Understanding Vector Spaces and Their Axioms

Vector Spaces

• A vector space is a set $V$.
• Vector addition: For $\textbf{u}, \textbf{v} \in V$, there is a unique element $\textbf{u} + \textbf{v} \in V$.
• Scalar multiplication: For $\textbf{u} \in V$ and scalar $c \in \mathbb{R}$, there is a unique element $c\textbf{u} \in V$.

Vector Space Axioms

• $\textbf{u} + \textbf{v} = \textbf{v} + \textbf{u}$ (Commutativity).
• $\textbf{u} + (\textbf{v} + \textbf{w}) = (\textbf{u} + \textbf{v}) + \textbf{w}$ (Associativity).
• There exists $\textbf{0} \in V$ such that $\textbf{u} + \textbf{0} = \textbf{u}$ (Additive Identity).
• For every $\textbf{u} \in V$, there exists $-\textbf{u} \in V$ such that $\textbf{u} + (-\textbf{u}) = \textbf{0}$ (Additive Inverse).
• $c(\textbf{u} + \textbf{v}) = c\textbf{u} + c\textbf{v}$ (Distributivity).
• $(c + d)\textbf{u} = c\textbf{u} + d\textbf{u}$ (Distributivity).
• $c(d\textbf{u}) = (cd)\textbf{u}$ (Associativity).
• $1\textbf{u} = \textbf{u}$ (Multiplicative Identity).

Vector Space Examples

• $\mathbb{R}^n$ is a vector space with the usual vector addition and scalar multiplication.
• $M_{m \times n}$, the set of all $m \times n$ matrices, is a vector space.
• $\mathbb{P}_n$, the set of all polynomials of degree at most $n$, is a vector space.
• $C[a, b]$, the set of all continuous functions on $[a, b]$, is a vector space.
• $\mathbb{R}^2 \nsubseteq \mathbb{R}^3$

Vector Space Theorem

• $V$ is a vector space, $\textbf{u} \in V$, and $c$ is a scalar.
• $0\textbf{u} = \textbf{0}$.
• $c\textbf{0} = \textbf{0}$.
• $(-1)\textbf{u} = -\textbf{u}$.

Subspace Definition

• A subspace $H$ of a vector space $V$ is a subset of $V$.

Subspace Properties

• The zero vector $\textbf{0}$ is in $H$.
• For each $\textbf{u}$ and $\textbf{v}$ in $H$, $\textbf{u} + \textbf{v}$ is in $H$ (Closed under addition).
• For each $\textbf{u}$ in $H$ and scalar $c$, $c\textbf{u}$ is in $H$ (Closed under scalar multiplication).
• A subspace $H$ of $V$ is itself a vector space.

Subspace Examples

• $H = {\begin{bmatrix} x \ y \end{bmatrix} : x = 2y }$ is a subspace of $\mathbb{R}^2$.
• $H = {\begin{bmatrix} x \ y \end{bmatrix} : x = y + 1 }$ is not a subspace of $\mathbb{R}^2$.
• $H = {\begin{bmatrix} x \ y \ z \end{bmatrix} : x - 3y + z = 0 }$ is a subspace of $\mathbb{R}^3$.
• $H = {\begin{bmatrix} x \ y \end{bmatrix} : x^2 + y^2 = 1 }$ is not a subspace of $\mathbb{R}^2$.

Subspace Theorem

• $V$ is a vector space and $\textbf{v}_1, \textbf{v}_2,..., \textbf{v}_p$ are vectors in $V$.
• $Span{\textbf{v}_1, \textbf{v}_2,..., \textbf{v}_p}$ is a subspace of $V$.

Chapitre 3 Suites Numériques

Suite Numérique

• It is a function defined on $\mathbb{N}$ or a part of $\mathbb{N}$ with values in $\mathbb{R}$.
• It is noted as $(U_n){n \in \mathbb{N}}$ or $(U_n){n \geq n_0}$ where $n_0 \in \mathbb{N}$.
• $U_n$ is the general term, and $U_0, U_1, U_2,...$ are the terms of the sequence.

