#### Advances in Pure Mathematics

Journal Information

ISSN / EISSN :
2160-0368 / 2160-0384

Current Publisher: Scientific Research Publishing, Inc. (10.4236)

Total articles ≅ 687

Archived in

SHERPA/ROMEO

Filter:

#### Latest articles in this journal

Advances in Pure Mathematics, Volume 11, pp 27-62; doi:10.4236/apm.2021.111004

**Abstract:**

The Number Theory comes back as the heart of unified Science, in a Computing Cosmos using the bases 2;3;5;7 whose two symmetric combinations explain the main lepton mass ratios. The corresponding Holic Principle induces a symmetry between the Newton and Planck constants which confirm the Permanent Sweeping Holography Bang Cosmology, with invariant baryon density 3/10, the dark baryons being dephased matter-antimatter oscillation. This implies the DNA bi-codon mean isotopic mass, confirming to 0.1 ppm the electron-based Topological Axis, whose terminal boson is the base 2 c-observable Universe in the base 3 Cosmos. The physical parameters involve the Euler idoneal numbers and the special Fermat primes of Wieferich (bases 2) and Mirimanoff (base 3). The prime numbers and crystallographic symmetries are related to the 4-fold structure of the DNA bi-codon. The forgotten Eddington’s proton-tau symmetry is rehabilitated, renewing the supersymmetry quest. This excludes the concepts of Multiverse, Continuum, Infinity, Locality and Zero-mass Particle, leading to stringent predictions in Cosmology, Particle Physics and Biology.

Advances in Pure Mathematics, Volume 11, pp 19-26; doi:10.4236/apm.2021.111003

**Abstract:**

Based on a node group , the Newman type rational operator is constructed in the paper. The convergence rate of approximation to a class of non-smooth functions is discussed, which is regarding to X. Moreover, if the operator is constructed based on further subdivision nodes, the convergence rate is . The result in this paper is superior to the approximation results based on equidistant nodes, Chebyshev nodes of the first kind and Chebyshev nodes of the second kind.

Advances in Pure Mathematics, Volume 11, pp 138-148; doi:10.4236/apm.2021.112009

**Abstract:**

In any completely close complex field C, generalized transcendental meromorphic functions may have some new properties. It is well known that a meromorphic function of characteristic zero is a rational function. This paper introduced some mathematical properties of the transcendental meromorphic function, which is generalized to the meromorphic function by multiplying and differentiating the generalized meromorphic function. The analysis shows that the difference between any non-zero constant and the derivative of the general meromorphic function has an infinite zero. In addition, for any natural number n, there are no practically exceptional values for the multiplication of the general meromorphic function and its derivative to the power of n.

Advances in Pure Mathematics, Volume 11, pp 12-18; doi:10.4236/apm.2021.111002

**Abstract:**

Prime numbers are the integers that cannot be divided exactly by another integer other than one and itself. Prime numbers are notoriously disobedient to rules: they seem to be randomly distributed among natural numbers with no laws except that of chance. Questions about prime numbers have been perplexing mathematicians over centuries. How to efficiently predict greater prime numbers has been a great challenge for many. Most of the previous studies focus on how many prime numbers there are in certain ranges or patterns of the first or last digits of prime numbers. Honestly, although these patterns are true, they help little with accurately predicting new prime numbers, as a deviation at any digit is enough to annihilate the primality of a number. The author demonstrates the periodicity and inter-relationship underlying all prime numbers that makes the occurrence of all prime numbers predictable. This knowledge helps to fish all prime numbers within one net and will help to speed up the related research.

Advances in Pure Mathematics, Volume 11, pp 101-108; doi:10.4236/apm.2021.111006

**Abstract:**

Exponential integral for real arguments is evaluated by employing a fast-converging power series originally developed for the resolution of Grandi’s paradox. Laguerre’s historic solution is first recapitulated and then the new solution method is described in detail. Numerical results obtained from the present series solution are compared with the tabulated values correct to nine decimal places. Finally, comments are made for the further use of the present approach for integrals involving definite functions in denominator.

Advances in Pure Mathematics, Volume 11, pp 109-120; doi:10.4236/apm.2021.112007

**Abstract:**

In this paper, we showed how groups are embedded into wreath products, we gave a simpler proof of the theorem by Audu (1991) (see [1]), also proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. We further showed that when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding becomes an isomorphism. Examples were given to justify the results.

Advances in Pure Mathematics, Volume 11, pp 63-100; doi:10.4236/apm.2021.111005

**Abstract:**

As starting point for patterns with seven-fold symmetry, we investigate the basic possibility to construct the regular heptagon by bicompasses and ruler. To cover the whole plane with elements of sevenfold symmetry is only possible by overlaps and (or) gaps between the building stones. Resecting small parts of overlaps and filling gaps between the heptagons, one may come to simple parqueting with only a few kinds of basic tiles related to sevenfold symmetry. This is appropriate for parqueting with a center of seven-fold symmetry that is illustrated by figures. Choosing from the basic patterns with sevenfold symmetry small parts as elementary stripes or elementary cells, one may form by their discrete translation in one or two different directions periodic bordures or tessellation of the whole plane but the sevenfold point-group symmetry of the whole plane is then lost and there remains only such symmetry in small neighborhoods around one or more centers. From periodic tiling, we make the transition to aperiodic tiling of the plane. This is analogous to Penrose tiling which is mostly demonstrated with basic elements of fivefold symmetry and we show that this is also possible with elements of sevenfold symmetry. The two possible regular star-heptagons and a semi-regular star-heptagon play here a basic role.

Advances in Pure Mathematics, Volume 11, pp 1-11; doi:10.4236/apm.2021.111001

**Abstract:**

Newton’s method is used to find the roots of a system of equations f (x) = 0. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton’s method was proposed by McDougall and Wotherspoon [1] with an order of convergence of 1+ √2. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier [2]. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.

Advances in Pure Mathematics, Volume 11, pp 149-161; doi:10.4236/apm.2021.112010

**Abstract:**

As a new dimension reduction method, the two-dimensional principal component (2DPCA) can be well applied in face recognition, but it is susceptible to outliers. Therefore, this paper proposes a new 2DPCA algorithm based on angel-2DPCA. To reduce the reconstruction error and maximize the variance simultaneously, we choose F norm as the measure and propose the Fp-2DPCA algorithm. Considering that the image has two dimensions, we offer the Fp-2DPCA algorithm based on bilateral. Experiments show that, compared with other algorithms, the Fp-2DPCA algorithm has a better dimensionality reduction effect and better robustness to outliers.

Advances in Pure Mathematics, Volume 10, pp 471-491; doi:10.4236/apm.2020.109029

**Abstract:**

Numerical simulations by means of the Monte Carlo Potts model have been provided to simulate grain structures in two-phase polycrystalline materials. The topological features in the simulated microstructure analyzed for different diffusion mechanisms over a broad range of volume fractions for both phases. The topological properties include the average number of sides, grain topology distribution and the topological size relation function. It is found that the average number of sides depends proportionally on the volume fraction. It increases as the volumes fraction increases and vice versa. Moreover, it is shown that the grain topology distribution in the self-similar growth regime can be described by time unchanged function of the relative grain size. Additionally, topological size function in the simulated microstructure can be evaluated by a quadratic function.