--- title: Various LaTeX Mathematical Expressions date: 2025-03-14 tags: - mathematics - latex - markdown - wip summary: This document includes a variety of LaTeX mathematical expressions for use in technical documentation, including basic notation, set theory, calculus, linear algebra, equations, systems, and advanced topics in mathematics and physics. published: true featured: false image: /assets/Latex.png thumbnail: /assets/Latex.png --- # Advanced Mathematical Expressions with LaTeX :::warning WIP This article is still a work in progress. ::: This document demonstrates the power of LaTeX for representing mathematical expressions in technical documentation, from basic notation to complex equations and proofs. ## Basic Mathematical Notation ### Arithmetic Operations $a + b - c \times d \div e = f$ $7 \times 8 = 56$ $\frac{20}{5} = 4$ ### Exponents and Subscripts $x^2 + y^2 = z^2$ $a_1 + a_2 + a_3 + \ldots + a_n = \sum_{i=1}^{n} a_i$ Chemical formulas: $H_2O$, $CO_2$ ### Greek Letters Lowercase: $\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \iota, \kappa, \lambda, \mu, \nu, \xi, \pi, \rho, \sigma, \tau, \upsilon, \phi, \chi, \psi, \omega$ Uppercase: $\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Upsilon, \Phi, \Psi, \Omega$ Variants: $\varepsilon, \vartheta, \varkappa, \varpi, \varrho, \varsigma, \varphi$ ## Set Theory and Logic ### Set Notation $A \cup B$ - Union of sets A and B $A \cap B$ - Intersection of sets A and B $A \setminus B$ - Set difference (elements in A but not in B) $A \subset B$ - A is a subset of B $A \subseteq B$ - A is a subset of or equal to B $x \in A$ - x is an element of set A $\emptyset$ - Empty set $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ - Number sets (natural, integer, rational, real, complex) ### Logic Operators $P \land Q$ - Logical AND $P \lor Q$ - Logical OR $\lnot P$ - Logical NOT $P \implies Q$ - Implication (if P then Q) $P \iff Q$ - Logical equivalence (P if and only if Q) $\forall x$ - Universal quantifier (for all x) $\exists x$ - Existential quantifier (there exists x) ## Calculus ### Limits $\lim_{x \to a} f(x) = L$ $\lim_{x \to \infty} \frac{1}{x} = 0$ $\lim_{x \to 0} \frac{\sin x}{x} = 1$ ### Derivatives $\frac{d}{dx}f(x) = f'(x)$ $\frac{d^2y}{dx^2} = f''(x)$ $\frac{\partial f}{\partial x}$ - Partial derivative $\nabla f$ - Gradient of f ### Integrals Indefinite integral: $\int f(x) \, dx = F(x) + C$ Definite integral: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ Double integral: $\iint_D f(x,y) \, dx \, dy$ Line integral: $\oint_C f(x,y) \, ds$ ## Linear Algebra ### Vectors and Matrices Column vector: $\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$ Row vector: $\vec{v} = \begin{pmatrix} v_1 & v_2 & \cdots & v_n \end{pmatrix}$ Matrix: $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$ Identity matrix: $I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ ### Matrix Operations Matrix multiplication: $C = AB = \begin{pmatrix} \sum_{k=1}^{n} a_{1k}b_{k1} & \sum_{k=1}^{n} a_{1k}b_{k2} & \cdots & \sum_{k=1}^{n} a_{1k}b_{kp} \\ \sum_{k=1}^{n} a_{2k}b_{k1} & \sum_{k=1}^{n} a_{2k}b_{k2} & \cdots & \sum_{k=1}^{n} a_{2k}b_{kp} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{k=1}^{n} a_{mk}b_{k1} & \sum_{k=1}^{n} a_{mk}b_{k2} & \cdots & \sum_{k=1}^{n} a_{mk}b_{kp} \end{pmatrix}$ Determinant: $\det(A)$ or $|A|$ Transpose: $A^T$ Inverse: $A^{-1}$ Trace: $\text{tr}(A) = \sum_{i=1}^{n} a_{ii}$ ## Equations and Systems ### Quadratic Formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2 + bx + c = 0$ ### Systems of Linear Equations $\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m \end{cases}$ In matrix form: $A\vec{x} = \vec{b}$ ## Advanced Calculus ### Taylor Series $f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$ $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$ ### Fourier Series $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]$ Where: $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$ $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$ ## Differential Equations ### Ordinary Differential Equations First-order linear ODE: $\frac{dy}{dx} + P(x)y = Q(x)$ Second-order linear ODE: $\frac{d^2y}{dx^2} + p(x)\frac{dy}{dx} + q(x)y = g(x)$ ### Partial Differential Equations Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ Heat equation: $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$ Laplace equation: $\nabla^2 \phi = 0$ or $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$ ## Probability and Statistics ### Probability Probability of event A: $P(A)$ Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ Bayes' theorem: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$ ### Distributions Binomial distribution: $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$ Normal distribution probability density function: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ Poisson distribution: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$ ### Statistical Measures Expected value: $E[X] = \sum_{i} x_i p_i$ or $E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$ Variance: $\text{Var}(X) = E[(X - \mu)^2] = E[X^2] - E[X]^2$ Covariance: $\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]$ Correlation coefficient: $\rho_{X,Y} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$ ## Number Theory ### Divisibility and Congruences $a \mid b$ means a divides b $a \equiv b \pmod{m}$ means a is congruent to b modulo m Euler's totient function: $\phi(n) = n \prod_{p \mid n} \left(1 - \frac{1}{p}\right)$ ### Prime Numbers Prime factorization: $n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$ The prime number theorem: $\pi(n) \sim \frac{n}{\ln n}$ ## Complex Analysis ### Complex Numbers $z = a + bi$ where $i^2 = -1$ $|z| = \sqrt{a^2 + b^2}$ - Modulus of z $\arg(z) = \text{arctan}\frac{b}{a}$ - Argument of z ### Complex Functions Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ Cauchy's integral formula: $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} \, dz$ Residue theorem: $\oint_C f(z) \, dz = 2\pi i \sum_{k=1}^{n} \text{Res}(f, z_k)$ ## Physics Equations ### Mechanics Newton's second law: $\vec{F} = m\vec{a}$ Kinetic energy: $K = \frac{1}{2}mv^2$ Potential energy (gravitational): $U = mgh$ ### Electromagnetism Maxwell's Equations: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$ (Gauss's law) $\nabla \cdot \vec{B} = 0$ (Gauss's law for magnetism) $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (Faraday's law) $\nabla \times \vec{B} = \mu_0\left(\vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t}\right)$ (Ampère's law with Maxwell's addition) ### Thermodynamics First law: $\Delta U = Q - W$ Entropy change: $\Delta S = \int \frac{dQ}{T}$ ### Quantum Mechanics Schrödinger equation: $i\hbar \frac{\partial}{\partial t}\Psi(\vec{r},t) = \hat{H}\Psi(\vec{r},t)$ Heisenberg uncertainty principle: $\Delta x \Delta p \geq \frac{\hbar}{2}$ ## Mathematical Proofs and Theorems ### Proof by Induction Base case: P(1) is true. Inductive step: Assume P(k) is true, show P(k+1) is true. Therefore, P(n) is true for all positive integers n. ### Mathematical Theorems **The Fundamental Theorem of Calculus:** $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$ Where F is an antiderivative of f. **Pythagorean Theorem:** For a right triangle with sides a, b and hypotenuse c: $$a^2 + b^2 = c^2$$ ## Commutative Diagrams $$ \begin{CD} A @>f>> B\\ @VgVV @VVhV\\ C @>>j> D \end{CD} $$ This diagram commutes if $h \circ f = j \circ g$ ## Advanced Mathematical Structures ### Groups, Rings, and Fields Group axioms for $(G, \cdot)$: 1. Closure: $\forall a,b \in G, a \cdot