Loopless generation of up–down permutations

An up–down permutation P=(p₁,p₂,…,p_n) is a permutation of the integers 1 to n which satisfies constraints specified by a sequence C=(c₁,c₂,…,c_n−1) of U's and D's of length n−1. If c_i is U then p_i<p_i+1 otherwise p_i−1>p_i. A loopless algorithm is developed for generating all the up–down permutations satisfying any sequence C. Ranking and unranking algorithms are discussed.     

A Helly theorem for geodesic convexity in strongly dismantlable graphs

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x₀,…, x so that, for each ordinal β < , there exists a strictly increasing finite sequence (i_j)_{0 j n} of ordinals such that i₀ = β, i_n = and x_ij+1 is adjacent with x_ij and with all neighbors of x_ij in the subgraph ofG induced by {x_y: β γ }. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.     

The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f₁,f₂,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size f_i. It is proved that there exist constants c=c(n) and k₀=k₀(n) such that f_k+1=f_k+f_k−c for all k>k₀, and lim_n→∞ c(n)/(nlogn)=1.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

Sum coloring and interval graphs: a tight upper bound for the minimum number of colors

The SUM COLORING problem consists of assigning a color c(v_i)Z⁺ to each vertex v_iV of a graph G=(V,E) so that adjacent nodes have different colors and the sum of the c(v_i)'s over all vertices v_iV is minimized. In this note we prove that the number of colors required to attain a minimum valued sum on arbitrary interval graphs does not exceed min{n;2χ(G)−1}. Examples from the papers [Discrete Math. 174 (1999) 125; Algorithmica 23 (1999) 109] show that the bound is tight.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Sur les ensembles représentés par les partitions d'un entier n

Let n = n₁ + n₂ + … + n_j a partition Π of n. One will say that this partition represents the integer a if there exists a subsum n_il + n_i2 + … + n_il equal to a. The set (Π) is defined as the set of all integers a represented by Π. Let be a subset of the set of positive integers. We denote by p( ,n) the number of partitions of n with parts in , and by (( ,n) the number of distinct sets represented by these partitions. Various estimates for ( ,n) are given. Two cases are more specially studied, when is the set {1, 2, 4, 8, 16, …} of powers of 2, and when is the set of all positive integers. Two partitions of n are said to be equivalent if they represent the same integers. We give some estimations for the minimal number of parts of a partition equivalent to a given partition.     

Controllability completion problems of partial upper triangular matrices

The main result of this paper states sufficient conditions for the existence of a completion A_c of an n × n partial upper triangular matrix A, such that the pair (A_cB) has prescribed controllability indices, being B an n×m matrix. If A is a partial Hessenberg matrix some conditions may be dropped. An algorithm that obtains a completion A_c of A such that pair (A_ce_k) is completely controllable, where e_k is a unit vector, is used to proof the results.     

Combinatorial theorems on contractive mappings in power sets

We prove that to every positive integer n there exists a positive integer h such that the following holds: If S is a set of h elements and ƒ a mapping of the power set of S into such that ƒ(T)T for all T , then there exists a strictly increasing sequence T₁T_n of subsets of S such that one of the following three possibilities holds: (a) all sets ƒ(T_i), i= 1,…,n, are equal; (b) for all i=1,…, n, we have ƒ(T_i)=T_i; (c) T_i=ƒ(T_i+1) for all i= 1,…,n-1. This theorem generalizes theorems of the author, Rado, and Leeb. It has applications for subtrees in power sets.     

On the q-analogues of the Zassenhaus formula for disentangling exponential operators

Katriel, Rasetti and Solomon introduced a q-analogue of the Zassenhaus formula written as e_q^(A+B)=e_q^Ae_q^Be_q^c₂e_q^c₃e_q^c₄e_q^c₅… , where A and B are two generally noncommuting operators and e_q^z is the Jackson q-exponential, and derived the expressions for c₂, c₃ and c₄. It is shown that one can also write Explicit expressions for , and are given.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

Convergent regular splittings for nonnegative matrices

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. This paper characterizes the nonnegative MP matrices and hence, determines when a nonnegative matrix A has a convergent regular splitting M—Q which induces the linear stationary iterative scheme x_k+1=M^-1Qx_k+M^-1b to solve Ax=b.     

