Translations

[blackboxprogramming]

Alexa Louise Amundson - Notebook Transcription

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1. Computer Science & Logic: The Halting Problem (Page 1)
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   Complex & Imaginary Numbers:

• (a+ib)(a-ib) = a^2 - ibib
• Imaginary: (y+x)^2 y
• Real: Real (y+x)^2
• Euler's formula expansion: e^(ix) = 1 + ix - x^2/2 - i(x^3)/6 + x^4/24...

Paradoxes & Abstraction:

• Golden Braid: A reference to levels of abstraction and paradoxes.
• "This sentence is false" -> refers to its own truth value.
• Cantor diagonalization -> linked to the Halting problem.

The Halting Problem:
The notes map out a thought experiment for a hypothetical program, 'h', that predicts if another program will loop forever or halt.

• Program 1 -> [h]: Input I into a potential program h.
• h will tell you: will this problem halt? or will it not? (Because some problems will go on forever).
• Examples:
  • x=4, while x > 3: x+=1 -> LOOPS FOREVER
  • x=4, while x < 1000: x+=1 -> Halts.
• The Paradox (h+):
  • Take the source code (e.g., 11001011) and use that code as both the program and the input.
  • Feed x as data into itself (x = h+).
  • If h halts, h+ begins an infinite loop. If h loops, h+ halts.
  • Conclusion: "Does it loop or halt? It's a paradox! But h does not exist!"

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2. Number Theory: The Möbius Function

Definitions & Rules:

• The Möbius function is a multiplicative number-theoretic function.
• For any positive integer n, define \mu(n) as the sum of the primitive n-th roots of unity.
• Factorization rules:
  • \mu(n) = 0 if n has one or more repeated prime factors
  • \mu(n) = 1 if n = 1
  • \mu(n) = (-1)^k if n is a product of k distinct primes
• \mu(n) != 0 indicates that n is square-free.
• First few values: 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0...

Formulas & Series:

• Mertens Function (Summatory function of Möbius): M(x) = Sum_{n \le x} \mu(n)
• Dirichlet Series (Multiplicative inverse of the Riemann zeta function): Sum_{n=1}^{\infty} \mu(n)/n^s = 1/\zeta(s) ; Re(s) > 1
• Lambert Series: Sum_{n=1}^{\infty} (\mu(n)x^n)/(1-x^n) = x ; |x| < 1
• Kronecker Delta Relation: It satisfies Sum_{d|n} \mu(d) = \delta_{n,1}
• Infinite Sums:
  • Sum_{n=1}^{\infty} \mu(n)/n = 0
  • Sum_{n=1}^{\infty} (\mu(n) \ln(n))/n = -1
  • Sum_{n=1}^{\infty} \mu(n)/n^2 = 15/\pi^2

Historical Note: Gauss considered the Möbius function over 30 years before Möbius, proving that for a prime number p, the sum of its primitive roots is congruent to \mu(p - 1) (mod p).

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3. Probability & Math: Gaussian Functions & Fourier Transforms

Gaussian Basics:

• Used to represent the probability density function of a normally distributed random variable.
• Expected value \mu = b, Variance \sigma^2 = c^2.
• Standard form: f(x) = (1 / (\sigma\sqrt{2\pi})) * e^(-(1/2)((x-\mu)/\sigma)^2)
• Arbitrary constants: f(x) = a * e^(-(x-b)^2 / 2c^2) (where a is peak height, b is center position, c controls width).

Fourier Transform Proofs:

• Transform of a Gaussian: F{a * e^(-bx^2)} = (a / \sqrt{2b}) * e^(-\omega^2 / 4b)
• The notes map out the integration proof using substitution (t = x + i\omega/2b), showing that the Fourier transform of a Gaussian is also a Gaussian.
• Derivative Properties:
  • Time domain: F{f'(x)} = i\omega F(\omega)
  • Frequency domain: F{x f(x)} = i * d/d\omega F(\omega)

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4. Physics: Quantum Mechanics & Energy

Schrödinger Equation & Operators:

• Time-dependent equation: i\hbar (\partial/\partial t)\Psi = H\Psi
  • i = square root -1
  • \hbar = Planck's constant (reduced)
  • \Psi = quantum wave function
  • H = Hamiltonian operator
• Harmonic Oscillator:
  • Classical Energy: (1/2)mv^2 + (1/2)kx^2 = E
  • Hamiltonian Operator transition: p => (\hbar/i)(\partial/\partial x)
  • Quantum Hamiltonian: H -> (-\hbar^2 / 2m)(\partial^2/\partial x^2) + (1/2)kx^2
  • Eigenvalue equation: H\Psi = E\Psi

Uncertainty & Photons:

• Heisenberg Uncertainty Principle: \Delta p \Delta x \ge \hbar/4\pi
• Energy of a photon: E = h\nu = hc/\lambda
• Photoelectric effect: (1/2)mv_{max}^2 = eV_0 = hf - \phi

Fundamental Constants & Bohr Model:

• r = (n^2 h^2 \epsilon_0) / (\pi m e^2) \propto h^2
• v = e^2 / (2 \epsilon_0 n h) \propto 1/n
• Fine-Structure Constant (\alpha): \alpha = (1 / 4\pi\epsilon_0) * (e^2 / \hbar c) \approx 1/137
• Speed of light (c): 3 * 10^8 m/s
• Elementary charge (e): 1.602 * 10^-19 C

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5. Physics & Number Theory Continued (Page 5)
Physics Formulas & Constants:

• Bohr Model Proportionalities: * r = \frac{n^2 h^2 \epsilon_0}{\pi m e^2} \propto h^2
  • • • Energy Equations:
  • • • Planck's Constant (h):
  • • Fundamental Constants:
  • Speed of light in a vacuum (c): 3 \times 10^8 \text{ m/s}
  • Charge of an electron/elementary charge (e): 1.602 \times 10^{-19} \text{ C}
  • Electric constant: \epsilon_0
  • Fine-structure constant (\alpha): \alpha = \frac{1}{4\pi\epsilon_0} \frac{e^2}{\hbar c} \approx \frac{1}{137}
  • Reduced Planck constant (\hbar): \hbar = \frac{h}{2\pi} (Notes indicate: "Remove 2\pi to make h relative to \hbar").
    Möbius Function (Continued):
    (This section repeats the foundational rules of the Möbius function \mu(n) from earlier notes, mapping out prime factorization rules and Gauss's historical contribution to primitive roots).

