New Type Of Diagram Research Articles

Stochastic quantization of Einstein gravity.

We determine a one-parameter family of covariant Langevin equations for the metric tensor of general relativity corresponding to DeWitt's one-parameter family of supermetrics. The stochastic source term in these equations can be expressed in terms of a Gaussian white noise upon the introduction of a stochastic tetrad field. The only physically acceptable resolution of a mathematical ambiguity in the ansatz for the source term is the adoption of Ito's calculus. By taking the formal equilibrium limit of the stochastic metric a one-parameter family of covariant path-integral measures for general relativity is obtained. There is a unique parameter value, distinguished by any one of the following three properties: (i) the metric is harmonic with respect to the supermetric, (ii) the path-integral measure is that of DeWitt, (iii) the supermetric governs the linearized Einstein dynamics. Moreover the Feynman propagator corresponding to this parameter is causal. Finally we show that a consistent stochastic perturbation theory gives rise to a new type of diagram containing ``stochastic vertices.''

Turing's conditions and the analysis of morphogenetic models

The conditions under which changes in parameters lead to changes in the pattern-generating behaviour of Turing's two-morphogen linear model can be expressed in terms of two reduced rate constants, k′ 1 and k′ 4 which represent autocatalytic and self-inhibitory rates in relation to cross-catalysis and cross-inhibition, and the ratio of the diffusivities for the two morphogens. This allows a new type of diagram to be drawn in which a two-dimensional k′ 1 k′ 4 space is divided by Turing's conditions into regions where the various morphogenetic behaviours occur. An analysis using this type of diagram is applied to the linear limit of two non-linear models, those of Gierer and Meinhardt and of Tyson's modification of the Brusselator, and is used to clarify what is happening in their non-linear development. Several possible applications are mentioned; to the stability of a simple binary pattern in slime moulds, to the development and decay of a succession of patterns in the imaginal wing discs of Drosophila as treated by Kauffman et al., and to the apparent ease of disturbing the sea urchin blastula to produce a three-part, rather than a two-part pattern.

An Analysis of the Relation between Grain Growth and Recrystallization Based on Experimental Boundary Mobilities

AbstractThe rate of migration of grain boundaries was studied in two copper-cadmium alloys during recrystallization as well as during normal grain growth. It was found that the two processes can be described using the same grain-boundary mobility. The recrystallization experiments were carried out with various degreesof deformation and with various annealing times and temperatures. Recrystallization diagrams can be rationalized using basic laws of recrystallizationand grain growth. A new type of diagram is presented in which a combination of the time and temperature is plotted along one of the axes. A characteristic degree of deformation is defined below which the stored energy is removed by grain growth rather than by recrystallization. This concept is related to but not identical with, that usually called the criticai deformation.

THE PALAEOECOLOGY OF THE FAUNA OVER THE PARK MINE (WESTPHALIAN B) NEAR WIGAN, LANCASHIRE

Summary The cyclothem above the Park Mine (Lower similis-pulchra Zone) at the Amberswood opencast site near Wigan is composed of mudstones which can be divided into two parts of almost equal thickness. The lower half, dark grey in colour, contains abundant bivalves, mostly referable to the genera Anthracosia and Naiadites. The upper part is light grey and contains macroscopic plant debris, bivalves only occurring at the base. The variation shown by six assemblages of Anthracosia is demonstrated by means of a new type of diagram. The degree of water turbulence is determined by studying the percentage disarticulation, the orientation of shells, and the lithology of the sediments. It is concluded that an alteration in the growth pattern of Anthracosia and a shift of the entire size range to smaller values correlate with the change from a sulphate zone at the base to a carbonate zone above, together with increasing coarseness in texture and reduced carbon content. All the changes in fauna and sediment may be brought about by the gradual filling and freshening of a lagoon.

Graphical Aids in Interpreting the Performance of Crystal Transducers

When the mechanical damping of a transducer is large, as by acoustic radiation from one or both faces into a liquid or solid, the circular diagram that represents its characteristics requires special treatment. As a background for this treatment, the uses and limitations of the conventional circle for a resonator with small losses is first reviewed. The problem of the transducer with large losses is then considered with special reference to the equations and graphs for a thickness-type transducer with unsymmetrical loading. For plane-wave transducers the expressions are exact for all loads and at all frequencies, including harmonics. Either the voltage or the current may be constant. From the admittance or impedance diagrams the magnitude and phase of current, voltage, particle velocity, and vibrational amplitude at any frequency can be obtained immediately. Similar results would be found with plates in lengthwise vibration. A new type of diagram is developed for representing vibrational amplitudes. As an illustration, the case of a quartz plate radiating into three liquids of widely different acoustic properties is treated. When the load is unsymmetrical, there is no true node anywhere in the crystal except when the load is zero or infinity. There is, however, a plane of minimal vibration, the amplitude and location of which are derived. The equations indicate certain peculiar effects when the specific acoustic resistance of the medium is just twice that of the crystal.

Our Present Dilemma Regarding the Values of the Natural Constantse,mandh. A New Graphical Method of Presentation

Fourteen experiments each of which determines some function of one or more of the constants $e$, $m$ and $h$, are classified, according to the function determined, into nine groups. The latter number is reduced to seven by the elimination of five of the thirteen experiments as unreliable in the light of our present knowledge. Three cardinally good experiments, the directly measured x-ray values of $e$, the many concordant measurements of $\frac{e}{m}$ and the values of $\frac{h}{e}$ from the limit-of the continuous x-ray spectrum are exhibited on a new type of diagram showing separately results of all independent reliable determinations to portray the consistencies graphically. The indubitable nature of the discrepancy is thus made evident and the remaining four experimental values also plotted on the diagram seem to give better support to the first two named cardinal experiments than to the last. The new type of diagram is an isometric projection of a cartesian coordinate system whose axes represent the relative deviations of $e$, $m$ and $h$ from conventionally assumed values. Planes in this space represent the different experimental determinations of functions of $e$, $m$ and $h$ and the discussion brings out the absolute geometric property that five of these planes are cozonal (parallel to a common axis). Though there are seven equations it thus appears that to solve for $e$, $m$ and $h$ either the direct $e$, equation or alternately the ${R}_{\ensuremath{\infty}}$ equation is indispensable.The results are briefly recounted of a careful examination both by the author and others of numerous possible experimental and theoretical sources of the discrepancy. Modified diagrams are shown for (a) the assumption that the velocity of propagation of radiation in vacuum suffers a slow decline with increasing frequency (the velocity for 20,000-volt x-rays is assumed to be \textonequarter{} percent lower than the low frequency optical value); (b) the alternative assumption that the Bohr-Rydberg formula must be revised to include a factor ($1\ensuremath{-}\ensuremath{\alpha}$). Both assumptions rectify the diagram as regards the three cardinal experiments but throw the remaining experiments out of line with them. Since the two assumptions named are about equally objectionable both on theoretical grounds and in their results on the diagram they are now rejected as unlikely. The author concludes that in all probability some unsuspected theoretical or experimental flaw in the determination of $\frac{h}{e}$ by the continuous x-ray spectrum method is the source of the discrepancy and emphasizes the need for renewed study of this experiment along with x-ray ionization and excitation potentials over a wider range of voltages with better spectral resolution. The weakness of low voltage measurements of ionization and excitation potentials is pointed out. It is emphasized that when systematic error is as glaringly evident as at present the temptation to obtain compromise or best values of the natural constants by least-squares methods should be strongly resisted for it is not inconceivable that the discrepancy may reveal some important error of principle or theory.

Societies and Academies

Societies and Academies