scholarly journals Platform-Independent Specification and Verification of the Standard Mathematical Square Root Function

2019 ◽  
Vol 53 (7) ◽  
pp. 595-616
Author(s):  
N. V. Shilov ◽  
D. A. Kondratyev ◽  
I. S. Anureev ◽  
E. V. Bodin ◽  
A. V. Promsky
Keyword(s):  
Root Function ◽  
Square Root ◽  
Specification And Verification

scholarly journals Platform-independent Specification and Verification of the Standard Mathematical Square Root Function

2018 ◽  
Vol 25 (6) ◽  
pp. 637-666
Author(s):  
Nikolay V. Shilov ◽  
Dmitry A. Kondratyev ◽  
Igor S. Anureev ◽  
Eugene V. Bodin ◽  
Alexei V. Promsky
Keyword(s):  
Root Function ◽  
Square Root ◽  
Combined Approach ◽  
Real Numbers ◽  
Mathematical Functions ◽  
Square Roots ◽  
Computer Aided ◽  
Platform Independence ◽  
Fix Point ◽  
Specification And Verification

The project “Platform-independent approach to formal specification and verification of standard mathematical functions” is aimed onto the development of incremental combined approach to specification and verification of standard Mathematical functions like sqrt, cos, sin, etc. Platform-independence means that we attempt to design a relatively simple axiomatization of the computer arithmetics in terms of real arithmetics (i.e. the field \(\mathbb{R}\) of real numbers) but do not specify neither base of the computer arithmetics, nor a format of numbers representation. Incrementality means that we start with the most straightforward specification of the simplest case to verify the algorithm in real numbers and finish with a realistic specification and a verification of the algorithm in computer arithmetics. We call our approach combined because we start with manual (pen-and-paper) verification of the algorithm in real numbers, then use this verification as proof-outlines for a manual verification of the algorithm in computer arithmetics, and finish with a computer-aided validation of the manual proofs with a proof-assistant system (to avoid appeals to “obviousness” that are common in human-carried proofs). In the paper, we apply our platform-independent incremental combined approach to specification and verification of the standard Mathematical square root function. Currently a computer-aided validation was carried for correctness (consistency) of our fix-point arithmetics and for the existence of a look-up table with the initial approximations of the square roots for fix-point numbers.

Correlation of Electrical, Structural, and Optical Properties of Erbium In Silicon

1993 ◽  
Vol 301 ◽  
Author(s):  
J. L. Benton ◽  
D. J. Eaglesham ◽  
M. Almonte ◽  
P. H. Citrin ◽  
M. A. Marcus ◽  
...  
Keyword(s):  
Root Function ◽  
Free Exciton ◽  
Excitation Power ◽  
Optically Active ◽  
Photoluminescence Intensity ◽  
Square Root ◽  
Structural And Optical Properties ◽  
X Ray ◽  
X Ray Absorption ◽  
Donor Activity

ABSTRACTAn understanding of the electrical, structural, and optical properites of Er in Si is necessary to evaluate this system as an opto-electronic material. Extended x-ray absorption fine structure, EXAFS, measurements of Er-implanted Si show that the optically active impurity complex is Er surrounded by an O cage of 6 atoms. The Er photoluminescence intensity is a square root function of excitation power, while the free exciton intensity increases linearly. The square root dependence of the 1.54μm-intensity is independent of measurement temperature and independent of co-implanted species. Ion-implantation of Er in Si introduces donor activity, but spreading resistance carrier concentration profiles indicate that these donors do not effect the optical activity of the Er.

Fast algorithm for the three-dimensional Poisson equation in infinite domains

2020 ◽  
Author(s):  
Chunxiong Zheng ◽  
Xiang Ma
Keyword(s):  
Poisson Equation ◽  
Fast Algorithm ◽  
Root Function ◽  
Computational Cost ◽  
Three Dimensional ◽  
Chebyshev Approximation ◽  
Laplacian Operator ◽  
Square Root ◽  
Infinite Domain ◽  
Infinite Domains