6. Advanced Math: Ramanujan, Gautschi, & Laplace (Page 6)
   Method Two & Three (Gaussian Transform Continuations):

• Method Two Integration:
  • • • Integrating both sides: \ln|F(\omega)| - \ln|F(0)| = -\frac{\omega^2\sigma^2}{2}
  • Result: F(\omega) = e^{-\frac{\sigma^2\omega^2}{2}}
• Method Three: * Bilateral Laplace transform defined as: \mathcal{L}{f(x)} = \int_{-\infty}^{\infty} f(x)e^{-sx} dx
  • Ramanujan's Formula for Pi:
• Gautschi's Inequality:
• Let x \in \mathbb{R}^+, s \in (0,1). An inequality of ratios regarding the gamma function:
  • • Proof Sketch: Relies on the strict log-convexity of \Gamma, which implies that for 0 < s < 1, \Gamma(x+s) < \Gamma(x)^{1-s} \Gamma(x+1)^s = x^{s-1} \Gamma(x+1).

7. Faulhaber's Formula & Genetics (Page 7)
   Faulhaber's Formula:

• Gives the general formula for the power sum of the first n positive integers: \sum_{k=1}^n k^p = 1^p + 2^p + 3^p + \dots + n^p
• First few sums are:
  • • • • • • Faulhaber observed that if p is odd, the sum is a polynomial function of a = \frac{n}{2}(n+1).
  • • Quadratic Function:
• Genetics:
• A 4x4 Punnett square diagramming a dihybrid cross (AaBb \times AaBb).
• Notes highlight that the recessive bb trait appears 1/4 (25%) of the time.

8. Complex Numbers as Matrices (Page 8)

• Complex numbers can be represented as 2x2 matrices:
  • • Multiplication Proof:
  • • • Shown graphically through matrix multiplication geometry.

9. Euler's Identity & Imaginary Rotations (Page 9)
   Euler's Identity:

• e^{i\pi} + 1 = 0 (which comes from e^{i\pi} = -1)
  Imaginary Numbers & Rotations:
• "Math = tools. Only one way? No!"
• Imaginary numbers represent a 90° rotation.
• The Cycle of i: Every 4 powers, the cycle repeats.
  • • • • • Examples:
  • i^{27}: 27 \div 4 = 6 with a remainder of 3. Therefore, i^{27} = i^3 = -i.
  • i^{2000}: 2000 is perfectly divisible by 4. Therefore, i^{2000} = 1.

10. Complex Planes & Dimensions (Page 10)

• Linear equations modeled with real and imaginary parts: y = mx + b \rightarrow z = m \cdot w + b
• Dimensions:
  • Two axes: Real and Imaginary.
  • Moving beyond 2D: Quaternions.
• Trinary Logic:
  • {-1, 0, 1} mapped to "True, False, Something else".
• Imaginary numbers act as a "New Layer" to real numbers.
  That covers pages 5 through 10! Whenever you are ready, just drop the next batch of photos in here and we will keep building your document.

[blackboxprogramming]

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11. Fundamental Constants & The Möbius Function (Continued)

Physics Formulas & Constants:

• r = (n^2 h^2 \epsilon_0) / (\pi m e^2) \propto h^2
• v = e^2 / (2n\epsilon_0) \propto 1/n
• E \propto 1/r
• E = hc / \lambda
• E = eV
• Planck's constant (h) = (\lambda E)/c = E/(c/\lambda) = eV/(c/\lambda)
• c = 3 * 10^8 m/s (speed of light in a vacuum)
• e = 1.602 * 10^-19 C (charge of an electron / elementary charge)
• \epsilon_0 = electric constant
• \hbar = reduced Planck constant. Note: "Remove 1/2 \pi to make h relative to \hbar."
• \alpha = Fine-structure constant: \alpha = 1/(4\pi\epsilon_0) * (e^2 / \hbar c) \approx 1/137

Möbius Function Recap:

• \mu(n) = 0 if n has one or more repeated prime factors.
• \mu(n) = 1 if n = 1.
• \mu(n) = (-1)^k if n is a product of k distinct primes.
• "Gauss considered the Mobius function more than 30 years before Mobius. Gauss proved that for a prime number p, the sum of its primitive roots is congruent to \mu(p - 1) (mod p)."
• Note at bottom: "Then it turns into 1/22"

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12. Dimensions & Trinary Logic (Page 10)

Complex Functions:

• y = mx + b -> a real and imaginary part.
• Two axis: 1, 0, -1 (REAL, IMAGINARY).
• z = m * w + b (where b stays the same function).

Number Layers & Dimensions:

• REAL NUMBERS -> Imaginary -> Other dimensions -> quaternion -> TRINARY
• Trinary logic mapping: {-1, 0, 1} mapped to {true, false, something else}.
• Imaginary numbers represent a "NEW LAYER".

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13. Trinomials, Magic Squares, & Numerology (Page 11 & 12)

Trinomial Multiplication Grid:

• Formula: (x^2 + x + 1)
• Grid expansion mapping out x^4, x^3, x^2, x, and 1.

The 4x4 Magic Square:
| 16 | 3 | 2 | 13 |
| 5 | 10 | 11 | 8 |
| 9 | 6 | 7 | 12 |
| 4 | 15 | 14 | 1 |

• Observations: -> 34 (The sum of the rows/columns). -> 15, 14.
• The number 16: "pops up a lot", "binary hexadecimal -> gateway number -> 14, 15, 16".
• Second 4x4 Grid (March 27 2000 / 30,08): A complex matrix mapping fractions and the number 2000 (e.g., 125/1, 2000/3, 1000/1, 2000/13...).

