Partial Differential Equations

• Partial Differential Equations (PDEs) describe relationships between functions and their partial derivatives.

Introduction

• PDEs appear in various fields to model physical phenomena.

Important Equations

The Heat Equation

• Describes heat flow over time: $\frac{\partial u}{\partial t} = k \nabla^2 u$
• $u(x, y, z, t)$: Temperature distribution
• $k$: Thermal diffusivity

Wave Equation

• Models wave propagation: $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$
• $u(x, y, z, t)$: Wave displacement
• $c$: Wave speed

Laplace's Equation

• Represents steady-state potential distributions: $\nabla^2 u = 0$
• $u(x, y, z)$: Potential at a point

Poisson's Equation

• Generalization of Laplace's equation with a source term: $\nabla^2 u = f$
• $u(x, y, z)$: Potential
• $f(x, y, z)$: Source term

Schrodinger's Equation

• Describes quantum system evolution: $i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi$
• $\Psi(x, y, z, t)$: Wave function
• $\hbar$: Reduced Planck constant
• $\hat{H}$: Hamiltonian operator

General Form of Linear Second Order PDE

• $A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u = G$

• A, B, C, D, E, F, G are functions of x and y.

• PDE type depends on $B^2 - 4AC$

• Elliptic: $B^2 - 4AC < 0$ (e.g., Laplace's equation)

• Parabolic: $B^2 - 4AC = 0$ (e.g., Heat equation)

• Hyperbolic: $B^2 - 4AC > 0$ (e.g., Wave equation)

Boundary Conditions

• Specify the behavior of the solution on the domain's boundary.
• Dirichlet BC: $u$ is specified on the boundary.
• Neumann BC: $\frac{\partial u}{\partial n}$ is specified on the boundary.
• Robin BC: A linear combination of $u$ and $\frac{\partial u}{\partial n}$ is specified on the boundary.

Matematicas

• Mathematics is the study of relationships between quantities, magnitudes, and properties.

Algebra

• Algebra uses letters and symbols to represent numbers and quantities in formulas and equations.

Algebraic Expressions

• These combine numbers, variables, and mathematical operations.
• Example: $3x + 2y - 5$

Equations

• Equations show equality between two algebraic expressions.
• Solving means finding variable value(s) that make the equality true.
• Example: $2x + 3 = 7$, solution is $x = 2$.

Systems of Equations

• Systems contain two or more equations sharing the same variables.
• Solving finds variable values satisfying all equations.
• Example: $x + y = 5$, $x - y = 1$, solution is $x = 3$ and $y = 2$.

Geometría

• Geometry studies properties and measures of figures in planes and spaces.

Geometric Figures

• Basic figures include points, lines, planes, triangles, squares, circles, cubes, and spheres.

Area and Volume

• Area measures the surface of a flat figure.
• Volume measures the space occupied by a body in three dimensions.
• Example: A square with side $l$ has an area of $l^2$, a cube with a side $l$ has a volume of $l^3$.

Cálculo

• Calculus studies change and accumulation.
• It's split into differential and integral calculus.

Differential Calculus

• Studies the rate of change of a function at a given point, also known as a derivative.

Integral Calculus

• Studies the accumulation of quantities which is a measure of the total quantity that accumulates as the independent variable varies.

Estadística

• Statistics involves the collection, analysis, interpretation, and presentation of data.

Types of Data

• Qualitative data describe qualities or characteristics.
• Quantitative data measure numerical quantities.

Measures of Central Tendency

• These represent the center of a dataset e.g., mean, median, and mode.

Measures of Dispersion

• These indicate the variability within a dataset, examples include standard deviation and range.

Prérequis

Divisibilité

• $b$ divides $a$ if there exists an integer $k$ such that $a = kb$, denoted $b \mid a$.

Nombre premier

• A prime number has exactly two distinct divisors: 1 and itself.

PGCD

• The greatest common divisor (PGCD) of two integers is the largest integer that divides both.

Nombres premiers entre eux

• Two numbers are coprime if their PGCD is equal to 1.

Division euclidienne

• For any integers $a$ and $b$ where $b \neq 0$, there exists a unique pair of integers $(q, r)$ such that $a = bq + r$ and $0 \leq r < |b|$.

Théorème de Bézout

• Integers $a$ and $b$ are coprime if and only if there exist integers $u$ and $v$ such that $au + bv = 1$.

Théorème de Gauss

• If $a \mid bc$ and $a$ and $b$ are coprime, then $a \mid c$.

Décomposition en facteurs premiers

• Every integer $n \geq 2$ can be uniquely decomposed into a product of prime factors.

