On Erdos' ten-point problem

Around 1994, Erdoset al. abstracted from their work the following problem: “Given ten pointsA _ij, 1≤i≤j≤5, on a plane and no three of them being collinear, if there are five pointsA _k, 1≤k≤5, on the plane, including points at infinity, with at least two points distinct, such thatA _i, A_j, A_ij are collinear, where 1≤i≤j≤5, is it true that there are only finitely many suchA _k's?” Erdoset al. obtained the result that generally there are at most 49 groups of suchA _k's. In this paper, using Clifford algebra and Wu's method, we obtain the results that generally there are at most 6 such groups ofA _k's.     

Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation

A matrix A∈R^n×n is called a bisymmetric matrix if its elements a_i,j satisfy the properties a_i,j=a_j,i and a_i,j=a_n-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.     

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.     

Symposium on combinatorial mathematics and optimal design

Let A₁,A₂,…,A_n be finite sets such that A_i?A_j for all i≠j. Let F be an intersecting family consisting of sets contained in some A_i, i=1,2,…,n. Chvátal conjectured that among the largest intersecting families, there is always a star. In this paper, we obtain another proof of a result of Schönheim: If A₁∩A₂∩?∩A_n≠?, then the conjecture is true. We also prove that if A_i∩A_jA_k = ? for all i≠j≠k≠i or if the independent system satisfies a hereditary tree structure, then the conjecture is also true.     

Approximating reversal distance for strings with bounded number of duplicates

For a string A=a₁…a_n, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A^′=a₁…a_i-1a_ja_j-1…a_ia_j+1…a_n, that is, the reversal ρ(i,j) reverses the order of symbols in the substring a_i…a_j of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k²)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm.     

Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

On the range maximum-sum segment query problem

The range minimum query problem, RMQ for short, is to preprocess a sequence of real numbers A[1…n] for subsequent queries of the form: “Given indices i, j, what is the index of the minimum value of A[i…j]?” This problem has been shown to be linearly equivalent to the LCA problem in which a tree is preprocessed for answering the lowest common ancestor of two nodes. It has also been shown that both the RMQ and LCA problems can be solved in linear preprocessing time and constant query time under the unit-cost RAM model. This paper studies a new query problem arising from the analysis of biological sequences. Specifically, we wish to answer queries of the form: “Given indices i and j, what is the maximum-sum segment of A[i…j]?” We establish the linear equivalence relation between RMQ and this new problem. As a consequence, we can solve the new query problem in linear preprocessing time and constant query time under the unit-cost RAM model. We then present alternative linear-time solutions for two other biological sequence analysis problems to demonstrate the utilities of the techniques developed in this paper.     

Continuity properties of the Drazin pseudoinverse

Let B^D denote that Drazin inverse of the n×n complex matrix B. Define the core-rank of B as rank (B^i(B)) where i(B) is the index of B. Let j = 1,2,…, and A_j and A be square matrices such that A_i converges to A with respect to some norm. The main result of this paper is that A_j^D converges to A^D if and only if there exist a j₀ such that core-rank A_j=core-rankA for j ? j₀.     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

Outsourcing and scheduling for two-machine ordered flow shop scheduling problems

This paper considers a two-machine ordered flow shop problem, where each job is processed through the in-house system or outsourced to a subcontractor. For in-house jobs, a schedule is constructed and its performance is measured by the makespan. Jobs processed by subcontractors require paying an outsourcing cost. The objective is to minimize the sum of the makespan and the total outsourcing cost. Since this problem is NP-hard, we present an approximation algorithm. Furthermore, we consider three special cases in which job j has a processing time requirement p_j, and machine i a characteristic q_i. The first case assumes the time job j occupies machine i is equal to the processing requirement divided by a characteristic value of machine i, that is, p_j/q_i. The second (third) case assumes that the time job j occupies machine i is equal to the maximum (minimum) of its processing requirement and a characteristic value of the machine, that is, max{p_j, q_i} (min{p_j, q_i}). We show that the first and the second cases are NP-hard and the third case is polynomially solvable.     

On quadruples of Griffiths points

Tabov (Math Mag 68:61–64, 1995) has proved the following theorem: if points A ₁, A ₂, A ₃, A ₄ are on a circle and a line l passes through the centre of the circle, then four Griffiths points G ₁, G ₂, G ₃, G ₄ corresponding to pairs (Δ _i ,l) are on a line (Δ _i denotes the triangle A _j A _k A _l , j,k,l ≠ i). In this paper we present a strong generalisation of the result of Tabov. An analogous property for four arbitrary points A ₁, A ₂, A ₃, A ₄, is proved, with the help of the computer program “Mathematica”.     

A minimum length covering subgraph of a network

This paper concerns the problem of locating a central facility on a connected networkN. The network,N, could be representative of a transport system, while the central facility takes the form of a connected subgraph ofN. The problem is to locate a central facility of minimum length so that each of several demand points onN is covered by the central facility: a demand point atv _i inN is covered by the central facility if the shortest path distance betweenv _i and the closest point in the central facility does not exceed a parameterr _i . This location problem is NP-hard, but for certain special cases, efficient solution methods are available.     

