On Some Bounds for the Exponential Integral Function

In 1934, Hopf established an elegant inequality bounding the exponential integral function. In 1959, Gautschi established an improvement of Hopf’s results. In 1969, Luke also established two inequalities with each improving Hopf’s results. In 1997, Alzer also established another improvement of Hopf’s results. In this paper, we provide two new proofs of Luke’s first inequality and as an application of this inequality, we provide a new proof and a generalization of Gautschi’s results. Furthermore, we establish some inequalities which are analogous to Luke’s second inequality and Alzer’s inequality. The techniques adopted in proving our results are simple and straightforward.

Proof the Skewes’ number is not an integer using lattice points and tangent line

Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

MODELING OF ASYMPTOTICALLY OPTIMAL PIECEWISE LINEAR INTERPOLATION OF PLANE PARAMETRIC CURVES

Context. Piecewise linear approximation of curves has a large number of applications in computer algorithms, as the reconstruction of objects of complex shapes on monitors, CNC machines and 3D printers. In many cases, it is required to have the smallest number of segments for a given accuracy. Objective. The objective of this paper is to improve the method of asymptotically optimal piecewise linear interpolation of plane parametric curves. This improvement is based to research influence of the method parameters and algorithms to distributions of approximation errors. Method. An asymptotically optimal method of curves interpolation is satisfied to the condition of minimum number of approximation units. Algorithms for obtaining the values of the sequence of approximation nodes are suggested. This algorithm is based on numerical integration of the nodes regulator function with linear and spline interpolation of its values. The method of estimating the results of the curve approximation based on statistical processing of line segments sequence of relative errors is substantiated. Modeling of real curves approximation is carried out and influence of the sampling degree of integral function – the nodes regulator on distribution parameters of errors is studied. The influence is depending on a method of integral function interpolation. Results. Research allows to define necessary the number of discretization nodes of the integral function in practical applications. There have been established that with enough sampling points the variance of the error’s distribution stabilizes and further increasing this number does not significantly increase the accuracy of the curve approximation. In the case of spline interpolation of the integral function, the values of the distribution parameters stabilized much faster, which allows to reduce the number of initial sampling nodes by 5–6 times having similar accuracy. Conclusions. Modelling of convex planar parametric curves reconstruction by an asymptotically optimal linear interpolation algorithm showed acceptable results without exceeding the maximum errors limit in cases of a sufficient discretization of the integral function. The prospect of further research is to reduce the computational complexity when calculating the values of the integral distribution function by numerical methods, and to use discrete analogues of derivatives in the expression of this function.

VPS35 Downregulation Alters Degradation Pathways in Neuronal Cells

Background: The vacuolar protein sorting 35 (VPS35) is the main component of the retromer recognition core complex system which regulates intracellular cargo protein sorting and trafficking. Downregulation of VPS35 has been linked to the pathogenesis of neurodegenerative disorders such Alzheimer’s and Parkinson’s diseases via endosome dysregulation. Objective: Here we show that the genetic manipulation of VPS35 affects intracellular degradation pathways. Methods: A neuronal cell line expressing human APP Swedish mutant was used. VPS35 silencing was performed treating cells with VPS35 siRNA or Ctr siRNA for 72 h. Results: Downregulation of VPS35 was associated with alteration of autophagy flux and intracellular accumulation of acidic and ubiquitinated aggregates suggesting that dysfunction of the retromer recognition core leads to a significant alteration in both pathways. Conclusion: Taken together, our data demonstrate that besides cargo sorting and trafficking, VPS35 by supporting the integral function of the retromer complex system plays an important role also as a critical regulator of intracellular degradation pathways.

A HARMONIC MEAN INEQUALITY CONCERNING THE GENERALIZED EXPONENTIAL INTEGRAL FUNCTION

In this paper, we prove that for $s\in(0,\infty)$, the harmonic mean of $E_k(s)$ and $E_k(1/s)$ is always less than or equal to $\Gamma(1-k,1)$. Where $E_k(s)$ is the generalized exponential integral function, $\Gamma(u,s)$ is the upper incomplete gamma function and $k\in \mathbb{N}$.

Numerical analysis approach to twin primes conjecture

The purpose of this paper is to demonstrate how the modified Sieve of Eratosthenes is used to have an approach to twin prime conjecture. If the Sieve is used in its basic form, it does not produce anything new. If it is used through the numerical analysis method explained in this paper, we obtain a specific counting of twin primes. This counting is based on the false assumption that distribution of primes follows punctually the Logarithmic Integral function denoted as Li(x) (see [5] and [10], pp. 174–176). It may be a starting point for future research based on this numerical analysis method technique that can be used in other mathematical branches.

High efficient method for calculating the Straton - Chu integral for problems of interaction of laser radiation with materials

Описан высокоэффективный метод вычисления интеграла Стрэттона-Чу для определения электромагнитного поля, создаваемого отражением плоскопараллельного лазерного импульса от параболического зеркала. Рассмотрены импульс с постоянной во времени амплитудой и импульс фемтосекундной длительности, с зависимостью от времени в виде функции Гаусса. Описанный метод актуален для решения задач о взаимодействии сильно сфокусированных лазерных импульсов с веществом The interaction of femtosecond laser pulses with materials is usually modelled within the approach of nonlinear Maxwell equations. The laser pulse in the calculations is initiated by specifying the conditions for the electric field at the boundary of the computational domain, which is located at a distance of 100-200 micron from the focus. The usually used Gaussian distribution for the radius and time in the boundary conditions are not applicable for such pulses. It is necessary to consider a real optic system, or a system, which can be realized. In the presented paper we address the situation in which the laser pulse is created by the reflection of parallel pulse from a parabolic mirror. To determine the field near the focus we use the Stratton-Chu integral (SCI), with fast oscillating sub-integral function. It makes very difficult to calculate SCI. In the presented paper a highly efficient method allows to overcome this difficulty. The main idea is that a change of variables is made in the integral, which, in the case of a short pulse being important for applications, allows calculating the integral over one of the variables once for all times. In addition, for this variable, the integrand is smooth and its calculation does not require large computational resources. The paper investigates the accuracy of calculating the SCI by the proposed method and demonstrates its high efficiency

The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

Cryptosystem based on a key function of a real variable

Using the function of a real variable in cryptosystems as a key allows you to increase its cryptographic strength, because it is more difficult to pick up such key. Therefore, the development of such systems is relevant. A cryptosystem with a symmetric key is offered. This key is some function of a real variable that satisfies some restrictions. It can be either continuous or discrete. The transmitting and receiving parties select the key-function, the first transmitted character or the first transmitted value for the analog message, the function area of the key function, and the step of changing the function argument. A Disproportion over first-order derivative is used to encrypt an analog message. The Cauchy problem is solving for decrypting this message. Discrete messages are encrypted using the first-order disproportionality integral function. Decryption is performed by the inverse transformation of the formula for integral disproportion. Algorithms for encrypting and decrypting messages are presented. The ability to encrypt and decrypt text information, 2D graphic images, as well as analog messages are shown. The examples show the complexity to pick up the key function and the cryptographic strength of the proposed cryptosystem. A cryptosystem, in which the function of a real variable is used as a key and as well as disproportion functions are used, is suitable for encryption of both discrete and continuous messages. To “crack” such a system, it is required to pick up the form of the key function and to find the values of its parameters with very high accuracy. That is, the system has high cryptographic strength.