessary and sufficient conditions for hooked sequences are given.     

The maximum distance problem and band sequences

We solve the following problem. For 1 ⩽ j, k ⩽ n and |j − k| ⩽ m, let a_jk be a given complex number with a_kj = ā_jk. We wish to find linearly independent vectors x₁,…,x_n such that 〈x_k, x_j〉 = a_jk for |j − k| ⩽ m and such that the distance from x_k to the linear span of x₁,…,x_k−1 is maximal for 2 ⩽ k ⩽ n. We construct and characterize all such sequences of vectors. Our solution leads naturally to the class of m-band sequences of vectors in an inner product space. We study these sequences and characterize their equivalence classes under unitary transformations.