Bernoulli's Principle and Airfoils

Bernoulli's Principle

• Developed by Daniel Bernoulli in the 18th century.
• For inviscid flow of a nonconducting fluid, an increase in fluid speed occurs simultaneously with a decrease in pressure or potential energy.

How Airplanes Fly

• Air flow faster over the wing's top leads to lower pressure, generating lift.
• Lift counteracts the plane's weight, enabling flight.

Airfoil

• Airfoil describes a wing's cross-sectional shape, designed to manipulate airflow for lift.

Pressure Distribution

• The pressure above the wing is less than the pressure below the wing.

Velocity

• Speed of air above the wing exceeds that beneath it.

Chapter 13: Competitive Markets for Factors of Production

13.1 Factor Markets

• Factors of production are demanded through Derived Demand, the demand for the factor is dependent on the company's output level and input costs.
  • Steel as auto input influences its demand.

Independent Decisions

• Firms decide on production quantity and method.

Demand for Factors When Only One Input is Variable

• Assumptions: Labor is the only variable input and a firm's output (Q) depends only on the amount of Labor (L) hired.
  • Q = F(L)

Marginal Revenue Product

• Marginal Product of Labor (MPL): Additional output produced when 1 additional unit of labor is employed.

$$ MP_L = \frac{\Delta Q}{\Delta L} $$

• Marginal Revenue (MR): Additional revenue from producing one more unit of output.

$$ MR = \frac{\Delta TR}{\Delta Q} $$

• Marginal Revenue Product (MRP): Additional revenue from employing one more unit of labor.

$$ MRP_L = MPL \times MR $$

• If the firm is in a competitive market:

$$ MRP_L = MPL \times P $$

Rule for Employing Labor

• Profitable to hire labor as long as the MRP exceeds the wage rate.
• Hire labor until MRP equals the wage rate.

$$ MRP_L = w $$

MRP and the Demand for Labor

• A firm's labor demand mirrors the MRP curve.
• Curve slopes down as MPL decreases with more labor.

13.2 Competitive Factor Markets

• Market demand comes from all firms' labor demand curves
  • It has a downward slope.
  • MPL declines as more labor is employed.
  • As wage rate falls, companies produce more, which leads to a lower product price.

The Market Supply Curve

• Labor supply slopes upwards.
• Higher wages can result in more labour.
• Market wage determined by labor supply/demand intersection.

Equilibrium in a Competitive Labor Market

• Equilibrium is the point where labor supply matches labor demand.
• All firms pay the same wage rate.
• Individual firms will hire until $MRP_L = w$.

13.3 Economic Rent

• Economic Rent: Payment made to a production factor minus the minimum needed to secure its use.

Example

• Football player ready to perform for $50,000 annually paid $1,000,000.
  • Economic rent = $950,000

Perfectly Inelastic Supply

• Payment is wholly economic rent when a factor's supply is perfectly inelastic.
• Example: Land.

13.4 Factor Shares and Labor Productivity

• National income encompasses payments to labor and capital.
  • Wages, Rent, Interest, and Profit

Allocation of National Income

• Depends on the the supply and the demand of each factor.

The Division of National Income between Labor and Capital

• Assumptions:
  • Output is produced with only capital (K) and labor (L).
  • Markets are competitive.
  • The economy is efficient.
• Under these assumptions, each factor is paid its marginal product.

$$ w = MP_L \ r = MP_K $$

• Where:
  • w = wage rate
  • r = rental rate of capital

Example

• Output (Q) is given by:

$$ Q = AK^\alpha L^{1-\alpha} $$

• Where:
  • A is a constant
  • $\alpha$ is a constant between 0 and 1
• The wage rate (w) is:

$$ w = (1 - \alpha) A K^\alpha L^{-\alpha} $$

• Total wages paid to labor (wL) is:

$$ wL = (1 - \alpha) A K^\alpha L^{1-\alpha} = (1 - \alpha)Q $$

• The total return to capital (rK) is:

$$ rK = \alpha Q $$

• The fraction of output paid to labor is $1 - \alpha$
• The fraction of output paid to capital is $\alpha$

Labor Productivity

• Labor productivity is the average product of labor:

$$ \frac{Q}{L} $$

• If capital grows faster than labor, labor productivity will increase.
• As labor becomes more productive, wages will rise.

