Mathematics

Scientific paper is devoted to research of T-matrix - matrix of composite numbers 6h+1 v 6h-1 in special view, and to her application in number theory.

A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral group of symmetry D5, special attention is paid to those that, can be obtained by one universal set, with consists five different tiles. Geometric characteristics of tiles from this set are determined. It is shown, by this set of tiles it is possible to carry out topologically different tilings of the plane, possessing rotating symmetries of both the fifth and tenth orders.

According to the Abel-Ruffini theorem [1] and Galois theory [2], there is no solution in finite radicals to the general quintic equation. This article takes a different approach and proposes a new method to solve the quintic by iteration of radicals. But, the most intriguing result is an accurate algebraic formula for absolute and relative root approximation: |formula - root| < 0.00432 and |formula/root - 1| < 0.0251. We then expand some of the geometric properties discussed to construct a trigonometric algorithm that derives all roots.

It is known that prime numbers occupy specific geometrical patterns or moduli when numbers from one to infinity are distributed around polygons having sides that are integer multiple of number 6. In this paper, we will show that not only prime numbers occupy these moduli, but non-prime numbers sharing these same moduli have unique prime-ness properties. When utilizing digital root methodologies, these non-prime numbers provide a novel method to accurately identify prime numbers and prime factors without trial division or probabilistic-based methods. We will also show that the icositetragon (24-sided regular polygon) is a unique polygon pertaining to prime numbers and their ultimate incidence and distribution.

From the more than two hundred partial orders for fuzzy numbers proposes in the literature, only a few are totals. In this paper, we introduce the notion of admissible orders for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, is given special attention when thr partial order is the proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order.

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field as principal bundle connection, and the other is to represent gauge field as affine connection. Poincar{é} gauge theory and metric-affine gauge theory adopt the first approach. This paper adopts the second. This theory has the following advantages. (i) Gauge field and gravitational field can both be represented by affine connection; they can be described by a unified spatial frame. (ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as a geometric theory of distribution of gradient directions. Hence, gravitational theory and quantum theory obtain the same view of time and space and a unified description of evolution in affine connection representation of gauge fields. (iii) Chiral asymmetry, coupling constants, MNS mixing and CKM mixing can appear spontaneously as geometric properties in affine connection representation, whereas in U(1)×SU(2)×SU(3) principal bundle connection representation they can just only be artificially set up. Some postulates of the Standard Model can be turned into theorems in affine connection representation, so they are not necessary to be regarded as postulates anymore. (iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem that a proton decays into a lepton. (v) There exists a possible geometric interpretation to the color confinement of quarks. The Standard Model is not possessed of the above advantages. In the affine connection representation, we can get better interpretations of these physical properties. This is probably a necessary step towards the ultimate theory of physics.

An algebraic approximation, of order K , of a polyhedron correlation function (CF) can be obtained from $\gamma\pp(r)$, its chord-length distribution (CLD), considering first, within the subinterval [ D i−1 , D i ] of the full range of distances, a polynomial in the two variables (r− D i−1 ) 1/2 and ( D i −r ) 1/2 such that its expansions around r= D i−1 and r= D i simultaneously coincide with left and the right expansions of $\gamma\pp(r)$ around D i−1 and D i up to the terms O(r− D i−1 ) K/2 and O( D i −r ) K/2 , respectively. Then, for each i , one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large q s, the asymptotic behaviour of the exact form factor up to the term O( q −(K/2+4) ) . For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.

We show that the definition of an algebraic basis for a vector space allows the construction of an isomorphism with the one here called Algebraic Vector Space. Although the concept does not bring anything new, we mention some of the problems that the language established here can inspire.

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods and results of this paper are based on well-known theorems on the number of zeros for complex value functions (Jensen, Titchmarsh, Rouche theorems), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta Function, except leveraging its implied symmetry for non-trivial zeros on the critical strip.

We determine a positive real number (weight) which corresponds to the intersection point (vertex) of two non-overlapping geodesic arcs, which depends on the two weights which correspond to two points of these geodesicarcs, respectively, and an infinitesimal number \epsilon. As a limiting case, for \epsilon \to 0,the triad of the corresponding weights yields a degenerate weighted Fermat-Torricelli tree which coincides with these two geodesic arcs. By applying this process for a geodesic triangle on a circular cone, we derive an \epsilon characterization of conical points in R^3.