Uniform Harmonic Approximation with Continuous Extension to the Boundary

1988 ◽  
Vol 40 (6) ◽  
pp. 1375-1388 ◽  
Author(s):  
M. Goldstein ◽  
W. H. Ow
Keyword(s):  
Complex Plane ◽  
Nonempty Subset ◽  
Harmonic Approximation ◽  
Continuous Extension ◽  
Partial Derivatives ◽  
Extended Plane ◽  
The One ◽  
Image Position ◽  
One Point Compactification

Let G be a domain in the complex plane and F a nonempty subset of G such that F is the closure in G of its interior F0. We will say f ∊ C1(F) if f is continuous on F and possesses continuous first partial derivatives in F which extend continuously to F as finite-valued functions. Let G* – F be connected and locally connected, f ∊ C1(F) be harmonic in F0, and E be a subset of ∂F ∩ ∂G (here G* denotes the one-point compactification of G and the boundaries ∂F, ∂G are taken in the extended plane). Suppose there is a sequence 〈hn〉 of functions harmonic in G such thatuniformly on F as n → ∞.

A further note on differentials

1939 ◽  
Vol 31 ◽  
pp. iii-iv
Author(s):  
E. G. Phillips
Keyword(s):  
Partial Derivatives ◽  
Independent Variables ◽  
The One ◽  
Image Position

Since the publication of my article on “The advantage of differentials in the technique of differentiation” both Dr H. A. Hayden and Prof. A. Oppenheim have kindly pointed out to me that there is a much shorter solution of the problem by partial derivatives than the one which I gave as Solution 2. The solution is as follows:—If y and z are the independent variables we have, sincewhich gives the result required.

Structure Information From Electron Diffraction Intensities

1990 ◽  
Vol 48 (2) ◽  
pp. 516-517
Author(s):  
J. Gjønnes ◽  
N. Bøe ◽  
K. Gjønnes
Keyword(s):  
Computational Effort ◽  
Unit Cell ◽  
Small Unit ◽  
Structure Information ◽  
Dispersion Surface ◽  
Integrated Intensity ◽  
The One ◽  
Image Position ◽  
Convergent Beam ◽  
Beam Patterns

Structure information of high precision can be extracted from intentsity details in convergent beam patterns like the one reproduced in Fig 1. From low order reflections for small unit cell crystals,bonding charges, ionicities and atomic parameters can be derived, (Zuo, Spence and O’Keefe, 1988; Zuo, Spence and Høier 1989; Gjønnes, Matsuhata and Taftø, 1989) , but extension to larger unit cell ma seem difficult. The disks must then be reduced in order to avoid overlap calculations will become more complex and intensity features often less distinct Several avenues may be then explored: increased computational effort in order to handle the necessary many-parameter dynamical calculations; use of zone axis intensities at symmetry positions within the CBED disks, as in Figure 2 measurement of integrated intensity across K-line segments. In the last case measurable quantities which are well defined also from a theoretical viewpoint can be related to a two-beam like expression for the intensity profile:With as an effective Fourier potential equated to a gap at the dispersion surface, this intensity can be integrated across the line, with kinematical and dynamical limits proportional to and at low and high thickness respctively (Blackman, 1939).

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

scholarly journals Tensile Strength of NaCl Ice

1962 ◽  
Vol 4 (31) ◽  
pp. 25-52 ◽  
Author(s):  
W. F. Weeks
Keyword(s):  
Tensile Strength ◽  
Sea Ice ◽  
Fresh Water ◽  
Eutectic Point ◽  
Water Ice ◽  
Average Value ◽  
Freshwater Ice ◽  
The One ◽  
Image Position ◽  
Bodies Of Water

AbstractTo resolve some of the factors causing strength variation in natural sea ice, fresh water and five different NaCl–H2O solutions were frozen in a tank designed to simulate the one-dimensional cooling of natural bodies of water. The resulting ice was structurally similar to lake and sea ice. The salinity of the salt ice varied from 1‰ to 22‰. Tables of brine volumes and densities were computed for these salinities in the temperature range 0° to −35° C. The ring-tensile strength σ of fresh-water ice was found to be essentially temperature independent from −10° to −30°C., with an average value of 29.6±8.5 kg./cm.2at −10° C. The strength of salt ice at temperatures above the eutectic point (–21.2° C.) significantly decreases with brine volumev;. The σ–axis intercept of this line is comparable to the a values determined for fresh ice indicating that there is little, if any, difference in stress concentration between sea and lake ice as a result of the presence of brine pockets. The strength of ice containing NaCl.2H2O is slightly less than the strength of freshwater ice and is independent of the volume of solid salt and the ice temperature. No evidence was found for the existence of either phase or geometric hysteresis in NaCl ice. The strength of ice at sub-eutectic temperatures, however, is decreased appreciably if the ice has been subjected to temperatures above the eutectic point; this is the result of the redistribution of brine during the warm-temperature period. Short-term cooling produces an appreciable (20 per cent) decrease in strength, in fresh-water and NaCl.2H2O ice. The present results are compared with tests on natural sea ice and it is suggested that the strength of freshwater ice is a limit which is approached but not exceeded by cold sea ice and that the reinforcement of brine pockets by Na2SO4.10H2O is either lacking or much less than previously assumed.

On BMOA for Riemann Surfaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1255-1260 ◽  
Author(s):  
Thomas A. Metzger
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Hardy Class ◽  
Image Position

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if(1)whereIt is known (see [3, p. 96]) that (1) is equivalent to(2)

scholarly journals Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fatemah Ayatollah Zadeh Shirazi ◽  
Meysam Miralaei ◽  
Fariba Zeinal Zadeh Farhadi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
The One ◽  
One Point Compactification

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P∈Ξ and X∈P, the one point compactification of topological space X belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T1, TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

Radicals of PID's and Dedekind Domains

1972 ◽  
Vol 24 (4) ◽  
pp. 566-572 ◽  
Author(s):  
R. E. Propes
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Radical Class ◽  
Dedekind Domains ◽  
Radical Ideals ◽  
The One ◽  
Image Position ◽  
Principal Ideal Domains

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, ifthen T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

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