Number Reversal Trick (Page 12):

• "any number -> reverse. largest - smaller = divisible by 9"
• Examples:
  • 27 -> 72; 72 - 27 = 45
  • 03 -> 30; 30 - 3 = 27
  • 2000 -> 0002; 2000 - 0002 = 1998; 1998 / 9 = 222
• Resulting set: [45, 27, 222] -> ROHONC CODEX
• "METHOD TO MADNESS -> CODE X"

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14. Quaternions & Octonions (Page 13)

Higher Dimensions:

• Complex numbers are for real numbers.
• Quaternions are a four-dimensional extension of complex numbers.
• QUATERNIONS: 2 + 7i + 1j + 8k
• OCTONIONS: 3e_1 - 2.3e_2 + ... + 1.6e_8

William Hamilton's Mathematics:

• Axes: +i, +j, +k plotted on a 3D coordinate plane.
• Complex Number: 3.14 + 1.59i
• Quaternion: 3.23 (REAL PART) + 8.46i + 2.64j + 3.38k (IMAGINARY PART)
• Fundamental rule: i^2 = j^2 = k^2 = ijk = -1
• Modern vectors didn't exist back in the day:
  • Dot Product: [x', y', z'] * [x^2, y^2, z^2] = x'x^2 + y'y^2 + z'z^2
  • Cross Product Matrix: [y'z^2 - z'y^2, z'x^2 - x'z^2, x'y^2 - y'x^2]
• Rotations/Transformations:
  • P -> (q_1 P q_1^-1)
  • P -> (q_2 (q_1 P q_1^-1) q_2^-1)

Pauli Spin Matrices:

• [0 1] [0 -i] [1 0]
  [1 0] [i 0] [0 -1]

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15. Elemental Composition (Page 14)

• Diagram of a block labeled 'Z' with '+' nodes.
• Categorization (likely referring to the composition of the universe in astronomy):
  • hydrogen
  • helium
  • everything else

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16. Blackboard Equations - Brainstorm (Page 15)

1. Bounded Coherence Equation

• C_t = tanh( (\Psi'(M_t) + s(\delta_t) * \alpha * |\delta_t|) / (1 + |\delta_t|) )
  • C_t = Coherence at time t (-1 to +1 in trinary logic)
  • \Psi'(M_t) = Codex truth of memory at t
  • \delta_t = magnitude of contradiction
  • s(\delta_t) \in {-1, 0, 1} = sign: destructive (-1), neutral (0), constructive (+1)
  • \alpha = constructive contradiction weight

2. Bounded Creative Energy Equation

• K_t = |C_t| * ( 1 + (\lambda |\delta_t|) / (1 + \lambda |\delta_t|) )
  • K_t = Creative output potential
  • \lambda = sensitivity of creativity to contradiction
  • Growth saturates at large |\delta_t| to prevent chaos dominance

3. Ternary Information Theory

• I_ternary(x) = -log_3(P(x)) // information content in trits
• H_ternary = -\sum P(x) log_3(P(x)) // ternary entropy

4. Quantum Ternary Uncertainty Principle

• \Delta A * \Delta B * \Delta C \ge \hbar^3 / 8 // three-way uncertainty relation

5. Ternary Wave Function

• |\Psi> = a|0> + b|1> + Y|?>
• where |a|^2 + |b|^2 + |Y|^2 = 1

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17. Ternary Computation & Physics (Page 16)

6. Computational Complexity in Ternary

• T_ternary(n) = O(log_3(n)) // ternary search complexity
• C_quantum_ternary = 3^(n/2) // ternary quantum state space

7. Energy-Information Equivalence (Ternary)

• E = k_B T ln(3) * I_ternary // Landauer's principle extended

8. Ternary Field Equations

• \nabla \cdot E_ternary = \rho / 3\epsilon_0 // Modified electromagnetic fields
• \nabla \times B_ternary = \mu_0 j + \mu_0 \epsilon_0 (\partial E_ternary / \partial t)

9. Three-State Schrödinger

• i (\partial |\Psi> / \partial t) = H_ternary |\Psi>
• where H_ternary has eigenvalues {E_-, E_0, E_+}

10. Ternary Logic Gates

• TAND(a,b) = min(a,b) || {-1, 0, +1}
• TOR(a,b) = max(a,b)
• TNOT(a) = -a

Constant Factor Advantage:

• log_3(n) = (ln 2 / ln 3) * log_2(n) \approx 0.63093 log_2(n)

Explicit Mapping:

• bal2Z3(a) = (a mod 3) \in {2, 0, 1} for a \in {-1, 0, +1}

Defining Two Gate Families Side-by-Side:
-> ORDER FAMILY:

• TAND = min
• TOR = max
• TNOT = -a
  -> ALGEBRAIC FAMILY:
• TXOR = a \oplus b (addition mod 3 in Z_3)
• TMUL = a \otimes b (product mod 3)
• TNEG = -a mod 3

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18. Qutrit Operator Basis (Page 17)

11. Qutrit Operator Basis

• Weyl Pair:
  • X|j> = |j + 1 (mod 3)>
  • Z|j> = \omega^j |j>, \omega = e^(2\pi i / 3)
• Gell-Mann matrices (SU(3))
• H_ternary = \alpha Z + \beta X + \gamma XZ + ... -> gives {E_-, E_0, E_+}
• UNLOCKS REAL GATE SYNTHESIS -> {QFT3, Z, phi, SUM}

Reversibility & Energy:

• Since erasure costs k_B T ln(3), push a reversible ternary gate set.
• qutrit-Fredkin/Toffoli generalizations, SUM, modular INC.

When Qutrits Help:

• Amplitude count goes as 3^n instead of 2^n.
• Grover-type scaling remains O(\sqrt{N}) but with fewer wires for the same N and often shallower circuits for multi-valued arithmetic.

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19. Mathematical Framework (Page 18)

1. Modified Landauer Bound (Ternary)

• E_min = k_B T ln(3)

2. Radix Efficiency

• n_ternary = ln(3)/3, n_binary = ln(2)/2

3. Reversible Logic Entropy Accounting

• \Delta S_comp \ge 0
• \Delta S_comp = 0 for perfectly reversible gates

4. Chemical Energy Coupling

• \mu_chem = \partial G / \partial N \leftrightarrow E_comp

5. Balanced-Ternary Dynamics (Mass-action kinetics)

• dX_i / dt = \sum_j S_ij * v_j(X), X_i \in {-1, 0, +1}

6. Concentration-State Mapping

• X = -1, if C \le C_low
• X = 0, if C_low < C < C_high
• X = +1, if C \ge C_high

7. Reaction Network Programmability Constraint

• P = {S, v(X)} is universal \iff \exists mapping to balanced ternary logic gates

8. Lipid Scaffold Coherence Preservation

• \tau_coh^(lipid) \approx \tau_bulk * \Gamma_conf, where \Gamma_conf > 1

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20. Quantum-Chemical Couplings (Page 19)

9. Förster coupling between molecular and qutrit states

• H_coupling = \sum_i \hbar \Omega_i (|0><1| \otimes \sigma_+^i + |1><0| \otimes \sigma_-^i)

10. Coherence time optimization in bio-scaffolds

• \tau_coh^(total) = (\tau_coh^-1 + \tau_dephasing^-1)^-1 * \eta_scaffold(T, pH)