Sperner''s theorem with constraints

A set X of subsets of an n-element set S is called an anti-chain if no two elements of X are related by set-wise inclusion. Sperner showed [8] that max |X|=(ⁿ_[n/2]), where |X| denotes the number of elements in X and the maximum is taken over all anti-chains of subsets of S.

Let non-negative integers i_o<n and m_io≠0, m_io+1,…m_n be given. In this paper we give an algorithm for calculating max |X| where the maximum is taken only over anti-chains containing exactly m_i i-element subsets of S for i_o i n.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

Maximal IM-unextendable graphs

A graph G is maximal IM-unextendable if G is not induced matching extendable and, for every two nonadjacent vertices x and y, G+xy is induced matching extendable. We show in this paper that a graph G is maximal IM-unextendable if and only if G is isomorphic to M_r(K_s(K_n1K_n2K_nt)), where M_r is an induced matching of size r, r1, t=s+2, and each n_i is odd.     

On the area of the intersection of disks in the plane

Let C₁,…, C_n and C′₁,…, C′_n be two collections of equal disks in the plane, with centers c₁,…, c_n and c′₁,…, c′_n. According to a well-known conjecture of Klee and Wagon (1991), if |c_i − c_j| ≥ |c′_i − c′_j| for all i, j, then Area(∩_i C_i) ≤ Area(∩_i C′_i).

We prove this statement in the special case when there is a continuous contraction of {c₁,…, c_n} onto {c′₁,…, c′_n}.     

An infinite family of cubic edge- but not vertex-transitive graphs

An infinite family of cubic edge- but not vertex-transitive graphs is constructed. The graphs are obtained as regular -covers of K_3,3 where n=p₁^e₁p₂^e₂p_k^e_k where p_i are distinct primes congruent to 1 modulo 3, and e_i1. Moreover, it is proved that the Gray graph (of order 54) is the smallest cubic edge- but not vertex-transitive graph.     

Embeddings of the base and bundle isomorphisms

Let π_i :E_i→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections s_i :M→E_i. Suppose that U is an open neighborhood of s₁(M) in E₁ and F :U→E₂ a smooth embedding so that π₂Fs₁ :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E₁ and the induced bundle f^*E₂ are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<kn we give examples of nonisomorphic oriented bundles E₁ and E₂ of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E₁ and f^*E₂ are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.     

Scheduling starting times for an active redundant system with non-identical lifetimes

Here we examine an active redundant system with scheduled starting times of the units. We assume availability of n non-identical, non-repairable units for replacement or support. The original unit starts its operation at time s₁ = 0 and each one of the (n − 1) standbys starts its operation at scheduled time s_i (i = 2, …, n) and works in parallel with those already introduced and not failed before s_i. The system is up at times s_i (i = 2, …, n), if and only if, there is at least one unit in operation. Thus, the system has the possibility to work with up to n units, in parallel structure. Unit-lifetimes T_i (i = 1, …, n) are independent with cdf F_i, respectively. The system has to operate without inspection for a fixed period of time c and it stops functioning when all available units fail before c. The probability that the system is functioning for the required period of time c depends on the distribution of the unit-lifetimes and on the scheduling of the starting times s_i. The reliability of the system is evaluated via a recursive relation as a function of the starting times s_i (i = 2, …, n). Maximizing with respect to the starting times we get the optimal ones. Analytical results are presented for some special distributions and moderate values of n.     

Duality in Padé-type approximation

Padé and Padé-type approximants are usually defined by replacing the function (1 − xt)⁻¹ by its Hermite (that is confluent) interpolation polynomial and then applying the functional c defined by c(xⁱ) = c_i where the c_i's are the coefficients of the series to be approximated. In this paper the functional d which, applied to (1 − xt)⁻¹, gives the same Padé or Padé-type approximant as before is studied. It can be considered as the dual of the interpolation operator applied to the functional c.