scholarly journals Uniform bounds on sup-norms of holomorphic forms of real weight

2016 ◽  
Vol 12 (05) ◽  
pp. 1163-1185 ◽  
Author(s):  
Raphael S. Steiner
Keyword(s):  
Modular Forms ◽  
Orthonormal Basis ◽  
Fourier Coefficients ◽  
Compact Set ◽  
Index Subgroup ◽  
Uniform Bounds ◽  
Order Of Magnitude ◽  
The Right ◽  
Finite Index Subgroup ◽  
Right Order

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight [Formula: see text] with respect to a finite index subgroup [Formula: see text] of SL2(ℤ). We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis [Formula: see text] and analyzing this sum by means of a Bergman kernel and the Fourier coefficients of Poincaré series. Under some weak assumptions, we further prove the right order of magnitude of [Formula: see text]. Our results are valid without any assumption that the forms are Hecke eigenfunctions.

scholarly journals Berechnung der Oberfläche von Kernmaterie mit Hilfe spezieller Einteilchen-Wellenfunktionen

1962 ◽  
Vol 17 (8) ◽  
pp. 640-649
Author(s):  
Franz Lanzl
Keyword(s):  
Surface Energy ◽  
Plane Waves ◽  
Yukawa Interaction ◽  
Range Interaction ◽  
Zero Range ◽  
Volume Energy ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order ◽  
Surface Thickness

Using a YUKAWA interaction between the nucleons the volume energy per particle and the specific surface energy are calculated. These expressions assume a very simple form for a zero range interaction. The space parts of the applied orthogonal single particle wave functions are products of plane waves and a momentum independent function which brings about the decrease in density within the surface layer. First the volume energy per particle as a function of the density in the interior of nuclear matter is minimized. The interaction parameters are so selected that the values of the volume energy per particle and the density at the minimum agree with the experimental data. Then the surface energy as a function of the surface thickness is minimized. The values of the surface energy and the surface thickness in the minimum are compared with the empirical data. They are found to be of the right order of magnitude

Nucleon-Nucleon Scattering Calculations in a Pole Regularized Spinor Theory

1974 ◽  
Vol 29 (7) ◽  
pp. 981-990
Author(s):  
K. Dammeier
Keyword(s):  
Quantum Theory ◽  
Cross Section ◽  
Test Object ◽  
Nucleon Nucleon ◽  
Coupling Constant ◽  
Nonlinear Spinor ◽  
Spinor Theory ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order

A pole regularized nonlinear spinor theory may be a suitable test object to compare scattering calculations of Stumpf's functional quantum theory with LSZ-results. To apply the LSZ-technique in this theory, a dressing of the occurring massless Green's function is necessary. It is shown which special approximations allow for this dressing. The renormalized nucleon-nucleon coupling constant yields the right order of magnitude for the elastic nucleon cross section.

scholarly journals Generation of Coronal Currents by the Solar Convection Zone

2001 ◽  
Vol 18 (4) ◽  
pp. 329-335 ◽  
Author(s):  
D. J. Galloway ◽  
Y. Uchida ◽  
N. O. Weiss
Keyword(s):  
Convection Zone ◽  
Energy State ◽  
Minimum Energy ◽  
Solar Convection Zone ◽  
Solar Convection ◽  
Poynting Flux ◽  
Available Energy ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order

AbstractSolar flares are thought to be caused by reconnection of magnetic fields and their associated electric currents in the solar corona. The currents have to be there to provide available energy over and above the current-free minimum energy state, but what generates them has been little discussed. This paper investigates the idea that twisting motions in the turbulent convection zone below may provide a natural source for the currents and explain some of their properties. The twists generate upward-propagating Alfvén waves with a Poynting flux of the right order of magnitude to power a flare. Depending on the depth it takes place, the twisting event that initiates a particular flare may occur hours, days or even months before the flare itself.

Dielectric relaxation of dihalobenzenes. I. Ortho and meta Cl2, Br2, and I2 in benzene solution

1968 ◽  
Vol 46 (24) ◽  
pp. 2745-2748 ◽  
Author(s):  
Abhai Mansingh ◽  
David B. McLay
Keyword(s):  
Benzene Solution ◽  
Relaxation Times ◽  
Dielectric Constants ◽  
Cole Plot ◽  
Molecular Reorientation ◽  
Dielectric Data ◽  
Distribution Parameters ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order

