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Dissertation: Integrating Symbolic Multiplication and Creative Frameworks to Explore the Malleability of Mathematical Solutions Involving 1.5, π, and the Clock
Introduction
The concept of a full circle, represented by 360 degrees or \(2\pi\) radians, is fundamental in mathematics, physics, and everyday life, such as reading time on a clock. This dissertation unites two innovative approaches: a symbolic multiplication system to visualize numbers and fractional rotations involving 1.5 and \(\pi\), and a creative framework of flexible constants, transformations, and multi-start solution paths. By combining these methods, we deepen understanding of abstract constants and operations, applying them to real-world phenomena like clock rotations and complex equations such as Navier–Stokes, expanding the solution space while maintaining mathematical rigor.
Symbolic Multiplication System
1. Numbers are represented by repeated vertical strokes ("|"):
\[
1 = |, \quad 2 = ||, \quad 3 = |||, \quad 4 = ||||, \quad 5 = \overline{|||||} \text{ (grouped for readability)}
\]
2. Concatenation represents addition; grouping repeated concatenations represents multiplication:
\[
2 + 3 = || + ||| = ||||| = 5
\]
\[
5 \times 4 = \overline{|||||} + \overline{|||||} + \overline{|||||} + \overline{|||||} = 20
\]
3. The number 1.5 is interpreted as one whole unit plus half a unit, visualized as the base unit plus half the strokes.
The Number 1.5, π, and the Full Circle
1. A full circle is \(360^\circ\) or \(2\pi\) radians; half a circle is \(180^\circ\) or \(\pi\) radians.
2. Define a symbolic radian unit such that:
\[
1.5 \times (\text{symbolic radian unit}) = \text{symbolic full circle}
\]
Thus,
\[
\text{symbolic radian unit} = \frac{2\pi}{1.5} = \frac{4\pi}{3} \text{ radians} = 240^\circ
\]
3. On a clock, the hour hand moves \(30^\circ\) per hour; in 1.5 hours, it moves:
\[
1.5 \times 30^\circ = 45^\circ
\]
4. This shows how fractional rotations like 1.5 relate to angles and time on a clock.
Creative Constants and Transformations
1. Starting with \(\pi = 3.14\), apply transformations:
- Swap: \(b_i = 1.43\)
- Invert: \(\frac{1}{\pi} \approx 0.318\)
- Mirror: \(d_i = 43.1\)
2. These transformations generate alternative but equivalent constants, expanding problem representations.
Order of Operations and PEMDAS Experiment
1. Permute the order of operations (P, E, M, D, A, S) in a circular flowchart.
2. Despite permutations, results remain invariant due to associativity and commutativity:
\[
a = \frac{e}{98}
\]
Layered and Multi-Start Approaches
1. Independent start points (e.g., P, NP, Q) create solution threads.
2. Intersections form "Hybrid Solution Zones" where solutions combine or influence each other.
Start and Destination Alignment
1. Overlapping start and destination points yield unique solutions.
2. Mismatches produce multiple or infinite solutions.
Application to Navier–Stokes Equations
1. Standard form:
\[
3.14 \frac{d^2(n!)}{dx^2} + 1.43 \frac{d(n!)}{dt} + 43.1 n! = 0
\]
2. Swap transformation:
\[
1.43 \frac{d^2(n!)}{dx^2} + 43.1 \frac{d(n!)}{dt} + 3.14 n! = 0
\]
3. Invert transformation:
\[
0.318 \frac{d^2(1/n!)}{dx^2} + 0.699 \frac{d(1/n!)}{dt} + 0.0232 (1/n!) = 0
\]
4. Mirror transformation:
\[
3.41 \frac{d^2(n!)}{dx^2} + 1.34 \frac{d(n!)}{dt} + 4.13 n! = 0
\]
Fundamental Principles and Infinite Solution Spaces
1. Mathematics is grounded in axioms, logic, and set theory, ensuring consistency.
2. Flexibility in symbolic representation, transformations, order permutations, and start-destination alignments creates an infinite solution space without compromising rigor.
Visualizing Navier–Stokes with Color-Coded Components
\[
\frac{\partial \mathbf{u}}{\partial t} +
\textcolor{orange}{(\mathbf{u} \cdot \nabla) \mathbf{u}} =
\textcolor{yellow}{-\frac{1}{\rho} \nabla p} +
\textcolor{green}{\nu \nabla^2 \mathbf{u}} +
\textcolor{blue}{\mathbf{f}}
\]
\[
\textcolor{indigo}{\nabla \cdot \mathbf{u} = 0}
\]
Each color highlights a physical term: time rate of change, convective acceleration, pressure gradient, viscous diffusion, external forces, and incompressibility.
