Perpetual Free-choice Petri Nets are Lucent Proof of a Theorem of van der Aalst Using CP-exhaustions

Van der Aalst’s theorem is an important result for the analysis and synthesis of process models. The paper proves the theorem by exhausting perpetual free-choice Petri nets by CP-subnets. The resulting T-systems are investigated by elementary methods.

Elementary Methods for Generating Three-Dimensional Coordinate Estimation and Image Reconstruction from Series of Two-Dimensional Images

The increase in computational power in recent years has opened a new door for image processing techniques. Three-dimensional object recognition, identification, pose estimation, and mapping are becoming popular. The need for real-world objects to be mapped into three-dimensional spatial representation is greatly increasing, especially considering the heap jump we obtained in the past decade in virtual reality and augmented reality. This paper discusses an algorithm to convert an array of captured images into estimated 3D coordinates of their external mappings. Elementary methods for generating three-dimensional models are also discussed. This framework will help the community in estimating three-dimensional coordinates of a convex-shaped object from a series of two-dimension images. The built model could be further processed for increasing the resemblance of the input object in terms of its shapes, contour, and texture.

A Note on the Primitive Roots and the Golomb Conjecture

In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p 1 / 2 + ɛ ≤ n < p , there must exist two primitive roots α and β modulo p with 1 < α , β ≤ n − 1 such that the equation n = α + β holds, where 0 < ɛ < 1 / 2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p .

On Pythagorean Triples and the Primitive Roots Modulo a Prime

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x , y , and z modulo p such that x 2 + y 2 = z 2 .

Projective MDS Codes Over GF(27)‎

MDS code is a linear code that achieves equality in the Singleton bound, and projective MDS (PG-MDS) is MDS code with independents property of any two columns of its generator matrix. In this paper, elementary methods for modifying a PG-MDS code of dimensions 2, 3, as extending and lengthening, in order to find new incomplete PG-MDS codes have been used over . Also, two complete PG-MDS codes over of length and 28 have been found.

On the Diophantine Equation Mp^x + (Mq + 1)^y = z^2

In this paper, we study and solve the exponential Diophantine equation of the formMxp + (Mq + 1)y = z2 for Mersenne primes Mp and Mq and non-negative integers x, y, and z. We use elementary methods, such as the factoring method and the modular arithmetic method, to prove our research results. Several illustrations are presented, as well as cases where solutions to the Diophantine equation do not exist.

The Legendre’s Symbol Modulo p and Its Some Elementary Results

The main purpose of this article is using the elementary methods and the properties of the character sums to study the calculating problem of the number of the solutions for one kind congruence equation modulo p (an odd prime) and give some interesting identities and asymptotic formulas for it.

On a Certain Quadratic Character Sums of Ternary Symmetry Polynomials mod p

In this article, we are using the elementary methods and the properties of the classical Gauss sums to study the calculating problem of a certain quadratic character sums of a ternary symmetry polynomials modulo p and obtain some interesting identities for them.

On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime p to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.