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preprint2025arXiv

Analyzing Dynamical Systems Inspired by Montgomery's Conjecture: Insights into Zeta Function Zeros and Chaos in Number Theory

In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(πu)}{πu} \right)^2 + δ(u)$. The recurrence $x_{n+1} = 1 - \left( \frac{\sin(π/x_n)}{π/x_n} \right)^2 + \frac{1}{x_n}$ emulates eigenvalue repulsion as a quantum operator analogue realizing the Pólya-Hilbert conjecture. Bifurcation analysis and Lyapunov exponents reveal quantum-like chaos: near $x=0$, linearized dynamics $f(x) = 1 - π^2 x^2$ yield Gaussian Lyapunov function $V(x) = C_1 e^{-π^2 x^3/3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth $λ_n \to \ln(π^2/6)$. Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual $γ_n$ achieves errors $<10^{-100}$, while spectral density $ρ(E) \sim \frac{\log E}{2π}$ matches zeta zero statistics. This bridges Montgomery pair correlation to quantum chaos, providing computational evidence for Riemann zero spacing distributions and supporting the quantum operator hypothesis for $ζ(1/2+it)$.

preprint2023arXiv

Higher Derivatives of the Tangent and Inverse Tangent Functions and Chebyshev Polynomials

The higher derivatives of the tangent and hyperbolic tangent functions are determined. Formulas for the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions as polynomials are stated and proved. Using another formula for the higher derivatives of the inverse tangent function from literature, two known formulas for the Chebyshev polynomials of the first and second kind are proved. From these formulas the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions in terms of the Chebyshev polynomial of the second kind are provided.

preprint2022arXiv

Evolution patterns in Collatz problem

The concept of evolution patterns is introduced for Collatz sequences and it is shown that any finite evolution pattern is implemented in some particular Collatz sequence.

preprint2022arXiv

Entropy, Symmetry, and the Difficulty of Self-Replication

The defining property of an artificial physical self-replicating system, such as a self-replicating robot, is that it has the ability to make copies of itself from basic parts. Three questions that immediately arises in the study of such systems are: 1) How complex is the whole robot in comparison to each basic part ? 2) How disordered can the parts be while having the robot successfully replicate ? 3) What design principles can enable complex self-replicating systems to function in disordered environments generation after generation ? Consequently, much of this article focuses on exploring different concepts of entropy as a measure of disorder, and how symmetries can help in reliable self replication, both at the level of assembly (by reducing the number of wrong ways that parts could be assembled), and also as a parity check when replicas manufacture parts generation after generation. The mathematics underpinning these principles that quantify artificial physical self-replicating systems are articulated here by integrating ideas from information theory, statistical mechanics, ergodic theory, group theory, and integral geometry.

preprint2025arXiv

Homogeneity

The four types of homogeneity -- additive, multiplicative, exponential, and logarithmic -- are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity functions capture different scaling behaviors and establish fundamental properties. Such properties include how homogeneity is preserved under function operations and how it determines the transformation behavior of related constructions like quotient functions. This framework extends the classical concept of homogeneity to a wider class of functional symmetries, providing a unified approach to analyzing scaling properties in various mathematical contexts.

preprint2010arXiv

Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane

This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three parameter families of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams are exhibited. Our main result describes the unfolding of the so called Fold-Saddle singularity.

preprint2011arXiv

A Logical Calculus To Intuitively And Logically Denote Number Systems

Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, we construct a logical calculus and deduce numbers to intuitively and logically denote number systems.

preprint2023arXiv

On the Regularity for 3D Navier-Stokes Equation

In this paper we will prove that the vorticity belongs to L1(0; T ; L2(Ω)) for 3D incompressible Navier-Stokes equation with periodic initial-boundary value conditions, then the existence of a global smooth solution is obtained. Our approach is to construct a set of auxiliary problems to approximate the original one of vorticity equation.

preprint2026arXiv

A geometry-dependent, force balance-driven model of Staphylococcus epidermidis biofilm cell cluster detachment