Thermodynamics

• Applies to various cycles, defining specific processes & heat engine types.

Energy Forms

• Heat (Q): Energy transfer due to temperature difference.
• Work (W): Energy transfer not due to temperature difference.
  • Displacement Work: $W = \int F \cdot dx$
  • Electrical Work: $W = V \cdot I$

1st Law of Thermodynamics

• Energy conservation principle: $\Delta E = Q - W$
  • $\Delta E$: Change in total energy
  • $Q$: Heat added to the system
  • $W$: Work done by the system
• For a cycle: $\oint dE = 0$, and $Q_{net} = W_{net}$

2nd Law of Thermodynamics

• Kelvin-Planck states that a cyclic engine cannot convert all heat into work.
• Clausius states that a device cannot transfer heat from a cold to a hot body without external work.

Thermodynamic Processes

• Isothermal: Constant temperature
• Isobaric: Constant pressure
• Isochoric/Isometric: Constant volume
• Adiabatic: No heat transfer

Thermodynamic Cycles

• Carnot Cycle: Theoretical cycle with maximum efficiency.
• Otto Cycle: Spark-ignition internal combustion engines.
• Diesel Cycle: Compression-ignition internal combustion engines.
• Rankine Cycle: Steam power plants.
• Brayton Cycle: Gas turbine engines.
• Refrigeration Cycle: Transferring heat from a cold reservoir to a hot reservoir.

Properties

• Intensive: Independent of mass (e.g., temperature, pressure, density).
• Extensive: Depends on mass (e.g., volume, energy).

Algorithmic Complexity

Time Complexity

• Measures how long an algorithm takes based on input size.

Space Complexity

• Measures how much memory an algorithm uses.

Importance

• Help comparing algorithms.
• Understand how the algorithm scales.

Big O Notation

• Expresses the upper bound of an algorithm's complexity.
• Describes the worst-case scenario.

Constant Complexity - O(1)

• Time is independent of input size.

Logarithmic Complexity - O(log n)

• Time increases logarithmically with input size.

Linear Complexity - O(n)

• Time increases linearly with input size.

Quadratic Complexity - O(n^2)

• Time increases quadratically with input size.

Exponential Complexity - O(2^n)

• Time doubles with input size.

Lab 1: Introduction to Vectors

• Discusses geometric and algebraic interpretations of vectors.

Definition of Vectors

• Geometric Definition: A directed line segment with magnitude and direction.
• Algebraic Definition: An n-tuple of real numbers represented as $\mathbf{v} = (v_1, v_2,..., v_n)$.

Examples

• 2-D vector: (2, 3)
• 3-D vector: (1, -1, 0)

Vector Operations

Addition

• Given $\mathbf{u}$ and $\mathbf{v}$, their sum is $\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2,..., u_n + v_n)$.

Scalar Multiplication

• Given a scalar $c$ and a vector $\mathbf{v}$, the scalar product is $c\mathbf{v} = (cv_1, cv_2,..., cv_n)$.

Dot Product

• Given $\mathbf{u}$ and $\mathbf{v}$, their dot product is $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 +... + u_nv_n$.

Cross Product (for 3-D vectors)

• Given $\mathbf{u}$ and $\mathbf{v}$, their cross product is $\mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)$.

Vector Spaces

• Set with addition and scalar multiplication operations.
  • Closure under addition/scalar multiplication
  • Commutativity/Associativity of addition
  • Existence of additive identity/inverse
  • Distributivity of scalar multiplication (w.r.t vector & scalar addition)
  • Associativity of scalar multiplication
  • Identity element of scalar multiplication

Examples

• $\mathbb{R}^n$
• $m \times n$ matrices with real entries
• Polynomials with real coefficients

Linear Independence

• The set of vectors ${\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_k}$ is linearly independent if the only solution to $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 +... + c_k\mathbf{v}_k = \mathbf{0}$ is $c_1 = c_2 =... = c_k = 0$.
  • In $\mathbb{R}^2$, (1, 0) and (0, 1) are linearly independent.
  • In $\mathbb{R}^3$, (1, 0, 0), (0, 1, 0), and (0, 0, 1) are linearly independent.

Basis and Dimension

• A basis of a vector space V consists of linearly independent vectors that span V.
• The dimension of V is the number of vectors in a basis.
  • The standard basis for $\mathbb{R}^n$ is ${(1, 0,..., 0), (0, 1,..., 0),..., (0, 0,..., 1)}$.

IFT 3355

• Examines formal languages, automata, and proofs related to the regularity of languages.

