Partial Differential Equations (PDEs)

Partial Differential Equations

• A PDE involves an unknown function with two or more variables and some of its partial derivatives.
• The order of a PDE corresponds to the highest order derivative in the equation.
• The general form for a PDE with two independent variables x and y is F(x, y, u, u_x, u_y, u_{xx}, u_{xy}, u_{yy},...) = 0 where u = u(x, y) is the dependent variable.
• A PDE solution is a function that satisfies the equation within a region of independent variables.

Linear PDE

• A linear PDE has a form in which the dependent variable and its derivatives appear linearly.
• The general form of a linear PDE is a(x, y)u_{xx} + b(x, y)u_{xy} + c(x, y)u_{yy} + d(x, y)u_x + e(x, y)u_y + f(x, y)u = g(x, y).

Examples of PDEs

• Heat Equation: u_t = ku_{xx}
• Wave Equation: u_{tt} = a^2u_{xx}
• Laplace's Equation: u_{xx} + u_{yy} = 0

Boundary and Initial Conditions

• Boundary conditions are required to specify a unique solution to a PDE. These are conditions that the solution must meet on the boundary of the region of interest.
• Initial conditions, which must be met at the initial time, are also necessary for time-dependent PDEs.

Física

• Vectors have magnitude and direction.

Vector Addition - Analytical Method

• To perform $\vec{R} = \vec{A} + \vec{B}$ where $\vec{A} = A_x\hat{i} + A_y\hat{j}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j}$, then
• $\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.
• Calculating the components $R_x = A_x + B_x$ and $R_y = A_y + B_y$.
• The magnitude is $R = \sqrt{R_x^2 + R_y^2}$ and the direction $\theta = arctan(\frac{R_y}{R_x})$.

Vector Example

• Given $\vec{A} = 2\hat{i} + 4\hat{j}$ and $\vec{B} = 1\hat{i} + 3\hat{j}$, then
• $\vec{R} = (2 + 1)\hat{i} + (4 + 3)\hat{j}$ leading to $\vec{R} = 3\hat{i} + 7\hat{j}$.
• Resulting in $R = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.61$ magnitude
• With a heading of $\theta = arctan(\frac{7}{3}) = 66.8^\circ$.

Algorithmic Game Theory

• Game theory is the formal study of strategic interaction.
• A game includes a set of players, possible actions for each player, and specification of payoffs tied to players’ actions.
• Game theory can be descriptive, to predict outcomes, or normative, to suggest optimal player behavior.

Matching Pennies

• Two players simultaneously place a penny, either heads or tails, on a table.
• Player 1 wins if both pennies match; player 2 wins if they differ.
• Payoff Structure:
  • Player 1 Heads, Player 2 Heads: Player 1 gets 1 and Player 2 gets -1
  • Player 1 Heads, Player 2 Tails: Player 1 gets -1 and Player 2 gets 1
  • Player 1 Tails, Player 2 Heads: Player 1 gets -1 and Player 2 gets 1
  • Player 1 Tails, Player 2 Tails: Player 1 gets 1 and Player 2 gets -1

Rock-Paper-Scissors

• Two players chooses rock, paper, or scissors simultaneously.
• Rock beats scissors, scissors beats paper, paper beats rock.
• Payoff Structure:
  • Player 1 Rock, Player 2 Rock: Players get 0
  • Player 1 Rock, Player 2 Paper: Player 1 gets -1, Player 2 gets 1
  • Player 1 Rock, Player 2 Scissors: Player 1 gets 1, Player 2 gets -1
  • Player 1 Paper, Player 2 Rock: Player 1 gets 1, Player 2 gets -1
  • Player 1 Paper, Player 2 Paper: Players get 0
  • Player 1 Paper, Player 2 Scissors: Player 1 gets -1, Player 2 gets 1
  • Player 1 Scissors, Player 2 Rock: Player 1 gets -1, Player 2 gets 1
  • Player 1 Scissors, Player 2 Paper: Player 1 gets 1, Player 2 gets -1
  • Player 1 Scissors, Player 2 Scissors: Players get 0

Nash Equilibrium

• In Nash Equilibrium, no player benefits by unilaterally changing strategy.
• Player i's strategy $s_i$ is a best response to strategies $s_{-i}$ of other players if $u_i(s_i, s_{-i}) \ge u_i(s'i, s{-i})$ for all strategies $s'_i$ of player $i$.
• All games with a finite number of players and action profiles have a Nash equilibrium (in mixed strategies). [Nash, 1950]

Prisoner's Dilemma

• Two suspects are arrested and interrogated separately.

• If one confesses and the other doesn't, the confessor is freed.

• If neither confesses, they get a light sentence, if both confess, they get a moderate sentence.

• Representation:

  	Prisoner 2: Confess	Prisoner 2: Don't
  Prisoner 1: Confess	-5, -5	0, -10
  Prisoner 1: Don't Confess	-10, 0	-1, -1

• Confessing is the dominant stragety.

• (Confess, Confess) is the only Nash Equilibrium.

Vektorer og geometri i rummet.

Definition of a Vector

• Vectors have magnitude and direction, being represented by a directed line segment.
• $\overrightarrow{AB}$ indicates the vector going from point A to B, while $\vec{a}$ denotes a general vector.

Vector Arithmetic

• Addition: $\vec{a} + \vec{b}$
• Subtraction: $\vec{a} - \vec{b}$
• Scalar Multiplication: $k\vec{a}$

Rules

• Commutative Law: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
• Associative Law: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
• Distributive Law: $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$
• Distributive Law: $(k_1 + k_2)\vec{a} = k_1\vec{a} + k_2\vec{a}$

Example Vector Calculation

• Given $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 5, 6)$, then
• $\vec{a} + \vec{b} = (1+4, 2+5, 3+6) = (5, 7, 9)$
• With Scalar Multiplication: $2\vec{a} = (2\cdot1, 2\cdot2, 2\cdot3) = (2, 4, 6)$

Dot Product (Skalarproduktet)

• $\vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos(\theta)$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

Dot Product Computation

• If $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, then:
• $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$

Properties of the Dot Product

• Commutative Law: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
• Distributive Law: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
• Scalar Association: $k(\vec{a} \cdot \vec{b}) = (k\vec{a}) \cdot \vec{b} = \vec{a} \cdot (k\vec{b})$
• Orthogonality: if $\vec{a} \cdot \vec{b} = 0$, then $\vec{a}$ and $\vec{b}$ are orthogonal.

Example Dot Product

• Given vectors $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 5, 6)$.
• Calculate:
• $\vec{a} \cdot \vec{b} = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32$

Cross Product (Vektorproduktet)

• The cross product of $\vec{a}$ and $\vec{b}$ produces a new vector $\vec{c}$ that is perpendicular to both input vectors.

Cross Product Calculation

• The magnitude of the new vector is given by $|\vec{c}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin(\theta)$, with $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
• If given $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, then:
• $\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$

Cross Product Properties

• Anticommutative Law: $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$
• Distributive Law: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$
• Scalar Association: $k(\vec{a} \times \vec{b}) = (k\vec{a}) \times \vec{b} = \vec{a} \times (k\vec{b})$
• Parallel Vectors: If $\vec{a} \times \vec{b} = \vec{0}$, $\vec{a}$ and $\vec{b}$ are parallel.

Example Cross Product

• Given $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 5, 6)$, then
• $\vec{a} \times \vec{b} = ((2)(6) - (3)(5), (3)(4) - (1)(6), (1)(5) - (2)(4)) = (12 - 15, 12 - 6, 5 - 8) = (-3, 6, -3)$

Parallelepiped Volume (Rumfang af et parallellepipedum)

• In 3D space, the volume (V) of a parallelepiped formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is determined by the absolute value of the scalar triple product:

Volume Calculation

• The formula is $V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$.
• Using coordinates, if $\vec{a} = (a_1, a_2, a_3)$, $\vec{b} = (b_1, b_2, b_3)$ and $\vec{c} = (c_1, c_2, c_3)$, the volume can be expanded as:
• $\vec{a} \cdot (\vec{b} \times \vec{c}) = a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1)$.
• With is formula it can also be calculaterd using the determinant:
• $V= \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}$

Example Volume Calculation

• For vectors $\vec{a} = (1, 2, 3)$, $\vec{b} = (4, 5, 6)$, and $\vec{c} = (7, 8, 9)$:

• First use cross product to calculate: $\vec{b} \times \vec{c} = ((5)(9) - (6)(8), (6)(7) - (4)(9), (4)(8) - (5)(7)) = (45 - 48, 42 - 36, 32 - 35) = (-3, 6, -3)$.

