Frozen section and cytology

Unidad 3: Derivadas

• Este capítulo explora el concepto de derivadas, sus reglas, cálculo, propiedades y aplicaciones.
• Incluye la interpretación geométrica, reglas de diferenciación, derivadas de orden superior, diferenciación implícita y aplicaciones específicas como rectas tangentes, razones de cambio relacionadas, máximos y mínimos y trazado de curvas.

3.1. Interpretación Geométrica de la Derivada

• La derivada f'(a) representa la pendiente de la recta tangente a la gráfica de la función f(x) en x = a.
• La pendiente se calcula mediante el límite: f'(a) = lim (h→0) [f(a+h) - f(a)] / h, si este existe.
• Ejemplo: Para f(x) = x², la pendiente en x = 2 se calcula como f'(2) = 4.

3.2. Reglas de Derivación

• Facilitan el cálculo de derivadas:
  • Regla de la Potencia: Si f(x) = xⁿ, entonces f'(x) = nxⁿ⁻¹.
  • Regla de la Constante: Si f(x) = c, entonces f'(x) = 0.
  • Regla del Producto: Si f(x) = u(x)v(x), entonces f'(x) = u'(x)v(x) + u(x)v'(x).
  • Regla del Cociente: Si f(x) = u(x) / v(x), entonces f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]².
  • Regla de la Cadena: Si f(x) = g(h(x)), entonces f'(x) = g'(h(x)) ⋅ h'(x).

3.3. Derivadas de Orden Superior

• Se obtienen derivando sucesivamente la derivada de una función.
• Notación: f'(x), f''(x), f'''(x), y f⁽ⁿ⁾(x) para la n-ésima derivada.
• Ejemplo: Para f(x) = x⁴ + 3x² - 2x + 1:
  • f'(x) = 4x³ + 6x - 2
  • f''(x) = 12x² + 6
  • f'''(x) = 24x
  • f⁽⁴⁾(x) = 24
  • f⁽⁵⁾(x) = 0

3.4. Derivación Implícita

• Se aplica cuando x e y están relacionados por una ecuación en lugar de una función explícita y = f(x).
• Se derivan ambos lados de la ecuación con respecto a x y se despeja dy/dx.
• Ejemplo: Para x² + y² = 25, la derivada dy/dx = -x/y.

3.5. Aplicaciones de la Derivada

3.5.1. Rectas Tangentes y Normales

• Recta Tangente: y - f(a) = f'(a)(x - a)
• Recta Normal: y - f(a) = -[1 / f'(a)](x - a)

3.5.2. Razones de Cambio Relacionadas

• Involucran variables con tasas de cambio interrelacionadas.
• Los pasos incluyen identificar variables, encontrar una ecuación que las relacione, derivar con respecto al tiempo, sustituir valores conocidos y resolver.
• Ejemplo: Un globo se infla a 100 cm³/s; cuando el radio es de 5 cm, la tasa de cambio del radio dr/dt es 1/π cm/s.

3.5.3. Máximos y Mínimos

• Los puntos críticos de una función son valores donde la derivada es cero o indefinida.
• Criterio de la primera derivada: Cambio de signo de la derivada en x = c indica un máximo local (de positivo a negativo) o un mínimo local (de negativo a positivo).
• Criterio de la segunda derivada: f'(c) = 0 y f''(c) > 0 implican un mínimo local, mientras que f'(c) = 0 y f''(c) < 0 implican un máximo local.

3.5.4. Concavidad y Puntos de Inflexión

• La segunda derivada determina la concavidad de la función.
• f''(x) > 0 indica concavidad hacia arriba, mientras que f''(x) < 0 indica concavidad hacia abajo.
• Un punto de inflexión ocurre donde cambia la concavidad (f''(x) = 0 o no existe).

3.5.5. Trazado de Curvas

• Utiliza información de la primera y segunda derivada para trazar la gráfica de una función.
• Los pasos incluyen encontrar el dominio, intersecciones, asíntotas, puntos críticos, intervalos de crecimiento/decrecimiento, concavidad y puntos de inflexión.

Matrices

Definition

• A matrix is a rectangular array (or field) of numbers (or more generally, elements of a field).
• Denoted as $A = (a_{ij}){m \times n}$ or $A = A{m,n}$, where $a_{ij} \in \mathbb{R}$ (or $\mathbb{K}$).
  • $m$: Rows
  • $n$: Columns
  • $a_{ij}$: Element in the $i$-th row and $j$-th column
• An $m \times n$ matrix has $m \cdot n$ elements.