Modes de Génération d'une Suite

• Explicite: $U_n = f(n)$ for all $n \in \mathbb{N}$.
  • Example: $U_n = n^2 + 1$, $U_n = \frac{2n}{n+1}$.
• Par récurrence: $U_0$ and a relation $U_{n+1} = f(U_n)$ for all $n \in \mathbb{N}$ are given.
  • Example: $U_0 = 2$ and $U_{n+1} = 3U_n - 1$.

Variations d'une Suite

• $(U_n)$ is increasing if $U_{n+1} \geq U_n$ for all $n \in \mathbb{N}$.
• $(U_n)$ is strictly increasing if $U_{n+1} > U_n$ for all $n \in \mathbb{N}$.
• $(U_n)$ is decreasing if $U_{n+1} \leq U_n$ for all $n \in \mathbb{N}$.
• $(U_n)$ is strictly decreasing if $U_{n+1} < U_n$ for all $n \in \mathbb{N}$.
• $(U_n)$ is constant if $U_{n+1} = U_n$ for all $n \in \mathbb{N}$.

Méthodes pour Étudier les Variations d'une Suite

• Calculate $U_{n+1} - U_n$ and study its sign.
• If $U_n > 0$, calculate $\frac{U_{n+1}}{U_n}$ and compare to 1.
• If $U_n = f(n)$, study the variations of the function $f$.

Suites Majorées, Minorées, Bornées

• $(U_n)$ is majorée if there exists a real $M$ such that $U_n \leq M$ for all $n \in \mathbb{N}$.
• $(U_n)$ is minorée if there exists a real $m$ such that $U_n \geq m$ for all $n \in \mathbb{N}$.
• $(U_n)$ is bornée if it is majorée and minorée.
• $(U_n)$ is bornée if and only if there exists a real $k > 0$ such that $|U_n| \leq k$ for all $n \in \mathbb{N}$.

Limites d'une Suite

• $\lim_{n \to +\infty} U_n = l$ if every open interval containing $l$ contains all values of $U_n$ from a certain rank.
• $\lim_{n \to +\infty} U_n = +\infty$ if every interval of the form $]A; +\infty[$ contains all values of $U_n$ from a certain rank.
• $\lim_{n \to +\infty} U_n = -\infty$ if every interval of the form $]-\infty; A[$ contains all values of $U_n$ from a certain rank.
• If a sequence $(U_n)$ has a limit, then that limit is unique.

Opérations sur les Limites

• If $\lim U_n = l$ and $\lim V_n = l'$ then $\lim (U_n + V_n) = l + l'$ and $\lim (U_n \times V_n) = l \times l'$
• If $\lim U_n = l$ and $\lim V_n = l'$ then $\lim (\frac{U_n}{V_n}) = \frac{l}{l'}$ if $l' \neq 0$.
• If $\lim U_n = l$ and $\lim V_n = +\infty$ then $\lim (U_n + V_n) = +\infty$
• If $\lim U_n = l$ and $\lim V_n = +\infty$ then $\lim (U_n \times V_n) = +\infty$ if $l > 0$ and $\lim (\frac{U_n}{V_n}) = 0$.
• If $\lim U_n = l$ and $\lim V_n = -\infty$ then $\lim (U_n + V_n) = -\infty$.
• If $\lim U_n = l$ and $\lim V_n = -\infty$ then $\lim (U_n \times V_n) = -\infty$ if $l > 0$ and $\lim (\frac{U_n}{V_n}) = 0$.

Limites et Comparaison

• Si $U_n \leq V_n$ et $\lim_{n \to +\infty} U_n = +\infty$, alors $\lim_{n \to +\infty} V_n = +\infty$.
• Si $U_n \geq V_n$ et $\lim_{n \to +\infty} U_n = -\infty$, alors $\lim_{n \to +\infty} V_n = -\infty$.
• Si $V_n \leq U_n \leq W_n$ et $\lim_{n \to +\infty} V_n = \lim_{n \to +\infty} W_n = l$, alors $\lim_{n \to +\infty} U_n = l$.
• Si $\lim_{n \to +\infty} U_n = l$ et $f$ is continue en $l$, alors $\lim_{n \to +\infty} f(U_n) = f(l).

