Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing conngura-tions of points from dissimilarity information about interpoint distances. A popular measure of the t of the constructed distances to the observed dissimilar-ities is the stress criterion, which must be minimized by numerical optimization. Empirical evidence concerning the existence of nonglobal...

* Correspondence: [email protected] Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, PR China Abstract In this article, we consider the uniqueness of positive radial solutions to the Dirichlet boundary value problem u + f (|x|, u) + g(|x|)x · ∇u = 0, x ∈ , u = 0, x ∈ ∂ , where Ω denotes an annulus in R (n ≥ 3). The uniqueness criterion is established by applyi...

We consider a parameter estimation problem with independent observations where one samples from a finite population of independent and identically distributed experimental conditions X. The size of the population is N but only n samples, a proportion α of N , can be used. The quality of a sample is measured by a regular optimality criterion φ(·) based on the information matrix, such as the D-cr...

1 Nonlinear Least Squares 1.1 Nonlinear Least Squares on Euclidean n-space We are interested in minimizing (locally) the following criterion function 4>: R" —• R defined by 1 1 m where ƒ : R " —* R m denotes the residual mapping with corresponding coordinate functions ( the so-called residuals) f' : R " —• R , (i = 1 , . . . , m ) , and || • || denotes the Euclidean norm (on R m ) . We assume ƒ...

We introduce an entropy of the form −Tr(ρnlogρ) to investigate quantum mechanical properties, like entanglement, for mixed states. We compare it to other quantum entropies where the power of the density matrix has been used to obtain more information than the Von Neumann entropy. We discuss both the pure state case and the mixed state case (Werner state) with special regard to separability. The...

A graph G = (V,E) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : V (G) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ V (G), xy ∈ E(G), and the total number of 0, 1 and 2 are balanced. That is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). In thi...

This paper establishes an apparently overlooked relationship between the iv yiV p'air of fourth order line,r equations y p(x)y 0 and + p(x)y 0, where p is , positive, continuous function defined on [O,m). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equatio...

We consider the algebraic curve defined by $y^m = f(x)$ where $m \geq 2$ and $f(x)$ is a rational function over $\mathbb{F}_q$. We extend the concept of pure gap to {\bf c}-gap and obtain a criterion to decide when an $s$-tuple is a {\bf c}-gap at $s$ rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

We consider the case where a parameter, ; is estimated by maximizing a criterion function, Q(X ; ). The estimate is then used to evaluate the criterion function with the same data, X , as well as with an independent data set, Y. The in-sample …t and out-of-sample …t relative to that of 0; the “true”parameter, are given by Tx;x = Q(X ; ̂x) Q(X ; 0) and Ty;x = Q(Y; ̂x) Q(Y; 0). We derive the limi...