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S. M. Abrarov

Publications and source records attributed to S. M. Abrarov.

At least 37 records · Page 2Linked to original sources
arXiv2016

A new application methodology of the Fourier transform for rational approximation of the complex error function

This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the obtained rational approximation of the complex error function provides the average accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000$ and ${10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications. Since the rational approximation does not contain trigonometric or exponential functions dependent upon the input parameters $x$ and $y$, it is rapid in computation. Such an example demonstrates that the considered methodology of the Fourier transform may be advantageous in practical applications.

math.GM↗
arXiv2015

Representation of the Fourier transform as a weighted sum of the complex error functions

In this paper we show that a methodology based on a sampling with the Gaussian function of kind $h\,{e^{ - {{\left( {t/c} \right)}^2}}}/\left( {{c}\sqrt π} \right)$, where ${c}$ and $h$ are some constants, leads to the Fourier transform that can be represented as a weighted sum of the complex error functions. Due to remarkable property of the complex error function, the Fourier transform based on the weighted sum can be significantly simplified and expressed in terms of a damping harmonic series. In contrast to the conventional discrete Fourier transform, this methodology results in a non-periodic wavelet approximation. Consequently, the proposed approach may be useful and convenient in algorithmic implementation.

math.GM↗
arXiv2015

A rational approximation for the Dawson's integral of real argument

We present a rational approximation for the Dawson's integral of real argument and show how it can be implemented for accurate and rapid computation of the Voigt function at small $y < < 1$. The algorithm based on this approach enables computation with accuracy exceeding ${10^{ - 10}}$ within the domain $0 \le x \le 15$ and $0 \le y \le {10^{ - 6}}$. Due to rapid performance the proposed rational approximation runs the algorithm without deceleration.

math.NA↗
arXiv2015

A rational approximation for efficient computation of the Voigt function in quantitative spectroscopy

We present a rational approximation for rapid and accurate computation of the Voigt function, obtained by residue calculus. The computational test reveals that with only $16$ summation terms this approximation provides average accuracy ${10^{- 14}}$ over a wide domain of practical interest $0 < x < 40,000$ and ${10^{- 4}} < y < {10^2}$ for applications using the HITRAN molecular spectroscopic database. The proposed rational approximation takes less than half the computation time of that required by Weideman's rational approximation. Algorithmic stability is achieved due to absence of the poles at $y \geqslant 0$ and $ - \infty < x < \infty $.

physics.data-an↗
arXiv2014

Accurate approximations for the complex error function with small imaginary argument

In this paper we present two efficient approximations for the complex error function $w \left( {z} \right)$ with small imaginary argument $\operatorname{Im}{\left[ { z } \right]} < < 1$ over the range $0 \le \operatorname{Re}{\left[ { z } \right]} \le 15$ that is commonly considered difficult for highly accurate and rapid computation. These approximations are expressed in terms of the Dawson's integral $F\left( x \right)$ of real argument $x$ that enables their efficient implementation in a rapid algorithm. The error analysis we performed using the random input numbers $x$ and $y$ reveals that in the real and imaginary parts the average accuracy of the first approximation exceeds ${10^{ - 9}}$ and ${10^{ - 14}}$, while the average accuracy of the second approximation exceeds ${10^{ - 13}}$ and ${10^{ - 14}}$, respectively. The first approximation is slightly faster in computation. However, the second approximation provides excellent high-accuracy coverage over the required domain.

math.NA↗
arXiv2014

Sampling by incomplete cosine expansion of the sinc function: application to the Voigt/complex error function

A new sampling methodology based on incomplete cosine expansion series is presented as an alternative to the traditional sinc function approach. Numerical integration shows that this methodology is efficient and practical. Applying the incomplete cosine expansion we obtain a rational approximation of the complex error function that with the same number of the summation terms provides an accuracy exceeding the Weideman\text{'}s approximation accuracy by several orders of the magnitude. Application of the expansion results in an integration consisting of elementary function terms only. Consequently, this approach can be advantageous for accurate and rapid computation.