Suites Monotones

• Toute suite croissante et majorée converge.
• Toute suite décroissante et minorée converge.
• Toute suite croissante non majorée diverge vers $+\infty$.
• Toute suite décroissante non minorée diverge vers $-\infty$.

Suites Arithmétiques

• $U_{n+1} = U_n + r$ where $r$ is the reason.
• $U_n = U_0 + nr$.
• $U_n = U_p + (n-p)r$.
• $S_n = U_0 + U_1 +... + U_n = (n+1) \frac{U_0 + U_n}{2}$.
• $S_n = U_1 + U_2 +... + U_n = n \frac{U_1 + U_n}{2}$.

Limites des Suites Arithmétiques

• Si $r > 0$, alors $\lim_{n \to +\infty} U_n = +\infty$.
• Si $r < 0$, alors $\lim_{n \to +\infty} U_n = -\infty$.
• Si $r = 0$, alors $\lim_{n \to +\infty} U_n = U_0$.

Suites géométriques

• $U_{n+1} = q U_n$ where $q$ is the reason.
• $U_n = U_0 q^n$.
• $U_n = U_p q^{n-p}$.
• $S_n = U_0 + U_1 +... + U_n = U_0 \frac{1 - q^{n+1}}{1 - q}$ si $q \neq 1$.
• $S_n = U_1 + U_2 +... + U_n = U_1 \frac{1 - q^n}{1 - q}$ si $q \neq 1$.

Limites des Suites géométriques

• Si $q > 1$, alors $\lim_{n \to +\infty} q^n = +\infty$.
• Si $q = 1$, alors $\lim_{n \to +\infty} q^n = 1$.
• Si $-1 < q < 1$, alors $\lim_{n \to +\infty} q^n = 0$.
• Si $q \leq -1$, la suite $(q^n)$ n'a pas de limite.

Chapter 4: Applications of Derivatives

4.1 Related Rates

• Strategy: Read, Draw, Introduce notation, Express given info/required rate in derivatives, Write equation, Chain Rule, Substitute given info

Examples

Ex 1.

• Air is pumped into a spherical balloon.
• Volume increases: $\frac{dV}{dt} = 100 \mathrm{~cm}^3/s$.
• Find radius increase $\frac{dr}{dt}$ when $d = 50$ cm ($r = 25$ cm).
• Equation: $V = \frac{4}{3} \pi r^3$.
• Derivative: $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$.
• Solution: $\frac{dr}{dt} = \frac{1}{25 \pi} \mathrm{cm}/s$.

Ex 2.

• Ladder 10 ft long against a wall.
• Bottom slides away: $\frac{dx}{dt} = 1$ ft/s.
• Find top sliding down $\frac{dy}{dt}$ when $x = 6$ ft.
• Equation: $x^2 + y^2 = 10^2$.
• Derivative: $x \frac{dx}{dt} + y \frac{dy}{dt} = 0$.
• When $x = 6$, $y = 8$.
• Solution: $\frac{dy}{dt} = -\frac{3}{4}$ ft/s.

Ex 3.

• Water tank is an inverted cone.
• Base radius 2 m, height 4 m.
• Water pumped in: $\frac{dV}{dt} = 2 \mathrm{~m}^3/\mathrm{min}$.
• Find water level rising $\frac{dh}{dt}$ when $h = 3$ m.
• Equation: $V = \frac{1}{3} \pi r^2 h$, $r = \frac{1}{2} h$, $V = \frac{\pi}{12} h^3$.
• Derivative: $\frac{dV}{dt} = \frac{\pi}{4} h^2 \frac{dh}{dt}$.
• Solution: $\frac{dh}{dt} = \frac{8}{9 \pi} \mathrm{m}/\mathrm{min}$.

Ex 4.

• Car A travels west: 50 mi/h ($\frac{da}{dt} = -50$).
• Car B travels north: 60 mi/h ($\frac{db}{dt} = -60$).
• Find approaching rate $\frac{dc}{dt}$ when $a = 0.3$ mi, $b = 0.4$ mi.
• Equation: $c^2 = a^2 + b^2$.
• Derivative: $c \frac{dc}{dt} = a \frac{da}{dt} + b \frac{db}{dt}$.
• When $a = 0.3$, $b = 0.4$, $c = 0.5$.
• Solution: $\frac{dc}{dt} = -78$ mi/h.

