Publications

In this paper, we introduce the Tri-Quarter Topological Duality Theorem, the foundation of a novel mathematical framework that unifies complex, Cartesian, and polar coordinate systems on the complex plane C while equipping the circle T_r of radius r > 0 with a new topological property. Our framework integrates a generalized coordinate system—where real and imaginary components are assigned unique phase pairs—with a structured orientation that elevates T_r to an active separator with intrinsic directional properties. We prove that T_r, as the boundary zone, exhibits topological duality with the inner zone X_-,r (||x|| < r) and outer zone X_+,r (||x|| > r), ensuring consistent separation between inner and outer radial directions across T_r with a phase pair map encoding additional information. We also introduce the Escher Tri-Quarter Reflective Duality Theorem, proving reflective duality across T_r via a circle inversion map that preserves phase pairs while swapping X_-,r and X_+,r. This approach offers insights into topological separation, orientation, and reflection, facilitating analysis of systems with circular symmetry, with potential applications in fields such as black hole physics, signal processing, and other areas reliant on complex domain partitioning. A case study on quadrant-based transformations demonstrates streamlined directional mappings, geometric elegance, unified classification, and computational efficiency in C. A software tool visualizes some of these concepts, with future work aimed at exploring practical implementations.

The event GW150914 was the first historical detection of gravitational waves (GWs). The emergence of this ground-breaking discovery came not only from incredibly innovative experimental work, but also from a centennial of theoretical analyses. Many such analyses were performed by pioneering scientists who had wandered through a wild territory of mathematical laws. We explore such wandering and explain how it may impact the grand goal of unification in physics.

A latin square of order-n is an n-by-n array over a set of n symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic structures. When an algebraic structure passes certain "latin square tests"​, it is a candidate for use in the construction of cryptographic systems. A transversal of a latin square is a list of n distinct symbols, one from each row and each column. The question regarding the existence of transversals in latin squares that encode the Cayley tables of finite groups is far from being resolved and is an area of active investigation. It is known that counting the pairs of permutations over a Galois field F_p^d whose point-wise sum is also a permutation is equivalent to counting the transversals of a latin square that encodes the addition group of F_p^d. We survey some recent results and conjectures pertaining to latin squares and transversals. We create software tools that generate latin squares and count their transversals. We confirm previous results that cyclic latin squares of prime order-p possess the maximum transversal counts for 3 ≤ p ≤ 9. Furthermore, we create a new algorithm that uses these prime order-p cyclic latin squares as "building blocks"​ to construct super-symmetric latin squares of prime power order-p^d with d > 0; using this algorithm we accurately predict that super-symmetric latin squares of order-p^d possess the confirmed maximum transversal counts for 3 ≤ p^d ≤ 9 and the estimated lower bound on the maximum transversal counts for 9 < p^d ≤ 17. Also, we give some conjectures regarding the number of transversals in a super-symmetric latin square. Lastly, we use the super-symmetric latin square for the additive group of the Galois field (F_3², +) to create a simplified version of Grøstl, an iterated hash function, where the compression function is built from two fixed, large, distinct permutations.

In a series of papers, Santilli and collaborators released various strong statements against the general theory of relativity (GTR) and the standard λCDM model of cosmology. In this paper we show that such claims are due to misunderstandings of basic concepts of gravitation and cosmology. In particular, we show that Santilli and collaborators demonstrated neither that the GTR is wrong, nor that the Universe is not expanding. We also show that the so-called iso-gravitation theory (IGT) of Santilli is in macroscopic contrast with geodesic motion and, in turn, with the Equivalence Principle (EP) and must therefore be ultimately rejected. Finally, we show that, although the so called iso-redshift could represent an interesting alternative (similar to the tired light theory historically proposed by Zwicky) to the Universe expansion from a qualitative point of view, it must be rejected from a quantitative point of view because the effect of iso-redshift is 10^-6 smaller than the effect requested to achieve the cosmological redshift.

In this short remark, we report on recent hypothetical work that aims to equip Santilli's magnecule model with topological deformation order parameters (OP) of fractional statistics to define a preliminary set of wave-packet wave-functions for the electron toroidal polarizations. The primary objective is to increase the representational precision and predictive accuracy of the magnecule model by exemplifying the fluidic characteristics for direct industrial application. In particular, the OPs are deployed to encode the spontaneous superfluidic gauge symmetry breaking (which may be restored at the iso-topic level) and correlated with Leggett's superfluid B phases to establish a long range constraint for the wave-functions. These new, developing, theoretical results may be significant because the OP configuration arms us with an extra degree of freedom for encoding a magnecule's states and transitions, which may reveal further insight into the underlying physical mechanisms and features associated with these state-of-the-art magnecular bonds.

