Bernoulli's Principle

Bernoulli's Principle

• States that fluid speed increases simultaneously with a decrease in pressure or potential energy.
• Formula: ( P + \frac{1}{2} \rho v^2 + \rho g h = constant )

Formula Components

• ( P ) is the static pressure of the fluid.
• ( \rho ) is the density of the fluid.
• ( v ) is the speed of the fluid.
• ( g ) is the acceleration due to gravity.
• ( h ) is the height of the fluid above a reference point.
• Dynamic Pressure: ( \frac{1}{2} \rho v^2 ) represents pressure from the fluid's kinetic energy; increases with speed.
• Hydrostatic Pressure: ( \rho g h ) represents pressure due to fluid weight at a certain height.

Implications

• As fluid speed ( v ) increases, static pressure ( P ) decreases if height ( h ) is constant.
• As fluid speed ( v ) decreases, static pressure ( P ) increases.

Applications

• Airplane Wings: Air flows faster over the top, creating lower pressure and upward lift.
• Venturi Meters: Measure fluid flow rate by measuring pressure differences in a pipe constriction.
• Carburetors: Use the principle to mix air and fuel in internal combustion engines.
• Chimneys: Taller chimneys enhance pressure differences for smoke and gas removal.

Example

• In a narrowing pipe, fluid speed ( v ) increases, and static pressure ( P ) decreases.

Algèbre Linéaire

Espaces Vectoriels (Vector Spaces)

Definition

• Vector space 'E' is a set equipped with vector addition and scalar multiplication operations.
• Vector addition: (u, v) ∈ E × E → u + v ∈ E
• Scalar multiplication: (λ, u) ∈ 𝕂 × E → λ.u ∈ E, where 𝕂 is a field (usually ℝ or ℂ).

Axioms

• Associativity of addition: ∀ u, v, w ∈ E, (u + v) + w = u + (v + w)
• Commutativity of addition: ∀ u, v ∈ E, u + v = v + u
• Existence of additive identity: ∃ 0_E ∈ E, ∀ u ∈ E, u + 0_E = u
• Existence of additive inverse: ∀ u ∈ E, ∃ -u ∈ E, u + (-u) = 0_E
• Compatibility of scalar multiplication: ∀ λ, μ ∈ 𝕂, ∀ u ∈ E, λ. (μ. u) = (λ μ). u
• Identity element for scalar multiplication: ∀ u ∈ E, 1_𝕂. u = u
• Distributivity of scalar multiplication over vector addition: ∀ λ ∈ 𝕂, ∀ u, v ∈ E, λ. (u + v) = λ. u + λ. v
• Distributivity of scalar multiplication over field addition: ∀ λ, μ ∈ 𝕂, ∀ u ∈ E, (λ + μ). u = λ. u + μ. u

Examples

• ℝ^n: n-tuples of real numbers.
• ℂ^n: n-tuples of complex numbers.
• ℳ_{n,m}(ℝ): n × m matrices with real coefficients.
• ℝ[X]: Polynomials with real coefficients.
• 𝓕(X, ℝ): Functions from X to ℝ.

Sous-espaces vectoriels (Vector Subspaces)

Requirements

• A subset F of a vector space E is a subspace if:
  • F is non-empty.
  • ∀ u, v ∈ F, u + v ∈ F (stable under addition).
  • ∀ λ ∈ 𝕂, ∀ u ∈ F, λ.u ∈ F (stable under scalar multiplication).
• $F$ is a subspace of $E$ if and only if $F \neq \varnothing$ and $\forall u, v \in F, \forall \lambda \in \mathbb{K}, \lambda u + v \in F$

Combinaison linéaire (Linear Combination)

• Vectors u_1,..., u_n in E, a linear combination is a vector of the form:
• λ_1 u_1 +...+ λ_n u_n, where λ_1,..., λ_n ∈ 𝕂.

Sous-espace vectoriel engendré (Subspace Spanned)

• The subspace spanned by vectors u_1,..., u_n is the set of all their linear combinations:
• Vect(u_1,..., u_n) = {λ_1 u_1 +...+ λ_n u_n | λ_1,..., λ_n ∈ 𝕂}.
• This is the smallest subspace containing u_1,..., u_n.

Famille génératrice (Generating Set)

• Vectors (u_1,..., u_n) generate E if Vect(u_1,..., u_n) = E.
• Every vector in E can be written as a linear combination of u_1,..., u_n.

