Journal of Mathematics Research, Issue: Vol.18, No.3

Applying Noether Theorems to Nontrivial Zeros of Riemann Zeta Function and Odd Primes From Sieve of Eratosthenes

Wed, 30 Sep 2026 05:07:56 +0000

We differentiate rigorous Statistical mathematical proofs from rigorous Non-statistical mathematical proofs. Statement 1: As faithfully present in self-dual L-function of Dirichlet eta function [that represents analytic continuation of Genus 0 curve Riemann zeta function], the entire Set Nontrivial Zeros containing infinitely many elements is only located on its Critical line. Statement 2: As faithfully computed using Sieve of Eratosthenes, the entire Set Gap n Odd Primes containing infinitely many elements is precisely constituted from Arbitrarily Large Number of Subsets Gap n = 2, 4, 6, 8, 10... Odd Primes with each Subset again containing infinitely many elements. Using correct and complete mathematical arguments, we show these two statements are true only if Noether theorems [involving Science of Symmetry and Law of Conservation] are fully complied with. As extra supplementary materials, we also discuss various types of higher Genus curves that relate Generalized Riemann hypothesis to Generalized Birch and Swinnerton-Dyer conjecture. We provide the rigorous proof for Gilbreath's conjecture "and beyond" in the Conclusions section of this paper.

Integro-differential Equation of a Risk Model Built from the Hawkes Process with a Dividend Payement Strategy at a Variable Threshold to Shareholders

Wed, 30 Sep 2026 05:09:37 +0000

This article advances the risk model constructed from the Hawkes process by incorporating a variable-threshold dividend payment strategy for shareholders, as well as a dependence structure between claim sizes and the inter-arrival times of claims through a Spearman copula. We determine the integro-differential equation associated with this risk model.

Performance Analysis of the Crank–Nicolson Scheme for Volterra Integro-Differential Equations

Wed, 30 Sep 2026 14:53:32 +0000

In this study, the Crank–Nicolson (CN) method is applied to solve higher-order Volterra integro-differential equations. The original integro-differential equation is first transformed into an equivalent system of first-order equations. The CN method is then used to discretize the differential terms, while the integral terms are approximated using the composite trapezoidal rule. To handle the nonlinear terms, a semi-implicit predictor–corrector scheme is employed. For the stability analysis, perturbation arguments and Grönwall's inequality are used to show that the numerical errors remain bounded under suitable conditions. Numerical experiments on both linear and nonlinear Volterra integro-differential equations demonstrate that the proposed method produces highly accurate solutions and achieves second-order convergence for linear problems. For nonlinear problems, however, a reduction in the convergence rate is observed.

Tribonacci Cordial Labeling on Composite Graphs of Cycles and Paths

Wed, 30 Sep 2026 14:14:35 +0000

This paper investigates Tribonacci cordial labeling on two classes of composite graphs formed from cycles and paths. While Fibonacci-based cordial labeling has been extensively studied, results involving higher-order recurrence sequences, particularly the Tribonacci sequence, remain limited—especially for graphs constructed by combining fundamental structures such as cycles and paths.

Motivated by this gap, we consider two families of graphs, namely G(C_n,C_m,P_k), obtained by connecting two cycle graphs via a path, and G_k(C_n), consisting of multiple cycle graphs sharing a common adjacent vertex. By exploiting the parity properties of the Tribonacci sequence and constructing appropriate injective vertex labelings, we establish that both graph classes admit Tribonacci cordial labeling for all integers n,m≥3,k≥2.

The results highlight the crucial role of the periodic parity pattern of the Tribonacci sequence in achieving balanced edge labelings and extend existing studies on Fibonacci cordial labeling to higher-order recurrence sequences. Several illustrative examples are provided to demonstrate the proposed constructions.

Step-Norm Thresholds and Prime-Walk Connectivity in the Eisenstein Integers

Wed, 30 Sep 2026 14:53:12 +0000

The moat problem asks whether one can move arbitrarily far from the origin using prime elements as stepping stones while keeping every Euclidean step below a fixed bound. We study finite-radius prime-walk graphs in the Gaussian and Eisenstein integers and introduce a discrete step-norm framework for the Eisenstein lattice. For radius $R$ and step bound k, prime elements are vertices, and two distinct vertices are adjacent when their Euclidean distance is at most k. A selected connected component is computed by breadth-first search from a deterministic minimum-norm prime. At k=2, the selected Gaussian component has 720 vertices for $R=50,100,150,200, whereas the selected Eisenstein component grows from 1{,}974$ to 4{,}186. The principal theoretical contribution is a threshold-invariance theorem: because squared Eisenstein displacements have the form a^2-ab+b^2, the finite graph can change only when k^2 crosses a represented norm. At R=200, the selected Eisenstein component grows from 23 vertices at k=1 to 1{,}402 at \sqrt3, 4{,}186 at 2, and 22{,}474 at \sqrt7; at k=4, all 22{,}810 enumerated prime vertices lie in one truncated component. These results identify a pronounced finite-radius transition without asserting an infinite walk or a global moat.