Lecture 16 - Sections 4.1, 4.3

Review

• Theorem: Suppose $f$ is continuous on $[a,b]$.
  1. If $F(x) = \int_{a}^{x} f(t)dt$, then $F'(x) = f(x)$.
  2. $\int_{a}^{b} f(x)dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$, that is $F' = f$.

Indefinite Integrals

• Indefinite Integrals $\int f(x)dx = F(x)$ means $F'(x) = f(x)$.
• The general indefinite integral is, $\int f(x)dx = F(x) + C$ where $C$ is an arbitrary constant.
• Example:*

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$

$\int \cos(x) dx = \sin(x) + C$

$\int \sin(x) dx = -\cos(x) + C$

$\int \sec^2(x) dx = \tan(x) + C$

$\int \frac{1}{x} dx = \ln |x| + C$

Table of Indefinite Integrals

• Example:* Find the general indefinite integral

$\int (10x^4 - 2\sec^2(x))dx = 10 \int x^4 dx - 2 \int \sec^2(x) dx = 10 \frac{x^5}{5} - 2\tan(x) + C = 2x^5 - 2\tan(x) + C$

• Example:* Evaluate $\int_{0}^{3} (x^3 - 6x)dx$

$\int_{0}^{3} (x^3 - 6x)dx = [\frac{x^4}{4} - \frac{6x^2}{2}]{0}^{3} = [\frac{x^4}{4} - 3x^2]{0}^{3} = (\frac{81}{4} - 27) - (0 - 0) = \frac{81 - 108}{4} = -\frac{27}{4}$

• Example:* Find $\int_{1}^{9} \frac{2x^2 + x^2\sqrt{x} - 1}{x^2} dx$

$\int_{1}^{9} (\frac{2x^2}{x^2} + \frac{x^2\sqrt{x}}{x^2} - \frac{1}{x^2})dx = \int_{1}^{9} (2 + \sqrt{x} - \frac{1}{x^2})dx = [2x + \frac{2}{3}x^{3/2} + \frac{1}{x}]_{1}^{9} = (18 + \frac{2}{3}(27) + \frac{1}{9}) - (2 + \frac{2}{3} + 1) = 18 + 18 + \frac{1}{9} - 3 - \frac{2}{3} = 33 + \frac{1}{9} - \frac{6}{9} = 33 - \frac{5}{9} = \frac{297 - 5}{9} = \frac{292}{9}$

• Net Change Theorem:* The integral of a rate of change is the net change

$\int_{a}^{b} F'(x)dx = F(b) - F(a)$

• Example:* A particle moves along a line so that its velocity at time $t$ is $v(t) = t^2 - t - 6$ (in meters per second). Find the displacement of the particle during the time period $1 \le t \le 4$.

Displacement = $\int_{1}^{4} (t^2 - t - 6)dt = [\frac{t^3}{3} - \frac{t^2}{2} - 6t]_{1}^{4} = (\frac{64}{3} - \frac{16}{2} - 24) - (\frac{1}{3} - \frac{1}{2} - 6) = \frac{63}{3} - 8 - 24 + \frac{1}{2} + 6 = 21 - 32 + \frac{1}{2} = -11 + \frac{1}{2} = -\frac{21}{2}$.

So the displacement is $-\frac{21}{2}$ m. Now find the distance traveled during this time period.

