On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space (X,d) has the de Groot property GPn if for any points x₀,x₁,…,xn+2∈X there are positive indices i,j,k?n+2 such that i≠j and d(xi,xj)?d(x₀,xk). If, in addition, k∈{i,j} then X is said to have the Nagata property NPn. It is known that a compact metrizable space X has dimension dim(X)?n iff X has an admissible GPn-metric iff X has an admissible NPn-metric.We prove that an embedding f:(0,1)→X of the interval (0,1)⊂R into a locally connected metric space X with property GP₁ (resp. NP₁) is open, provided f is an isometric embedding (resp. f has distortion Dist(f)=‖fLip_‖⋅‖f−1Lip_‖<2). This implies that the Euclidean metric cannot be extended from the interval [−1,1] to an admissible GP₁-metric on the triode T=[−1,1]∪[0,i]. Another corollary says that a topologically homogeneous GP₁-space cannot contain an isometric copy of the interval (0,1) and a topological copy of the triode T simultaneously. Also we prove that a GP₁-metric space X containing an isometric copy of each compact NP₁-metric space has density ?c.     

Eigenvalue Intervals for 2mth Order Sturm—Liouville Boundary Value Problems

Values of λ are determined for which there exist positive solutions of the 2mth order differential equation on a measure chain, (-1)^m x ^?2m (t)=λa(t)f(u(σ(t))), y? [0,1], satisfying α _i+1 u ^?21(0)+0, γ _i+1 u ^?21(σ(1))=0, 0≤i≤m?1 with αi,βi,γi,δi≥0, where a and f are positive valued, and both lim _x-0+ (f(x)/x) and lim _x-0+ (f(x)/x) exist.     

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.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Domains with growth conditions for the quasihyperbolic metric

For a given growth functionH, we say that a domainD ?R ⁿ is anH-domain if δ_D x≤δ_D(x ₀)H(k _D(x,x ₀)),x ∈D, where δ_D(x)=d(x?D) andk _D denotes the quasihyperbolic distance. We show that ifH satisfiesH(0)=1, |H'|≤H, andH"≤H, then there exists an extremalH-domain. Using this fact, we investigate some fundamental properties ofH-domains.     

A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

The two dimensional diffusion equation of the form is considered in this paper. We try a bi-cubic spline function of the form as its solution. The initial coefficients C_i,j(0) are computed simply by applying a collocation method; C_i,j = f(x_i, y_j) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients C_i,j(t) are computed by X(t) = e^tQX(0) where X(t) = (C_0,1, C_0,1, C_0,2, … , C_0,N, C_1,0, … , C_N,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.     

Prophet Regions for Discounted,Uniformly Bounded Random Variables

Abstract

Let X ₁, X ₂,… be any sequence of [0,1]-valued random variables. A complete comparison is made between the expected maximum E(max _j≤n Y _j ) and the stop rule supremum sup _t E Y _t for two types of discounted sequences: (i) Y _j  = b _j X _j , where {b _j } is a nonincreasing sequence of positive numbers with b ₁ = 1; and (ii) Y _j  = B ₁… B _j?1 X _j , where B ₁, B ₂,… are independent [0,1]-valued random variables that are independent of the X _j , having a common mean β. For instance, it is shown that the set of points {(x, y): x = sup _t E Y _{{(x, y): x=sup t E Y} and y = E(max _j≤n Y _j ), for some sequence X ₁,…,X _n and Y _j  = b _j X _j }, is precisely the convex closure of the union of the sets {(b _j x, b _j y): (x, y) ∈ C _j }, j = 1,…,n, where C _j  = {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ x[1 + (j ? 1)(1 ? x ^1/(j?1))]} is the prophet region for undiscounted random variables given by Hill and Kertz [8 Hill , T.P. , and R.P. Kertz . 1983 . Stop rule inequalities for uniformly bounded sequences of random variables . Trans. Amer. Math. Soc. 278 : 197 – 207 . [CSA]  [Google Scholar]]. As a special case, it is shown that the maximum possible difference E(max _j≤n β ^j?1 X _j ) ? sup _t E(β ^t?1 X _t ) is attained by independent random variables when β ≤ 27/32, but by a martingale-like sequence when β > 27/32. Prophet regions for infinite sequences are given also.     