Préparation à l'Agrégation Externe de Mathématiques

Analyse

Exercice 1

• Soit $f \in C([0, 1], \mathbb{R})$.
• Calculer $\lim_{n \to \infty} \int_{0}^{1} f(t^n) dt$.

Exercice 2

• Soit $f : \mathbb{R} \to \mathbb{R}$ uniformément continue.
• Montrer que : $\lim_{n \to \infty} \int_{0}^{1} f(x + n) dx = 0$.

Exercice 3

• Soit $f : [0, +\infty[ \to \mathbb{R}$ continue et bornée.
• Montrer que : $\lim_{x \to +\infty} \frac{1}{x} \int_{0}^{x} f(t) dt$ existe.

Exercice 4

• Étudier la convergence de l'intégrale $\int_{0}^{+\infty} \frac{\sin(t)}{t} dt$.

Exercice 5

• Étudier la convergence de l'intégrale $\int_{0}^{+\infty} \sin(t^2) dt$.

Exercice 6

• Montrer que $\int_{0}^{+\infty} e^{-t^2} dt = \frac{\sqrt{\pi}}{2}$.

Exercice 7

• Soit $f : \mathbb{R} \to \mathbb{R}$ continue telle que $f(x+1) = f(x)$ pour tout $x \in \mathbb{R}$.
• Montrer que : $\lim_{n \to \infty} \int_{0}^{1} f(nx) dx = \int_{0}^{1} f(x) dx$.

Exercice 8

• Soit $f : [0, 1] \to \mathbb{R}$ continue.
• Calculer $\lim_{n \to \infty} \int_{0}^{1} x^n f(x) dx$.

Exercice 9

• Soit $f : [0, 1] \to \mathbb{R}$ continue.
• Calculer $\lim_{n \to \infty} n \int_{0}^{1} x^n f(x) dx$.

Exercice 10

• Soit $f : [0, 1] \to \mathbb{R}$ continue.
• Calculer $\lim_{n \to \infty} \int_{0}^{1} \frac{f(x)}{1 + x^n} dx$

Lecture 11 - Secure Computation

Definition

• Alice has input x, Bob has input y, computing f(x, y).
• Correctness: Discovering correct function value.
• Privacy: Minimal Input Information Leakage

Example

• Alice & Bob - Who's richer? - without disclosing income

Yao's Protocol

Securely compute functions using boolean circuits, avoiding input reveals.

• Secure against semi-honest adversaries.
• Alice and Bob want to compute $f(x, y)$, where x is Alice's input, y is Bob's input

Alice's part

• Express $f$ as boolean circuit $C$
• For wire $i$, choose two random keys $K_{i}^{0}, K_{i}^{1}$ = values 0, 1 of wire $i$.
• Alice encrypts the truth table of the gate, using the keys corresponding to the gate's input wires to encrypt the keys corresponding to the gate's output wire.
• Alice sends the encrypted circuit to Bob, as well as the keys corresponding to her input wires.

Bob's part

• Getting keys corresponding to the inputs to Alice.
• Gate Evaluations: The preservation of privacy comes from knowing a key

Yao's Protocol - Example

• $f(x, y) = x \land y$
  • AND Gate Circuit
• Use the alice part protocol discussed before

Homework

• Prove the protocol is secure against semi-honest adversaries
• Consider $f(x_1, x_2,..., x_n) = \sum_{i=1}^{n} x_i \mod 2$
  • Construct the computing circle
  • determine the circuit size = function n

Static Electricity

Electric Charge

• Two Types of Charge: Positive and Negative
• Like Charges Repel.
• Opposite Charges Attract.

