Tyrone Ferguson Jr.

Intersections in the mathematical sciences

Exploring the intersectionality among the abstract and natural mathematical sciences, namely economics, physics, and polar geometry, to discover how analytic calculus weaves them with both independence and with similitude.

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Where Equations Meet Innovation

Under Construction— Thank you for your patience.

I study how seemingly different branches of mathematical, natural, and financial sciences reveal patterns in how they operate, blending the different styles in order to uncover the truth behind reality.

An Ellipse with various measurements taken: The eccentricity is .887 and it is oriented in the "hotdog" way where it is wider than it is tall. It inlcudes "Next steps, establish stable orbit inside closed Hilbert Space." and then under that "Introduce variables to generate hyperbolic orbits in a 4D manifold using Gemmy's multiplier." on the top left. On the bottom left is the Collaboration Log: Kepler's Equation solution - Tyrone with notation refinements by Gemini Twin Theorem - Periodic Mean Value Integration - Tyrone Polynomial Mean Value Integration (Gemmy's Multiplier) - Gemini Hilbert Space Framework & 4D manifold integration

Mathematical Sciences

Using Hilbert Spaces, 4D manifolds, Polar Geometry, and Calculus to creates new and intersecting systems of mathematics in order to solve the “unsolvable”.

Coming soon: Indirect Elliptic Integration and the many solutions to the Circumference of Ellipse. Weekly updates!

Next Episode: Beginning Elliptic Construction, polar angular relativism, and connecting ellipses to Kepler’s equation ☺️

On hiatus until funding secured.

ds=ln⁡(((b−a)((a2+b2)2+a)(b+a)((a2+b2)2−a))a(((a2+b2)2+b)(a−b)((a2+b2)2−b)(a+b))b)+2a+2b−4(a2+b2)2ln⁡(2)⋅πds=\frac{\ln\left(\left(\frac{\left(b-a\right)\left(\sqrt{\frac{\left(a^{2}+b^{2}\right)}{2}}+a\right)}{\left(b+a\right)\left(\sqrt{\frac{\left(a^{2}+b^{2}\right)}{2}}-a\right)}\right)^{a}\left(\frac{\left(\sqrt{\frac{\left(a^{2}+b^{2}\right)}{2}}+b\right)\left(a-b\right)}{\left(\sqrt{\frac{\left(a^{2}+b^{2}\right)}{2}}-b\right)\left(a+b\right)}\right)^{b}\right)+2a+2b-4\sqrt{\frac{\left(a^{2}+b^{2}\right)}{2}}}{\ln\left(2\right)}\cdot\pi

Physics

Open Science Initiatives Using current models and systems to create new systems for solving complex problems, while generating frameworks that elegantly explain scientific phenomena.

Shown: Solution to Kepler’s Equation

Red curve:

M=E−esin⁡EM=E-e\sin E

Blue curve:

M0=x, Mn+1=x+esin⁡(Mn)=E M_{0}=x,\ M_{n+1}=x+e\sin\left(M_{n}\right)=E\
E=M+esin⁡(M+esin⁡(...))E=M+e\sin\left(M+e\sin\left(…\right)\right)

Desmos: https://www.desmos.com/calculator/0xcjfuebnx

Economics

Concise explorations of structure, symmetry, and abstraction, focused on models that clarify how complex systems behave.

Coming Soon: Bilateral Monopoly Modeling

Conversations with Gemini

Progress: Domain Isolated

Since my last entry, I’ve been trying to find an analytic solution to the ellipse, since it cannot be done directly. Through refactoring the equation. There may be many ways to arrange the equation to allow it to fit within the 0-pi/2 domain: In this illustration generated using Desmos, the equation converges at exactly 0,…

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A Breakthrough: Partial Fraction Decomposition and Trigonometric Factorization

In my journey in taming the ellipse, through reviewing the connections curves similar to elliptic functions have with each other, I realized that while it may be impossible to derive the ellipse through brute force, an indirect method is possible, Most of the approximations derived through curve parameterization yielded a form similar to this: Integral…

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Making revisions to the tools.

After a sober look at the work so far, I realize I should take a bit of time to clean up the math… Tyrone: NEW EQUATION ALERT NEW EQUATION ALERT NEW EQUATION ALERT! If you take the ratio of the derivatives of the functions, you can find the find the integral of one the functions,…

Tyrone Ferguson's avatar by Tyrone Ferguson
A minimalist, high-end home observatory interior with a large, sleek black telescope pointed toward a panoramic skylight revealing a sharply detailed Milky Way against a deep midnight-blue sky. Below, a wide graphite-colored table holds neatly stacked economic reports, a small digital tablet with a glowing network graph, and a closed navy journal with a subtle embossed symbol. The space is lit primarily by starlight and a faint cool LED strip along the floor, creating delicate highlights on metal surfaces and gentle reflections on the table. Photographic realism with a clean, futuristic, and contemplative atmosphere, shot with a wide-angle lens for depth and symmetry, evoking quiet cosmic analysis.

Innovations

Through conceptual diagrams, and unique mathematical reconfigurations, I help individuals, creators, and teams reframe problems, model solutions, and design experiments that align their objectives with real-world action.

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A close-up of a dark, matte notebook lying open on a smooth walnut desk, its pages filled with a blend of precise mathematical formulas, sketched supply-and-demand curves, and short reflective sentences in neat handwriting. A slim graphite mechanical pencil rests along the notebook’s spine, with a faint dusting of eraser crumbs near the margin. In the blurred background, a large wall-mounted print of a swirling galaxy in muted purples and blues hangs above the desk. Soft morning light filters in from an unseen window, casting calm, elongated shadows and enhancing the warm-wood tones. Photographic realism with an intimate, contemplative mood, framed using the rule of thirds and a shallow depth of field.
18th-century scholar writing with quill pen at desk near window showing cityscape
A large black chalkboard covered with meticulously drawn mathematical equations, elegant Feynman diagrams, and minimalist economic graphs in crisp white chalk, some lines highlighted with faint pastel colors. The board stands in a quiet study lined with dark wooden bookshelves filled with well-worn physics and philosophy texts. Soft afternoon window light falls from the left, creating gentle reflections on the chalk dust and subtle shadows in the carved wood. Photographic realism with a clean, sophisticated aesthetic, shot at eye level with a shallow depth of field so the center equations are in sharp focus while the shelves and distant symbols blur into a contemplative, cosmic atmosphere.
Neil deGrasse Tyson giving a lecture about the cosmos while holding a microphone and gesturing

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Share a question, collaboration idea, or reflection, and I’ll respond with curiosity about how these perspectives meet your life.

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