Biofilms, bacteria cells surrounded by a self-produced polymeric matrix, are common on medical devices and lead to many hospital infections. The biofilm lifecycle includes disassembly and dispersion, where bacteria clusters detach from the biofilm, circulate in the bloodstream, and potentially colonize secondary infection sites. Existing models often simplify detachment to a function of biofilm thickness or extracellular polymeric substance (EPS) density, without tracking properties of detached clusters that impact their biological fate, including cluster size and morphology. Addressing this gap, our detachment model accounts for drag and adhesion in tagged sections of the biofilm determined by the cluster geometry and local arrangement of bacteria and EPS. A stickiness parameter controls local EPS adhesion strength, which is modulated to disrupt (or compromise) EPS biomass. We specifically model the detachment of clusters from a Staphylococcus epidermidis biofilm grown for 24 hours. Experimental data for biofilm microstructural features are utilized to benchmark the simulated biofilm, which is then subjected to different EPS disruption levels. We examine parameters that influence detached biofilm cell cluster frequency, size, and shape, providing mechanistic insights into how compromised EPS influences detachment dynamics. This integrated modeling framework is a significant advance in the predictive capabilities for biofilm detachment processes.

preprint2022arXiv

On $\mathbf{K}$-reflections of Scott spaces

In this paper, for a full subcategory $\mathbf{K}$ of the category of all $T_0$ spaces with continuous mappings, we investigate the questions under what conditions the $\mathbf{K}$-reflection of a Scott space is still a Scott space and under what conditions the Scott $\mathbf{K}$-completion of a poset exists. Some necessary and sufficient conditions for the $\mathbf{K}$-reflection of a Scott space to be a Scott space and for the existence of Scott $\mathbf{K}$-completion of a poset are established, respectively. It is shown that neither the sobrification nor the well-filtered reflection of the Johnstone space is a Scott space. The $\mathbf{K}$-reflections of Alexandroff spaces and the $\mathbf{K}$-completions of posets are also discussed.

preprint2024arXiv

Golden ratios, Lucas Sequences and the Quadratic Family

It is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $φ^1$, and the next golden ratio $φ^2$. Moreover, the graphic depiction of those ratios show some overlap with the quadratic family, and some numerical evidence suggest that everyone of those ratios in the finite set obtained, belong to at least one quadratic family, and finally a proof is presented that the converging sequence of some generalized Fibonacci ratios belong to at least one quadratic family.

preprint2022arXiv

A note on $LP$-Kenmotsu manifolds admitting Ricci-Yamabe solitons

In the current note, we study Lorentzian para-Kenmotsu (in brief, $LP$-Kenmotsu) manifolds admitting Ricci-Yamabe solitons (RYS) and gradient Ricci-Yamabe soliton (gradient RYS). At last by constructing a 5-dimensional non-trivial example we illustrate our result.

preprint2022arXiv

Maurer-Cartan characterization, cohomology and deformations of equivariant Lie superalgebras

In this article, we give Maurer-Cartan characterizations of equivariant Lie superalgebra structures. We introduce equivariant cohomology and equivariant formal deformation theory of Lie superalgebras. As an application of equivariant cohomology we study the equivariant formal deformation theory of Lie superalgebras. As another application we characterize equivariant central extensions of Lie superalgebras using second equivariant cohomology. We give some examples of Lie superalgebras with an action of a group and equivariant formal deformations of a classical Lie superalgebras.

preprint2024arXiv

The novel Tauberian conditions associated with the $(\overline{N},p,q)$ summability of double sequences

In this paper, our primary objective is to provide a fresh perspective on the relationship between the $(\overline{N},p,q)$ method, which is a product of relevant one-dimensional summability methods, and $P$-convergence for double sequences. To accomplish this objective, we establish certain Tauberian conditions that control the behavior of a double sequence in terms of both $O_L$-oscillation and $O$-oscillation in certain senses, building a bridge between $(\overline{N},p,q)$ summability and $P$-convergence, while imposing certain restrictions on the weight sequences. As special circumstances of our findings, we demonstrate that Landau-type $O_L$ condition with respect to $(P_m)$ and $(Q_n),$ as well as Hardy-type $O$ condition with respect to $(P_m)$ and $(Q_n),$ serve as Tauberian conditions for $(\overline{N},p,q)$ summability under particular additional conditions. Consequently, these results encompass all classical Tauberian theorems, including conditions such as slow decrease or slow oscillation in certain senses.

