We have obtained chopped near-infrared maps at 1.25, 1.65 and 2.2 micron in the Serpens dark cloud (S68) centered on IRS3. They show unresolved structure, most likely due to illumination of scattering grains, out to 1´ from the center. The combination of high reddening (and 3.07 m absorption) and asymmetrical optical reflection points to a non-spherical dust distribution typical for
bi-polar flow sources. In addition to the maps at 3,H,K, which consist of 19 by 47 pixells spaced by 2, we are using (Gunn r. and z) CCD images taken at Palomar by Bel Campbell. We are developing software to process these and similar intrared maps using a modified maximum entropy technique. The modifications have been necessary, to avoid singularities inherent in the standard maximum entropy algorithm when it is applied to data with zerovolume point-spread functions.
An operator T acting on a Banach space X possesses property (gw) if σa(T) \ σSBF− + (T) = E(T), where σa(T) is the approximate point spectrum of T, σSBF− + (T) is the essential semi-B-Fredholm spectrum of T and E(T) is the set of all isolated eigenvalues of T. In this paper we introduce and study two new properties (b) and (gb) in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if T is a bounded linear operator acting on a Banach space X, then property (gw) holds for T if and only if property (gb) holds for T and E(T) = Π(T), where Π(T) is the set of all poles of the resolvent of T.
It is shown that $n$ times Peano differentiable functions defined on a closed subset of $\mathbb{R}^m$ and satisfying a certain condition on that set can be extended to $n$ times Peano differentiable functions defined on $\mathbb{R}^m$ if and only if the $n$th order Peano derivatives are Baire class one functions.
An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau $-extending module, where $\tau $ is a hereditary torsion theory on $\mathop {\text{Mod}}$-$R$. An $R$-module $M$ is called type 2 $\tau $-extending if every type 2 $\tau $-closed submodule of $M$ is a direct summand of $M$. If $\tau _I$ is the torsion theory on $\mathop {\text{Mod}}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 $\tau _I$-extending $R$-module, then the question of whether or not $M/MI$ is an extending $R/I$-module is investigated. In particular, for the Goldie torsion theory $\tau _G$ we give an example of a module that is type 2 ${\tau }_G$-extending but not extending.
We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.
In this paper we contrast linear parametric estimation with non-parametric non-linear neural estimation of the reversion speed a, in the context of the Vasicek model, which is routinely being used for deriving the term structure of the short rate. The sampling parameters of the short-rate, even its realization, were varied widely. Neural regression was employed in an attempt to identify a possibly non-linear relationship, and from that to extract a measure of instantaneous reversion speed (a local equivalent of reversion speed). Neural network models outperformed consistently the linear estimator in ternis of explained variability by more than 10%, indicating a degree of non-linearity in the underlying relationship.
We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. As an application, we prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.
We present a categorical approach to the extension of probabilities, i.e. normed σ-additive measures. J. Novák showed that each bounded σ-additive measure on a ring of sets A is sequentially continuous and pointed out the topological aspects of the extension of such measures on A over the generated σ-ring σ(A): it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space X over its Čech-Stone compactification βX (or as the extension of continuous functions on X over its Hewitt realcompactification υX). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that σ(A) is the sequential envelope of A with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category ID of D-posets of fuzzy sets (such D-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on A over σ(A) is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems.