Lorentz Transformation

Lecture 22: The Lorentz Transformation

• Inertial Reference Frame (IRF) is one where Newton's First Law applies.
• A spaceship at constant velocity relative to a distant star exemplifies an IRF.
• An accelerating is not an example of an IRF.
• Frame S' moves at velocity ( v ) in the ( x ) direction relative to frame S, with origins overlapping at ( t=0 ).
• The Galilean Transformation inaccurately relates coordinates between moving reference frames.
• Galilean Transformation fails to preserve the speed of light.
• The Galilean Transformation does not reflect reality, and Maxwell's equations are not invariant under it.

The Lorentz Transformation

• Asserts that physics laws are identical in all inertial reference frames.
• Postulates a constant vacuum speed of light for every observer, no matter the light source's motion.
• Lorentz Transformation equations relate space and time coordinates between two inertial reference frames.
• ( \gamma ) approaches 1 at low speeds, turning the Lorentz transformation into the Galilean transformation.
• The Lorentz transformation upholds the speed of light.
• This transformation holds for speeds ( v < c ).

Inverse Lorentz Transformation

• Used to determine original coordinates ( (x,y,z,t) ) from transformed coordinates ( (x',y',z',t') ).
• Switch primed and unprimed coordinates and replace ( v ) with ( -v ) to derive the inverse Lorentz transformation.

Consequences

• Length contraction describes how an object appears shorter when moving versus at rest.
• The equation ( L = \frac{L_0}{\gamma} ) shows the relationship between observed length and proper length.
• Time dilation explains that moving clocks tick slower relative to stationary ones.
• Use the equation ( \Delta t = \gamma \Delta t_0 ) to relate time intervals in different reference frames.

Algorithmic Game Theory

• It is the study of multi-agent decision problems.
• Scenarios include auctions, network formation, selfish routing, social networks, elections.

Selfish Routing

• A network of roads where cost depends on congestion.
• Drivers are players aiming for the quickest route.
• Strategies involve choosing the path with the least cost.
• The problem is understanding the dynamics when all drivers choose routes selfishly.

Key Questions in Algorithmic Game Theory

• Does the system stabilize into an equilibrium?
• How does equilibrium compare to a centrally planned solution?

Congestion Games

• Defined by players, resources, and cost functions for each resource.
• Players are represented by ( N={1,..,n} ).
• The set of resources is represented by ( E ).
• ( \mathcal{S}_i \subseteq 2^E ) is the set of strategies for each player ( i ).
• ( c_e : \mathbb{N} \rightarrow \mathbb{R} ) is a cost function for resource ( e ), setting costs based on usage.

Strategy Profile

• Comprises a vector ( S=(S_1,..,S_n) ) where each ( S_i ) is a strategy for player ( i ).
• ( n_e(S)=|{i : e \in S_i}| ) indicates players using resource ( e ) in profile ( S ).
• Player i's cost is ( C_i(S)=\sum_{e \in S_i} c_e(n_e(S)) ).

Nash Equilibrium

• Exists when no player benefits from changing strategies alone.
• For Nash equilibrium, for all ( i ) and ( S'_i \in \mathcal{S}_i ), ( C_i(S) \le C_i(S'i, S{-i}) ) which means no unilateral deviation lowers cost.
• Denotes other players' strategies as ( (S'i, S{-i}) = (S_1,..,S_{i-1}, S'i, S{i+1},..,S_n) ).
• Every congestion game has a Nash equilibrium.

Proof of Nash Equilibrium

• A potential function ( \Phi(S) = \sum_{e \in E} \sum_{j=1}^{n_e(S)} c_e(j) ) is defined.
• When player ( i ) switches to ( S'_i ), cost changes and ( C_i(S) - C_i(S'i, S{-i}) ) equals the change in the potential function ( \Phi(S) - \Phi(S'i, S{-i}) ).
• The game reaches Nash equilibrium as the number of profiles is finite and the potential function decreases with beneficial deviations.