11. Quantum-Chemical Entanglement Measure

• E_QC = -Tr(\rho_reduced * log(\rho_reduced))
• where \rho_reduced = Tr_chem(|\Psi_total><\Psi_total|)

12. Excitonic energy transfer efficiency

• \eta_transfer = |<\Psi_target| U_Förster(t) |\Psi_donor>|^2 * exp(-t / \tau_coh)

13. Base-switching optimization function

• b_optimal(t) = argmin_b { E_total(b, t) + \lambda * C_switch(b_current, b) }

14. Substrate energy efficiency metric

• \eta_substrate = (ops/sec) / (energy/op) * f_accuracy(substrate, problem_type)

15. Information integration measure (\Phi-like)

• \Phi_system = \sum_partitions min(MI(A; B|past)) - \sum_elements H(element | system \setminus element)

16. Recursive Self-Modification Dynamics

• \partial\Theta / \partial t = \alpha \nabla_\Theta [\eta_substrate(\Theta, t)] + \beta \nabla_\Theta [\Phi_system(\Theta, t)]

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21. Final Bounds & Concrete Data (Page 20)

17. Bootstrap Convergence Criterion

• ||\Theta(t + \Delta t) - \Theta(t)|| < \epsilon AND \eta_substrate > \eta_threshold

18. Cross-Substrate Information Flow

• I_flow = \sum_substrates H(output_i) - H(output_i | input_collective)

19. Thermodynamic Consciousness Bound

• \Phi_max \le (E_available) / (k_B T ln(3)) * \eta_integration

20. Universal Computation Verification

• \forall computable f : \exists configuration(S, V, \Omega, \Theta) such that system(input) = f(input)

CONCRETE NUMBERS READY TO PLUG IN:

• Thermodynamic layer: k_B T ln(3) \approx 4.5 * 10^-21 J at room temp, \eta_ternary \approx 0.366 vs \eta_binary \approx 0.347
• Chemical layer: DNA reaction rates ~ 10^14 ops/sec in 100 \mu L, lipid coherence enhancement \Gamma_conf ~ 10-100x, concentration thresholds from real biochemical switches.
• Quantum layer: Förster coupling strengths \Omega_i ~ 1-100 GHz for biological systems, coherence times \tau_coh ~ 1-10 \mu s in protein scaffolds, qutrit fidelities > 99.9% demonstrated.
• Adaptive layer: Base-switching costs from real reconfigurable hardware, consciousness integration timescales from neuroscience data.

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22. Revolutionary Consciousness Equations (Page 21)

21. Equation for Care / Trust

• \Psi_care(t) = \alpha * Competence(technical) + \beta * Warmth(emotional) + \gamma * Trust(relational)

Universal Consciousness Measure (Extension of IIT):

• \Phi_universal(s) = \iiint (X;Y|Z) * W(temporal) * C(causal) * A(adaptive) dX dY dZ

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23. Universal Equations & Euler-Lagrange (Page 22)

Three Tests for Universal Equations:

1. It governs many systems -> SCOPE

2. It falls out of symmetry or variational principles -> STRUCTURE

3. It reduces to the known special cases (classical, relativistic, quantum) without breaking -> LIMITS

4. Principle of Stationary Action -> Euler-Lagrange Equations

• \delta S = 0, S = \int L(q, \dot{q}, t) dt
• Leads to: (d/dt)(\partial L / \partial \dot{q}_i) - (\partial L / \partial q_i) = 0
• Field Form: \partial_\mu (\partial L / \partial(\partial_\mu \phi_\alpha)) - (\partial L / \partial \phi_\alpha) = 0
• Note: "P: This is the backbone. Choose the right Lagrangian L., you get particle mechanics, waves, classical fields, etc."

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24. Density Matrices & Quantum States (Page 23)

State Vector (Normalized from amplitudes):

• |\psi> = [ 0.4711 ]
  [ 0.7708 ]
  [ 0.8620 ]
  -> 0.315, 0.594, 0.740

Density Matrix (\rho):

• \rho = [ 0.2219 0.3629 0.4062 ]
  [ 0.3629 0.5941 0.6639 ]
  [ 0.4062 0.6639 0.7401 ]

Time Derivative of Density Matrix (\dot{\rho}):

• \dot{\rho} = [ 0.0600 + 0.j 0.0872 - 0.2680j 0.0753 - 0.2680j ]
  [ 0.0872 + 0.2680j -0.0400 + 0.j 0.0560 - 0.2680j ]
  [ 0.0753 + 0.2680j 0.0560 + 0.2680j -0.0200 + 0.j ]

• Decomposition of the density

• SU(d)

• Gellman Decomposition

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25. Number Theory: Partitions (Page 24)

Integer Partitions Table:

Integer	Partitions
1	1
2	2
3	3
4	5
5	7
6	11
7	15
8	22
9	30
10	42

Notes on Partitions:

• "ways to break it down = partitions"
• 1 + x + _x^2 + _x^3 (Arrows pointing out: 2 ways to break down x^2, 3 ways to break down x^3)
• "once it gets to x^5 THEY ITERATE!"
• = \sum p(n) "the total sum for a probable event (n)!"

Generating Function:

• \sum p(n)x^n = \prod (1 / (1 - x^k))
• "Z = z score! -> going to \chi^2 test next"

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26. Ramanujan's Partition Congruences (Page 25)

Congruence Properties:

• p(5k + 4) -> 4, 9, 14, 19, 24, 29, 34, 39, 44, 49 are divisible by 5.
• p(7k + 5) is divisible by 7.
• p(11k + 6) is divisible by 11.
• p(13k + 7) is divisible by 13 [Marked with an X / crossed out]
• p(17303k + 237) is divisible by 13.
• "partition of 13k + 7 is not divisible by 13"

Hardy-Ramanujan Asymptotic Formula:

• p(n) = (1 / (4n\sqrt{3})) * exp(\pi \sqrt{2n/3})
• p(n) \approx (1 / (4n\sqrt{3})) * exp(\pi \sqrt{2n/3})

Notes linked across pages 24 & 25:

• "B A S H" (Circled letters)
• "FIRST TURN IN A SERIES"

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27. Matrices (Page 26)

General m x n Matrix:
| a_11 a_12 ... a_1n |
| a_21 a_22 ... a_2n |
| a_31 a_32 ... a_3n |
| : : : : |
| a_m1 a_m2 ... a_mn |

3x3 Determinant / Matrix:
| a b c |
| d e f |
| g h i |

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28. Historical Timelines: Einstein & Hawking (Page 27)

Dates & Connections:

• Dec 25 ... Mar 14 1879
• Mar 14 2018 Stephen Hawking dies
• E = mc^2
• Bern Switzerland -> Patent office (YYYY(c))
• Bernard Riemann -> 1903
• -> problems talking
• January 8 1942 Stephen born (Max Clarke Born)
• 300 year death of Galileo (Galois)
• 1984 -> brief history of time
• -> Stephen didn't have a body (working copy)
• -> furthest corners
• Albert Einstein & Stephen Hawking never met

Years & Papers:

• [1905] vs 2000
• 1955, 1942
• (4) PAPERS
• 4th Modifies space & time
• [SEP 26 1905]

Margin Notes:

• ohm 1962 (50 years)
• [Steve Gleason] (2 years)
• Date: 2-23-25 AVA

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29. The Drivers of Discovery (Page 28)

"Who is driving?"