Abstract This paper is concerned with a fast finite element method for the three-dimensional Poisson equation in infinite domains. Both the exterior problem and the strip-tail problem are considered. Exact Dirichlet-to-Neumann (DtN)-type artificial boundary conditions (ABCs) are derived to reduce the original infinite-domain problems to suitable truncated-domain problems. Based on the best relative Chebyshev approximation for the square-root function, a fast algorithm is developed to approximate exact ABCs. One remarkable advantage is that one need not compute the full eigensystem associated with the surface Laplacian operator on artificial boundaries. In addition, compared with the modal expansion method and the method based on Pad$\acute{\textrm{e}}$ approximation for the square-root function, the computational cost of the DtN mapping is further reduced. An error analysis is performed and numerical examples are presented to demonstrate the efficiency of the proposed method.

scholarly journals On Square Root Function over ℚpand its Application

2017 ◽  
Vol 819 ◽  
pp. 012028
Author(s):  
Mansoor Saburov ◽  
Muhammad Jundullah bin Ismail
Keyword(s):  
Root Function ◽  
Square Root

Use of various functions to analyse the fertiliser responses of maize ( Zea mays L.) hybrids in long-term experiments

2006 ◽  
Vol 54 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Z. Berzsenyi ◽  
Q. L. Dang
Keyword(s):  
Analysis Of Variance ◽  
Response Pattern ◽  
Root Function ◽  
Quadratic Function ◽  
Maize Yield ◽  
Maximum Point ◽  
Square Root ◽  
Maize Hybrids ◽  
Year Effect

The effect of various fertiliser treatments on the yield of maize hybrids was studied on the basis of 26 years of data obtained in a long-term bifactorial split-plot experiment set up in 1967. The seven treatments (NPK ratio 2:1:1) applied were as follows (rates per hectare): 1. Control (no fertiliser), 2. 100 kg NPK, 3. 200 kg NPK, 4. 300 kg NPK, 5. 400 kg NPK, 6. 600 kg NPK, 7. 800 kg NPK. The maize was grown with the conventional cultivation techniques in continuous cropping. The results of analyses carried out with three different methods (analysis of variance, cumulative yield analysis and regression analysis) all indicated that under the given conditions the yield of maize hybrids was highest at an NPK fertiliser rate of 200-400 kg ha -1 . The effect of fertilisation on the maize yield was significant in 21 of the 26 years. Combined analysis of variance for the years showed that the year effect (quantity of rainfall) had the greatest effect on the maize yield, but although the year effect had a fundamental effect on the yield level it did not influence the fertiliser response pattern. The fertiliser responses of the maize hybrids were described by fitting four types of functions (quadratic, square root, inverse exponential, linear-plateau) to the yield data. It was found that when selecting the best function a consideration of the regression deviations (measured yield - calculated yield) was just as important as the coefficient of determination (R 2 ). In 12 of the 26 years the fitting of the quadratic function was not significant and overestimated the fertilisation optimum. The fertiliser response curve generally has a broad maximum which is far better described by the square root function than by the quadratic. If the fertiliser response pattern includes a depressive phase, a square root function should definitely be used in place of the quadratic function. If the maximum of the response surface forms a plateau (as opposed to a maximum point) a linear-plateau function or an inverse exponential function can be recommended. In the present work the linear-plateau function gave the best results.

A Study of the Abney Effect in Some Colour Atlases: Designing Opponent Variables and Hue Functions

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 254-254
Author(s):  
F Martínez-Verdú ◽  
J Pujol ◽  
J M Artigas
Keyword(s):  
Root Function ◽  
Photon Absorption ◽  
Square Root ◽  
Nonlinear Functions ◽  
Dominant Wavelength ◽  
Intensity Response ◽  
Colour System ◽  
Abney Effect ◽  
Straight Lines ◽  
Colour Appearance

The Abney effect in colour-appearance systems (Munsell, NCS) means that the lines for identical apparent hue (at constant lightness) do not coincide with the straight lines for constant dominant wavelength. The curvature of constant-hue lines in chromaticity diagrams reflects the fact that cone signals are nonlinear functions of the rate of photon absorption. The most widely used nonlinear intensity-response function in vision is the Naka - Rushton function, which in an intermediate range can be approximated by a square-root function. Our purpose has been to study the Abney effect in the Munsell and NCS colour atlases in order to develop a mathematical-physiological description on this basis, designing the redness - greenness and yellowness - blueness perceptual variables and the perceptual hue function in each colour system. The description is applicable to both biological and machine-vision systems.

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