b \in G$ 2. Associativity: $\forall a,b,c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c)$ 3. Identity: $\exists e \in G, \forall a \in G, e \cdot a = a \cdot e = a$ 4. Inverse: $\forall a \in G, \exists a^{-1} \in G, a \cdot a^{-1} = a^{-1} \cdot a = e$ ### Transformations and Mappings Linear transformation: $T(\alpha \vec{v} + \beta \vec{w}) = \alpha T(\vec{v}) + \beta T(\vec{w})$ Eigenvalue equation: $T(\vec{v}) = \lambda \vec{v}$ where $\lambda$ is the eigenvalue ## Notable Equations in Science ### Einstein's Mass-Energy Equivalence $$E = mc^2$$ ### Euler's Identity $$e^{i\pi} + 1 = 0$$ ### Boltzmann's Entropy Formula $$S = k_B \ln W$$ ### The Standard Model Lagrangian (Simplified) $$\mathcal{L}_{SM} = \mathcal{L}_{gauge} + \mathcal{L}_{fermion} + \mathcal{L}_{Higgs} + \mathcal{L}_{Yukawa}$$ ## Combinatorial Mathematics ### Permutations and Combinations Number of permutations: $P(n,r) = \frac{n!}{(n-r)!}$ Number of combinations: $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ ### Binomial Theorem $$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$ ## Optimization Problems ### Lagrange Multipliers To maximize/minimize f(x,y) subject to g(x,y) = c: $$\nabla f(x,y) = \lambda \nabla g(x,y)$$ ## Multi-Dimensional Calculus ### Multiple Integrals Triple integral in rectangular coordinates: $$\iiint_E f(x,y,z) \, dx \, dy \, dz$$ Triple integral in spherical coordinates: $$\iiint_E f(\rho,\theta,\phi) \, \rho^2\sin\phi \, d\rho \, d\theta \, d\phi$$ ### Stokes' Theorem $$\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$$ ### Divergence Theorem $$\iiint_V (\nabla \cdot \vec{F}) \, dV = \oiint_S \vec{F} \cdot d\vec{S}$$ ## Advanced Theoretical Mathematics ### The Riemann Hypothesis One of the most famous unsolved problems in mathematics, the Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function have real part $\frac{1}{2}$. The Riemann zeta function is defined as: $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \quad \text{for } \Re(s) > 1$$ And it can be analytically continued to the entire complex plane except for a simple pole at $s=1$: $$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$$ The Riemann Hypothesis conjectures that: $$\zeta(\sigma + it) = 0 \Rightarrow \sigma = \frac{1}{2}$$ The zeros of the zeta function are closely related to the distribution of prime numbers through the explicit formula: $$\psi(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2}\log\left(1-\frac{1}{x^2}\right)$$ Where $\rho$ runs through the non-trivial zeros of the zeta function. ### Mandelbrot Set and Complex Dynamics The Mandelbrot set $M$ is defined as the set of complex numbers $c$ for which the sequence defined by the recurrence relation: $$z_{n+1} = z_n^2 + c \quad \text{with } z_0 = 0$$ remains bounded. Mathematically: $$M = \{c \in \mathbb{C} : \lim_{n \to \infty} |z_n| \not\to \infty \}$$ The boundary of the Mandelbrot set exhibits infinite complexity with self-similar structures at all scales. The "bulb" regions correspond to different periods in the orbit dynamics: $$\text{Period-}p \text{ bulb: } z^p + c_p = z \text{ has an attracting fixed point}$$ The Mandelbrot set can also be characterized using the potential function: $$G(c) = \lim_{n \to \infty} \frac{1}{2^n} \log|z_n|$$ where $G(c) = 0$ if $c \in M$ and $G(c) > 0$ otherwise. ## Euler's Remarkable Identities ### The Most Beautiful Equation in Mathematics Euler's identity elegantly connects five fundamental mathematical constants: $$e^{i\pi} + 1 = 0$$ This can be derived from the more general Euler's formula: $$e^{ix} = \cos x + i \sin x$$ Setting $x = \pi$ yields: $$e^{i\pi} = \cos \pi + i \sin \pi = -1 + 0i = -1$$ ### Extended Euler's Formula Applications For any complex number $z = x + iy$: $$e^z = e^{x+iy} = e^x \cdot e^{iy} = e^x(\cos y + i\sin y)$$ This