Question 1

• Defines languages $L$ over $\Sigma = {a, b}$ based on conditions about the number of $a$'s and $b$'s.
  • Example: $L = {a^n b^m \mid n+m \text{ is even}}$

Question 2

• Describes regular expressions for languages over $\Sigma = {a, b}$.
  • Example: $(a \cup b)^* a (a \cup b)(a \cup b)$ represents strings with at least one $a$ and with length at least 3.

Question 3

• Automata state transition table.

Question 4

• Presents NFAs recognizing specific languages.
  • Example: Automaton accepting strings containing "01".

Question 5

• Refers back to solutions from Questions 3a and 4b.

Question 6

• If L is regular, then L* is regular is proven.

Question 7

• States that if L is regular, the language L' = {xy | x ∈ L and y ∉ L} is regular is proven.

Algorithmic Trading

Algorithmic Trading (Algo Trading)

• Executes trades via automated, pre-programmed instructions accounting for price, timing, and volume.
• Generates profit at high speeds and frequencies.

Benefits

• Reduces costs and emotional influence.
• Improved order execution.
• Enables backtesting and diversification.

Challenges

• Requires continuous monitoring, and can cause unexpected outcomes.

Types of Strategies

Trend Following Strategies

• Exploit opportunities caused by herd behavior using price levels, channel breakouts, moving averages.

Arbitrage Opportunities

• Profit from price differences of identical or similar assets.

Mathematical Model-Based Strategies

• Employ stochastic calculus, linear algebra, numerical methods, etc.,.

Execution Algorithms

• Efficiently execute large orders without significant price impact using VWAP, TWAP, etc.,.

Statistical Arbitrage

• Exploit pricing anomalies using pairs trading, mean reversion, index arbitrage strategies.

Important Technical Indicators

Momentum Indicators

• Determine trend strength and are used to find potential overbought or oversold conditions.
  • Moving Average Convergence Divergence (MACD): $MACD = 12-period EMA - 26-period EMA$, and the $Signal Line = 9-period EMA of MACD$.
  • Relative Strength Index (RSI): $RSI = 100 - [100 / (1 + (Average Gain / Average Loss))]$.

Volume Indicators

• Evaluate trend by relating price and volume with the On Balance Volume (OBV).

OBV Formula

• If today's closing price > yesterday's closing price, then $OBV = Yesterday's OBV + Today's Volume$
• If today's closing price < yesterday's closing price, then $OBV = Yesterday's OBV - Today's Volume$
• If today's closing price = yesterday's closing price, then $OBV = Yesterday's OBV$.

Volatility Indicators

• Measure variation of a trading price series over time.
  • Average True Range (ATR): $TR = Max[(High - Low), abs(High - Previous Close), abs(Low - Previous Close)]$
  • $ATR = Average of TR over a period (e.g., 14 days)$

Fonction Logarithme Népérien

Definition

• ln(x) is the primitive of x ↦ 1/x that equals 0 at 1.
• Defined on $]0; +\infty[$.

Properties

Algebraic

• $\ln(1) = 0$
• $\ln(e) = 1$
• $\ln(ab) = \ln(a) + \ln(b)$
• $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
• $\ln(\frac{1}{b}) = -\ln(b)$
• $\ln(a^n) = n\ln(a)$
• $\ln(\sqrt{a}) = \frac{1}{2}\ln(a)$

Function Study

• ln is differentiable on $]0; +\infty[$ with derivative $(\ln(x))' = \frac{1}{x}$
• ln is strictly increasing on $]0; +\infty[$
• $\lim\limits_{x \to +\infty} \ln(x) = +\infty$
• $\lim\limits_{x \to 0} \ln(x) = -\infty$

Consequence

• For all $a,b>0$: $\ln(a) = \ln(b) \iff a = b$.
• For all $a,b>0$: $\ln(a) < \ln(b) \iff a < b$

Derivatives

• If $u$ is differentiable and positive on I, then $\ln(u)$ is differentiable on I with $(\ln(u))' = \frac{u'}{u}$.

Limits

Growth comparisons

• $\lim\limits_{x \to +\infty} \frac{\ln(x)}{x} = 0$
• $\lim\limits_{x \to +\infty} \frac{\ln(x)}{x^n} = 0$
• $\lim\limits_{x \to 0} x\ln(x) = 0$

Other important limits

• $\lim\limits_{h \to 0} \frac{\ln(1+h)}{h} = 1$
• $\lim\limits_{x \to +\infty} \ln(x+1) - \ln(x) = 0$