In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci’s numbers and (the complex moduli of) Mandelbrot’s numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin’s holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli’s iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.

In this work, we upgrade the Inopin-Schmidt quark confinement and baryon-antibaryon duality proof with Santilli's new iso-mathematics. For a baryon-antibaryon pair confined to the six-coloring kagome lattice of the Inopin Holographic Confinement Ring (IHCR), we construct a cutting-edge procedure that iso-topically lifts the antisymmetric wavefunctions and matrices to iso-wavefunctions and iso-matrices, respectively. The initial results support our hypothesis that transitions between the energy and resonance states of the hadronic spectra may be rigorously characterized by properly-calibrated iso-topic liftings. In total, these rich developments suggest a promising future for this emerging iso-confinement framework, which must be subjected to additional scientific inquiry, scrutiny, and exploration.

We propose a preliminary framework that engages iso-triplex numbers and deformation order parameters to encode the spatial states of Iso-Open Topological Strings (Iso-OTS) for fermions and the temporal states of Iso-Closed Topological Strings (Iso-CTS) for bosons, where space and time are iso-dual. The objective is to introduce an elementary Topological Iso-String Theory (TIST) that complies with the holographic principle and fundamentally represents the twisting, winding, and deforming of helical, spiral, and vortical information structures—by default—for attacking superfluidic motion patterns and energy states with iso-topic lifting. In general, these preliminary results indicate a cutting-edge, flexible, consistent, and powerful iso-mathematical framework with considerable representational capability that warrants further examination, collaboration, construction, and discipline.

In this preliminary work, we focus on a particular iso-geometrical, iso-topological facet of iso-mathematics by suggesting a developing, generalized approach for encoding the states and transitions of spherically-symmetric structures that vary in size. In particular, we introduce the notion of "effective iso-radius" to facilitate a heightened characterization of dynamic iso-sphere Inopin holographic rings (IHR) as they undergo "iso-transitions" between "iso-states". In essence, we propose the existence of "effective dynamic iso-sphere IHRs". In turn, this emergence drives the construction of a new "effective iso-state" platform to encode the generalized dynamics of such iso-complex, non-linear systems in a relatively straightforward approach of spherical-based iso-topic liftings. The initial results of this analysis are significant because they lead to alternative modes of research and application, and thereby pose the question: do these effective dynamic iso-sphere IHRs have application in physics and chemistry? Our hypothesis is: yes. To answer this inquiry and assess this conjecture, this developing work should be subjected to further scrutiny, collaboration, improvement, and hard work via the scientific method in order to advance it as such.

In this preliminary paper, we apply the Riemannian dual (fractional quantum Hall superfluidic) space-time topology and the six-coloring Gribov vacuum to protium and antiprotium. The results suggest that it may be possible to generalize this framework to all atomic elements. Therefore, this subject warrants further scrutiny, collaboration, refinement, and investigation.

Black hole (BH) area quantization may be the key to unlocking a unifying theory of quantum gravity (QG). Surmounting evidence in the field of BH research continues to support a horizon (surface) area with a discrete and uniformly spaced spectrum, but there is still no general agreement on the level spacing. In the specialized and important BH case study, our objective is to report and examine the pertinent groundbreaking work of the strictly thermal and nonstrictly thermal spectrum level spacing of the BH horizon area quantization with included entropy calculations, which aims to tackle this gigantic problem. In particular, such work exemplifies a series of imperative corrections that eventually permits a BH’s horizon area spectrum to be generalized from strictly thermal to nonstrictly thermal with entropy results, thereby capturing multiple preceding developments by launching an effective unification between them. Moreover, the results are significant because quasi-normal modes (QNM) and "effective states" characterize the transitions between the established levels of the nonstrictly thermal spectrum.

In a preliminary assessment, we begin to apply Santilli's iso-mathematics to triplex numbers, Euclidean triplex space, triplex fractals, and Inopin's 2-sphere holographic ring (HR) topology. In doing so, we successfully identify and define iso-triplex numbers for iso-fractal geometry in a Euclidean iso-triplex space that is iso-metrically equipped with an iso-2-sphere HR topology. As a result, we state a series of lemmas that aim to characterize these emerging iso-mathematical structures. These initial outcomes indicate that it may be feasible to engage this encoding framework to systematically attack a broad range of problems in the disciplines of science and mathematics, but a thorough, rigorous, and collaborative investigation should be in order to challenge, refine, upgrade, and implement these ideas.