Dielectric data have been measured for the dilute solutions in benzene at 20.0 °C of the ortho- and meta-isomers of dichlorobenzene, dibromobenzene, and diiodobenzene. The "static" dielectric constants have been measured at 100 kHz, the dielectric constants and losses have been measured at both 9.06 and 21.00 GHz, and the refractive indices have been measured at optical wavelengths. Cole–Cole plots can be fitted to the data to yield mean relaxation times τ0 and distribution parameters α. The values of the relaxation times in the ortho-isomers are 11.8, 14.9, and 20.6 ps for dichlorobenzene, dibromobenzene, and diiodobenzene respectively. The corresponding values for the meta-isomers are 8.6, 10.7, and 13.5 ps respectively, values which increase in the same direction with halogen substituent but which are significantly smaller than the relaxation times for the other isomers. All of these times are of the right order of magnitude for molecular reorientation and there is no evidence for dipole–dipole interactions. Although the nonzero values of the distribution parameters will allow mathematical descriptions in terms of two relaxation times τ1 and τ2, the values derived from two such descriptions yield two unrealistic relaxation times for each molecule. It is concluded that the analysis based on the Cole–Cole plot gives the most meaningful results.

scholarly journals POTENTIAL, IMPEDANCE, AND RECTIFICATION IN MEMBRANES

1943 ◽  
Vol 27 (1) ◽  
pp. 37-60 ◽  
Author(s):  
David E. Goldman
Keyword(s):  
Membrane Potential ◽  
Constant Electric Field ◽  
Kinetic Equations ◽  
Giant Axon ◽  
Membrane Structures ◽  
Ion Motion ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order ◽  
Good Agreement

Impedance and potential measurements have been made on a number of artificial membranes. Impedance changes were determined as functions of current and of the composition of the environmental solutions. It was shown that rectification is present in asymmetrical systems and that it increases with the membrane potential. The behavior in pairs of solutions of the same salt at different concentrations has formed the basis for the studies although a few experiments with different salts at the same concentrations gave results consistent with the conclusions drawn. A theoretical picture has been presented based on the use of the general kinetic equations for ion motion under the influence of diffusion and electrical forces and on a consideration of possible membrane structures. The equations have been solved for two very simple cases; one based on the assumption of microscopic electroneutrality, and the other on the assumption of a constant electric field. The latter was found to give better results than the former in interpreting the data on potentials and rectification, showing agreement, however, of the right order of magnitude only. Although the indications are that a careful treatment of boundary conditions may result in better agreement with experiment, no attempt has been made to carry this through since the data now available are not sufficiently complete or reproducible. Applications of the second theoretical case to the squid giant axon have been made showing qualitative agreement with the rectification properties and very good agreement with the membrane potential data.

Fourier coefficients of cusp forms

1986 ◽  
Vol 100 (1) ◽  
pp. 5-29 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Field Of Study ◽  
Important Paper ◽  
Present Survey ◽  
Arithmetical Functions ◽  
Survey Article ◽  
Order Of Magnitude

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society's Proceedings [32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, the Transactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in the Proceedings, it seems appropriate to include further developments in this field of study in the present survey.

The Coastal Sea Breeze in Relation to Diurnal Temperature Changes in the Lower Atmosphere (Southern California)

1947 ◽  
Vol 28 (8) ◽  
pp. 371-380
Author(s):  
Luna B. Leopold ◽  
Charles G. P. Beer
Keyword(s):  
Los Angeles ◽  
Vertical Motion ◽  
Sea Breeze ◽  
Lower Atmosphere ◽  
Los Angeles Basin ◽  
Temperature Changes ◽  
Coastal Sea ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order

A diurnal variation of temperature at constant levels in the lower atmosphere was observed at all stations in the Los Angeles basin area. The amplitude of the variation increases with height. This variation has been studied as a diurnal oscillation in the height of isentropic surfaces. The rise and fall of isentropic surfaces may indicate an appreciable diurnal change of vertical motion, but in the higher levels, are at least partly explained by errors in measurement of temperatures by radiosondes. In levels below 4,000 feet msl vertical motion of the right order of magnitude could be caused by the diurnal accumulation and depletion of air as a result of the sea-breeze regime.

scholarly journals The constants of the magnetic dispersion of light

1927 ◽  
Vol 114 (768) ◽  
pp. 474-490 ◽  
Keyword(s):  
Wave Length ◽  
Spectroscopic Behaviour ◽  
Optical Effects ◽  
Other Substances ◽  
Order Of Magnitude ◽  
University Of Edinburgh ◽  
The Individual ◽  
The Right ◽  
The University ◽  
Right Order