Famille libre (Linearly Independent Set)

• Vectors (u_1,..., u_n) are linearly independent if:
• λ_1 u_1 +...+ λ_n u_n = 0_E ⇒ λ_1 =... = λ_n = 0.
• No vector can be written as a linear combination of the others.

Base (Basis)

• A basis of E is a set that is both linearly independent and generating.

Dimension (Dimension)

• If E has a finite basis, any bases of E have the same number of elements. This is the dimension of E, denoted dim(E). If E does not have a finite basis, E is of infinite dimension.

Théorèmes importants (Important Theorems)

• Théorème de la base incomplète: Any linearly independent set can be extended to a basis.

Dans un espace de dimension finie n / In a space of dimension n

• A linearly independent set of n vectors is a basis.
• A generating set of n vectors is a basis.

Somme de sous-espaces vectoriels (Sum of Subspaces)

• For subspaces F and G of E, the sum is:
• $F + G = {u + v \mid u \in F, v \in G}$.
• F + G is a subspace of E.

Somme directe (Direct Sum)

• The sum F + G is direct if F ∩ G = {0_E}. This is written F ⊕ G.
• In this case, every vector of F + G can be uniquely written as a sum of a vector from F and a vector from G.

Espaces supplémentaires (Complementary Subspaces)

• Subspaces F and G are complementary in E if $E = F \oplus G$.
• Every vector of E can be uniquely written as a sum of a vector from F and a vector from G.

Formule de Grassmann (Grassmann's Formula)

• If E is finite-dimensional and F, G are subspaces of E, then:
• $dim(F + G) = dim(F) + dim(G) - dim(F \cap G)$.
• In particular, if F and G are complementary, $dim(E) = dim(F) + dim(G)$.

Applications Linéaires (Linear Transformations)

Définition (Definition)

• Given vector spaces E and F over the same field 𝕂, a map f : E → F is linear if:
• $\forall u, v \in E, f(u + v) = f(u) + f(v)$ -$\forall \lambda \in \mathbb{K}, \forall u \in E, f(\lambda u) = \lambda f(u)$
• Equivalently: f is linear if and only if $\forall u, v \in E, \forall \lambda \in \mathbb{K}, f(\lambda u + v) = \lambda f(u) + f(v)$.

Exemples (Examples)

• The zero map: $f(u) = 0_F, \forall u \in E$
• The identity: $f(u) = u, \forall u \in E$
• Differentiation: $f(P) = P'$, where P is a polynomial.
• Integration: $f(g)(x) = \int_a^x g(t) dt$, where g is a continuous function.
• Multiplication by a matrix: $f(x) = Ax$, where A is a matrix.

Noyau et Image (Kernel and Image)

• Let $f : E \rightarrow F$ be a linear map.
  • Noyau (Kernel): $Ker(f) = {u \in E \mid f(u) = 0_F}$. $Ker(f)$ is a subspace of $E$.
  • Image (Image): $Im(f) = {f(u) \mid u \in E}$. $Im(f)$ is a subspace of $F$.

Injectivité, Surjectivité, Bijectivité (Injectivity, Surjectivity, Bijectivity)

• $f$ is injective if $Ker(f) = {0_E}$.
• $f$ is surjective if $Im(f) = F$.
• $f$ is bijective if it is injective and surjective. In this case, the inverse $f^{-1}$ is also linear.

Théorème du rang (Rank-Nullity Theorem)

• If E is finite-dimensional and $f : E \rightarrow F$ is a linear map, then:
• $dim(E) = dim(Ker(f)) + dim(Im(f))$.
• $dim(Im(f))$ is called the rank of f, denoted $rg(f)$.

Isomorphisme (Isomorphism)

• An isomorphism is a bijective linear map. If $f : E \rightarrow F$ is an isomorphism, E and F are isomorphic.

Lab 4: Binary Classification, Model Selection

Introduction

• Explores binary classification using logistic regression.
• Focuses on model selection, specifically regularization (L1 and L2) with cross-validation.
• Dataset relates brain activity and visual stimulus.

Data

• stimuli.csv: Visual stimulus labels (0 or 1).
• fmri.csv: fMRI data, each row is a time point, each column a voxel.

Packages

• Includes NumPy, Pandas, Matplotlib, Seaborn, and scikit-learn modules.
• Utilizes LogisticRegression, cross_validate, KFold, roc_auc_score, roc_curve, and StandardScaler from scikit-learn.

Data Preparation

Load Data

• Loads stimuli.csv and fmri.csv into Pandas DataFrames.
• Verifies the shapes of the DataFrames.