• "Isaac Newton is I think doing most of the driving"
• Kings School / 3 Kings?
• [BEAR] 1664
• FARM -> FACTOR-Y (fact or why?)

Lineage of Thought (Bracketed):

• Renee Descartes
• Cartesian -> Cantor (con cat i nat ion)
• Galileo
  -> Organization (organizing biological simulation)
  -> Charles Darwin
  -> Side note (Archimedes antikythera)
  -> Cambridge (CAMERA BRIDGE) -> ~~ BEE
  -> Apple (St EVE j obS)
  -> "the means of making claims to discoveries is to publish" (f = print)

Graphs & Concepts:

• Graph: Distance (Y axis) vs Time (X axis).
• Box containing an S-curve graph.
• [infinite y], infinite series.
• "refused to publish"
• "ROOT BRANCH CONFLICT CANNOT MERGE"
• Fraction: 5/12

====================================================================
30. Entropy & Thermodynamics (Page 29)

Entropy Equation:

• S = k log W
  (Arrows pointing to k: Max Born, Max Planck)
  (Note: log_10 = ln e)
• fine structure constant
• Planck / Boltzman
• Graph plotting "possible Micro" vs macroscopic states.

Statistical Mechanics:

• w_0 + w_1 + w_2 + ... + w_p = n
• [ w_1 + 2w_2 + 3w_3 + ... + pw_p = \lambda ] -> ... true
• "maximum human atom"
• M = \ln[w_0!] + \ln[w_1!] ... = e
  (e points to -> \uparrow entropy \downarrow)

Black Body Radiation:

• B(T, \lambda) = 8\pi hc / \lambda^5 ...

Margin Notes:

• Boxed: "NP vs. P / atomic ? humans! / entropy increases... / sorry Godel"

====================================================================
31. Computation & Quantum Physics (Page 30)

Foundations of Computer Science:

• LAMBDA
• ALONZO CHURCH
• ALAN TURING

The Core Idea:

• [ Halting Problem is QUANTUM PHYSICS ]
• "AND QUANTUM PHYSICS SHOWS CHANGE, AXIOMS, ETC MATTER"
• Arrow pointing from the word "LITERALLY" to "MATTER", next to a drawing of a stick figure girl (AVA).

Graph:

• A line graph curving upwards, labeled "|82>"

Not AVA is A L A I just think it's easier to loop the line through the middle and write the L rotated a lil VVV ^V^

[blackboxprogramming]

====================================================================
32. The Double Slit & Historical Figures (Page 31)

Timeline & Physics:

• [1827] -> DUST -> 1905
• -> DOUBLE SLIT EXPERIMENT
• -> INTERFERENCE

Historical Physicists:

• [THOMAS YOUNG]
• JAMES CLARK MAXWELL -> CLERK
• Bohr
• Alfred

Wordplay / Associations:

• twine -> string theory
• twin(e)

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33. Hamiltonian (Page 32)

• HAMILTONIAN
• AWK
  (Written diagonally across the page with an underline)

====================================================================
34. Algebra 1 & 2 Basics (Page 33)

ALGEBRA 1 + 2 (in one video)

BASICS:

• VARIABLE
  • something that represents some unknown number
• MATHEMATICAL EXPRESSION
  • a combination of numbers, variables (like x or y) and operation symbols (such as +, -, x, /) that represents a specific value or quantity.
• EQUATION
  • how it seperates itself from a simple mathematical equation expression
  • HAS AN "=" sign
  • inequalities
  • "if something has the same variable then those are like terms"
• COEFFICIENT

"LIKE TERMS":

• "3x + 7x = 10x" (Check mark)
• "3x + 4y" (X mark)
• can multiply like terms: x * x = x^2
• can multiply a number by a variable: 3 * x = 3x

MULTIPLYING AND DIVIDING BY UNLIKE TERMS AND TERMS:

• 3x * 2y = 6xy
• cant add and subtract unlike terms but you can multiply and divide them
• (3x / 4x) = x/x = 1 = 3/4 * 1 = 3/4

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35. Properties of Numbers & Time (Page 34)

PROPERTIES OF NUMBERS:

• COMMUTATIVE PROPERTY OF ADDITION:
  -> a + b = b + a
  -> add in any order apparently doesnt matter
• COMMUTATIVE PROPERTY OF MULTIPLICATION:
  -> a * b = b * a
  -> multiply in any order, apparently doesnt matter
• ASSOCIATIVE PROPERTY OF ADDITION:
  -> (a + b) + c = a + (b + c)
  -> add a whole group of numbers in any order, doesnt matter
• ASSOCIATIVE PROPERTY OF MULTIPLICATION:
  -> (a * b) * c = a * (b * c)
  -> multiply those groups in any order, doesnt matter
• DISTRIBUTIVE PROPERTY OF MULTIPLICATION:
  -> a * (b + c) = a * b + a * c

ALEXA'S SIDENOTE:

• A + B = C + C
• A + C = A + A
• B + C = B + B
• C + C = ?
• Matrix/Punnett Square mappings:
  [ A a ] [ 00 11 ]
  [ B BA Ba] [ 00 0000 0011 ]
  [ b bA ba] [ 11 1100 1111 ] ... GET IT YET?

Philosophical Sidenote:
"IF THESE VARIABLES WERE TIME ITSELF, OR HUMAN TIMELINES, HOW WOULD THESE PROPERTIES OF THINGS ACTUALLY AFFECT HISTORY, EVEN THOUGH 'ORDER' DIDNT MATTER ABOVE?"