Special Matrices

• Nullmatrix: $A = 0 = (0)_{m \times n}$
• Quadratische Matrix: $m = n$, $A = A_n = (a_{ij})_{n \times n}$
  • Diagonal matrix: $a_{ij} = 0$ for $i \neq j$
  • Einheitsmatrix: $I_n$ (Identity matrix)
  • Dreiecksmatrix (Triangular matrix):
    • Obere Dreiecksmatrix (Upper triangular matrix): $a_{ij} = 0$ for $i > j$
    • Untere Dreiecksmatrix (Lower triangular matrix): $a_{ij} = 0$ for $i < j$
• Vektor (Vector):
  • Zeilenvektor (Row vector): $m = 1$, $A = (a_{11}, \cdots, a_{1n})$
  • Spaltenvektor (Column vector): $n = 1$, $A = \begin{pmatrix} a_{11} \ \vdots \ a_{m1} \end{pmatrix}$

Matrix Operations

• Addition:
  • If $A = (a_{ij}){m \times n}$ and $B = (b{ij}){m \times n}$, then $A + B = (a{ij} + b_{ij})_{m \times n}$
• Scalar Multiplication:
  • If $A = (a_{ij}){m \times n}$ and $\lambda \in \mathbb{R}$ (or $\mathbb{K}$), then $\lambda \cdot A = (\lambda \cdot a{ij})_{m \times n}$
• Matrix Multiplication:
  • If $A = (a_{ij}){m \times n}$ and $B = (b{jk}){n \times p}$, then $C = A \cdot B = (c{ik}){m \times p}$ with $c{ik} = \sum_{j=1}^{n} a_{ij} \cdot b_{jk}$
  • $A \cdot B \neq B \cdot A$ (generally)

Algorithmic Complexity

• Algorithmic complexity measures the time (time complexity) and space (space complexity) an algorithm requires for an input of size n.
• It helps compare the efficiency of different solutions by focusing on the rate of growth.
• It is expressed using Big O notation.

Big O Notation

• O(1): Constant time (Excellent) - Number of operations is constant.
• O(log n): Logarithmic time (Great) - Number of operations increases logarithmically (e.g., Binary Search).
• O(n): Linear time (Good) - Number of operations increases linearly (e.g., printing each element in a list).
• O(n log n): Log-linear time (Fair) - Number of operations increases by n * log(n) (e.g., Merge Sort, Quick Sort).
• O(n²): Quadratic time (Bad) - Number of operations increases quadratically, as in comparing all pairs in a list.
• O(2ⁿ): Exponential time (Horrible) - Number of operations doubles as the input grows (e.g., Fibonacci recursive).
• O(n!): Factorial time (Nightmare)

Eigenvalues and Eigenvectors

• Example 1: For matrix $A = \begin{bmatrix} 2 & 1 \ 1 & 2 \end{bmatrix}$.
  • Eigenvalues are $\lambda_1 = 1$ and $\lambda_2 = 3$.
  • Eigenvectors are $V_1 = \begin{bmatrix} -1 \ 1 \end{bmatrix}$ for $\lambda_1 = 1$ and $V_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix}$ for $\lambda_2 = 3$.
  • Diagnolization: $P = \begin{bmatrix} -1 & 1 \ 1 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 \ 0 & 3 \end{bmatrix}$, $P^{-1} = \begin{bmatrix} -1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$,and therefore $A = PDP^{-1}$.
• Example 2: For matrix $A = \begin{bmatrix} 0.8 & 0.3 \ 0.2 & 0.7 \end{bmatrix}$.
  • Eigenvalues are $\lambda_1 = 0.5$ and $\lambda_2 = 1$.
  • Eigenvectors are $V_1 = \begin{bmatrix} -1 \ 1 \end{bmatrix}$ for $\lambda_1 = 0.5$ and $V_2 = \begin{bmatrix} 3 \ 2 \end{bmatrix}$ for $\lambda_2 = 1$.
  • Diagnolization: $P = \begin{bmatrix} -1 & 3 \ 1 & 2 \end{bmatrix}$, $D = \begin{bmatrix} 0.5 & 0 \ 0 & 1 \end{bmatrix}$, $P^{-1} = \begin{bmatrix} -2/5 & 3/5 \ 1/5 & 1/5 \end{bmatrix}$,and therefore $A = PDP^{-1}$.

Resumen Ejecutivo

• The report presents key research results on factors influencing consumer purchase decisions.
• A mixed methodology of surveys and data analysis identified the most relevant variables affecting consumer behavior.
• The survey involved 500 consumers, with statistical techniques used for analysis.

Key findings

Demographic Factors

• Age: Younger consumers prioritize innovation and sustainability, while older ones value quality and durability.
• Income: Higher-income consumers are willing to pay more for premium products.
• Gender: Women are more influenced by recommendations.