Pigou's Game

• Illustrates traffic routing between start S and target T with varying costs.
• In Nash equilibrium, all traffic uses the lower edge because it's perceived as the lowest cost route.
• Social optimum involves minimizing total cost by optimizing traffic distribution.
• It's determined by minimizing ( C(x) = x^2 - x + 1 ), leading to ( x = \frac{1}{2} ).

Price of Anarchy (PoA)

• A measurement of social welfare loss due to selfish players.
• The formula to calculate this is (PoA = \frac{\text{Cost of worst Nash equilibrium}}{\text{Cost of social optimum}}).

Braess's Paradox

• Even adding roads can worsen network efficiency.
• The Nash equilibrium in the original network involves traffic splitting evenly.
• Adding a zero-cost road changes the Nash equilibrium where all traffic uses the new, worsened route.

Fonction Logarithme Népérien

• The natural logarithm function, written as ln, is defined on ( ]0; +\infty[ ).
• The natural logarithm is the antiderivative of ( x \mapsto \frac{1}{x} ) that is zero at 1.
• ln is differentiable on ( ]0; +\infty[ ) and for all ( x > 0 ), ( ln'(x) = \frac{1}{x} ).
• ln is continuous on ( ]0; +\infty[ ).
• ( ln(1) = 0 ) and ( ln(e) = 1 ).

Algebraic Properties

• For all strictly positive real numbers a and b, and for any relative integer n, the following formulas are applicable.
• ( ln(ab) = ln(a) + ln(b) )
• ( ln(\frac{1}{a}) = -ln(a) )
• ( ln(\frac{a}{b}) = ln(a) - ln(b) )
• ( ln(a^n) = n \cdot ln(a) )
• ( ln(\sqrt{a}) = \frac{1}{2}ln(a) )

Natural Logarithm Study

• The ln function is strictly increasing on ( ]0; +\infty[ ).
• For all strictly positive real numbers a and b, ( ln(a) = ln(b) \Leftrightarrow a = b ).
• For all strictly positive real numbers a and b, ( ln(a) < ln(b) \Leftrightarrow a < b ).

Estadística Descriptiva

• Is a branch of statistics that focuses on summarizing and presenting data informatively.
• Aims to describe the key characteristics of a data set without generalizing to a broader population.

Descriptive Statistical Methods

• Frequency tables organize data into categories, displaying how often each appears.
• Graphs like histograms and bar graphs visually represent data distributions and category comparisons.
• Pie charts show the proportions of each category relative to the total.
• Scatter plots are used to examine the relationship between variables.

Measures of Central Tendency

• The formula to find the arithmetic mean is ( \mu = \frac{\sum_{i=1}^{N} X_i}{N} )
• Provide insight into the typical value of the data.
• The central value in an ordered data is the median.
• The most frequently occurring value is the mode.

Measures of Dispersion

• The distance between maximum and minimum values indicates the range.
• Variance measures dispersion around the mean, using the formula ( \sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N} ).
• Standard deviation is the square root of variance that is described by the formula, ( \sigma = \sqrt{\frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}} )

Measures of Position

• Quartiles (Q1, Q2, Q3) divide data into four equal parts.
• Percentiles segment data into one hundred equal parts.

Static Equilibrium

• Requires both zero net force and zero net torque.
• Equations are ( \sum \vec{F} = 0 ) and ( \sum \vec{\tau}=0 ).
• These expand to ( \sum F_x = 0 ), ( \sum F_y = 0 ), and ( \sum \tau_z = 0 ).
• Center of gravity is the balance point of an object's weight.
• For homogeneous objects, it's at the geometric center.

Problem-Solving Strategy

• Resolve forces into x and y components after drawing a free-body diagram.
• You then apply the equilibrium conditions.
• You should check the work to logically make sure answers are reasonable.