Preprocessing

• Drops the first 4 rows from both DataFrames
• Standardizes fMRI data using StandardScaler.

Logistic Regression and Cross-Validation

Basic Logistic Regression

• Initializes a Logistic Regression model without regularization (penalty=None).
• Performs 5-fold cross-validation using cross_validate.
• Uses 'roc_auc' as the scoring metric.
• Examines cross-validation AUC scores and their mean.

L2 Regularization (Ridge)

• Defines a range of L2 regularization parameters using np.logspace.
• Iterates through the regularization parameters, training and evaluating a Logistic Regression model with L2 penalty for each.
• Stores the mean and standard deviation of the AUC scores for each penalty.

L1 Regularization (Lasso)

• Defines a range of L1 regularization parameters using np.logspace.
• Iterates through the regularization parameters, training and evaluating a Logistic Regression model with L1 penalty for each.
• Requires the 'liblinear' solver for L1 regularization.
• Stores the mean and standard deviation of the AUC scores for each penalty.

Visualization

Plotting

• Defines a function plot_regularization_path to plot the mean AUC scores and their standard deviations against the regularization parameters.
• Plots the L2 regularization path.
• Plots the L1 regularization path.

Best Model Selection

• Manual Selection (L2)
  • Manually selects the best L2 penalty based on the plot of the regularization path.
• K-Fold Cross-Validation (Nested CV for Robustness)
  • Defines a function nested_cross_validation to perform nested cross-validation for robust model selection.
  • Performs K-fold cross validation to find the best auc scores

Model Evaluation and Visualization

• ROC Curve (Best L2 Model)
  • Refits the best L2 model on the entire dataset.
  • Predicts probabilities for the test set.
  • Calculates and plots the ROC curve.
• ROC Curve (Best model)
  • Refits the best model on the entire dataset
  • Predict probabilities for the test set
  • Calculates and plots the ROC curve

Coefficient Analysis

Visualization

• Gets the coefficients from the best L2 model.
• Creates a bar plot of the coefficients.

Top Coefficients

• Finds the top N coefficients with the largest absolute values.
• Prints the indices and values of the top coefficients.

UNIDAD 5: INTEGRALES (INTEGRALS)

5.1. LA INTEGRAL INDEFINIDA (INDEFINITE INTEGRAL)

DEFINICIÓN (DEFINITION)

• A primitive (antiderivative) of a function (f(x)) is another function (F(x)) such that (F'(x) = f(x)).
• [ \int f(x) dx = F(x) + C \Leftrightarrow F'(x) = f(x) ]
• The indefinite integral of a function is the set of all its primitives represented by ( \int f(x) dx ).
• (f(x)) is the integrand.
• (dx) is the variable of integration.
• (C) is the constant of integration.
• (F(x)) is an antiderivative of (f(x)).

PROPIEDADES (PROPERTIES)

• ( \int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx )
• ( \int k \cdot f(x) dx = k \cdot \int f(x) dx )

INTEGRALES INMEDIATAS (IMMEDIATE INTEGRALS)

Integrals defined in a table format

• Table provides immediate integrals for various functions including constants, powers of x, reciprocals, exponential and trigonometric functions, along with their respective antiderivatives

EJEMPLOS (EXAMPLES)

• Integrals are explicitly computed for polynomial, rational, exponential, trigonometric, and square root functions, demonstrating direct application of immediate integral formulas.
• (\int (5x^3 - 3x^2 + x - 7) dx = \frac{5}{4} x^4 - x^3 + \frac{1}{2} x^2 - 7x + C)
• (\int (\frac{5}{x} + 3e^x) dx = 5 \ln|x| + 3e^x + C)
• (\int \frac{x^2 - x + 5}{x} dx = \frac{x^2}{2} - x + 5 \ln|x| + C)
• (\int (x - 5)^2 dx = \frac{x^3}{3} - 5x^2 + 25x + C)
• (\int \sqrt[3]{x^2} dx = \frac{3}{5} \sqrt[3]{x^5} + C)
• (\int \frac{3}{x^5} dx = \frac{-3}{4x^4} + C)
• (\int \frac{5}{x^2 + 1} dx = 5 \arctan x + C)
• (\int \frac{5}{\sqrt{1 - x^2}} dx = 5 \arcsin x + C)
• (\int 5 \cdot \sin x dx = -5 \cos x + C)
• (\int 3 \cdot \cos x dx = 3 \sin x + C)
• (\int 5 \cdot 2^x dx = 5 \frac{2^x}{\ln 2} + C)
• (\int \frac{7}{\cos^2 x} dx = 7 \tan x + C)

5.2. MÉTODOS DE INTEGRACIÓN (INTEGRATION METHODS)

5.2.1. MÉTODO DE SUSTITUCIÓN (SUBSTITUTION METHOD)

• Substitution method involves changing variables to simplify an integral.

