Legendre Functions of Large Degree: Real Arguments

1997 ◽  
pp. 481-487
Keyword(s):  
Large Degree ◽  
Legendre Functions

The asymptotic expansion of legendre function of large degree and order

1957 ◽  
Vol 249 (971) ◽  
pp. 597-620 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Variable ◽  
Bessel Functions ◽  
Legendre Function ◽  
Large Degree ◽  
Legendre Functions ◽  
Airy Functions ◽  
Exponential Functions ◽  
The Third ◽  
Shaped Domain

New expansions for the Legendre functions and are obtained; m and n are large positive numbers, is kept fixed as is an unrestricted complex variable. Three groups of expansions are obtained. The first is in terms of exponential functions. These expansions are uniformly valid as with respect to z for all z lying in except for the strips given by. The second set of expansions is in terms of Airy functions. These expansions are uniformly valid with respect to z throughout the whole z plane cut from except for a pear-shaped domain surrounding the point z = — 1 and a strip lying immediately below the real z axis for which . The third group of expansions is in terms of Bessel functions of order m . These expansions are valid uniformly with respect to z over the whole cut z plane except for the pear-shaped domain surrounding z = — 1. No expansions have been given before for the Legendre functions of large degree and order.

Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells

1965 ◽  
Vol 32 (3) ◽  
pp. 553-561 ◽  
Author(s):  
E. W. Ross
Keyword(s):  
Natural Frequencies ◽  
Large Degree ◽  
Mode Shapes ◽  
Asymptotic Formulas ◽  
Axisymmetric Vibration ◽  
Legendre Functions ◽  
Approximate Analysis ◽  
Axially Symmetric ◽  
Spherical Shells ◽  
Approximate Formulas

This paper contains the results of an approximate analysis of the axially symmetric vibrations of deep spherical elastic shells. The approximation is based on the asymptotic formulas for Legendre functions of large degree. The results are relatively simple approximate formulas for the natural frequencies and mode shapes under a variety of boundary conditions at the shell edge. The results agree very well with values obtained by Kalnins, using numerical methods.

Integrals with a large parameter: Legendre functions of large degree and fixed order

1984 ◽  
Vol 95 (2) ◽  
pp. 367-380 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Asymptotic Approximation ◽  
Large Degree ◽  
Legendre Functions ◽  
Large Parameter ◽  
Numerical Work ◽  
Fixed Order ◽  
The Difference

AbstractSuppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.

Characterization of homoepitaxial diamond films by Electron microscopy

1991 ◽  
Vol 49 ◽  
pp. 880-881
Author(s):  
D.P. Malta ◽  
S.A. Willard ◽  
R.A. Rudder ◽  
G.C. Hudson ◽  
J.B. Posthill ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Surface Defects ◽  
Substrate Surface ◽  
Diamond Films ◽  
Chemical Vapor ◽  
Polycrystalline Diamond ◽  
Large Degree ◽  
Rf Plasma ◽  
Semiconducting Diamond

Semiconducting diamond films have the potential for use as a material in which to build active electronic devices capable of operating at high temperatures or in high radiation environments. A major goal of current device-related diamond research is to achieve a high quality epitaxial film on an inexpensive, readily available, non-native substrate. One step in the process of achieving this goal is understanding the nucleation and growth processes of diamond films on diamond substrates. Electron microscopy has already proven invaluable for assessing polycrystalline diamond films grown on nonnative surfaces.The quality of the grown diamond film depends on several factors, one of which is the quality of the diamond substrate. Substrates commercially available today have often been found to have scratched surfaces resulting from the polishing process (Fig. 1a). Electron beam-induced current (EBIC) imaging shows that electrically active sub-surface defects can be present to a large degree (Fig. 1c). Growth of homoepitaxial diamond films by rf plasma-enhanced chemical vapor deposition (PECVD) has been found to planarize the scratched substrate surface (Fig. 1b).

Using Convolutional Neural Networks to measure the contribution of visual features to the representation of object animacy in the brain

2019 ◽  
Author(s):  
Sushrut Thorat
Keyword(s):  
Neural Networks ◽  
Convolutional Neural Networks ◽  
Temporal Cortex ◽  
Large Degree ◽  
Visual Features ◽  
Visual Feature ◽  
Visual Stream ◽  
Feature Information ◽  
Ventral Visual Stream ◽  
Animal Images

A mediolateral gradation in neural responses for images spanning animals to artificial objects is observed in the ventral temporal cortex (VTC). Which information streams drive this organisation is an ongoing debate. Recently, in Proklova et al. (2016), the visual shape and category (“animacy”) dimensions in a set of stimuli were dissociated using a behavioural measure of visual feature information. fMRI responses revealed a neural cluster (extra-visual animacy cluster - xVAC) which encoded category information unexplained by visual feature information, suggesting extra-visual contributions to the organisation in the ventral visual stream. We reassess these findings using Convolutional Neural Networks (CNNs) as models for the ventral visual stream. The visual features developed in the CNN layers can categorise the shape-matched stimuli from Proklova et al. (2016) in contrast to the behavioural measures used in the study. The category organisations in xVAC and VTC are explained to a large degree by the CNN visual feature differences, casting doubt over the suggestion that visual feature differences cannot account for the animacy organisation. To inform the debate further, we designed a set of stimuli with animal images to dissociate the animacy organisation driven by the CNN visual features from the degree of familiarity and agency (thoughtfulness and feelings). Preliminary results from a new fMRI experiment designed to understand the contribution of these non-visual features are presented.

