Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

Functional estimation of activity criticality indices and sensitivity analysis of expected project completion time

For a PERT network, a new method is developed for estimating the criticality index of activity i (ACI _i ) as a function of the expected duration of activity i (μ _i ) and for the sensitivity analysis of the expected project completion time (μ _T ) with respect to μ _i . The proposed method evaluates the frequency of activity i being on the critical path, and thereby its ACI _i using Monte Carlo simulation or a Taguchi orthogonal array experiment at several values of μ _i , fits a logistic regression model for estimating ACI _i as a function of μ _i , and then, using the estimated ACI _i function, evaluates the amount of change in μ _T when μ _i is changed by a given amount. Unlike the previous works, the proposed method models ACI _i as a nonlinear (ie, logistic) function of μ _i , which can be used to estimate the amount of change in μ _T for a variety of changes in μ _i . Computational results indicate that the performance of the proposed method is comparable to that of direct Monte Carlo simulation.     

Existence of a positive solution to a system of discrete fractional boundary value problems

We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by -Δ^ν_iy_i(t)=λ_ia_i(t+ν_i-1)f_i(y₁(t+ν₁-1),y₂(t+ν₂-1)), for t∈[0,b]_N0, subject to y_i(ν_i − 2) = ψ_i(y_i) and y_i(ν_i + b) = ?_i(y_i), for i = 1, 2, where ψ_i,?_i:R^b+3→R are given functionals. We also assume that ν_i ∈ (1, 2], for each i. Although we assume that both a_i and f_i(y₁, y₂) are nonnegative for each i, we do not necessarily presume that each ψ_i(y_i) and ?_i(y_i) is nonnegative for each i and each y_i ? 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case.     

On some combinatorial questions in finite-dimensional spaces

Bounds are given for supinf∥Σ^p_i=1ν_i∥, where sup is taken over all set systems V₁,…, V_p of Rⁿ with 0∈∩^p_i=1convV_i and sup_v∈Vi∥ν∥?1 for i=1,…, p, and inf is taken over all possible choices ν_i∈V_i for i=1,…, p. Another similar problem is considered. The bounds are sharp.     

Positive periodic solutions for an integrodifferential model of mutualism

Sufficient conditions are obtained for the existence of positive periodic solutions of the following integrodifferential model of mutualism:

where r_i, K_i, α_i, σ_i, i = 1,2 are positive continuous ω-periodic functions, α_i > K_i, i = 1, 2, J_i ϵ C([0, ∞], [0, ∞)), and ∫^∞₀ J_i(s) ds = 1, i = 1, 2.     

An ABS algorithm for a class of systems of stochastic linear equations

This paper is to explore a model of the ABS algorithms dealing with the solution of a class of systems of linear stochastic equations Aξ=η when η is a m-dimensional normal distribution. It is shown that the stepsize α _i is distributed as N(u _i ,σ _i ) (being u _i the expected value of α _i and σ _i its variance) and the approximation to the solutions ξ _i is distributed as N _n (U _i ,Σ _i ) (being U _i the expected value of ξ _i and Σ _i its variance), for this algorithm model.     

A distance-labelling problem for hypercubes

Let i₁≥i₂≥i₃≥1 be integers. An L(i₁,i₂,i₃)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…} such that |?(u)−?(v)|≥i_t for any u,v∈V with d(u,v)=t, t=1,2,3, where d(u,v) is the distance in G between u and v. The integer ?(v) is called the label assigned to v under ?, and the difference between the largest and the smallest labels is called the span of ?. The problem of finding the minimum span, λ_i1,i2,i3(G), over all L(i₁,i₂,i₃)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i₁,i₂,i₃)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i₁,i₂,i₃)-labelling problem for hypercubes Q_d (d≥3) and obtain upper and lower bounds on λ_i1,i2,i3(Q_d) for any (i₁,i₂,i₃).     

Maximizing the probability of correctly ordering random variables using linear predictors

Let (T₁, x₁), (T₂, x₂), …, (T_n, x_n) be a sample from a multivariate normal distribution where T_i are (unobservable) random variables and x_i are random vectors in R^k. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T_i} is maximized by ranking according to the order of the best linear predictors {E(T_i|x_i)}. Furthermore, it orderings are chosen according to linear functions {b′x_i} then the conditional probability of correct order given (T_i = t₁; i = 1, …, n) is maximized when b′x_i is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {T_i} if {(T_i, x_i)} are independent but not identically distributed, or if the distributions are not normal.     

Sensitivity,specificity, and accuracy of predictive models on phenols toxicity

The investigation of how the influential affect the metrics and predictivity of multiple linear regressions on a set of phenolic compounds with toxicity on Tetrahymena pyriformis is presented. The investigation of influential was conducted using standardized residuals (r_i-model) and Cook's distance (D_i-model) approaches. The applied approaches let to improvement of model's metrics, robustness and accuracy on the investigated sample. Overall, the r_i-model proved higher accuracy and robustness in terms of sensitivity while D_i-model proved robustness in terms of specificity. Characterization of the withdrawn compounds is essential for advance in developing models for the toxicity of phenols.     

