Beurling's criterion and extremal metrics for Fuglede modulus

We formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede p-modulus of a system of measures. When p=2, this characterization generalizes Beurling's criterion, a sufficient condition for an admissible metric to be extremal for the extremal length of a planar curve family. In addition, we prove that every non-negative Borel function in R^n with positive and finite p-norm is extremal for the p-modulus of some curve family.Comment: 12 pages, 2 figures (version 4: minor improvements, updated numbering and corrected typos, final version

A new sputtering apparatus

The writer, in sputtering a large number of mirrors, has found that the apparatus and methods to be described have several advantages, particularly where very large and perfect mirrors are desired. The accompanying diagram will be self explanatory, and the dimensions given have been found convenient for mirrors up to 10 cm diameter

Note on the Spectra of the Disubstituted Acetylenes and of the Mustard Oils

It has been observed that in the Raman spectra of many disubstituted acetylenes the line corresponding to the so-called C≡C frequency is split into two components of not greatly different intensity (1), though this is not the case in the monosubstituted compounds. In the mustard oils the C≡N Raman line is double (2), and in allyl mustard oil (3), the one substance which appears to have been studied with sufficient dispersion, the corresponding infrared band shows definite indications of splitting into two components. This doubling is not observed in the organic thiocyanates. These facts have caused some speculation so far appear never to have received a satisfactory explanation. The presence of two fundamentals in the 2200 cm^-1 region appears quite impossible for most of the molecules showing the doubling, and on the other hand the coexistence of two forms of molecule with slightly different frequencies appears unlikely, especially in the case of the acetylenes. Consequently the author proposes an explanation involving a Fermi resonance interaction between the C≡C or C≡N fundamental, as the case may be, and the overtone of a certain other vibration of approximately one-half or one-third the frequency

The Relation Between the Energy of a Hydrogen Bond and the Frequencies of the O[Single Bond]H Bands

It has previously been pointed out (1) that there appears to be a relation between the energy of a hydrogen bond and the shift of the frequency of the O-H bands which accompanies the formation of the bond. Recently additional data have been obtained in this laboratory (2,3) which confirm the existence of this useful relation and it seems worth while to discuss them in this connection. The data here presented all relate to linkages in which a hydroxyl hydrogen is concerned, though the atom to which it is weakly bound may be oxygen, chlorine, or carbon in an aromatic ring. It is surprising that the same connection between energy and frequency shift should be found in these various cases but such appears to be the fact. For other types of bond, for example those involving the hydrogen of an amino group, a slightly different relation may well be expected

Two devices facilitating spectrometry in the far infrared

In spectrometric investigations in the infrared two principal difficulties are the extremely small energy available in the long waves, and the relatively great intensity of the near infrared. This latter is of particular importance in using an echlette grating which reflects the short waves with great intensity in high orders. The arrangement here described was devised to overcome both of these inconveniences

Some Aspects of Printing Quality

The object of this experiment was to find the effect of various pressures upon the ink coverage and the visual appearance of the half-tone print. A specially designed plate with four evenly spaced elevations was used with a Vandercook proof press. Various grades of paper were used. Results showed: a variation in the range of ink coverage due to the grade of paper and the number of lines of the half-tone; within the area of good printing a small change in pressure caused a large change in ink coverage; the change in ink coverage varied from elevation to elevation; an excess of ink was used in this experiment; the optimum printing pressure might have fallen between elevations; and the nature and/or quality of the data gathered did not at this time lend itself to mathematical analysis

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^n$ ($n\geq 4$) has upper Minkowski dimension at most $n-4$.Comment: 52 pages, 5 figure

Multiscale analysis of 1-rectifiable measures II: characterizations

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an $L^2$ gauge the extent to which $\mu$ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between $\mu$ and 1-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an $L^2$ variant of P. Jones' traveling salesman construction, which is of independent interest.Comment: 47 pages, 4 figures (v3: added/updated figures, new Remarks 2.1, 4.6, 5.8, minor improvements, final version