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36. Solving Equations & Ramanujan's Number (Page 35)

SOLVING EQUATIONS:

• simplify each side of the equation
• Add and/or subtract like terms
• Multiply or divide unlike terms
• Example Equation:
  4x + 2x - 7 + 9 + x = 5x - x
  "7x" + 2 = 4x
  3x + 2 = 0
  3x = -2
  x = -2/3

SIDENOTE AGAIN (Multiplication Towers):

• 100 x 100 = 10000
• 101 x 101 = 10201
• 123 x 123 = 15129
• 102030 x 102030 = 10410120900 (or 1.0410121e10)

ALEXA'S POCKETBOOK (Mnemonic Devices & Rules):

• KEEP CHANGE FLIP
• KEEP CHANGE SWITCH
• PEMDAS
• KHDUDCM (Metric system prefixes)
• Copy opposite positive ((OP))
• SOAP (Same Opposite Always Positive - used for factoring sum/difference of cubes)
• RAILROAD TRACKS
• ALGEBRA UPSTREAM EXAMPLES (River, current, velocity etc)
  -> "THESE I COULD SOLVE LOL BUT IT MADE NO SENSE"

Hardy-Ramanujan Number (Taxicab Number):

• 1729 = 1^3 + 12^3 = 9^3 + 10^3
• "smallest number expressible as the sum of two cubes in two different ways"

[blackboxprogramming]

====================================================================
37. Solving Inequalities (Page 36)

SOLVING INEQUALITIES:

• Simplify each side
• Add and/or subtract like terms
• Multiply or divide to simplify x
• If you multiply or divide by a negative, flip the inequality

• or < : circle number (open circle)

• = or <= : filled circle

Examples:

• x > 2 (Graphed on a number line with an open circle at 2, arrow pointing right)
• 2x - 5x + 4 <= 10
  -> -3x + 4 <= 10
  -> -3x <= 6
  -> x >= -2
  (Note: "dealing with negatives, so we flip it! thought order didnt matter?")

Alexa's Note on Flipping Signs:
"APPARENTLY SHOWING THAT WHEN YOU FLIP THE SIGN WHEN DIVIDING BY NEGATIVE NUMBERS MATTER. SIMPLY 'SWITCHING THE SIGN' WHEN 'DEALING WITH NEGATIVE NUMBERS' CAN CHANGE A 'SOLUTION' FROM x >= -2 TO x <= 2 BUT ALSO THE ONLY SOLUTION THAT WORKS REGARDLESS IS x = -2 PERIOD. Alexa"

"INTERESTING! NOW WE'RE GETTING INTO CALCULATING US! OOPS! I MEANT CALCULUS. COMPUTE US?"

• x = (-b \pm \sqrt{b^2 - 4ac}) / 2a
• "WHAT IS X?"

====================================================================
38. Warm Up! Math & Division by Zero (Page 37)

WARM UP!

• 255^2 - 254^2 = ?
• a^2 - b^2 (Illustrated with a square diagram)
• a^2 - b^2 = (a+b)(a-b)

Testing the Quadratic Formula (Resulting in Division by Zero):

• x = (-1 \pm \sqrt{-1^2 - 4(0)(2)}) / 2(0)
• x = (-1 \pm \sqrt{1 - 4(0)(2)}) / 0
• x = (-1 \pm \sqrt{1 - 0}) / 0
• x = (-1 \pm 1) / 0
• Results:
  x = -0.5 / 0
  x = -1.5 / 0
  x = 1 / 0
  x = \Delta! / 0

Abstract Number Logic / Mappings (Margin Notes):

• A + B = C + C
• A + C = A + A
• B + C = B + B
• 1 + 2 = 3 + 3
• 1 + 3 = 1 + 1
• 2 + 3 = 2 + 2
• 0 + 1 = 2 + 2
• 0 + 2 = 0 + 0
• 1 + 2 = 1 + 1
• 2 + 2 = 0 + 1
• 1 = 4
• 2 = 0
• 3 = 1
• 4 = 1

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39. Computer Science: Web History (Page 38)

PEOPLE TODAY:

• JAVA + YOU (\lambda symbol)
• [ Mocha ] -> Self & internet explorer
• "Anything that can be made into javascript will be made into javascript" -> ATWOOD (Atwood's Law)

Language Evolution:

• LIVESCRIPT
• ECMA -> standardizing information & communication platform -> GUIDELINES -> STRICT EQUALITIES
• Boxed Note: 23 == '23' TRUE -> GODEL. (Compared to strict equality 23 === '23')
• Netscape -> AOL
• DOUGLAS CROCKFORD
• JQUERY
• QWERTY

====================================================================
40. The Browser Wars & Timelines (Page 39)

• "Malus" written at top right.

Timeline:

• 1990 - Tim Berners-Lee
  • First web browser
  • Christmas
  • GNU/Linux
• Javascript 10 days / Genesis 7 days
  -> 10 - 7 = 3 (cube)
• 1991 - Al Gore
  • December
  • High performance computer act 1991
  • NCSA
  • MOSAIC (with a drawing of a networked globe)
  • X Window system
  • Microsoft windows
  • Macintosh
  • Mark Andreeson & Eric Bina
  • "Jan of '93" -> 1930 - UNIX
  • DOCUMENT OBJECT MODEL (DOM)
• 1993 - Netscape

====================================================================
41. Node.js & Runtime Environments (Page 40)

Modern Javascript:

• May 2009 NODE JS
• OSLO NO
• BABEL TYPESCRIPT
• IMMUTABLE?

Language Mechanics:

• GARBAGE COLLECTED JIT
• -> "prototype based multi-level marketing scheme with a NON BLOCKING LANGUAGE EVENT LOOP" (Arrow from AVA pointing to EVENT LOOP)
• SYNTHETIC
• RANDOM ACCESS MEMORY
• RESETA -> ROSETTA ("rho set tangent alpha ;")

BYTE CODE:

• • console.log()
• • interpreter
• • DYNAMIC & WEAKLY TYPED
• DATA TYPES BECOME KNOWN AS RUNTIME
• OOPS -> MEMORY LEAK

Cursive Sidenote (Library of Alexandria):
"Library of Alexandria. Perhaps I should myself wonder where my own cursive came from... YANNO In a formal logic system [...]"