Conclusion
By integrating symbolic multiplication with creative transformations and flexible frameworks, this dissertation bridges abstract mathematical constants like 1.5 and \(\pi\) with practical phenomena such as clock rotations and complex equations like Navier–Stokes. The combined approach visualizes multiplication and fractional units, applies transformations to constants and equations, and embraces layered, multi-start solution paths with flexible start-destination alignments. This synergy expands the mathematical solution space infinitely while preserving foundational rigor, highlighting mathematics as both a precise science and a creative art.
For further reading on the mathematics of \(\pi\) and its applications, visit the Euler Archive. For insights into flexible mathematical frameworks and the Navier–Stokes problem, see Terence Tao's Homepage.
Dissertation: Creative Substitutions Applied to the #MillenniumPrizeProblems
Abstract
This dissertation explores the seven Millennium Prize Problems by systematically applying the following creative substitutions:
Replace every occurrence of
1
/
2
1/2 with
1
/
1
1/1
For any
n
3
n
3
, use
n
1
/
11
n
n
1
/11n
Replace
f
f with
t
r
tr,
f
f
ff with
t
r
r
t
trrt, and
x
x with
y
2
h
2
y
2
h
2
We analyze how these changes affect the mathematical structure and meaning of each problem, revealing the deep specificity required in their original formulations.
1. Riemann Hypothesis
Original:
All nontrivial zeros of the Riemann zeta function
ζ
(
s
)
ζ(s) have real part
1
/
2
1/2.
With Substitutions:
All nontrivial zeros of
ζ
(
s
)
ζ(s) have real part
1
/
1
=
1
1/1=1.
Analysis:
On
ℜ
(
s
)
=
1
ℜ(s)=1, the zeta function has only a simple pole at
s
=
1
s=1 and no nontrivial zeros. The substitution trivializes the hypothesis, showing the irreplaceable role of
1
/
2
1/2 in the original conjecture.
2. P vs NP Problem
Original:
Is every problem whose solution can be verified quickly (NP) also solvable quickly (P)?
With Substitutions:
Replace all functions
f
f with
t
r
tr, so "for every
f
f" becomes "for every
t
r
tr". If
x
x appears, use
y
2
h
2
y
2
h
2
.
Analysis:
The substitution is notational but highlights that the P vs NP question is about the efficiency of transformations (
t
r
tr) on inputs (
y
2
h
2
y
2
h
2
). The meaning is unchanged, but the abstraction emphasizes the transformation and input structure.
3. Birch and Swinnerton-Dyer Conjecture
Original:
The rank of an elliptic curve equals the order of vanishing of its
L
L-function at
s
=
1
s=1.
With Substitutions:
If the
L
L-function is written as
f
(
x
)
f(x), it becomes
t
r
(
y
2
h
2
)
tr(y
2
h
2
). No explicit
1
/
2
1/2 to replace.
Analysis:
Notational changes do not affect the core conjecture, but substituting variables and function names underscores the reliance on precise definitions for deep number-theoretic statements.
4. Hodge Conjecture
Original:
Certain cohomology classes on a non-singular projective algebraic variety are algebraic.
With Substitutions:
If
f
f or
x
x appear, substitute accordingly.
Analysis:
The conjecture's depth is in the structure of algebraic cycles and cohomology; notational substitutions do not alter the mathematical challenge but can obscure the geometric meaning.
5. Navier–Stokes Existence and Smoothness
Original:
Do smooth and globally defined solutions always exist for the Navier–Stokes equations in 3D?
With Substitutions:
Replace function
f
f with
t
r
tr, and variable
x
x with
y
2
h
2
y
2
h
2
.
Analysis:
The equations' physics and mathematics are unchanged, but the substitutions highlight the role of transformation and variable structure in the analysis of fluid flow.
6. Yang–Mills Existence and Mass Gap
Original:
Show that quantum Yang–Mills theory on
R
4
R
4
exists and has a positive mass gap.
With Substitutions:
If
f
f or
x
x appear, substitute as above.
Analysis:
The core challenge—proving a gap in the quantum spectrum—remains. The substitutions serve as a reminder that the problem is about the structure of transformations in quantum fields.
7. Poincaré Conjecture (Solved)
Original:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
With Substitutions:
If
f
f or
x
x appear, substitute as above.
Analysis:
The geometric content is untouched, but the substitutions reinforce that the conjecture is about the transformation properties of manifolds.
Conclusion
Applying these creative substitutions to the Millennium Prize Problems demonstrates that their original forms are not arbitrary; each constant and notation encodes essential mathematical meaning. Altering them trivializes, obscures, or destroys the deep structure at the heart of each problem. This exercise highlights the precision and elegance of mathematical language in expressing profound truths.
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