Are local losses of biodiversity causing degraded ecosystem function?

2017 ◽  
Author(s):  
Mark Vellend
Keyword(s):  
Ecosystem Function ◽  
Species Interactions ◽  
Regional Scale ◽  
Temporal Trends ◽  
Biodiversity Loss ◽  
Large Degree ◽  
Global Scale ◽  
Degraded Ecosystem ◽  
Biodiversity Change ◽  
Local Extirpation

This chapter highlights the scale dependence of biodiversity change over time and its consequences for arguments about the instrumental value of biodiversity. While biodiversity is in decline on a global scale, the temporal trends on regional and local scales include cases of biodiversity increase, no change, and decline. Environmental change, anthropogenic or otherwise, causes both local extirpation and colonization of species, and thus turnover in species composition, but not necessarily declines in biodiversity. In some situations, such as plants at the regional scale, human-mediated colonizations have greatly outnumbered extinctions, thus causing a marked increase in species richness. Since the potential influence of biodiversity on ecosystem function and services is mediated to a large degree by local or neighborhood species interactions, these results challenge the generality of the argument that biodiversity loss is putting at risk the ecosystem service benefits people receive from nature.

At the Simla Conference and in the General Elections

2018 ◽  
Author(s):  
J.S. Grewal
Keyword(s):  
Large Degree ◽  
Large Majority ◽  
Degree Of Polarization ◽  
Considerable Proportion ◽  
Executive Council ◽  
Crucial Importance ◽  
The Future ◽  
General Elections ◽  
The Right

On Lord Wavell’s initiative, the Simla Conference was held to discuss the proposal of a new executive council and a new constitution for India after the war. Master Tara Singh represented the Sikh community at the Simla Conference. The Conference failed due to Jinnah’s insistence that the Muslim League alone had the right to nominate Muslim representatives on the Executive Council. The failure of the Conference made the general elections of 1945–6 all the more important. The general elections resulted in a large degree of polarization between the Hindus, Muslims, and Sikhs. The Muslim League got Muslim mandate in favour of Pakistan. The Congress got a mandate of Hindus and a considerable proportion of the Sikhs for independence without partition. The Akalis got support of the large majority of the Sikhs for an independent political entity of the Sikhs. This polarization was of crucial importance for the future.

scholarly journals Optimized CSIDH Implementation Using a 2-Torsion Point

Cryptography ◽  
2020 ◽  
Vol 4 (3) ◽  
pp. 20 ◽  
Author(s):  
Donghoe Heo ◽  
Suhri Kim ◽  
Kisoon Yoon ◽  
Young-Ho Park ◽  
Seokhie Hong
Keyword(s):  
Elliptic Curve ◽  
Group Action ◽  
Large Degree ◽  
Transitive Group ◽  
Optimization Method ◽  
Torsion Point ◽  
Torsion Points ◽  
Diffie Hellman ◽  
Odd Degree ◽  
Montgomery Curve

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

scholarly journals Temperature chaos is present in off-equilibrium spin-glass dynamics

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Marco Baity-Jesi ◽  
Enrico Calore ◽  
Andrés Cruz ◽  
Luis Antonio Fernandez ◽  
José Miguel Gil-Narvion ◽  
...  
Keyword(s):  
Spin Glass ◽  
Dynamic Effect ◽  
Large Degree ◽  
Equilibrium Temperature ◽  
Experimental Conditions ◽  
Temperature Changes ◽  
Equilibrium Dynamics ◽  
Glass Dynamics ◽  
Extreme Sensitivity ◽  
Non Equilibrium

AbstractExperiments featuring non-equilibrium glassy dynamics under temperature changes still await interpretation. There is a widespread feeling that temperature chaos (an extreme sensitivity of the glass to temperature changes) should play a major role but, up to now, this phenomenon has been investigated solely under equilibrium conditions. In fact, the very existence of a chaotic effect in the non-equilibrium dynamics is yet to be established. In this article, we tackle this problem through a large simulation of the 3D Edwards-Anderson model, carried out on the Janus II supercomputer. We find a dynamic effect that closely parallels equilibrium temperature chaos. This dynamic temperature-chaos effect is spatially heterogeneous to a large degree and turns out to be controlled by the spin-glass coherence length ξ. Indeed, an emerging length-scale ξ* rules the crossover from weak (at ξ ≪ ξ*) to strong chaos (ξ ≫ ξ*). Extrapolations of ξ* to relevant experimental conditions are provided.

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