Mathematics

This paper gives a structural explanation for the Z-relation by modelling pitch-class sets as complete weighted graphs and encoding their interval content in a composition of $n$ via an additivity rule. We introduce the realization number…

This article explores several fundamental aspects of fuzzy $\mathscr{F}$-metric spaces and their applications in mathematical analysis. We investigate some essential properties concerning compactness and total boundedness in fuzzy…

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.

We develop a unified analytical and dynamical framework for the qualitative study of the one-parameter family of generalized Dirichlet eta functions $\eta_{a}(t)=\sum_{m\ge0}(-1)^{m}(am+1)^{-t}$, $a>0$, $t>0$, which includes the classical…

In this article we solve the Cauchy problem for the relaxation equation posed in a framework of variable order fractional calculus. After introducing some general mathematical theory we establish concepts of Scarpi derivative and transition…

The graph Fourier transform (GFT) is a fundamental tool in graph signal processing and has recently been extended to the graph fractional Fourier transform (GFRFT). Existing sampling methods in the GFRFT domain are primarily designed to…

For $p,q\in\mathbb{N}$ and $\alpha,\beta\in\mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cos\alpha x-\cos\beta x)^p}{x^q}dx.\] We establish a complete classification of the parameter ranges $(p, q;…

This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$…

We study equivariant perfect matchings on the Boolean hypercube $\B^6$ under the Klein four-group $K_4 = \langle \comp, \rev \rangle$ generated by bitwise complement and reversal. Among matchings using only $\comp$ or $\rev$ pairings, there…

This note studies a concrete bitwise and triangular coordinate model for the central product of n copies of the quaternion group Q8. The positive basis elements are words of length n in the alphabet {1, 2, 4, 7}, identified with i, j, k,…

The prime numbers and the non-trivial zeros of the Riemann zeta function are globally linked by the explicit formula of analytic number theory. Whether they share a hidden, scale-by-scale geometric symmetry has remained unexplored. We…

This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime…

We introduce a novel deterministic fractal set PF in the unit interval whose construction is driven by the sequence of prime numbers modulo 16. At each step of the recursive construction, two subintervals are retained based on the residues…

This paper presents a path to proving the Four-Color Theorem that differs from the traditional "reducible configuration" method. By introducing concepts such as "outer boundary," "primitive set," "Property A," "knot," "valid pair group,"…

We present a branch-consistent framework for integrals involving quadratic radicals by expressing exponentials of principal inverse trigonometric functions in algebraic form. Two identities for $e^{\pm i\cos^{-1}(y)}$ and $e^{\pm…

This paper collects polynomial Diophantine equations that are simple to state but apparently difficult to solve.

Several identities of the cosh-weighted finite Hilbert Transform and the Bertola-Katsevich-Tovbis inversion formulas are rederived by the Sokhotski-Plemelj formula and the Poincare-Bertrand formula. The explicit formulas are derived for the…

It is generally accepted that the incompleteness of first-order number theory (PA) is established by an application of Godel's proof. This paper shows that the arithmetization of the syntax of PA implies that the hypothesised class of PA…

We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual…

We present a complete computational tabulation of all 961,619,972 P-positions in 4xn Chomp for n <= 3000, obtained via a new O(n^4) shadow-array sieve that replaces the O(n^5) hash-set approach of prior work. Three structural results are…