In this preliminary work, we use a dynamic iso-unit function to iso-topically lift the "static" Inopin holographic ring (IHR) of the unit sphere to an interconnected pair of "dynamic iso-sphere IHRs" (iso-DIHR), where the IHR is simultaneously iso-dual to both a magnified "exterior iso-DIHR" and de-magnified "interior iso-DIHR". For both the continuously-varying and discretely-varying cases, we define the dynamic iso-amplitude-radius of one iso-DIHR as being equivalent to the dynamic iso-amplitude-curvature of its counterpart, and conversely. These initial results support the hypothesis that a new IHR-based mode of iso-geometry and iso-topology may be in order, which is significant because the interior and exterior zones delineated by the IHR are fundamentally "iso-dual inverses" and may be inferred from one another.

In this preliminary work, we propose a hypothesis and initiate a step-by-step, systematic upgrade to the cutting-edge iso-electronium model by further equipping it with order parameters of fractional statistics to encode the topological deformations, spontaneous superfluidic gauge symmetry breaking, correlated helices with long range order, and wavepacket wavefunctions for the toroidal polarizations. For this initial case, we consider the singlet planar coupling of two hydrogen atoms that are interlocked with a Santilli-Shillady strong valence bond to form a molecule with iso-electronium. The enhancement results support our hypothesis and are significant because the order parameters arm the iso-electronium model an extra degree of freedom to work with, which may authorize us to further decode and comprehend the underlying physical mechanisms and features associated with the configuration of the toroidal polarizations. Thus, these outcomes should be subjected to additional rigorous scrutiny and improvement via the scientific method.

It is known that the nonstrictly thermal character of the Hawking radiation spectrum harmonizes Hawking radiation with black hole (BH) quasi-normal modes (QNM). This paramount issue has been recently analyzed in the framework of both Schwarzschild BHs (SBH) and Kerr BHs (KBH). In this assignment, we generalize the analysis to the framework of nonextremal Reissner-Nordström BHs (RNBH). Such a generalization is important because in both Schwarzschild and Kerr BHs an absorbed (emitted) particle has only mass. Instead, in RNBHs the particle has charge as well as mass. In doing so, we expose that, for the RNBH, QNMs can be naturally interpreted in terms of quantum levels for both particle emission and absorption. Conjointly, we generalize some concepts concerning the RNBH's "effective states."

In this work, we deploy Santilli's iso-dual iso-topic lifting and Inopin's holographic ring (IHR) topology as a platform to introduce and assemble a tesseract from two inter-locking, iso-morphic, iso-dual cubes in Euclidean triplex space. For this, we prove that such an "iso-dual tesseract" can be constructed by following a procedure of simple, flexible, topologically-preserving instructions. Moreover, these novel results are significant because the tesseract's state and structure are directly inferred from the one initial cube (rather than two distinct cubes), which identifies a new iso-geometrical inter-connection between Santilli's exterior and interior dynamical systems.

In this exploration, we introduce and define "dynamic iso-spaces", which are cutting-edge iso-mathematical constructions that are built with "dynamic iso-topic liftings" for "dynamic iso-unit functions". For this, we consider both the continuous and discrete cases. Subsequently, we engineer two simple examples that engage Fibonacci's sequence and Mandelbrot's set to define a "Fibonacci dynamic iso-space" and a "Mandelbrot dynamic iso-space", respectively. In total, this array of resulting iso-structures indicates that a new branch of iso-mathematics may be in order.

The non-strictly continuous character of the Hawking radiation spectrum generates a natural correspondence between Hawking radiation and black hole (BH) quasi-normal modes (QNM). In this work, we generalize recent results on this important issue to the framework of Kerr BHs (KBH). We show that also for the KBH, QNMs can be naturally interpreted in terms of quantum levels. Thus, the emission or absorption of a particle is in turn interpreted in terms of a transition between two different levels. At the end of the paper, we also generalize some concepts concerning the "effective state" of a KBH.