Psychological Factors

• Motivation: Self-expression and social belonging drive purchases (fashion/tech).
• Perception: Quality and value influence durable goods purchase decisions.
• Learning: Brand experience and advertising affect consumer attitudes.

Sociocultural Factors

• Culture: Cultural values impact food, clothing, and personal goods choices.
• Reference Groups: Family and opinion leaders influence purchase decisions.
• Social Class: Affects lifestyle and brand preferences.

Conclusion

• Demographic, psychological, and sociocultural factors interact to influence consumer decision-making
• Companies should tailor strategies for different target markets.

Recommendations

• Segmentation: Customize offers based on demographics and behavior.
• Communication: Adapt advertising messages to the segment´s value.
• Improve customer experience across all touchpoints.
• Innovation: Invest in R&D.
• Sustainability: Adopt sustainable practices.

Next steps

• Conduct in-depth qualitative study.

Algorithmic Game Theory - Assignment 1

• This is an assignment with five problems, each with specified point values, focused on game theory concepts.
• Students are instructed to submit their solutions individually through Brightspace, ensuring solutions are rigorous and concise while writing independently.
• They must acknowledge peer discussions, not searching the internet for solutions or using AI tools like ChatGPT.
• Latex must be used for written solutions, correctness and running time must be argued when designing an algorithm.

Problems

• Problem 1: Finding the number to announce in a game with Alice and Bob to maximize payoff. In this case, naming the same number would result in dollars payed to Bob.
• Problem 2: Analyzing pure-strategy Nash equilibria and computing mixed-strategy Nash equilibria for a given normal form game with payoff values for L, R, U, M and D.
• Problem 3: Computing the mixed-strategy Nash equilibrium for Rock-Paper-Scissors
• Problem 4: Identifying the Nash Equilibria for a game where n players announce real numbers from [0, 100]. Payoff function: $u_i(x_1, \dots, x_n) = 100 - (x_i - \bar{x})^2$.
• Problem 5: Analyzing a problem in directed graph with k Players wanting to travel from $s_i$ to $t_i$, analyzing pure-strategy nash equilibriums, social cost differences, and upper bounds on the price of anarchy.

Thermodynamics

Thermodynamic Properties

• Intensive Properties: Do not depend on mass (Temperature, Pressure, Density)
• Extensive Properties: Depend on mass (Mass, Volume, Energy)
• Specific Properties: Extensive properties per unit mass (Specific volume $\nu = \frac{V}{m}$, Specific energy $e = \frac{E}{m}$)

State of a System

• Defined by its properties with all properties having fixed values, or will change if one property does
  • Independent Properties: Properties that can be varied independently
  • Dependent Properties: Properties the change with independent properties.
  • State Postulate: The state of a simple compressible system is completely specified by two independent, intensive properties

Processes

• Any change a system undergoes from one state to another.

  • Path through a series of states
  • Describe a process with initial and final states, the path, and the interactions with the surroundings.

• Types of Processes*

  • Isothermal Process: Constant temperature (T)
  • Isobaric Process: Constant pressure (P)
  • Isochoric: Constant specific volume ($\nu$)
  • Cycle: Initial and final states are identical

Equilibrium

• Thermal Equilibrium: Temperature is constant
• Mechanical Equilibrium: No change in pressure
• Phase Equilibrium: Mass of each phase is constant
• Chemical Equilibrium: Chemical composition is constant
  • Thermodynamic Equilibrium: thermal, mechanical, phase, and chemical equiliubrium
  • Quasi-Equilibrium Process: very slow process, system always in constant thermal equilibrium

Termperature Scales

• Celsius Scale: $T(^{\circ}C) = T(K) - 273.15$
• Fahrenheit Scale: $T(^{\circ}F) = 1.8T(^{\circ}C) + 32$
• Rankine Scale: $T(R) = 1.8T(K)$
  • $\Delta T(K) = \Delta T(^{\circ}C)$
  • $\Delta T(R) = \Delta T(^{\circ}F)$
  • $\Delta T(R) = 1.8\Delta T(K)$

Préparation à l'agrégation interne: Exam Prep

Rappels de topologie générale (General Topology Reminders)

• Adhérence/Closure - $\overline{A} = {x \in E, \forall \epsilon > 0, B(x, \epsilon) \cap A \neq \emptyset}$, which means Closure of A includes all points in E where every neighborhood of x intersects with A.
• Intérieur/Interior - $\mathring{A} = {x \in E, \exists \epsilon > 0, B(x, \epsilon) \subset A}$, which means Interior of A includes all points where there is a neighborhood of those points fully contained in A.
• Properties - $\overline{A}$ is the smallest closed set containing A. - $\mathring{A}$ is the largest open sets contained in A.
• Densité/Density - A is dense in E if $\overline{A} = E$.
• Example - $\mathbb{Q}$ is dense in $\mathbb{R}$.