preprint2008arXiv

The wave equation on static singular space-times

The first part of my thesis lays the foundations to generalized Lorentz geometry. The basic algebraic structure of finite-dimensional modules over the ring of generalized numbers is investigated. The motivation for this part of my thesis evolved from the main topic, the wave equation on singular space-times. The second and main part of my thesis is devoted to establishing a local existence and uniqueness theorem for the wave equation on singular space-times. The singular Lorentz metric subject to our discussion is modeled within the special algebra on manifolds in the sense of Colombeau. Inspired by an approach to generalized hyperbolicity of conical-space times due to Vickers and Wilson, we succeed in establishing certain energy estimates, which by a further elaborated equivalence of energy integrals and Sobolev norms allow us to prove existence and uniqueness of local generalized solutions of the wave equation with respect to a wide class of generalized metrics. The third part of my thesis treats three different point value resp. uniqueness questions in algebras of generalized functions

preprint2021arXiv

Pluralist-Monism. Derived Category Theory as the Grammar of n-Awareness

In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness ("n-awareness"). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the mathematical formalism of higher category theory. The beauty of higher category theory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and $(\infty, 1)$-topoi which will give rise to our meta-language. We then posit a "grammar" ("n-declension") which could express n-awareness, accompanied by a new temporal ontology ("n-time"). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of "personal identity over time" is thus introduced.

preprint2023arXiv

Introducing the Role of Shaping Order K in Set Shaping Theory

Set Shaping Theory, an emerging area of study, delves into the transformation of data sets via bijection functions. Central to this theory is the parameter $K$, which determines the extent of transformation, essentially reshaping the data. This article introduces the pivotal role of $K$ in the Set Shaping Theory, shedding light on its implications for data compression and transformation dynamics

preprint2022arXiv

On the definition of neutrosophic logic

Smarandache (2003) introduced a new set-valued fuzzy logic called (nonstandard) neutrosophic logic by using Robinson's nonstandard analysis. However, its definition involved many errors including the illegal use of nonstandard analysis. In this paper, we provide a rigorous definition of neutrosophic logic. All the errors in the original definition are addressed. We then point out some paradoxes of neutrosophic logic. Finally we formulate neutrosophic logic with no use of nonstandard analysis.

preprint2023arXiv

A Reduced Forward Collatz Algorithm: How Binary Strings Change Their Length Under 3x+1

We developed an algorithm that easily goes from one odd number to the next odd number in binary representation for the reduced forward Collatz map (Syracuse function). The algorithm indicates when an odd number can grow or shrink to the next odd number based on the pattern of binary digits. The algorithm is also used to provide a simpler method for determining the change in binary string length for the reduced map than one found in the literature. Accordingly, an inspection of the binary digits for an odd number can determine the number of binary digits of the subsequent odd number. We also show some simple results for what the smallest number could be for a counterexample to the Collatz conjecture.

preprint2023arXiv

The Product $e π$ Is Irrational

This note shows that the product $e π$ of the natural base $e$ and the circle number $π$ is an irrational number.

preprint2023arXiv

On Hilfer-Prabhakar fractional derivatives Sawi transform and its applications to fractional differential equations

The goal of this paper is to study the Sawi transform and its relationship to Hilfer-Prabhakar and regularized Hilfer-Prabhakar fractional derivatives, as well as to present some lemmas related to the Sawi transform. Additionally, the paper aims to find solutions for Cauchy type fractional differential equations using Hilfer-Prabhakar fractional derivatives, by utilizing the Sawi and Fourier transforms, and involving the three-parameter Mittag-Leffler function.

preprint2023arXiv

The Disproof of the Riemann Hypothesis

We show that there is a contradiction between the Riemann's Hypothesis and some form of the theorem on the universality of the zeta function.

preprint2023arXiv

Fractional Quantization of Singular Lagrangian systems with Second-Order Derivatives Using WKB Approximation

In this paper, the theory of the fractional singular Lagrangian systems is investigated with second order derivatives. The fractional quantization for these systems is examined using the WKB approximation. The Hamilton Jacobi treatment can be applied for these systems. The equations of motion are obtained. The Hamilton Jacobi partial differential equations are solved to determine the action function. By finding the action function, the wave function for these systems is constructed. We achieved that the quantum results are agreement with the classical results. Besides, to demonstrate the theory; two mathematical examples are examined.

preprint2023arXiv

Finding meaningful and workable applied mathematics problems in science

In this short review, I will summarize my research experience in three fields in applied mathematics: mathematical biology, applied probability, and applied discrete mathematics. Specifically, I will show how each project was initiated, and what wrong approaches were applied. Such details are important in learning how to do research, but they cannot be read out from research papers. I wish that students and junior researchers in applied mathematics could learn a lesson from this summary.