Distance = $\int_{1}^{4} |t^2 - t - 6|dt$

$t^2 - t - 6 = (t - 3)(t + 2)$ $t^2 - t - 6 = 0$ when $t = 3$ and $t = -2$. So $t^2 - t - 6 < 0$ on $(-2, 3)$ and $t^2 - t - 6 > 0$ when $t > 3$ or $t < -2$.

$\int_{1}^{4} |t^2 - t - 6|dt = \int_{1}^{3} -(t^2 - t - 6)dt + \int_{3}^{4} (t^2 - t - 6)dt = [-\frac{t^3}{3} + \frac{t^2}{2} + 6t]{1}^{3} + [\frac{t^3}{3} - \frac{t^2}{2} - 6t]{3}^{4} = (-\frac{27}{3} + \frac{9}{2} + 18) - (-\frac{1}{3} + \frac{1}{2} + 6) + (\frac{64}{3} - \frac{16}{2} - 24) - (\frac{27}{3} - \frac{9}{2} - 18) = -9 + \frac{9}{2} + 18 + \frac{1}{3} - \frac{1}{2} - 6 + \frac{64}{3} - 8 - 24 - 9 + \frac{9}{2} + 18 = 9 + \frac{8}{2} - 6 + \frac{65}{3} - 32 - 9 + 18 = 13 - 6 + \frac{65}{3} - 23 = -16 + \frac{65}{3} = \frac{-48 + 65}{3} = \frac{17}{3}$.

So the distance traveled is $\frac{17}{3} m$.

• Example:* A honeybee population starts with 100 bees and increases at a rate of $n'(t)$ bees per week. What does $100 + \int_{0}^{15} n'(t)dt$ represent? It represents the number of bees after 15 weeks.

Introduction to CFD

• Numerical prediction of fluid flow, heat transfer, mass transfer, chemical reactions, and related phenomena.

How CFD Works

• Discretizing domain into a grid.
• Solve governing equations (Navier-Stokes, energy, species, etc.) at each grid point.

Steps in a CFD Simulation

1. Problem Statement: Define the problem and goals.
2. Geometry: Create the computational domain.
3. Mesh Generation: Divide the domain into discrete cells.
4. Physics: Define physical phenomena, boundary conditions, and material properties.
5. Solving: Solve the equations numerically.
6. Post-processing: Analyze and visualize the results.
7. Validation: Compare the results to experimental data or other reliable sources.

Governing Equations

• Navier-Stokes equations describe conservation of mass, momentum, and energy.
  • Continuity Equation (Mass conservation):

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \overrightarrow{v}) = 0$$

- **Momentum Equation**:

$$\frac{\partial (\rho \overrightarrow{v})}{\partial t} + \nabla \cdot (\rho \overrightarrow{v} \overrightarrow{v}) = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \overrightarrow{g} + \overrightarrow{F}$$

• Energy Equation:

$$\frac{\partial (\rho h)}{\partial t} + \nabla \cdot (\rho \overrightarrow{v}h) = \nabla \cdot (k\nabla T) + S_{h}$$

- Where:
    -   $\rho$ = Density
    -   $\overrightarrow{v}$ = Velocity vector
    -   $p$ = Pressure
    -   $\overline{\overline{\tau}}$ = Viscous stress tensor
    -   $\overrightarrow{g}$ = Gravitational acceleration
    -   $\overrightarrow{F}$ = External body forces
    -   $h$ = Enthalpy
    -   $k$ = Thermal conductivity
    -   $T$ = Temperature
    -   $S_{h}$ = Source term for energy

Turbulence Modeling

• Approximates the effects of turbulence on the mean flow.

Common Turbulence Models

• Spalart-Allmaras
• k-ε (k-epsilon)
• k-ω (k-omega)
• Reynolds Stress Model (RSM)
• Large Eddy Simulation (LES)
• Detached Eddy Simulation (DES)

Boundary Conditions

• Flow variables on the borders of the computational area are defined through boundary conditions.