math.NA↗
arXiv2013

A rapid and highly accurate approximation for the error function of complex argument

We present efficient approximation of the error function obtained by Fourier expansion of the exponential function $\exp [{- {(t - 2 σ)^2}/4}]$. The error analysis reveals that it is highly accurate and can generate numbers that match up to the last decimal digits with reference values. Due to simple representation the proposed error function approximation can be utilized in a rapid algorithm.

math.NA↗
arXiv2012

Efficient application of the Chiarella and Reichel series approximation of the complex error function

Using the theorem of residues Chiarella and Reichel derived a series that can be represented in terms of the complex error function (CEF). Here we show a simple derivation of this CEF series by Fourier expansion of the exponential function $\exp ({- {τ^2}/4})$. Such approach explains the existence of the lower bound for the input parameter $y = \operatorname{Im} [z]$ restricting the application of the CEF approximation. An algorithm resolving this problem for accelerated computation of the CEF with sustained high accuracy is proposed.

math.GM↗
arXiv2012

On the Fourier expansion method for highly accurate computation of the Voigt/complex error function in a rapid algorithm

In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified representation of the proposed complex error function approximation makes possible further algorithmic optimization resulting in a considerable computational acceleration without compromise on accuracy.

math.NA↗
arXiv2011

Properties and applications of the prime detecting function: infinitude of twin primes, asymptotic law of distribution of prime pairs differing by an even number

The prime detecting function (PDF) approach can be effective instrument in the investigation of numbers. The PDF is constructed by recurrence sequence - each successive prime adds a sieving factor in the form of PDF. With built-in prime sieving features and properties such as simplicity, integro-differentiability and configurable capability for a wide variety of problems, the application of PDF leads to new interesting results. As an example, in this exposition we present proofs of the infinitude of twin primes and the first Hardy-Littlewood conjecture for prime pairs (the twin prime number theorem). On this example one can see that application of PDF is especially effective in investigation of asymptotic problems in combination with the proposed method of test and probe functions.

math.GM↗
arXiv2011

Sieve Procedure for the Möbius prime-functions, the Infinitude of Primes and the Prime Number Theorem

Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding Möbius prime-function, which in the limit of infinitely large primes becomes identical to the original Möbius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.

math.GM↗
arXiv2010

Probabilistic interpretation of the Möbius function identity and the Riemann Hypothesis

We obtained the probabilities for the values of the Möbius function for arbitrary numbers and found that the asymptotic densities of the squarefree integers among the odd and even numbers are $8/π^2$ and $4/π^2$, respectively. It is determined that statistics of successive outcomes of the Möbius function for very large squarefree odd and even numbers behaves similar to statistics of heads and tails of two flipping coins. These preliminary results are giving arguments supporting the Riemann Hypothesis. Its plausibility is based on statistical phenomena for integers.

math.GM↗
arXiv2009

Formulas for Positive, Negative and Zero Values of the Möbius Function

We obtained the formulas for the quantities of positive, negative and zero values of the Mobius function for any real x in terms of the Mobius function values for square root of x - similar to the identities we found earlier for the Mertens function [1]. Using the Dirac delta function approach [3, 2] we propose the equations showing how the Mertens and related functions can generate the output values.

math.NT↗
arXiv2008

Riemann Hypothesis may be proved by induction

The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At least one of these identities may be applied to prove the Riemann Hypothesis by induction. Additionally using this approach, the new series for Euler's constant gamma has been found.

math.GM↗
arXiv2007

Broadening of band-gap in photonic crystals with optically saturated media

Due to strong absorption of the incident light, the media with high refractive index are considered restrictive for applications in photonic crystals (PhCs). The possibility to resolve this problem by optical saturation effectively minimizing the absorption of the PhC medium is discussed. Such approach might be promising for the significant broadening of the photonic band-gap.

physics.optics↗