Global integration of differential equations through Lobatto quadrature

For the numerical solution of the initial value problemy=f(x,y), –1x1;y(–1)=y ₀ a global integration method is derived and studied. The method goes as follows.At first the system of nonlinear equations is solved. The matrix (A _i,k ⁽ⁿ⁾ ) of quadrature coefficients is nearly lower left triangular and the pointsx _k,n ,k=1,2,...,n are the zeros ofP _n –P _n–2, whereP _n is the Legendre polynomial of degreen. It is showed that the errors From the valuesf(x _i,n ,y _i,n ),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small.     

Finding normal solutions in piecewise linear programming

Letf: ⁿ (–, ] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min _x ¦x¦²/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} _t 0, minimizingf over an ellipsoid, and choosing penalty parameters inL ₁QP methods for strictly convex QP.This research was supported by the State Committee for Scientific Research under Grant 8S50502206.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory

A basic integral equation of random fields estimation theory by the criterion of minimum of variance of the estimation error is of the form Rh = f, where and R(x, y) is a covariance function.The singular perturbation problem we study consists of finding the asymptotic behavior of the solution to the equation as 0.$$" align="middle" border="0"> The domain D can be an interval or a domain in Rⁿ, n > 1. The class of operators R is defined by the class of their kernels R(x,y) which solve the equation Q(x, D_x)R(x, y) = P(x, D_x)δ(x − y), where Q(x, D_x) and Px, D_x) are elliptic differential operators.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f(x,y) and g(x,y) on D when n →∞, provided f(x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipαM(0 ＜α≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipαM and f(x,y) belongs to C1 (D), the inner product between the partial derivative of Hakopian interpolation polynomial (6)/(6)xHn(f;x,y) and g(x,y) on D converges to that between (6)/(6)xf(x,y) and g(x,y) on D when n →∞.     

Coconvex Pointwise Approximation

Assume that a function f C[–1, 1] changes its convexity at a finite collection Y := {y ₁, ... y _s} of s points y _i (–1, 1). For each n > N(Y), we construct an algebraic polynomial P _n of degree n that is coconvex with f, i.e., it changes its convexity at the same points y _i as f and

Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems

Let (P) denote the vector maximization problem

Formulas for best extrapolation

We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x _i ),i=1(1)n, –1x _i 1, including non-distinct pointsx _i in confluent formulas involving derivatives. The problem is to find the pointsx _i that minimize the factor in the remainderP _n (x)f ⁽ⁿ⁾()/n, –1<<x subject to the condition|P _n (x)|M, –1x1,2^–n+1M2 ⁿ . The solution is significant only when a single set of pointsx _i suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal , 0 rn, whererr(M) is a non-decreasing function ofM, P _n (–1)=(–1) ⁿ M, and for 0rn–2, thej-th extremumP _n (x _{e, j} )=(–1) ^n–j M,j=1(1)n–r–1 (except forM=M _r ,r=1(1)n–1, whenj=1(1)n–r).     

Note on the solution of a certain boundary-value problem

A simple algorithm is described for inverting the operatorD _x D _y (D _x andD _y here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationD _x D _y f (x, y)=g (x, y, f, D _x f, D _x ² f,D _x D _y f, D _y ² f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).Communication M. R. 43 of the Computation Department of the Mathematical Centre, Amsterdam.     

Divisibility of matrices associated with multiplicative functions

Let S={x₁,…,x_n} be a set of n distinct positive integers. For x,y∈S and y<x, we say the y is a greatest-type divisor of x in S if y∣x and it can be deduced that z=y from y∣z,z∣x,z<x and z∈S. For x∈S, let G_S(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(x_i,x_j)) denote the n×n matrix having f evaluated at the greatest common divisor (x_i,x_j) of x_i and x_j as its i,j-entry and let (f[x_i,x_j]) denote the n×n matrix having f evaluated at the least common multiple [x_i,x_j] of x_i and x_j as its i,j-entry. In this paper, we assume that S is a gcd-closed set and . We show that if f is a multiplicative function such that (f∗μ)(d)∈Z whenever and f(a)|f(b) whenever a|b and a,b∈S and (f(x_i,x_j)) is nonsingular, then the matrix (f(x_i,x_j)) divides the matrix (f[x_i,x_j]) in the ring M_n(Z) of n×n matrices over the integers. As a consequence, we show that (f(x_i,x_j)) divides (f[x_i,x_j]) in the ring M_n(Z) if (f∗μ)(d)∈Z whenever and f is a completely multiplicative function such that (f(x_i,x_j)) is nonsingular. This confirms a conjecture of Hong raised in 2004.     