Types of Boundary Conditions

• Inlet: Specifies the flow properties entering the domain (e.g., velocity, pressure, temperature).
• Outlet: Specifies the flow properties leaving the domain (e.g., pressure, outflow).
• Wall: Specifies the conditions on solid surfaces (e.g., no-slip, slip, wall function).
• Symmetry: Specifies symmetry conditions for symmetric geometries.
• Periodic: Specifies periodic conditions for repeating geometries.

Meshing

• Meshing is the process of dividing the computational domain into discrete cells or elements.
• The accuracy and stability of the CFD solution depend on the quality of the mesh.

Types of Meshes

• Structured Mesh: Regular grid with hexahedral or quadrilateral elements.
• Unstructured Mesh: Irregular grid with tetrahedral, triangular, or polyhedral elements.
• Hybrid Mesh: Combination of structured and unstructured meshes.

Mesh Quality Metrics

• Skewness: Measures the deviation of a cell from an ideal shape.
• Aspect Ratio: Ratio of the longest to shortest side of a cell.
• Orthogonality: Measures the angle between cell faces and cell centers.

Numerical Methods

• Solving discretized governing equations.

Common Numerical Methods

• Finite Difference Method (FDM): Approximates derivatives using difference quotients.
• Finite Volume Method (FVM): Integrates the governing equations over control volumes.
• Finite Element Method (FEM): Divides the domain into finite elements and approximates the solution using basis functions.

Applications of CFD

• Aerospace: Aircraft design, aerodynamics
• Automotive: Vehicle design, engine simulation
• Chemical Engineering: Reactor design, mixing
• Civil Engineering: Bridge design, wind loading
• Mechanical Engineering: Heat exchangers, turbomachinery
• Environmental Engineering: Pollution dispersion, weather forecasting
• Biomedical Engineering: Blood flow simulation, drug delivery

Algorithmic Complexity

What?

• Measures the efficiency of an algorithm.
• Describes how the number of steps grows as the input size grows.
• The focus is typically on large input sizes ($n \rightarrow \infty$).

Why?

• Used to compare different algorithms.
• Helps understand bottlenecks.
• Predicts performance.

How?

• Count operations as a function of input size $n$
• Express the result using Big-O notation, $O(f(n))$, where $f(n)$ represents the function of $n$.

Common complexities

• $O(1)$ - constant time
• $O(log n)$ - logarithmic time
• $O(n)$ - linear time
• $O(n log n)$ - "linearithmic"
• $O(n^2)$ - quadratic time
• $O(2^n)$ - exponential time

Example

• Consider searching for a value in an array.
• n represents the size of the array.

Linear Search

• Examines each element one by one until the target value is found.
• In the worst-case scenario, every element is checked.
• Complexity: $O(n)$

Binary Search

• Requires a sorted array, begins by examining the middle element.
• If the target value is less than the middle element, the search continues in the left half.
• Otherwise, the search occurs in the right half.
• Each step halves the search space.
• Complexity: $O(log n)$

Visual comparison

• The time or space requirements grow with input size for different algorithmic complexities.

The Foreign Exchange Market

Exchange Rate

• Price of one currency in another.

• Two expressions:

  • Units home currency per foreign unit (e.g., $1.50/€).
  • Units foreign currency per home unit (e.g., €0.67/$).

• Rates are reciprocals.

  $$\frac{1}{1.50} = €0.67/$$

Exchange Rate Regimes

• Floating exchange rate regime: Supply and demand decide rate.
• Fixed exchange rate regime: Government sets rate.
• Pegged exchange rate regime: Rate fixed against another currency/basket.
• Managed floating exchange rate regime: Rate fluctuates within limits.