Path Ramsey numbers in multicolorings

In this paper we consider the general Ramsey number problem for paths when the complete graph is colored with k colors. Specifically, given paths P_i1, P_i2,…, P_ik with i₁, i₂,…, i_k vertices, we determine for certain i_j (1 ≤ j ≤ k) the smallest positive integer n such that a k coloring of the complete graph K_n contains, for some l, a P_il in the lth color. For k = 3, given i₂, i₃, the problem is solved for all but a finite number of values of i₁. The procedure used in the proof uses an improvement of an extremal theorem for paths by P. Erdös and T. Gallai.     

Simple continued fractions for the Fredholm numbers

Let {a_i} with a₁ ≥ 2 be an infinite bounded sequence of positive integers, and d₁ = 1, d_i = ±1 for i = 2, 3,…. Let {Q_i} be another sequence defined by the recursion Q₁ = 1, Q_i = a_i?1Q_i?1^k for i = 2, 3,…, where k ≥ 2 an integer. Put C_k(a) = Σ_{i = 1}^∞d_iQ_i^?1. In this paper we shall determine the simple continued fraction expansion for the real numbers C_k(a).     

Enumeration of rectangular arrays of zeros and ones

For r=1,2 the rectangular arrays of zeros and ones with r rows and n columns, with m_i zeros and r_i changes in the ith row, and with s_i vertical changes between the ith and (i+1)st rows, i=1,…,r-1, are enumerated. The number of arrays of zeros and ones with 3 rows and n columns, with r_i changes in the ith row, i=1,2,3, and with s_i vertical changes between the ith and (i+1)st rows, i=1,2, is also determined.     

Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.     

The structure of 0-bisimple strongly E *-unitary inverse monoids

We introduce a class of strongly E ^*-unitary inverse semigroups S _i (G, P) (i = 1,2) determined by a group G and a submonoid P of G and give an embedding theorem for S _i (G, P). Moreover we characterize 0-bisimple strongly E ^*-unitary inverse monoids and 0-bisimple strongly F ^*-inverse monoids by using S _i (G, P).     

Class G,and a unification problem

In 1958, Karl Goldberg proved the following: Theorem G. Suppose A=(a_ij) is an n×n matrix over the complex field, with the following properties: (1) a_ija_ji?0 for i,j=1,2,…,n, and (2) a_i1i2a_i2i3?a_isii=a_i2i1a_i3i2?a_i1is for all s=1,2,…,n and i_t=1,2,…,n. Then A has only real characteristic values. Definition. Let G_n denote the class of n×n matrices over $\text{C}$, the complex field, which satisfy the Goldberg conditions (1) and (2). We investigate some properties of class G related to the following topics: Schur complements, weak sign symmetry, and inequalities due to Oppenheim for positive definite matrices, and an analogue due to Markham for tridiagonal, oscillatory matrices.     

The k-factor conjecture is true

A sequence 〈d_i〉, 1≤i≤n, is called graphical if there exists a graph whose i^th vertex has degree d_i for all i. It is shown that the sequences 〈d_i〉 and 〈d_i-k〉 are graphical only if there exists a graph G whose degree sequence is 〈d_i〉 and which has a regular subgraph with k lines at each vertex. Similar theorems have been obtained for digraphs. These theorems resolve comjectures given by A.R. Rao and S.B. Rao, and by B. Grünbaum.     

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.     

The inertia of a Hermitian matrix having prescribed diagonal blocks

For i=1,…,m let H_i be an n_i×n_i Hermitian matrix with inertia In(H_i)= (π_i, ν_i, δ_i). We characterize in terms of the π_i, ν_i, δ_i the range of In(H) where H varies over all Hermitian matrices which have a block decomposition H= (X_ij)_{i, j=1,…, m} in which X_ij is n_i×n_j and X_ii=H_i.     

A resolution method for multiobjective problems

A resolution method for multiobjective problems, based on a maximin criterion, is developed. Given the multiobjective problem Max_{Ax=b,x?0}{c_ix; i = 1,2,…,k}, we suppose that the decisor can construct, for each i, a function h_i:$\text{R}$ → $\text{R}$ (or h_i:$\text{R}$ⁿ→-$\text{R}$), such that h_i(c_i,x) is his satisfaction degree produced by the value c_ix, and we substitute the original problem by Max_{Ax=b,x?0～min_i{h_i(c_ix)}. We analize its resolution and basic properties.     

Ki-covers I: Complexity and polytopes

A K_i in a graph is a complete subgraph of size i. A K_i-cover of a graph G(V, E is a set C of K_{i − 1}'s of G such that every K_i in G contains at least one K_{i − 1} in C. Thus a K₂-cover is a vertex cover. The problem of determining whether a graph has a K_i-cover (i ⩾ 2) of cardinality ⩽k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the K_i-cover polytope. Further results of this type are also produced for the K₃-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the K_i-cover polytope.