leads to the hyperbolic functions: $$\cosh z = \frac{e^z + e^{-z}}{2} \quad \sinh z = \frac{e^z - e^{-z}}{2}$$ And the remarkable identity: $$\cos z = \cosh(iz) \quad \sin z = -i\sinh(iz)$$ ### Euler's Infinite Series $$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$$ $$\sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots$$ $$\cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots$$ ## Advanced Trigonometry ### Trigonometric Identities The double angle formulas: $$\sin 2\theta = 2\sin \theta \cos \theta$$ $$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$$ $$\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$$ The sum and difference formulas: $$\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$ $$\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$$ $$\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}$$ ### Hyperbolic Trigonometry The hyperbolic functions and their relationship to the exponential function: $$\sinh x = \frac{e^x - e^{-x}}{2} \quad \cosh x = \frac{e^x + e^{-x}}{2} \quad \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$ Fundamental hyperbolic identity: $$\cosh^2 x - \sinh^2 x = 1$$ ### Complex Trigonometry Extended complex trigonometric identities: $$\sin(x+iy) = \sin x \cosh y + i\cos x \sinh y$$ $$\cos(x+iy) = \cos x \cosh y - i\sin x \sinh y$$ $$\tan(x+iy) = \frac{\sin(x+iy)}{\cos(x+iy)} = \frac{\sin x \cosh y + i\cos x \sinh y}{\cos x \cosh y - i\sin x \sinh y}$$ ## Mathematical Visualizations (TikZ-Style) ### Vector Field Representation The visualization of a vector field $\vec{F}(x,y) = (y, -x)$ representing rotational motion: $$\vec{F}(x,y) = \begin{pmatrix} y \\ -x \end{pmatrix}$$ This field has curl: $$\nabla \times \vec{F} = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = -1 - 1 = -2$$ And divergence: $$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = 0 + 0 = 0$$ ### Parametric Surface Plotting A torus with major radius $R$ and minor radius $r$ can be parametrized as: $$\vec{r}(u,v) = \begin{pmatrix} (R + r\cos v)\cos u \\ (R + r\cos v)\sin u \\ r\sin v \end{pmatrix}$$ where $u, v \in [0, 2\pi]$. The surface area of this torus is: $$A = \iint_D \left| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \right| \,du\,dv = \iint_D (R + r\cos v) \cdot r \,du\,dv = 4\pi^2 Rr$$ ### Complex Function Visualization For a complex function $f(z) = z^2$, the conformal mapping of a grid in the complex plane: If $z = x + iy$, then $f(z) = (x^2-y^2) + i(2xy)$ The Jacobian of this transformation is: $$J = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} = \begin{vmatrix} 2x & -2y \\ 2y & 2x \end{vmatrix} = 4x^2 + 4y^2 = 4|z|^2$$ ## Quantum Mechanics and Advanced Physics ### Schrödinger's Wave Equation (Time-Dependent) $$i\hbar \frac{\partial}{\partial t} \Psi(\vec{r},t) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r},t) \right] \Psi(\vec{r},t)$$ For a particle in a 1D infinite potential well of width $L$, the energy eigenfunctions are: $$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$$ with energy eigenvalues: $$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$$ ### Dirac Equation The Dirac equation for a relativistic electron: $$i\hbar \frac{\partial}{\partial t}\psi = \left(c\vec{\alpha} \cdot \vec{p} + \beta mc^2\right) \psi$$ Where $\vec{\alpha}$ and $\beta$ are the Dirac matrices: $$\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}$$ And $\sigma_i$ are the Pauli matrices: $$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ ### Path Integrals in Quantum Field Theory Feynman path integral formulation of quantum mechanics: $$\langle x_b, t_b | x_a, t_a \rangle = \int_{x(t_a)=x_a}^{x(t_b)=x_b} \mathcal{D}[x(t)] \, e^{\frac{i}{\hbar}S[x(t)]}$$ Where $S[x(t)]$ is the action: $$S[x(t)] = \int_{t_a}^{t_b} L(x(t), \dot{x}(t), t) \, dt$$ For a non-relativistic free