We propose a preliminary algorithm which is designed to reduce aspects of the n-body problem to a 2-body problem for holographic principle compliance. The objective is to share an alternative view-point on the n-body problem to try and generate a simpler solution in the future. The algorithm operates 2D and 3D data structures to initiate the encoding of the chaotic dynamical system equipped with modified superfluid order parameter fields in both 3D and 4D versions of the Inopin holographic ring (IHR) topology. For the algorithm, we arbitrarily select one point-mass to be the origin and, from that reference frame, we subsequently engage a series of instructions to consolidate the residual (n-1)-bodies to the IHR. Through a step-by-step example, we demonstrate that the algorithm yields "IHR effective" (IHRE) net quantities that enable us to hypothetically define an IHRE potential, kinetic, and Lagrangian.

In this introductory work, we use Santilli's iso-topic lifting as a cutting-edge platform to explore Mandelbrot's set. The objective is to upgrade Mandelbrot's complex quadratic polynomial with iso-multiplication and then computationally probe the effects on this revolutionary fractal. For this, we define the "iso-complex quadratic polynomial" and engage it to generate a locally iso-morphic array of "Mandelbrot iso-sets" by varying the iso-unit, where the connectedness property is topologically preserved in each case. The iso-unit broadens and strengthens the chaotic analysis, and authorizes an enhanced classification and demystification such complex systems because it equips us with an additional degree of freedom: the new Mandelbrot iso-set array is an improvement over the traditional Mandelbrot set because it is significantly more general. In total, the experimental results exemplify dynamic iso-spaces and indicate two modes of topological effects: scale-deformation and boundary-deformation. Ultimately, these new and preliminary developments spark further insight into the emerging realm of iso-fractals.

In this work, we forge a powerful, easy-to-visualize, flexible, consistent, and disciplined abstract vector framework for particle and astro physics that is compliant with the holographic principle. We demonstrate that the structural properties of the complex number and the sphere enable us to introduce and define the triplex number—an influential information structure that is similar to the 3D hyper-complex number by D. White and P. Nylander—which identifies a 3D analogue of (2D) complex space. Consequently, we engage the complex and triplex numbers as abstract vectors to systematically encode the state space of the Riemannian dual 3D and 4D space-time topologies, where space and time are dual and interconnected; we use the triplex numbers (with triplex multiplication) to extend 1D and 2D algebraic systems to 3D and 4D configurations. In doing so, we equip space-time with order parameter fields for topological deformations. Finally, to exemplify our motivation, we provide three example applications for this framework.

We prove quark (and antiquark) confinement for a baryon-antibaryon pair and design a well-defined, easy-to-visualize, and simplified mathematical framework for particle and astro physics based on experimental data. From scratch, we assemble a dual 4D space-time topology and generalized coordinate system for the Schwarzschild metric. Space-time is equipped with "fractional quantum number order parameter fields" and topological defects for the simultaneous and spontaneous breaking of several symmetries, which are used to construct the baryon wavefunction and its corresponding antisymmetric tensor. The confined baryon-antibaryon pair is directly connected to skyrmions with "massive 'Higgs-like' scalar amplitude-excitations" and "massless Nambu-Goldstone pseudo-scalar phase-excitations". Newton's second law and Einstein's relativity are combined to define a Lagrangian with effective potential and effective kinetic. We prove that our theory upgrades the prediction precision and accuracy of QCD/QED and general relativity, implements 4D versions of string theory and Witten's M-theory, and exemplifies M.C. Escher's duality.

A regular expression and region-specific filtering system for biological records at the National Center for Biotechnology database is integrated into an object-oriented sequence counting application, and a statistical software suite is designed and deployed to interpret the resulting k-mer frequencies—with a priority focus on nullomers. The proteome k-mer frequency spectra of ten model organisms and the genome k-mer frequency spectra of two bacteria and virus strains for the coding and non-coding regions are comparatively scrutinized. We observe that the naturally-evolved (NCBI/organism) and the artificially-biased (randomly-generated) sequences exhibit a clear deviation from the artificially-unbiased (randomly-generated) histogram distributions. Furthermore, a preliminary assessment of prime predictability is conducted on chronologically ordered NCBI genome snapshots over an 18-month period using an artificial neural network; three distinct supervised machine learning algorithms are used to train and test the system on customized NCBI data sets to forecast future prime states—revealing that, to a modest degree, it is feasible to make such predictions.

Our objective is to promote scientific research among undergraduate students at Eastern Oregon University through the development of mobile robots. Simulating mobile robots in research is not a novel technique, but using simulated robots that utilize virtual physics or "Physicomimetics" [2] as a tool to promote undergraduate research is unique. This paper addresses the undergraduate research experience of designing and implementing a simulated mobile robot environment using a physics-based control algorithm.