Spot and Forward Exchange Rates

• Spot exchange rate: Rate for immediate currency delivery.
• Forward exchange rate: Rate for future currency delivery.
  • usually 30, 90, or 180 days forward.
• Forward premium/discount: Difference between forward/spot rates.

$$\frac{F-S}{S} * \frac{360}{n}$$

-   Where:
    -   F = forward exchange rate
    -   S = spot exchange rate
    -   n = number of days in the forward period

Example of Forward Premium/Discount

• Spot: $1.50/€; 90-day forward: $1.52/€. Calculate annualized forward premium/discount.

$$\frac{1.52-1.50}{1.50} * \frac{360}{90} = 0.0533 = 5.33%$$

• The euro is trading at an annualized forward premium of 5.33% relative to the U.S. dollar.

IFT 3913: Théorie de l'information (Hiver 2024, Notes #1)

Plan

1. Basic definitions and motivation
2. Entropy
3. Mutual Information
4. Kullback-Leibler (KL) Divergence
5. Fano's Inequality

1. Motivation et définitions de base

Objective:

• Quantify information.
  • Primary focus will be on probabilistic information tied to random variables.

Exemples d'applications

• Compression of data (e.g., Huffman's algorithm).
• Learning theory (e.g., information bottleneck).
• Cryptography.

Basic définitions

• Variable aléatoire (v. a.): A variable whose value is the numerical result of an unpredictable event.

  • $X \in {\text{heads}, \text{tails}}$ (Bernoulli)
  • $Y \in {1, 2, 3, 4, 5, 6}$ (6-sided die).

• Distribution de probabilité: Probability of observing each possible value of a v.a.

  • $P(X=x)$ is the probability that $X$ takes the value $x$.
    • Must satisfy $0 \leq P(X=x) \leq 1$ and $\sum_x P(X=x) = 1$ for discrete variables.
    • $\int_x P(X=x) dx = 1$ for continuous variables.
  • $P_X(x)$ can be written for the distribution of $X$.
  • Often $P(x)$ is written when context is clear.

Basic Definitions (Continued)

• Support of $P_X(x)$: Set of $x$ values for which $P_X(x) > 0$.
• Discrete v.a.: Support is countable.
• Continuous v.a.: Support is an interval in $\mathbb{R}$.
• Joint distribution: Likelihood of observing combinations of values from many v.a.
  • $P(X=x, Y=y)$ is the probability $X$ is $x$ and $Y$ is $y$.
  • $P_{X,Y}(x, y)$ can be used for the joint distribution.
• Marginal distribution: The probability of observing the possible values from one v.a., while ignoring the others.
  • Ex: $P(X=x) = \sum_y P(X=x, Y=y)$ (if $Y$ is discrete).
  • "Marginal" because of sum/integration.

Basic Definitions (Continued)

• Conditional distribution: The probability of watching one value of a v.a. given with the value of a secondary v.a.

  • $P(X=x \mid Y=y)$ is the prob. of $X$ being $x$ given that $Y$ is $y$.
  • $P_{X \mid Y}(x \mid y)$ can be used to emphasize that this is a conditional distribution.
  • Defined as $P(X=x \mid Y=y) = \frac{P(X=x, Y=y)}{P(Y=y)}$ if $P(Y=y) > 0$.
  • Thus, $P(X=x, Y=y) = P(X=x \mid Y=y) P(Y=y)$

• Independence: Two v.a. ($X$ and $Y$) are independent if their joint distribution is the product of their marginal distributions: $P(X=x, Y=y) = P(X=x) P(Y=y)$.

  • $X \perp Y$ represents independence of $X$ and $Y$.
  • Equivalent to $P(X=x \mid Y=y) = P(X=x)$.

2. Entropy

Definition:

• Associated with the quantity of uncertainty of a v.a.
  • For a discrete v.a. $X$, its entropy $H(X)$ is: $H(X) = - \sum_x P(X=x) \log_2 P(X=x)$
  • Base-2 log gives units in bits.
  • Also expressable as: $H(X) = - \mathbb{E}[\log_2 P(X=x)]$
    • $\mathbb{E}$ represents the mean with respect to $X$'s distribution.
  • $0 \log 0 = 0$ by convention.