[\int f(x) dx = \begin{cases} x = g(t) \ dx = g'(t) dt \end{cases} = \int f[g(t)] \cdot g'(t) dt = F(t) + C = F[g^{-1}(x)] + C]

EJEMPLOS (EXAMPLES)

• Examples provided show using substitution to solve more complex integrals by simplifying the integrand.
• (\int (x^2 + 5)^3 \cdot 2x dx = \frac{(x^2 + 5)^4}{4} + C)
• (\int \frac{2x}{x^2 + 1} dx = \ln|x^2 + 1| + C)
• (\int e^{x^2 + 1} \cdot x dx = \frac{1}{2} e^{x^2 + 1} + C)
• (\int \frac{1}{x \cdot \ln x} dx = \ln|\ln x| + C)
• (\int \tan x dx = -\ln|\cos x| + C)

5.2.2. INTEGRACIÓN POR PARTES (INTEGRATION BY PARTS)

• Integration by parts is used when the integrand is a product of two functions.

[ \int u dv = u \cdot v - \int v du ]

• Proper selection of (u) and (dv) simplifies the integral.
• "alpes" mnemonic helps choose (u):
• (A \rightarrow \arcsin, \arccos, \arctan)
• (L \rightarrow logarithmic)
• (P \rightarrow polynomial)
• (E \rightarrow exponential)
• (S \rightarrow sine, cosine)

EJEMPLOS (EXAMPLES)

• Examples demonstrate using integration by parts to effectively solve integrals by breaking them into manageable parts.
• (\int x \cdot e^x dx = e^x (x - 1) + C)
• (\int x \cdot \sin x dx = -x \cos x + \sin x + C)
• (\int \ln x dx = x(\ln x - 1) + C)
• (\int \arctan x dx = x \arctan x - \frac{1}{2} \ln|1 + x^2| + C)

5.2.3. INTEGRALES RACIONALES (RATIONAL INTEGRALS)

• Integrals that involve integrating rational functions, where the integrand is a ratio of two polynomials ( \int \frac{P(x)}{Q(x)} dx ).
• If ( \text{degree}(P(x)) \ge \text{degree}(Q(x)) ), divide (P(x)) by (Q(x)):

[ \frac{P(x)}{Q(x)} = C(x) + \frac{R(x)}{Q(x)} ]

• If ( \text{degree}(P(x)) < \text{degree}(Q(x)) ), factorize (Q(x)) and decompose the fraction into simpler fractions.

CASOS (CASES)

CASO 1: RAÍCES REALES SIMPLES (SIMPLE REAL ROOTS)

[ \text{If } Q(x) = (x - a_1)(x - a_2) \cdots (x - a_n), \text{ then: } ]

[ \frac{P(x)}{Q(x)} = \frac{A_1}{x - a_1} + \frac{A_2}{x - a_2} + \cdots + \frac{A_n}{x - a_n} ]

CASO 2: RAÍCES REALES MÚLTIPLES (MULTIPLE REAL ROOTS)

If ( Q(x) = (x - a)^n ), then:

[ \frac{P(x)}{Q(x)} = \frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_n}{(x - a)^n} ]

CASO 3: RAÍCES COMPLEJAS SIMPLES (SIMPLE COMPLEX ROOTS)

If ( Q(x) = (x^2 + bx + c) ), where ( x^2 + bx + c = 0 ) has no real roots, then:

[ \frac{P(x)}{Q(x)} = \frac{Ax + B}{x^2 + bx + c} ]

CASO 4: RAÍCES COMPLEJAS MÚLTIPLES (MULTIPLE COMPLEX ROOTS)

If ( Q(x) = (x^2 + bx + c)^n ), where ( x^2 + bx + c = 0 ) has no real roots, then:

[ \frac{P(x)}{Q(x)} = \frac{A_1 x + B_1}{x^2 + bx + c} + \frac{A_2 x + B_2}{(x^2 + bx + c)^2} + \cdots + \frac{A_n x + B_n}{(x^2 + bx + c)^n} ]

EJEMPLOS (EXAMPLES)

Illustrates the method of partial fractions to solve rational integrals

• (\int \frac{x + 1}{x^2 + 5x + 6} dx = -\ln|x + 2| + 2 \ln|x + 3| + C)
• (\int \frac{x^2 + 1}{x^2 - 1} dx = x + \ln|x - 1| - \ln|x + 1| + C)