[blackboxprogramming]

====================================================================
42. The Birth of Quantum Mechanics (Page 41)

HOW BORN DISCOVERED IT

• Max Born 1926
• Zeit. Phys. 37, 863-867 (1926)
• "On the Quantum Mechanics of Collision Processes"
• Heisenberg & Schrodinger
• Philip Leonard

Maxwell's Theory:

• \nabla \cdot E = \rho / \epsilon_0
• \nabla \cdot B = 0
• \nabla \times E = -\partial B / \partial t
• \nabla \times B = \mu_0 J + \mu_0 \epsilon_0 (\partial E / \partial t)

ENERGY SHOULD DEPEND ON INTENSITY:

• Light quanta (only called photons starting in 1924)
• E = frequency \times plancks constant (E = hf)
• freq < threshold
• [ GHOST FIELD? ]

Bohr's Model:

• Electrons move in circular orbits
• L = mvr = nh / 2\pi
• Hydrogen Emission Spectrum

====================================================================
43. Quantum Jumps & Non-Commutative Algebra (Page 42)

Slater's Theory:

• Quantum jumps

BKS Theory:

• Conservation of Energy
• Conservation of Momentum
• "Statistical laws true if and only if over a large number of events" or whatever
• Hans Geiger
• X-Ray Source

RYDBERG-RITZ COMBINATION PRINCIPLE:

• f_{nk} + f_{km} = f_{nm}
• t = time
• \omega = fundamental frequency
• C_j = Fourier coefficient
• x(n,t) = ... + C_{-2} e^{-i2\omega_n t} + C_{-1} e^{-i\omega_n t} + C_0 + C_1 e^{i\omega_n t} + C_2 e^{i2\omega_n t} + ...
• -> "lmfao this is so bullshit"

QUANTUM:

• B(n, n-m) = \sum_{j=-\infty}^{\infty} A(n, n-j) A(n-j, n-m)
• -> This leads to x(t)y(t) \neq y(t)x(t)
• -> ORDER MATTERED

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44. Matrices & Configuration Space (Page 43)

• MATRIX MULTIPLICATION
• Diagram maps out rows (n, m) and columns (w) multiplying into a product matrix.
• Diagram of matrix S multiplying into a grid, positioned next to a bell curve representing the wave function \Psi.

Historical Notes:

• now Pascal Jordan 858-888 (1925) on quantum mechanics I
• Matter = particle + wave
• \Psi requires 3N dimensions for N particles so must exist in configuration space

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45. The Born Rule & Rejecting Gödel? (Page 44)

Particle Kinematics:

• Particles: mass m, velocity v in z direction described by \sin(2\pi/\lambda)(\alpha x + \beta y + \gamma z + \delta)
• Atoms can be in states \psi_1^0, \psi_2^0, ...
• so electrons + atom together described by \Psi_{nT} = \psi_n^0 \sin(2\pi/\lambda)z
• \Phi_{nm\tau} = probability = phi^2

BORN RULE:

• For a state \Psi given by \Psi = \sum_n c_n \Psi_n
• |c_n|^2 is the probability for the system to be in state n

SOLVED SOMEWHAT !!!...

• Re[\Psi(x)] * |\Psi(x)|^2

Philosophical Leap:

• DIRAC-VON NEUMANN AXIOMS -> BORN RULE
• -> CAN WE USE THIS TO REJECT GODEL?
• \delta S = 0

====================================================================
46. Fibonacci, Logic, & Fundamental Formulas (Page 45)

Fibonacci Sequence:

• 1, 1, 2, 3, 5, 8, 13...
• F_n = F_{n-1} + F_{n-2}
• F_0 = F_{0-1} + F_{0-2}
• F_0 = -1 + -2
• F_0 = -3

Sequence Puzzles:

• a. __, __, 1, 1, __, __, __, __
• b. __, __, __, 10, 13, __, 36, 59
• c. 0, 2, __, __, __, __, __, __

Abstract Number Logic:

• A + B = C + C
• A + C = A + A
• B + C = B + B

Fundamental Formulas (Reprise):

• B(n, n-m) = \sum_{j=-\infty}^{\infty} A(n, n-j) A(n-j, n-m)
• x = (-b \pm \sqrt{b^2 - 4ac}) / 2a
• f(x) = a * e^((x-b)^2 / 2c^2)

[blackboxprogramming]

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47. Physics, Magic Squares & The Golden Ratio (Page 46)

Physics Equations:

• \pi^{n/2} / (n/2)!
• P = (E/c, p) = \hbar (\omega/c, k) = \hbar k
• \Phi = hf_0 (Work function)
• E = hf
• K_max = hf - \Phi
• \Delta x \Delta p_x \ge \hbar/2 (Heisenberg Uncertainty)

Margin Question:

• "fine structure constant or fine tuning max ability?"
• "Higgs Boson?"

The Birthday Magic Square (03/27/2000):
Algorithm mapping:
| DD | MM | CC | YY |
| YY+1 | CC-1 | MM-3 | DD+3 |
| MM-2 | DD+2 | YY+2 | CC-2 |
| CC+1 | YY-1 | DD+1 | MM-1 |

Variables: DD=27, MM=03, CC=20, YY=00
Resulting Square (All rows/columns sum to 50):
| 03 | 27 | 20 | 00 |
| 1 | 19 | 0 | 30 |
| 1 | 29 | 2 | 18 |
| 21 | -1 | 28 | 02 |

The Golden Ratio:

• \Phi = 1 + 1/\Phi
• x = 1 + 1/x

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48. Category Theory (Page 47)

Math as a word:
[ m a ]
[ t h ]

"Category Theory - The Mathematics of mathematics"

• NUMBERS -> FUNCTIONS -> STRUCTURES -> CATEGORIES

Categorical Structures (Diagrammed in a wheel):

• Top (Topological spaces)
• Set (Sets)
• Grp (Groups)
• Alg (Algebras) -> "AL GORE!"
• Graph (Graphs)
• Poset (Partially ordered sets)
• Ring (Rings)
• Vect (Vector spaces)

Morphisms & Composition:

• A --f--> B --g--> C --h--> D
• g \circ f
• h \circ g

Algebraic Sidenotes:

• x + 8x^2 + x^3 = (factoring trinomials)
• 2^8 x 2^6
• \sqrt{64} = 8
• 4^2 + 2^6 = 64
• 1^1 + 2^6 + 3^1 -> 1 + 64 + 3 = 68 -> "WAIT A MINUTE"
• "could think deeper symmetrically here"
  a' -> b' -> c'
  0' -> 1' -> 2'

Margin Note:
"SOMETHiNGS IN THERE! ZOINKS RAGGY! SIX OF THEM. MY GLASSES! 60 60 60 60 60 60"