Conductors

• Materials that allow electrons to move freely are conductors
  • Examples: Metals like copper, aluminum, and gold
  • Also includes: Salt water

Insulators

• Materials that do not allow electrons to move freely are insulators.
  • Examples: Rubber, Glass, Wood, Plastic

Charging

Friction

• Rubbing two neutral objects together facilitating electron transfer

Conduction

• Charging an object with existing charge through touching

Induction

• Charging a neutral object by bringing near a charged one

Electric Force

Coulomb's Law

• $F$ = Electric Force (N)
  • $k$ = Coulomb's Constant $(9.0 \times 10^9 \frac{N \cdot m^2}{C^2})$
  • $q_1$ = Charge of object 1 (C)
  • $q_2$ = Charge of object 2 (C)
  • $d$ = distance between objects (m)
• The electric force between two objects is proportional to the product of the charges and inversely proportional to the square of the distance.

Electric Fields

Definition

A region around a charged object where a force would be exerted on another charged object.

Direction

• The direction of the force on a positive test charge.

Field Lines

 - Positive = Point away
 - Negative = Point towards
 - Closer lines = Stronger field

Física

Vectores

Definición

• Un vector es un segmento de recta orientado.
  • Módulo: Es la longitud del vector.
  • Dirección: Es la recta que contiene al vector.
  • Sentido: Es la orientación del vector dentro de la dirección.
  • Punto de aplicación: Es el origen del vector.

Tipos de vectores

• Vectores iguales: Mismo módulo, dirección y sentido.
• Vectores opuestos: Mismo módulo y dirección, pero sentido contrario.
• Vectores unitarios: Su módulo es igual a 1.
• Vectores concurrentes: Sus rectas de acción se cortan en un punto.
• Vectores coplanarios: Están contenidos en el mismo plano.
• Vectores colineales: Están contenidos en la misma recta.

Operaciones con vectores

Suma de vectores

Método gráfico

• Método del triángulo: Unir origen del primer vector con extremo del último.
• Método del paralelogramo: Diagonal del paralelogramo que parte del origen.
• Método del polígono: Unir origen del primer vector con extremo del último.

Método analítico

• $\vec{A} = (A_x, A_y)$
• $\vec{B} = (B_x, B_y)$
• $\vec{A} + \vec{B} = (A_x + B_x, A_y + B_y)$

Resta de vectores

• $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Producto de un escalar por un vector

• $k \cdot \vec{A} = (k \cdot A_x, k \cdot A_y)$

Producto escalar de dos vectores

• $\vec{A} \cdot \vec{B} = |\vec{A}| \cdot |\vec{B}| \cdot \cos{\theta}$
• $\vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y$

Producto vectorial de dos vectores

• $|\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin{\theta}$
• $\vec{A} \times \vec{B} = (A_y \cdot B_z - A_z \cdot B_y, A_z \cdot B_x - A_x \cdot B_z, A_x \cdot B_y - A_y \cdot B_x)$

Versores

• Vectores unitarios que indican la dirección de los ejes de coordenadas.
  • $\hat{i}$: Versor del eje x.
  • $\hat{j}$: Versor del eje y.
  • $\hat{k}$: Versor del eje z.
• $\vec{A} = A_x \cdot \hat{i} + A_y \cdot \hat{j} + A_z \cdot \hat{k}$

Cinemática

Definiciones

• Posición: Lugar que ocupa un cuerpo en el espacio.
• Trayectoria: Conjunto de posiciones que ocupa un cuerpo.
• Desplazamiento: Variación de la posición de un cuerpo.
• Velocidad: Variación de la posición en función del tiempo.
• Aceleración: Variación de la velocidad en función del tiempo.

Tipos de movimiento

• Movimiento rectilíneo uniforme (MRU): Velocidad constante.
  • $v = \frac{\Delta x}{\Delta t}$
  • $x = x_0 + v \cdot t$
• Movimiento rectilíneo uniformemente variado (MRUV): Aceleración constante.
  • $a = \frac{\Delta v}{\Delta t}$
  • $v = v_0 + a \cdot t$
  • $x = x_0 + v_0 \cdot t + \frac{1}{2} \cdot a \cdot t^2$
  • $v^2 = v_0^2 + 2 \cdot a \cdot \Delta x$
• Movimiento circular uniforme (MCU): Velocidad angular constante.
  • $\omega = \frac{\Delta \theta}{\Delta t}$
  • $\theta = \theta_0 + \omega \cdot t$
  • $v = \omega \cdot r$
  • $a_c = \frac{v^2}{r} = \omega^2 \cdot r$
• Movimiento armónico simple (MAS): Movimiento periódico con variación sinusoidal.
  • $x = A \cdot \cos{(\omega \cdot t + \phi)}$
  • $v = -A \cdot \omega \cdot \sin{(\omega \cdot t + \phi)}$
  • $a = -A \cdot \omega^2 \cdot \cos{(\omega \cdot t + \phi)}$
  • $\omega = \sqrt{\frac{k}{m}}$
  • $T = \frac{2\pi}{\omega}$
  • $f = \frac{1}{T}$