5.3. LA INTEGRAL DEFINIDA (DEFINITE INTEGRAL)

DEFINICIÓN (DEFINITION)

• If (f(x)) is continuous on ( [a, b] ), the definite integral of (f(x)) from (a) to (b) is:
• [\int_{a}^{b} f(x) dx = [F(x)]_{a}^{b} = F(b) - F(a)]
• (F(x)) is any antiderivative of (f(x)), where (F'(x) = f(x)).
• (a) is the lower limit of integration.
• (b) is the upper limit of integration

PROPIEDADES (PROPERTIES)

• (\int_{a}^{a} f(x) dx = 0)
• (\int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx)
• (\int_{a}^{b} k \cdot f(x) dx = k \cdot \int_{a}^{b} f(x) dx)
• (\int_{a}^{b} [f(x) + g(x)] dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx)
• (\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx), where ( a < c < b )

TEOREMA FUNDAMENTAL DEL CÁLCULO INTEGRAL (FUNDAMENTAL THEOREM OF CALCULUS)

• If (f(x)) is continuous on ( [a, b] ) and (F(x) = \int_{a}^{x} f(t) dt), then (F'(x) = f(x)).

REGLA DE BARROW (BARROW'S RULE)

If (f(x)) is continuous on ( [a, b] ) and (F(x)) is an antiderivative of (f(x)), then:

[ \int_{a}^{b} f(x) dx = F(b) - F(a) ]

EJEMPLOS (EXAMPLES)

Calculates definite integrals using the evaluated antiderivative at the integration limits.

• (\int_{1}^{2} x^2 dx = \frac{7}{3})
• (\int_{0}^{\pi} \sin x dx = 2)
• (\int_{1}^{e} \frac{1}{x} dx = 1)
• (\int_{0}^{1} e^x dx = e - 1)

5.4. APLICACIONES DE LA INTEGRAL DEFINIDA (APPLICATIONS OF THE DEFINITE INTEGRAL)

5.4.1. CÁLCULO DE ÁREAS (AREA CALCULATION)

Calculates areas using definite integrals

ÁREA BAJO UNA CURVA (AREA UNDER A CURVE)

• The area bounded by the curve ( y = f(x) ), the x-axis, and the lines ( x = a ) and ( x = b ) is:

[ A = \int_{a}^{b} |f(x)| dx ]

• If ( f(x) \ge 0 ) on ( [a, b] ), then ( A = \int_{a}^{b} f(x) dx )
• If ( f(x) \le 0 ) on ( [a, b] ), then ( A = - \int_{a}^{b} f(x) dx )

ÁREA ENTRE DOS CURVAS (AREA BETWEEN TWO CURVES)

• The area bounded by the curves ( y = f(x) ) and ( y = g(x) ) and the lines ( x = a ) and ( x = b ) is:

[ A = \int_{a}^{b} |f(x) - g(x)| dx ]

• To calculate the area:
  • Find the intersections between the curves ( f(x) ) and ( g(x) )
• Divide the interval ( [a, b] ) into subintervals where ( f(x) \ge g(x) ) or ( f(x) \le g(x) )
• Calculate the definite integral in each subinterval and sum the absolute values of the integrals

EJEMPLOS (EXAMPLES)

Illustrates how to compute areas under curves and between curves using definite integration

• Calculate the area bounded by the curve ( y = x^2 ), the x-axis, and the lines ( x = 1 ) and ( x = 3 )
• ( A = \frac{26}{3} )
• Calculate the area bounded by curves ( y = x^2 ) and ( y = x )
  • ( A = \frac{1}{6} )

5.4.2. CÁLCULO DE VOLÚMENES (VOLUME CALCULATION)

Calculates volumes using definite integrals

VOLUMEN DE UN CUERPO DE REVOLUCIÓN (VOLUME OF A SOLID OF REVOLUTION)

• Revolving a function ( f(x) ) around the x-axis from ( x = a ) to ( x = b ) results in a solid of revolution with volume:

[ V = \pi \int_{a}^{b} [f(x)]^2 dx ]

EJEMPLOS (EXAMPLES)

• Calculates the volume of a solid formed by rotating a function around the x-axis
• Calculate the volume of the solid of revolution obtained by rotating the function ( y = \sqrt{x} ) around the x-axis between ( x = 0 ) and ( x = 1 )
• ( V = \frac{\pi}{2} )