arXiv subjects

Ahmad Sabihi

Publications and source records attributed to Ahmad Sabihi.

8 recordsLinked to original sources

arXiv2023

On optimization and solution of roots of a function using Taylor's expansion and fractional derivatives

A method is given for finding roots of a one-variable function using Taylor's expansion of that function and fractional derivative calculated at a suitable tangent point without using Newton's method, but is regarded as a variant of Halley and Newton's one. Several examples regarding polynomials are stated as well. Then, the given method is generalized to functions of several variables belonging to an $n$-dimensional space and one example is given for optimization and solution of a nonlinear system of equations by both our method and Gradient Descent one. A comparison of our method is made with Gradient descent one for a system of the functions of three variables. Our given method seems to be much more rapidly than the Newton's one since by finding a suitable point on the function's curve, the number of iterations is to be much less than Newton's iterative steps. We also find order of fractional derivative, which corresponds to equation's found root and compare tangent lines drawn at the root by both fractional and classical derivatives. The methods given in this paper can be used for optimization of function via fractional derivatives of order $β$.

math.OC↗

arXiv2022

Fourier Series in Fractional Dimensional Space

In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;α)$. We call it fractional Fourier series of the order $α$. Extending the basis functions of the linear space into fractional one, by rotation transformation, we define a real and complex Fourier series and obtain their coefficients. It is also shown that the fractional derivative of a periodic function can be realized through (fractional) Fourier series with modified coefficients.

math.GM↗

arXiv2019

A Novel Method for Drawing a Circle Tangent to Three Circles Lying on a Plane by Straightedge,Compass,and Inversion Circles

In this paper, we present a novel method to draw a circle tangent to three given circles lying on a plane. Using the analytic geometry and inversion (reflection) theorems, the center and radius of the inversion circle are obtained. Inside any one of the three given circles, a circle of the similar radius and concentric with its own corresponding original circle is drawn.The tangent circle to these three similar circles is obtained. Then the inverted circles of the three similar circles and the tangent circle regarding an obtainable point and a computable power of inversion (reflection) constant are obtained. These circles (three inverted circles and an inverted tangent circle)will be tangent together.Just,we obtain another reflection point and power of inversion so that those three reflected circles (inversions of three similar circles) can be reflections of three original circles, respectively. In such a case,the reflected circle tangent to three reflected circles regarding same new inversion system will be tangent to the three original ones. This circle is our desirable circle. A drawing algorithm is also given for drawing desirable circle by straightedge and compass. A survey of conformal mapping theory and inversion in higher dimensions is also accomplished. Although, Laguerre transformation might be used for solution of this problem, but we do not make use of this method. Our novelty is just for drawing a circle tangent to three given circles applying a tangent circle to three identical circles concentric with three given ones and then inverting them as original ones by compass and straightedge not any thing else.

math.HO↗

arXiv2018

On solutions of some of unsolved problems in number theory, specifically on the distribution of primes

We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cramér's, Shanks', and five Smarandache's conjectures. We make use of both Firoozbakht's conjecture (which recently proved by the author) and Kourbatov's theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks' conjecture, we make use of Panaitopol's asymptotic formula for $π(x)$ as well.

math.GM↗

arXiv2017

An approach towards the proof of the strong Goldbach's conjecture for sufficiently large even integers

We approach a new proof of the strong Goldbach's conjecture for sufficiently large even integers by applying the Dirichlet's series. Using the Perron formula and the Residue Theorem in complex variable integration, one could show that any large even integer is demonstrated as a sum of two primes. In this paper,the Riemann Hypothesis is assumed to be true in throughout the paper. A novel function is defined on the natural numbers set.This function is a typical sieve function.Then based on this function,several new functions are represented and using the Prime Number Theorem,Sabihi's theorem, and the Sabihi's second conjecture,the strong Goldbach's conjecture is proved for sufficiently large even integers.