Example 1

• A fair coin flip results in a variable that it is in {\text{head}, \text{tails}}, $P(X=\text{head}) = P(X=\text{tail}) = \frac{1}{2}$, hence its entropy is $H(X) = - \frac{1}{2} \log_2 \frac{1}{2} - \frac{1}{2} \log_2 \frac{1}{2} = - \log_2 \frac{1}{2} = 1 bit$.
• Non-symetric case gives P(X=pile) = .9 and similar equations lead to $H(X) = - 0.9 \log_2 0.9 - 0.1 \log_2 0.1 \approx 0.469 bits$, showing less uncertainty,

Example 2

• Die with six side gives entropy: $H(X) = - \sum_{i=1}^6 \frac{1}{6} \log_2 \frac{1}{6} = - \log_2 \frac{1}{6} = \log_2 6 \approx 2.585 bits$

Joint Entropy

• Discrete v.a. $X$ and $Y$'s joint entropy is $H(X, Y) = - \sum_x \sum_y P(X=x, Y=y) \log_2 P(X=x, Y=y)$.
  • The quantity of uncertainty of the pari (X,Y) is equal to the joint entropy.

Conditional entropy

$H(X \mid Y) = - \sum_x \sum_y P(X=x, Y=y) \log_2 P(X=x \mid Y=y)$, where,

expressing Y as the anticipation of X given Y.

Chain Rule for Entropy

• $H(X, Y) = H(X) + H(Y \mid X) = H(Y) + H(X \mid Y)$ is a chain entropy equation.
• Chain equations can be generalized to n variables.

Properties of Entropy

1. $H(X) \geq 0$.
2. $H(X) \leq \log_2 |\mathcal{X}|$, where $\mathcal{X}$ is the support of $X$.
   • Equality holds iff $P(X=x) = \frac{1}{|\mathcal{X}|}$. The entropy is also maximum in this position
3. $H(X, Y) \geq H(X)$ and $H(X, Y) \geq H(Y)$.
   • Uncertainty of $(X, Y)$ is at least Uncertainty of $X$ or $Y$.
4. $H(X \mid Y) \leq H(X)$.
   • Uncertainty of $X$ cannot become greator the Uncertainty of Y knowing X. -Equality indicates independance between X and Y.

3. Mutual Information

Definition:

• How much one v.a. says about another.

Its formulation makes use of the expected value to give a double log for X and Y.

• It can also be written as $I(X; Y) = H(X) - H(X \mid Y) = H(Y) - H(Y \mid X)$.
  • Showing syymetry between each variables with I(X; Y) = I(Y; X) and also independent variable with.
• Can be written in terms of $H(X) + H(Y) - H(X, Y)$.

Venn diagram

• Shows mutual information as intersected portions of variables.

4. Kullback-Leibler Divergence

Definition:

• Similarity measure between probability distributions.
  • KL, $D_{KL}(P||Q) = \sum_x P(x) \log_2 \frac{P(x)}{Q(x)}$
• It can be read to say how much aproximation affects a distribution.
• Is not a distance as mathematical as is in not symetrical and does not satisfy triangle form.
• Greater or equal than $0$ due to gibbs form. The equality holds in cases of symetry.
• Support of $P$ included in $Q$ is nessesary.

Connect it and mutual information

• $I(X; Y) = D_{KL}(P(X, Y) || P(X) P(Y))$. It also shows how independent is a variable based on their mutual information.

5. Fano's Inequality

Statement:

• Low bound to error cases.
• $X$: Variable to predict
• Observation that X have, Variable $Y$,
• A $Y$ gives a best prediction $X$, hence $P(\hat{X} \neq X)$.

Stating:

$H(X \mid Y) \leq H(P_e) + P_e \log_2 |\mathcal{X}|$

• Can be used to found a lower bound betwen learning algoritm.