Teorema de Bayes

• Relates the conditional and marginal probabilities of random events.
• ( P(A|B) = \frac{P(B|A)P(A)}{P(B)} )

Derivation of Bayes' Theorem

• Probability of A given B = probability of intersection of A and B divided by the probability of B.
• Solving these equations leads to Bayes' Theorem.

Example of Bayes' Theorem

• To determine the probability of having a rare disease given a positive test result, use the law of total probability to find ( P(+) ).
• Bayes' Theorem computes the likelihood with considerations for true positives, false positives, and disease rarity.

Lecture 1 - Probabilistic ML

• It uses probability to handle uncertainty, learn, and predict.
• It is used because uncertainty is unavoidable and it handles missing data.
• It uses a principled method to combine data and prior knowledge : ( P(\theta \mid \mathcal{D})=\frac{P(\mathcal{D} \mid \theta) P(\theta)}{P(\mathcal{D})} )
• It relies on probability, statistics, linear algebra, and optimization.
• Models include linear and logistic regression, Gaussian mixtures, and hidden Markov models.
• Algorithms include Expectation Maximization, Monte Carlo, and Variational Inference.

Probability

• Sample space is all possible experiment outcomes. An event space is a subset of outcomes.
• A probability function assigns a probability to each event with axioms.

Random Variable

• Maps outcomes to real numbers. Discrete random variables have finite values.
• Bernoulli, binomial, and Poisson distributions are examples of discrete random variables.
• The probability mass function (PMF) defines probabilities for discrete random variables with specific properties.

Continuous Random Variable

• Continuous random variables can take any value in a continuous range.
• Examples include Gaussian, exponential, and uniform distributions.
• Instead of probability, continuous variables use probability density functions (PDFs).

Devoir 1 - IFT 2420

• This is an assignment due on Friday, January 26, at 23:59.
• It can be done individually or in teams of 2.
• Code must be commented, compiled, and tested and include the student's name and NI.

Question 1

• The task is to implement functions in 32-bit RISC-V assembly including finding string length, comparing strings, and copying strings.

Question 2

• Code (in C) must be translated into RISC-V, illustrating the stack during loop execution.

Question 3

• Design a C program to dynamically allocate an integer array, populate it with user inputs, and compute the average of even numbers, using separate functions for allocation, input, calculation, and deallocation.

Question 4

• Write a program that creates an array of 5 students, takes inputs for each student, and calculates the average grade of all students, and sorts the students by grade.

Machine Learning: A Probabilistic Perspective

• Machine learning develops algorithms for computers to learn from data.
• Data mining finds patterns in large datasets.
• The difference between machine learning and data mining is ML focuses on prediction and DM focuses on knowledge discovery.

Types of Learning

• Supervised learning: predict output ( y ) from input ( \mathbf{x} ) using labeled data ( (\mathbf{x}_i, y_i) ).
  • Classification: ( y_i ) is discrete and in ( {1, \dots, C} ).
  • Regression: ( y_i ) is continuous and in ( \mathbb{R} ).
• Unsupervised learning: discover structure from unlabeled data ( \mathbf{x}_i ).
  • Clustering: group similar examples.
  • Dimensionality reduction: find lower-dimensional representations.
• Reinforcement learning: select actions to maximize a reward.

Probabilistic Perspective

• Uses probability theory to model uncertainty.
• Prediction is obtained via ( y^* = \arg \max_y p(y \mid \mathbf{x}^*) ) using Bayes' Rule.

Supervised Learning

• Task is to learn a function that maps inputs 𝐱 to outputs y from a supervised dataset.
• Algorithms such as email spam detectors are used.

Unsupervised Learning

• The task is to learn the underlying data structure ( \mathcal{D} = {\mathbf{x}i}{i=1}^N ).
• Algorithms such as customer segmentation are used.

Reinforcement Learning

• Task is to learn action selection for maximum reward.