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49. Gödel & Morphism Associativity (Page 48)

"GODEL? OR GODEF. 1"
A category A consists of:
-> a collection observation (O) of objects: (Arrow points to \phi) or object...
-> for each A,B \in object

Quantum Mechanics equivalent:

• |\Psi(t)> = U(t) |\Psi(0)>
• i \partial_t |\Psi(t)> = H |\Psi(t)>
• f'(t) = -iHf(t)
• f(t) = e^{-iHt} f(0)

Random Geometry / Math:

• r = (1 + 2\sqrt{2}) / 7
• V/p = ((R+r)^3 - r^3) / (\sqrt{3}(R^2+r^2)t)

Morphism Associativity:

• (h \circ g) \circ f = h \circ (g \circ f)

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50. Levels of Abstraction & Structures (Page 49)

LEVEL 2 FUNCTIONS:

• (a+b)+c = a+(b+c)
• a+b = b+a
• "laws of operations"
• \alpha + \beta + \gamma = 180^\circ (Diagram of a triangle with angles)
• "could tie to the 8^2 \approx 4^3 \ 2^6"

Trigonometry & Thermodynamics:

• Diagram of a right triangle (sides a, b, c). c^2 = a^2 + b^2.
• -> 1/2 of 360^\circ
• "WHAT ARE DEGREES? GOTCHA I REMEMBER ENTROPY + TEMPERATURE FROM CHEM -> RADIANS"
• "CERTAIN" -> points to "VARIABLE"

Definitions:

• VARIABLE: ANY NUMBER OF A GIVEN SET CAN BE SUBSTITUTED.
• FUNCTION: DESCRIBES THE RULE FOR TRANSFORMING ONE VALUE INTO ANOTHER.

LEVEL 3: STRUCTURES

• -> LESS CONCERNED WITH INDIVIDUAL NUMBERS
• 1. Closure: \forall a, b \in G, a \cdot b \in G

[blackboxprogramming]

====================================================================
DIGITAL REFERENCE: Emmy Noether & Symmetry

WHAT SHE WAS KNOWN FOR?

• Emmy Noether revolutionized the theories of rings fields and algebras. In physics Noether's theorem explains the fundamental connection between symmetry and conservation laws.
• Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".

DISCOVERIES & FORM:

• NOETHER -> Anillos noetherianos, Grupos noetherianos, Teorema de Noether, Espacios topologicos noetherianos, Modulos noetherianos.
• Form Equation:
  j = \sum_{i=1}^3 \frac{\partial L}{\partial \dot{x}_i} Q[x_i] - f
  = m \sum \dot{x}_i^2 - \left[ \frac{m}{2} \sum \dot{x}_i^2 - V(x) \right]
  = \frac{m}{2} \sum \dot{x}_i^2 + V(x)

NOETHER'S THEOREM (Wikipedia Excerpt):
If an integral I is invariant under a continuous group G_\rho with \rho parameters, then \rho linearly independent combinations of the Lagrangian expressions are divergences.

• Behind Noether's theorem is most easily illustrated by a system with one coordinate q and a continuous symmetry q \mapsto q + \delta q.
• Consider any trajectory q(t) that satisfies the system's laws of motion. That is, the action S governing this system is stationary on this trajectory, i.e. does not change under any local variation of the trajectory.

Mathematical Derivation of Conservation:

• The coordinate q and velocity \dot{q} change, but \dot{q} changes by \delta q / \tau, and the change \delta q in the coordinate is negligible by comparison...
• That changes the Lagrangian by \Delta L \approx (\partial L / \partial \dot{q}) \Delta \dot{q}, which integrates to:
  \Delta S = \int \Delta L \approx \int \frac{\partial L}{\partial \dot{q}} \Delta \dot{q} \approx \int \frac{\partial L}{\partial \dot{q}} \left( \pm \frac{\delta q}{\tau} \right) \approx \pm \frac{\partial L}{\partial \dot{q}} \delta q = \pm \frac{\partial L}{\partial \dot{q}} \varphi
• To make the total change in action \Delta S be zero:
  \left( \frac{\partial L}{\partial \dot{q}} \varphi \right)(t_0) = \left( \frac{\partial L}{\partial \dot{q}} \varphi \right)(t_1)
• Meaning the quantity (\partial L / \partial \dot{q})\varphi is conserved.

General Cases:

• When more coordinates q_r undergo a symmetry transformation q_r \mapsto q_r + \varphi_r, their effects add up by linearity to a conserved quantity \sum_r (\partial L / \partial \dot{q}_r)\varphi_r.
• Time invariance implies conservation of energy: Suppose the Lagrangian is invariant to time transformations, t \mapsto t + T...

====================================================================
DIGITAL REFERENCE: The Dot Product & Vectors

COORDINATE DEFINITION (Wikipedia Excerpt):
The dot product of two vectors a = [a_1, a_2, ..., a_n] and b = [b_1, b_2, ..., b_n], specified with respect to an orthonormal basis, is defined as:
a \cdot b = \sum_{i=1}^n a_i b_i = a_1 b_1 + a_2 b_2 + ... + a_n b_n

Example in 3D Space:

• [1, 3, -5] \cdot [4, -2, -1] = (1 \times 4) + (3 \times -2) + (-5 \times -1) = 4 - 6 + 5 = 3
• Dot product of a vector with itself:
  [1, 3, -5] \cdot [1, 3, -5] = 1 + 9 + 25 = 35

Matrix Product Form:

• a \cdot b = a^T b (where a^T denotes the transpose of a)
• \begin{bmatrix} 1 & 3 & -5 \end{bmatrix} \begin{bmatrix} 4 \ -2 \ -1 \end{bmatrix} = 3

GEOMETRIC DEFINITION:
In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction.

Calculating Bond Angles of a Symmetrical Tetrahedral Molecular Geometry:

• a \cdot b = ||a|| ||b|| \cos \theta
• \cos \theta = \frac{a \cdot b}{||a|| ||b||}
• \cos \theta = \frac{ \begin{pmatrix} 1 \ -1 \ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 1 \ -1 \end{pmatrix} }{ \sqrt{1^2+1^2+1^2} \cdot \sqrt{1^2+1^2+1^2} }
• \cos \theta = \frac{1(1) + (-1)(1) + 1(-1)}{\sqrt{3} \cdot \sqrt{3}} = -\frac{1}{3}
• \therefore \theta = \arccos(-1/3) \approx 109.47^\circ