Tiro oblicuo

• MRU en el eje x y MRUV en el eje y.
  • $v_{0x} = v_0 \cdot \cos{\alpha}$
  • $v_{0y} = v_0 \cdot \sin{\alpha}$
  • $x = v_{0x} \cdot t$
  • $y = v_{0y} \cdot t - \frac{1}{2} \cdot g \cdot t^2$
  • $t_{max} = \frac{v_{0y}}{g}$
  • $x_{max} = \frac{v_0^2 \cdot \sin{2\alpha}}{g}$
  • $y_{max} = \frac{v_{0y}^2}{2g}$

Dinámica

Leyes de Newton

• Primera ley: Inercia.
• Segunda ley: $\sum \vec{F} = m \cdot \vec{a}$
• Tercera ley: Acción y reacción.

Tipos de fuerzas

• Peso: $P = m \cdot g$
• Normal: Fuerza de una superficie sobre un cuerpo.
• Tensión: Fuerza de una cuerda o cable.
• Fuerza de rozamiento: $F_r = \mu \cdot N$
• Fuerza elástica: $F = k \cdot \Delta x$

Trabajo

• $W = \vec{F} \cdot \vec{d} = |\vec{F}| \cdot |\vec{d}| \cdot \cos{\theta}$

Energía

• Energía cinética: $E_c = \frac{1}{2} \cdot m \cdot v^2$
• Energía potencial gravitatoria: $E_p = m \cdot g \cdot h$
• Energía potencial elástica: $E_p = \frac{1}{2} \cdot k \cdot \Delta x^2$
• Energía mecánica: $E_m = E_c + E_p$

Teorema del trabajo y la energía

• $W = \Delta E_c$

Conservación de la energía mecánica

• $\Delta E_m = 0$

Potencia

• $P = \frac{W}{\Delta t}$

Estática

Definiciones

• Momento de una fuerza: $M = \vec{r} \times \vec{F} = |\vec{r}| \cdot |\vec{F}| \cdot \sin{\theta}$
• Cupla: Sistema de dos fuerzas iguales, opuestas y paralelas.

Condiciones de equilibrio

• $\sum \vec{F} = 0$
• $\sum \vec{M} = 0$

Hidrostática

Definiciones

• Presión: $P = \frac{F}{A}$
• Densidad: $\rho = \frac{m}{V}$
• Peso específico: $\gamma = \frac{P}{V} = \rho \cdot g$

Teorema fundamental de la hidrostática

• $P_2 - P_1 = \gamma \cdot (h_1 - h_2)$

Presión atmosférica

• $P_{atm} = 101325 Pa = 1 atm$

Principio de Arquímedes

• $E = \gamma \cdot V_{sumergido}$

Termodinámica

Definiciones

• Temperatura: Energía interna de un sistema.
• Calor: Transferencia de energía térmica.
• Trabajo: Transferencia de energía por desplazamiento.
• Energía interna: Energía total de un sistema.
• Entalpía: Energía que un sistema puede intercambiar con su entorno.
• Entropía: Medida del desorden de un sistema.

Leyes de la termodinámica

• Ley cero: Equilibrio térmico.
• Primera ley: $\Delta U = Q - W$
• Segunda ley: La entropía siempre aumenta.
• Tercera ley: La entropía tiende a un mínimo a medida que T se aproxima al cero absoluto.

Tipos de procesos termodinámicos

• Proceso isotérmico: Temperatura constante.
• Proceso isobárico: Presión constante.
• Proceso isocórico: Volumen constante.
• Proceso adiabático: Sin transferencia de calor.

Gases ideales

• $P \cdot V = n \cdot R \cdot T$