Mathematics

The manuscripts tabulates arc lists of the 1, 1, 3, 8, 25, 85, 397 ... unlabeled 2-regular digraphs on n=0, 1, 2, ..., 9 nodes, including disconnected graphs, graphs with multiarcs and/or graphs with loops. Each of these graphs represents one term of the Lagrangian of Lovelock's type -- a contraction of a product of n Riemann tensors -- once the 2 covariant and 2 contravariant indices of a tensor are associated with the in-edges and out-edges of a node.

We study the 2N-dimensional canonical systems and discuss some properties of its fundamental solution. We then discuss the Floquet theory of periodic canonical systems and observe the asymptotic behavior of its solution. Some important physical applications of the systems are also discussed: linear stability of periodic Hamiltonian systems, position-dependent effective mass, pseudo-periodic nonlinear water waves, and Dirac systems.

This article is a survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a new generalization of the binary/"strong" Goldbach's Conjecture (GC) briefly called "the Vertical Goldbach's Conjecture" (VGC), which is essentially a meta-conjecture, as VGC states an in finite number of Goldbach-like conjectures stronger than GC, which all apply on "iterative" primes with recursive prime indexes (named "i-primes"). VGC was discovered by the author of this paper in 2007, after which it was improved and extended (by computational verifications) until the present (2020). VGC distinguishes as a very important "meta-conjecture" of primes because it states a new class containing an infinite number of conjectures stronger/stricter than GC. VGC has great potential importance in the optimization of the GC experimental verification (including other possible theoretical and practical applications in mathematics and physics). VGC can be also regarded as a very special self-similar property of the distribution of the primes. This present survey contains some new results on VGC.

We present some recent results in Fibrewise General Topology with special regard to the theory of Tychonoff compactifications of mappings. Several open problems are also proposed.

The introduction of strings into the study of the Riemann Hypothesis provides a visualization of the genesis of zeros for the Zeta function. The method is heuristic and when originally introduced suggested strong visual evidence for the truth of the Riemann Hypothesis. Some sort of organizing principle for strings with similar t values is observed and points towards a region between (1, 0) and (0, 0) on the abscissa, and within order unity along the ordinate. Progress in understanding these observations has been made by expanding the domain of sigma from the critical strip, [0, 1], to the half-line [0, infinity]. The nature of the organizing principle is explained. A generic structure for strings over the expanded domain is proffered. New perspective is gained regarding the truth of the Riemann Hypothesis.

We prove that there is no non-trivial integral positive solution to the generalized Fermat equation.

Using a left-to-right "sweeping" algorithm, we define the \emph{Gauche basis} for the column space of a matrix M . By means of the Gauche basis we interpret the row reduced echelon form of M , and give a direct proof of its uniqueness. We conclude with pedagogical reflections.

Andrei et al. have shown in 2000 that the graph C of the Collatz function starting with root 8 after the initial loop is an infinite binary tree A(8) . According to their result they gave a reformulated version of the Collatz conjecture: the vertex set V(A(8))= Z + . In this paper an inverse Collatz function C → with eliminated initial loop is used as generating function of a Collatz graph C C → . This graph can be considered as the union of one forest that stems from sequences of powers of 2 with odd start values and a second forest that is based on branch values y=6k+4 where two Collatz sequences meet. A proof that the graph C C → (1) is an infinite binary tree A C → with vertex set V( A C → (1))= Z + completes the paper.

In this paper we prove that the Navier-Stokes initial value problem (1) has a unique smooth local strong solution and if the following condition are satisfied (1) and is Hölder continuous about on, (2) The initial value

A study of 7-fold tilings that use a set of three proto-rhombs in a substitution scheme to tile a large area. A set is discovered that is thought to be the most minimal or smallest one. The scheme uses 11 proto-rhombs to tile the next generation of inflated tiles. The general form of 7-fold substitutions is shown and the role of the 7-fold magic number phi is derived. The figures include a number of newly discovered 7-fold tilings.