1. The rotation of the plane of polarisation of light by a magnetic field provides perhaps one of the easiest approaches to a study of the spectroscopic behaviour of ordinary substances. The present work is an analysis of the available measures of the dispersion of this magnetic gyration. Several formulæ have been proposed for it, and these will be reviewed below, but it has proved most convenient to take one of them and use that; afterwards testing, to the rather limited extent possible, how far others would fit the facts. The test is made with the formula given by Becquerel in 1897, V = e /2 mc 2 λ dn/d λ , (1.1) where n is the refractive index, λ the wave-length and V is Verdet’s constant, the rotation of the plane of polarisation per centimetre per gauss. If this formula fits, and we shall find that it does, it should give the value of e/m . A few values were worked out by Becquerel himself on the rather meagre data available at that time, and later Siertsema obtained more accurate values for some other substances. All these gave e/m roughly of the right order of magnitude, and this fact was duly noted in the text-books and has been copied from one to another ever since, but usually without giving any numerical values at all. Since Larmor’s theorem fails to hold for molecules we should hardly expect to find the ordinary value, but nevertheless it seemed useful to analyse all the experimental measures available, so as to discover if any regularity would emerge. In the physical journals there are several results of this type for particular substances, but they are very much scattered, and it should prove convenient to collect together an analysis of all the substances for which the gyration has been measured. We are not attempting any deep theory of the matter, but merely a convenient summary which may prove useful when the time comes for a proper theory of the spectroscopic behaviour of ordinary substances of the type that has been so successful for monatomic gases and vapours. It is outside the scope of the present work to discuss the behaviour of the gyration of light of frequency very close to opaque bands ; this has been the subject of many experiments, but they are not by any means concordant, and take us deeper into the unknown theory than it is possible at present to go. To avoid this trouble we have limited ourselves to transparent substances, that is, to regions of the spectrum far from the bands which cause the optical effects. The data have been extracted from Landolt and Bornstein’s tables (edition 1921), in some cases supplemented from the original sources, and it has not, of course, been possible to assess the merit of each of the individual measures recorded. In a few cases there are measures of the gyration but not of the refraction; and we are greatly indebted to Dr. I. C. Somerville of the Chemical Department of the University of Edinburgh for measuring some of these refractive indices for us.

scholarly journals Comparison of Methods for Quantification of Topologically Close-Packed Phases in Ni-Based Superalloys

2019 ◽  
Vol 50 (12) ◽  
pp. 5925-5934 ◽  
Author(s):  
A. S. Wilson ◽  
K. A. Christofidou ◽  
A. Evans ◽  
M. C. Hardy ◽  
H. J. Stone
Keyword(s):  
X Ray Diffraction ◽  
Solid Samples ◽  
X Ray ◽  
Topologically Close Packed ◽  
Polycrystalline Alloy ◽  
Order Of Magnitude ◽  
Comparison Of The Results ◽  
The Right ◽  
Right Order

Abstract The ability to quantify accurately the formation of topologically close-packed phases in nickel-based superalloys is key to assessing their thermal stability and ensuring that their performances will not deteriorate during long-term exposure at high temperatures. To investigate the effectiveness of synchrotron XRD for the detection of such minority phases in Ni-based superalloys, the commercial polycrystalline alloy RR1000 was analyzed following exposures of varying times at 800 °C. Data were collected from both solid samples and extracted residues, and additional laboratory X-ray diffraction was performed on the residues. The minor phases were successfully detected in solid samples using synchrotron radiation, and a comparison of the results from these quantification methods shows that the extraction method gives results of the right order of magnitude to reflect the phase quantities that are present in the alloy. However, the results indicate that the synchrotron route is not a suitable method for the quantification of phases present in quantities less than approximately 0.3 wt pct.

scholarly journals Wave resistance: the effect of varying draught

1925 ◽  
Vol 108 (748) ◽  
pp. 582-591 ◽  
Keyword(s):  
Wave Resistance ◽  
Plane Section ◽  
Experimental Results ◽  
Interference Effects ◽  
Different Types ◽  
Order Of Magnitude ◽  
The Right ◽  
Right Order ◽  
Considerable Range ◽  
Simplified Form

1, In previous studies in the theory of wave resistance, while the water-plane section of the model was of a reasonably ship-like form, the draught was assumed to be infinite. In the following paper the model has the same simple lines and has vertical sides, but the draught is finite. The investigation shows how the resistance at different speeds depends on the draught, but it was undertaken specially for other reasons. In view of certain applications, it was important to find how the interference effects due to bow and stern waves are affected by varying draught. It is shown now that these become less prominent with diminishing draught, but the maxima and minima occur at practically the same positions. Further, when the ratio of draught to length is of the order of the values in actual ship models, one is in a position to attempt a comparison between the absolute values of theoretical and experimental results. Curves are shown in fig. 2 (p. 590) for the variation of resistance with velocity in three cases—when the draught is infinite, and when it is one-tenth and one-twentieth of the length of the model. The latter values cover approximately the usual ratios in practice. On the same diagram are reproduced experimental curves for three models of different types, the data being reduced to the same non-dimensional co-ordinates. Making allowance for the differences of form between these models and for the simplified form for which the calculations have been made, the results show that the calculated values are of the right order of magnitude over a considerable range of velocity. Differences in the two sets of curves, such as the greater prominence of interference effects in the theoretical curves, are discussed.

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