Sets in a euclidean space which are not a countable union of convex subsets

We prove that if a closed planar setS is not a countable union of convex subsets, then exactly one of the following holds:

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and

Singular Nonlinear (n − 1,1) Conjugate Boundary Value Problems

Solutions are obtained for the boundary value problem, y ⁽ⁿ⁾ + f(x,y) = 0, y ⁽ⁱ⁾(0) = y(1) = 0, 0 i n – 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

An iff fixed point criterion for continuous self-mappings on a complete metric space

Summary Letf be a self-map on a metric space (X, d). We give necessary and sufficient conditions for the sequences {f ⁿ x} (x X) to be equivalent Cauchy. As a typical application we get the following result. Letf be continuous and (X, d) be complete. If, for anyx, y X d(f ⁿ x, f ⁿ y) 0 and for somec > 0, this convergence is uniform for allx, y inX withd(x, y) c thenf has a unique fixed pointp, andf ⁿ x p, for eachx inX. This theorem includes among others results of Angelov, Browder, Edelstein, Hicks and Matkowski.     

Finding connected components of a semialgebraic set in subexponential time

Let a semialgebraic set be defined by a quantifier-free formula with atomic subformulas of the form f_i>0, 0,1 i where the polynomials f_i[X₁,..., X_n] of degree deg (f_i)<d have absolute value of the coefficients at most 2^M. An algorithm is constructed which finds the connected components of the semialgebraic set (i.e., giving formulas for them) in a time that is polynomial inM (kd) ^ro(1). Collins' previously known method has a bound which is polynomial inM (kd) ^ro(1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 3–46, 1991.     

The reverse spelling of an FPrt-universal word in two letters

ForX a set the expression Prt(X) denotes the composition monoid of all functionsf X ×X. Fork a positive integer the letterk denotes also the set of all nonnegative integers less thank. Whenk > 1 the expression r_k denotes the connected injective element {<i, i + 1>i k – 1} in Prt (k). We show for every word w=w(x,y) in a two-letter alphabet that if the equation w(x, y)=r_k has a solution =y) ²Prt(k) then ¯w(x,y)=r_k also has a solution in²Prt(k), where ¯w is the word obtained by spelling the wordw backwards. It is a consequence of this theorem that if for every finite setX and for everyf Prt(X) the equation w(x,y)=f has a solution in²Prt(X) then for every suchX andf the equation ¯w(x, y)=f has a solution in²Prt(X).Presented by J. Mycielski.     

Automorphisms and isotone self-maps of ordered sets with top and bottom

For an ordered setP letP ^P denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatxy impliesf(x)f(y), and let Aut (P) the set of all automorphisms onP, that is, all bijective isotone self-maps inP ^P . We establish an inequality relating ¦P ^P ¦ and ¦Aut(P)¦ in terms of the irreducibles ofP. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship.     

A polynomial-time algorithm for a class of linear complementarity problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x 0,y 0,x _i y _i = 0 (i = 1, 2,,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n ³ L) arithmetic operations by tracing the path of centers,{(x, y) S: x _i y _i = (i = 1, 2,,n) for some > 0} of the feasible regionS = {(x, y) 0:y = Mx + q}, whereL denotes the size of the input data of the problem.     

A note on certain functional determinants

Summary We consider the problem when a scalar function ofn variables can be represented in the form of a determinant det(f _i (x _j )), the so-called Casorati determinant off ₁,f ₂,,f _n . The result is applied to the solution of some functional equations with unknown functionsH of two variables that involve determinants det(H(x _i ,x _j )).