Convex interpolating splines of arbitrary degree II

For given data {(x _i ,y _i )} _i=0 ⁿ , (x ₀<x ₁<...<x _n ) we consider the possibility of finding a spline functions of arbitrary degreek+1 (k 1) with preassigned smoothnessl, where 1 l [(k+1)/2]. The splines should be such thats(x _i )=y _i ,i=0, 1,...,n ands is increasing and convex on [x ₀,x _n ]. Sufficient conditions which guarantee the existence ofs and an explicit formula for this function are derived.     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, 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.     

On a conjecture of R.F. Scott (1881)

A formula is given for the permanent of a general Cauchy matrix ((x_i?y_j)^?1). In a special case, where the x_i and the y_j are the distinct nth roots of 1 and ?1 respectively, a formula for the permanent, conjectured by R. F. Scott, is proved by computing the eigenvalues of related circulants.     

A parallel selection algorithm

The problem of selecting thekth largest or smallest element of {x _i +y _j |x _i X andy _j Y i,j} whereX=(x ₁,x ₂, ..,x _n ) andY=(y ₁,y ₂, ...,y _n ) are two arrays ofn elements each, is considered. Certain improvements to an existing algorithm are proposed. An algorithm requiringO(logk·logn) units of time on a Shared Memory Model of a parallel computer havingO(n ^1+1/) processors is presented where is a pre-assigned constant lying between 1 and 2.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

The optimal base of a matroid/with tree-type constraints

Let M be a matroid defined on a weighted finite set E=(e_1,…,e_n).l(e) is the weight of e∈E.P (X_1,…,X_m) is a set of subsets of E.X_i,X_j∈P,if X_i∩X_j≠(the empty set),then either X_i X_j or X_jX_i.For each X_i∈P,there are two associate nonnegative integers a_i and b_i with o_i≤b_i≤|X_i|.We call a base T of M a feasible base with respect to P(or simply call it a feasible base of M),if X_i∈P,a_i≤|X_i∩T|≤b_i.A base T' is called optimal if:i) This feasible,In this paper we present a polynomial algorithm to solve the optimal base problem.     

Computer difference scheme for a singularly perturbed convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ? (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an \(\mathcal{O}\) (δ _st ) rate, where δ _st = (? + N ^?1)^?1 N ^?1 and N + 1 is the number of grid nodes; the scheme is not ?-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ? down to its complete loss for small ? (namely, for ? = \(\mathcal{O}\) (δ^?2max _i,j |δa _i ^j | + δ^?1 max _{i, j} |δb _i ^j |), where δ = δ _st and δa _i ^j , δb _i ^j are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ? ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.     

More on the generalized macaulay theorem — II

Let k₁ ? k₂? ? ? k_n be given positive integers and let S denote the set of vectors x = (x₁, x₂, … ,x_n) with integer components satisfying 0 ? x₁ ? k_ni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l.     

New algorithm for computing the Hermite interpolation polynomial

Let x ₀, x ₁,? , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,? , n, and k = 0 ,1 ,? , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N ? 1}(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n _i . P _N?1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,? , n, k = 0,1,? , n _i }. The polynomial p _N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.     

On the rational approximation ofe −x on [0, ∞)

In recent years the problem of uniform approximation ofe ^?x on [0, ∞) by rational functions has found wide interest. In this paper we present a method for the construction of rational approximations toe ^?x that seem to come arbitrarily near to the asymptotically best one. We start with a generalization of the integral form of the Padé approximant by introducing certain real parametersa _j ,b _i ,k and?. The corresponding error function has again an integral representation and is estimated for everyx∈[0,∞) by the Laplace method. This leads to the consideration of a finite number of new error functionsφ _i·j whose maxima are determined by a set of nonlinear equations and, under some restrictions on thea _j ,b _i ,k, and?, are also unique. Variation of these parameters according to a special algorithm and computation of the corresponding maxima of the functionsφ _i·j shows that forn→∞ the order of rational approximation ofe ^?x on [0, ∞) is at least 9.03^?n .     

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 〈?〉.     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.