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).     

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).     

Reversible-shop scheduling

Consider n jobs (J₁,…,J_n) and m machines (M₁,…,M_m). The route by which a job passes through the machines is either M₁ → M₂ → … → M_m or M_m → M_m−1 → … M₁, i.e. reversible-shop. It is proved that for three machines, the problem of minimizing the schedule length is NP-hard. Efficient algorithms are developed for the following special cases: three machines of which one or more are dominated; arbitrary number of machines where all operations have equal processing times; three machines, route-dependent reversible-shop where the second machine dominates the other two.     

Job shop scheduling with unit time operations under resource constraints and release dates

We consider the problem of job shop scheduling with m machines and n jobs J_i, each consisting of l_i unit time operations. There are s distinct resources R_h and a quantity q_h available of each one. The execution of the j-th operation of J_i requires the presence of u_ijh units of R_h, 1 ≤i≤n, 1 ≤j≤l_i, and 1 ≤h≤s. In addition, each J_i has a release date r_i, that is J_i cannot start before time r_i. We describe algorithms for finding schedules having minimum length or sum of completion times of the jobs. Let l=max{l_i} and u=|{u_ijh}|. If m, u and l are fixed, then both algorithms terminate within polynomial time.     

On the Wick theorem for mixtures of centered Gaussian distributions

We consider centered conditionally Gaussian d-dimensional vectors X with random covariance matrix Ξ having an arbitrary probability distribution law on the set of nonnegative definite symmetric d × d matrices M _d ⁺. The paper deals with the evaluation problem of mean values \( E\left[ {\prod\nolimits_{i = 1}^{2n} {\left( {{c_i},X} \right)} } \right] \) for c _i ∈ ? ^d , i = 1, …, 2n, extending the Wick theorem for a wide class of non-Gaussian distributions. We discuss in more detail the cases where the probability law ?(Ξ) is infinitely divisible, the Wishart distribution, or the inverse Wishart distribution. An example with Ξ \( = \sum\nolimits_{j = 1}^m {{Z_j}{\sum_j}} \), where random variables Z _j , j = 1, …, m, are nonnegative, and Σ _j ∈ M _d ⁺, j = 1, …, m, are fixed, includes recent results from Vignat and Bhatnagar, 2008.     

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.     

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.     

On discount residue classes

It is shown that, whenever m₁, m₂,…, m_n are natural numbers such that the pairwise greatest common divisors, d_ij=(m_i, m_j), i≠j are distinct and different from 1, then there exist integers a₁, a₂,…,a_n such that the solution sets of the congruences x≡_i (modm_i), i= 1,2,…,n are disjoint.     

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.     

Efficient scheduling algorithms for a single batch processing machine

Scheduling problems of a batch processing machine are solved by efficient algorithms. On a batch processing machine, multiple jobs can be processed simultaneously in a batch form. We call the number of jobs in the batch the batch size, which can be any integer between 1 and k, a predetermined integer of the maximum batch size. The process time of a batch is constant and independent of the batch size. Preemption is not allowed. Given n jobs with release times r₁ and due dates d_i, i = 1…., n, we give efficient algorithms to find a feasible schedule, if any, which minimizes the final completion time under the assumption such that for r_i > r_j, d_i ⩾ d_j. Some industrial applications are discussed.     

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.     

Δ-Monotone arrangements of real numbers

Given a set of M × N real numbers, can these always be labeled as x_i,j; i = 1,…, M; j = 1,…, N; such that x_i+1,j+1 ? x_i+1,j ? x_i,j+1 + x_ij ≥ 0, for every (i, j) where 1 ≤ i ≤ M ? 1, 1 ≤ j ≤ N ? 1? For M = N = 3, or smaller values of M, N it is shown that there is a “uniform” rule. However, for max(M, N) > 3 and min(M, N) ≥ 3, it is proved that no uniform rule can be given. For M = 3, N = 4 a way of labeling is demonstrated. For general M, N the problem is still open although, for a special case where all the numbers are 0's and 1's, a solution is given.     

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).     

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

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

.     

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′}.     

An optimal online algorithm for two-machine open shop preemptive scheduling with bounded processing times

This paper deals with a two-machine open shop scheduling problem. The objective is to minimize the makespan. Jobs arrive over time. We study preemption-resume model, i.e., the currently processed job may be preempted at any moment in necessary and be resumed some time later. Let p _{1, j} and p _{2, j} denote the processing time of a job J _j on the two machines M ₁ and M ₂, respectively. Bounded processing times mean that 1 ≤ p _{i, j}  ≤ α (i = 1, 2) for each job J _j , where α ≥ 1 is a constant number. We propose an optimal online algorithm with a competitive ratio ${\frac{5\alpha-1}{4\alpha}}$ .     

Complementary bases of a matroid

Let e₁, e′₁, e₂, e′₂, …, e_n, e′_n be the elements of matroid M. Suppose that {e₁, e₂, …;, e_n} is a base of M and that every circuit of M contains at least m + 1 elements. We prove that there exist at least 2^m bases, called complementary bases, of M with the property that only one of each complementary pair e_j, e′_j is contained in any base.We also prove an analogous result for the case where E is partitioned into E₁, E₂, …, E_n and the initial base contains |E_j| ? 1 elements from partition E_j.     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

On hypergraphs without two edges intersecting in a given number of vertices

Let X be a finite set of n-melements and suppose t ? 0 is an integer. In 1975, P. Erdös asked for the determination of the maximum number of sets in a family $\text{F}$ = {F₁,…, F_m}, F_i ? X, such that ∥F_i ∩ F_j∥ ≠ t for 1 ? i ≠ j ? m. This problem is solved for n ? n₀(t). Let us mention that the case t = 0 is trivial, the answer being 2^{n ? 1}. For t = 1 the problem was solved in [3]. For the proof a result of independent interest (Theorem 1.5) is used, which exhibits connections between linear algebra and extremal set theory.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).