math.GM↗

arXiv2016

An analytical proof for Lehmer's totient conjecture using Mertens' theorems

We make an analytical proof for Lehmer's totient conjecture. Lehmer conjectured that there is no solution for the congruence equation $n-1\equiv 0~(mod~ϕ(n))$ with composite integers,$n$, where $ϕ(n)$ denotes Euler's totient function. He also showed that if the equation has any composite solutions, $n$ must be odd, square-free, and divisible by at least 7 primes. Several people have obtained conditions on values ,$n$, and number of square-free primes constructing $n$ if the equation can have composite solutions. Using Mertens' theorems, we show that it is impossible that the equation can have any composite solution and implies that the conjecture should be true for all the positively composite numbers.

math.GM↗

arXiv2016

On the Firoozbakht's conjecture

This paper proves Firoozbakht's conjecture using Rosser and Schoenfelds' inequality on the distribution of primes. This inequality is valid for all natural numbers ${n\geq 21}$. Firoozbakht's conjecture states that if $ {p_{n}}$ and ${p_{(n+1)}}$ are consecutive prime numbers, then ${p_{(n+1)}^{1/(n+1)}< p_{n}^{1/n}}$ for every ${n\geq 1}$. Rosser's inequality for the ${n}$th and ${(n+1)}$th roots, changes from strictly increasing to strictly decreasing for ${n\geq 21}$. The inequality is considered for ${n>e^{e^{3/2}}}$, i.e., ${n\geq 89}$, but since the inequalities for ${n\geq 195340>e^{e^{5/2}}}$, are also required, these inequalities are explicitly proven as well. Silva has already verified Firoozbakht's conjecture up to ${p_{n}<4 \times 10^{18}}$, and the additional theorem is proven here that there is the smallest natural number, ${m>n\geq 1}$ and ${p_{m}^{1/m}< p_{n}^{1/n}}$. It is also shown that there is a unique one to one function, which maps each element ${p_n}$ to each element ${p_{n}^{1/n}}$ for every ${n\geq 1}$ and ${1<p_{n}^{1/n}\leq 2}$. Finally, it is proved that there is a strictly decreasing sequence and Firoozbakht's conjecture would be true for all ${n\geq 1}$.

math.GM↗

arXiv2016

Robin's inequality,Lagarias' criterion, and Riemann hypothesis

In this paper, we make use of Robin and Lagarias' criteria to prove Riemann hypothesis. The goal is, using Lagarias criterion for $n\geq 1$ since Lagarias criterion states that Riemann hypothesis holds if and only if the inequality $\sum_{d|n}d\leq H_{n}+\exp(H_{n})\log(H_{n})$ holds for all $n\geq 1$. Although, Robin's criterion is used as well. Our approach breaks up the set of the natural numbers into the two main subsets. The first subset is $\{n\in \mathbb{N}| ~ 1\leq n\leq 2(3\times5\dots\times331)^{2}\}$. The second one is $\{n\in \mathbb{N}| ~ n\geq 2(3\times5\dots\times331)^{2}\}$. In our proof, the second subset is decomposed again into the three sub-subsets including odd numbers and the two groups of the even numbers. Then,each group of the even numbers is expressed by an odd integer class number set. Finally, mathematical arguments are stated for each odd integer class number set. Odd integer class number set is introduced in this paper. Since the Lagarias criterion holds for the first subset regarding computer aided computations and Thomas Morrill's paper, we do prove it for the second subset using both Lagarias and Robin's criteria and mathematical arguments. It then follows that Riemann hypothesis holds as well. Essential keys of the proof for large numbers are theorem 1 proving $\sigma(m)<\frac{1}{2}e^{\gamma}m \log\log(2m)$ for odd numbers $m\geq (3\times5\dots\times331)^{2}$, lemma9 and lemma 10 proving $e^{\gamma}(1-\frac{1}{p_{1}})\dots (1-\frac{1}{p_{n}})\log\log(2p_{1}\dots p_{n})<2$ for $n\geq 1$ and $e^{\gamma}(1-\frac{1}{p_{1}})\dots (1-\frac{1}{p_{n}})\log\log(2p^{2}_{1}\dots p^{2}_{n})>2$ for $n\geq 66$.

math.GM↗