Suites/Sequences

• Definitions - Bounded Sequence: $\exists M \in \mathbb{R}, \forall n \in \mathbb{N}, |u_n| \leq M$ - Convergent Sequence: $\exists l \in \mathbb{R}, \forall \epsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, |u_n - l| < \epsilon$ - Cauchy Sequence: $\forall \epsilon > 0, \exists N \in \mathbb{N}, \forall n, m \geq N, |u_n - u_m| < \epsilon$
• Theorems - Every convergent sequences is bounded - Every convergent sequences is Cauchy - In ℝ , every Cauchy sequence is convergent (_ℝ is complete)
• Suites extraites/SubSequeunces
• Soit $(u_n){n \in \mathbb{N}}$ une suite et $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ strictement croissante/Suppose ( u**n )n∈ℕ is a sequence and φ : ℕ → ℕ is strictly increasing. La suite $(u_{\varphi(n)}){n \in \mathbb{N}}$ est une suite extraite de $(u_n){n \in \mathbb{N}}$/The sequence $(u_{\varphi(n)}){n \in \mathbb{N}}$ is a subsequence of $(u**n )n∈ℕ*.
• Properties - If $(u_n){n \in \mathbb{N}}$ converges to $l$, then any subsequence of $(u**n )n∈ℕ* converges to l.
• Théorème de Bolzano-Weierstrass/Bolzano-Weierstrass Theorem - From every bounded sequence, one can extract a convergent subsequence.

Fonctions continues/Continuous Functions

• Definition - Soit $f : E \rightarrow F$, où $E$ et $F$ sont des espaces vectoriels normés/Suppose f : E → F, where E and F are normed vector spaces. - $f$ est continue en $x_0 \in E$/f is continuous at $x_0 \in E$ si $\forall \epsilon > 0, \exists \alpha > 0, \forall x \in E, ||x - x_0|| < \alpha \Rightarrow ||f(x) - f(x_0)|| < \epsilon$.
• Definition équivalente/Equivalent definition - $f$ est continue en $x_0 \in E$ if for any sequence $(x_n){n \in \mathbb{N}}$ that converges to $x_0$, the sequence $(f(x_n)){n \in \mathbb{N}}$ converges a function $(f(x**0))$.
• Continuité uniforme/Uniform continuity - $f : E \rightarrow F$ est uniformément continue/f : E → F is uniformly continuous si $\forall \epsilon > 0, \exists \alpha > 0, \forall x, y \in E, ||x - y|| < \alpha \Rightarrow ||f(x) - f(y)|| < \epsilon$.
• Théorème de Heine/Heine’s Theorem - If f : [a, b] → ℝ is continuous, then f is uniformly continuous.

Compacité/Compactness

• Definition - An normed vector space space is compact if from every sequence of E, one can find a convergent subsequence.
• Theorem - In finite dimensions, a set is compact if and only if it is closed and bounded.

Complex Numbers

• A complex number has the form z = a + bi, where a and b are real numbers and i is the imaginary unit (i = √(-1)).
• a is the real part and b is the imaginary part of z.

Geometric Interpretation

• A complex number z = a + bi is a point (a, b) in the complex plane.
  • The horizontal axis is the real axis.
  • The vertical axis is the imaginary axis.

Polar Form

• A complex number $z = a + bi$ or $z = r(\cos \theta + i \sin \theta)$.
  • $r = |z| = \sqrt{a^2 + b^2}$ is the magnitude of z.
  • theta = arg(z) = arctan (b/a) is the argument of z.

Comlex Number Operatinos

• Addition: $z_1 + z_2 = (a + c) + (b + d)i$
• Subtraction: $z_1 - z_2 = (a - c) + (b - d)i$
• Multiplication: $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$
• Division: $\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$
• Complex Conjugate:
  • Definition $\bar{z} = a - bi$
  • z ⋅ z̄ = |z|² = a² + b²
  • $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$
  • $\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}$

Euler's Formula

• With formula of e^(iθ) = cos θ + i sin θ, the formula of a complex number can be rewritten as z = re^(iθ).

De Moivre's Theorem

• As formula of z^(n) = r^n (cos(nθ) + i sin(nθ)), using Euler's formula and any integer n, can be expressed to (re^(iθ))^n = r^n e^(inθ).