Farm and home—jams and jellies for exhibition

Most housewives make excellent jams and jellies, but when it comes to exhibiting J-Ti they find there are many finer points which have to be considered. Taken separately these points are relatively unimportant but taken collectively they are sufficient to prevent an exhibitor from gaining an award

Approximating the inverse of a symmetric positive definite matrix

AbstractIt is shown for an n × n symmetric positive definite matrix T = (ti, j with negative off-diagonal elements, positive row sums and satisfying certain bounding conditions that its inverse is well approximated, uniformly to order l/n2, by a matrix S = (si, j), where si,j = δi,j/ti,j + 1/t.., δi,j being the Kronecker delta function, and t.. being the sum of the elements of T. An explicit bound on the approximation error is provided

Uhors'ke skladene majbutnê v areal'nomu aspektì

The рарег provides additional arguments in favor of the assumption cropped up in the scientific literature that the Hungarian periphrastic future with the auxiliary verb fog ‘take’ is of contact origin and was calqued of a similar construction in the Ukrainian and some other Slavic languages and dialects with verbs continuing Common Slavic *(j)?ti ‘take’

Asymptotic law of likelihood ratio for multilayer perceptron models

We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The data are assumed to be generated by a true MLP model and the estimation of the parameters of the MLP is done by maximizing the likelihood of the model. When the number of hidden units of the true model is known, the asymptotic distribution of the maximum likelihood estimator (MLE) and the likelihood ratio (LR) statistic is easy to compute and converge to a $\chi^2$ law. However, if the number of hidden unit is over-estimated the Fischer information matrix of the model is singular and the asymptotic behavior of the MLE is unknown. This paper deals with this case, and gives the exact asymptotic law of the LR statistics. Namely, if the parameters of the MLP lie in a suitable compact set, we show that the LR statistics is the supremum of the square of a Gaussian process indexed by a class of limit score functions.Comment: 19 page

On McMullen-like mappings

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of $f_{\lambda}$ belong to the trap door. We generalize this behavior defining a McMullen-like mapping as a rational map $f$ associated to a hyperbolic postcritically finite polynomial $P$ and a pole data $\mathcal{D}$ where we encode, basically, the location of every pole of $f$ and the local degree at each pole. In the McMullen family, the polynomial $P$ is $z\mapsto z^n$ and the pole data $\mathcal{D}$ is the pole located at the origin that maps to infinity with local degree $d$. As in the McMullen family $f_{\lambda}$, we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial $P$ and the pole data $\mathcal{D}$. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery.Comment: 21 pages, 2 figure

Anisotropic diffusion in continuum relaxation of stepped crystal surfaces

We study the continuum limit in 2+1 dimensions of nanoscale anisotropic diffusion processes on crystal surfaces relaxing to become flat below roughening. Our main result is a continuum law for the surface flux in terms of a new continuum-scale tensor mobility. The starting point is the Burton, Cabrera and Frank (BCF) theory, which offers a discrete scheme for atomic steps whose motion drives surface evolution. Our derivation is based on the separation of local space variables into fast and slow. The model includes: (i) anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps; (ii) diffusion of atoms along step edges; and (iii) attachment-detachment of atoms at step edges. We derive a parabolic fourth-order, fully nonlinear partial differential equation (PDE) for the continuum surface height profile. An ingredient of this PDE is the surface mobility for the adatom flux, which is a nontrivial extension of the tensor mobility for isotropic terrace diffusion derived previously by Margetis and Kohn. Approximate, separable solutions of the PDE are discussed.Comment: 14 pages, 1 figur