particle: $$S[x(t)] = \int_{t_a}^{t_b} \frac{1}{2}m\dot{x}^2(t) \, dt$$ ### Einstein Field Equations $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$ The Schwarzschild solution for a non-rotating spherical mass: $$ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta \, d\phi^2)$$ ## Fractals and Iterative Systems ### Julia Sets For a complex function $f_c(z) = z^2 + c$, the Julia set $J_c$ is the boundary of the set of points that escape to infinity under iteration: $$J_c = \partial\{z \in \mathbb{C} : \lim_{n\to\infty} |f_c^n(z)| = \infty\}$$ The dynamics near fixed points $z_0$ where $f_c(z_0) = z_0$ are determined by the multiplier: $$\lambda = f'_c(z_0)$$ - If $|\lambda| < 1$, the fixed point is attracting - If $|\lambda| > 1$, the fixed point is repelling - If $|\lambda| = 1$, the fixed point is indifferent ### Iterated Function Systems (IFS) For a set of contractive mappings $\{f_1, f_2, \ldots, f_n\}$ in a metric space, the attractor $A$ satisfies: $$A = \bigcup_{i=1}^n f_i(A)$$ The Hausdorff dimension $d$ of a self-similar fractal created by $n$ copies scaled by factor $r$ satisfies: $$n \cdot r^d = 1$$ For example, the Sierpinski triangle has $n=3$ and $r=\frac{1}{2}$, so: $$3 \cdot \left(\frac{1}{2}\right)^d = 1 \implies d = \frac{\log 3}{\log 2} \approx 1.585$$ ### The Logistic Map and Chaos The logistic map is defined by the recurrence relation: $$x_{n+1} = rx_n(1-x_n)$$ The orbit diagram shows period doubling bifurcations leading to chaos. The Feigenbaum constant: $$\delta = \lim_{n\to\infty} \frac{r_n - r_{n-1}}{r_{n+1} - r_n} \approx 4.669201609102990671853203821578$$ Where $r_n$ is the parameter value at which the period-$2^n$ orbit appears. ## Non-Euclidean Geometry ### Hyperbolic Geometry In the Poincaré disk model, the hyperbolic distance between points $z_1$ and $z_2$ is: $$d_H(z_1, z_2) = 2\tanh^{-1}\left|\frac{z_1-z_2}{1-\bar{z_1}z_2}\right|$$ The area of a hyperbolic triangle with angles $\alpha, \beta, \gamma$ is: $$A = \pi - (\alpha + \beta + \gamma)$$ ### Riemannian Geometry The Riemann curvature tensor measures how the geometry deviates from Euclidean space: $$R^{\rho}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}_{\mu\sigma} + \Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma} - \Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}$$ Where $\Gamma^{\rho}_{\mu\nu}$ are the Christoffel symbols: $$\Gamma^{\rho}_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}\left(\partial_{\mu}g_{\nu\sigma} + \partial_{\nu}g_{\mu\sigma} - \partial_{\sigma}g_{\mu\nu}\right)$$ ## Category Theory ### Functors and Natural Transformations For categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F: \mathcal{C} \rightarrow \mathcal{D}$ maps: - Objects $X$ in $\mathcal{C}$ to objects $F(X)$ in $\mathcal{D}$ - Morphisms $f: X \rightarrow Y$ in $\mathcal{C}$ to morphisms $F(f): F(X) \rightarrow F(Y)$ in $\mathcal{D}$ Preserving composition: $F(g \circ f) = F(g) \circ F(f)$ A natural transformation $\eta: F \Rightarrow G$ between functors $F, G: \mathcal{C} \rightarrow \mathcal{D}$ consists of a family of morphisms $\eta_X: F(X) \rightarrow G(X)$ such that for any $f: X \rightarrow Y$ in $\mathcal{C}$: $$\eta_Y \circ F(f) = G(f) \circ \eta_X$$ This is captured by the commutative diagram: $$ \begin{CD} F(X) @>F(f)>> F(Y) \\ @V\eta_X VV @VV\eta_Y V \\ G(X) @>>G(f)> G(Y) \end{CD} $$ ### Yoneda Lemma For a locally small category $\mathcal{C}$ and an object $A$ in $\mathcal{C}$, the Yoneda embedding: $$y_A = \text{Hom}_{\mathcal{C}}(A, -): \mathcal{C} \rightarrow \text{Set}$$ The Yoneda lemma states that for any functor $F: \mathcal{C} \rightarrow \text{Set}$: $$\text{Nat}(y_A, F) \cong F(A)$$ --- _This showcase represents only a subset of LaTeX's mathematical capabilities. 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