Distance matrices for points on a line,on a circle,and at the vertices of ann-dimensional cube

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ ^m , the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995.     

Extremal Systems with Upper-Bounded Odd Intersections

In this paper we determine the maximum cardinality m of a family A= {A ₁,..., A _m} of subsets of a set of n elements with the property that for each A _i there are at most s A _j such that |A _i ∩ A _j | is odd (even). A tight upper bound is given in the case s < c(2 ^n,2/n). In the remaining cases we give an asymptotically tight upper bound. As an application we give a tight upper-bound for the cardinality of a family with even multi-intersection. Both results generalize a result of Berlekamp and Graver.     

The complexity of generalized clique packing

The K_i - j packing problem P_{i, j} is defined as follows: Given a graph G and integer k does there exist a set of at least kK_i's in G such that no two of these K_i's intersect in more than j nodes. This problem includes such problems as matching, vertex partitioning into complete subgraphs and edge partitioning into complete subgraphs. In this paper it is shown thhat for i ? 3 and 0?j?i ?2 the P_{i, j} problems is NP-complete. Furthermore, the problems remains NP-complete for i?3 and 1?j?i ?2 for chordal graphs.     

Sperner systems consisting of pairs of complementary subsets

It was proved by Erdös, Ko, and Radó (Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser.12 (1961), 313–320.) that if $\text{A}$ = {;A₁,…, A_l}; consists of k-subsets of a set with n > 2k elements such that A_i ∩ A_j ≠ ? for all i, j then l ? (_k?1^n?1). Schönheim proved that if A₁, …, A_l are subsets of a set S with n elements such that A_i ? A_j, A_i ∩ A_j ≠ ø and A_i ∪ A_j ≠ S for all i ≠ j then $\text{l ? (}{}_{\mathrm{<mtext>[</mtext><mtext>n</mtext><mtext>2</mtext><mtext>]\; ?\; 1</mtext>}}{}^{\mathrm{n\; ?\; 1}}\text{)}$. In this note we prove a common strengthening of these results.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Regularized semigroups,iterated Cauchy problems and equipartition of energy

The iterated Cauchy problem under consideration is $$\Pi _{k = 1}^n (d/dt - A_k )u(t) = 0(t \geqslant 0).(*)$$ Here {A ₁,..., A_n} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When eachA _k is a (C ₀) semigroup generator, this semigroup is of class (C ₀) and was studied by J. T. Sandefur [26]. This result is extended to the case when eachA _k generates aC-regularized semigroup (withC independent ofk). This means one can solveu′=Au, u(0)=f∈C (Dom (A)) and getu(t)→0 wheneverC ^?1f→0; hereC is bounded and injective. When theA _k are commuting generator withA _k-A_j injective fork≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form \(u = \sum\nolimits_{i = 1}^n {u_i } \) whereu _i is a solution ofu′ _i=A_iu_i. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.     

An algorithm for associative bilinear forms

We establish the equivalence between the problem of existence of associative bilinear forms and the problem of solvability in commutative power-associative finite-dimensional nil-algebras. We use the tensor product to find sufficient and necessary conditions to assure the existence of associative bilinear forms in an algebra A. The result gives us an algorithm to describe the space of associative bilinear forms for an algebra when its constants of structure γ_i,j,k for i,j,k=1,…,n are known.     

Nash-solvable two-person symmetric cycle game forms

A two-person positional game form g (with perfect information and without moves of chance) is modeled by a finite directed graph (digraph) whose vertices and arcs are interpreted as positions and moves, respectively. All simple directed cycles of this digraph together with its terminal positions form the set A of the outcomes. Each non-terminal position j is controlled by one of two players i∈I={1,2}. A strategy x_i of a player i∈I involves selecting a move (j,j^′) in each position j controlled by i. We restrict both players to their pure positional strategies; in other words, a move (j,j^′) in a position j is deterministic (not random) and it can depend only on j (not on preceding positions or moves or on their numbers). For every pair of strategies (x₁,x₂), the selected moves uniquely define a play, that is, a directed path form a given initial position j₀ to an outcome (a directed cycle or terminal vertex). This outcome a∈A is the result of the game corresponding to the chosen strategies, a=a(x₁,x₂). Furthermore, each player i∈I={1,2} has a real-valued utility function u_i over A. Standardly, a game form g is called Nash-solvable if for every u=(u₁,u₂) the obtained game (g,u) has a Nash equilibrium (in pure positional strategies).A digraph (and the corresponding game form) is called symmetric if (j,j^′) is its arc whenever (j^′,j) is. In this paper we obtain necessary and sufficient conditions for Nash-solvability of symmetric cycle two-person game forms and show that these conditions can be verified in linear time in the size of the digraph.