• Then use dot prouct to calculate: $\vec{a} \cdot (\vec{b} \times \vec{c}) = (1)(-3) + (2)(6) + (3)(-3) = -3 + 12 - 9 = 0 $.

• To get the volume $V = |0| = 0$

Algorithmic Complexity

• Algorithmic complexity measures how many resources (time and memory) an algorithm needs as the input size increase.

Big O Notation

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

Big Omega Notation

• Describes the lower bound, or best-case scenario.

Big Theta Notation

• Describes the tight bound, or average-case scenario.

Why Big O Matters

• Algorithmic complexity affects performance greatly.
• Using the right data structures optimizies performance.
• Analysis helps write efficient code.
• Always consider the worst-case scenario when analyzing algorithms.

Resumen Ejecutivo

• Offers consulting services for digital transformations and custom software development.
• Organizations struggle adapting to new technology and software development.
• Consultints offer a solution for transformation and customer software creation.
• Focus on companies of certain sectors looking to increase their efficiency and competition.
• Primary income comes from consulting and software licencing.
• Team has experience in software development and digital transformation.
• Seeking $500,000 in financing to expand the team, develop new software products, and expand marketing.
• Expect to reach $2 million in revenue in three years, with a 20% profit margin.
• Measuring success by number of customers, customer satisfaction, and reaccuring revenue.
• Primary exit strategy revolves around buy outs from larger consulting or software companies.

Lecture 18 - October 29, 2012

Black-Scholes Assumptions

• Stock prices follow a geometric Brownian motion or $dS = \mu S dt + \sigma S dZ$
• Where $\mu$ and $\sigma$ are constant
• No dividends.
• No arbitrage
• Markets are complete
• Risk free rate ($r$) and transaction costs or taxes, all securities are perfectly divisible, and the world is risk neutral.

Black-Scholes Formula

• $C = S_0N(d_1) - e^{-rT}KN(d_2)$
• where $d_1 = \frac{ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}$
• $d_2 = d_1 - \sigma \sqrt{T}$

Implied Volatility

• Volatility is the only non-directly observableBlack-Scholes parameter
• Implied volatility can be used to derive the market price of an option (IV)
• IV is the value of $\sigma$ that equates the Black-Scholes price to the market price
• Inconsistent with Black-Scholes, because options on the same assets with different strike prices ot times until maturity will have different implied volatilities
• The graph of IV is a function of strike price
• Volatility smiles are the graph of IV

Delta

• Delta ($\Delta$) is the sensitivity of the option price to changes in the underlying asset price:
• $\Delta = \frac{\partial C}{\partial S}$ for a call option
• $\Delta = \frac{\partial P}{\partial S}$ for a put option
• Black Scholes deltas are:
• $\Delta = N(d_1)$ for call options
• $\Delta = N(d_1) - 1$ for put options

Delta Hedging

• Delta neutral portfolios are not sensitive to small price changes of the underlying asset.
• Short call option positions are often hedged by buying $N(d_1)$ shares of shares.
• Black-Scholes helps define the hede cost

Chemical Kinetics

Reaction Rate

• $aA + bB \rightarrow cC + dD$
• Where $Rate = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$
• The value of the rate is a function of Reactant concentration, Temperature, Surface area, or the presence of a catalyst.

Rate Law

• $aA + bB \rightarrow cC + dD$
• $Rate = k[A]^x[B]^y$
• k is the rate constant
• x is the order with respect to A
• y is the order with respect to B
• x + y is the overall order of the reaction

Integrated Rate Laws

Collision Theory

• Collision theory relies on reactant molecules must to collide with Proper orientation and Sufficient energy

Activation Energy

• The minimum energy required for a reaction.

Arrhenius Equation

• $k = Ae^{\frac{-E_a}{RT}}$
• $ln(k) = \frac{-E_a}{R}\frac{1}{T} + ln(A)$
• $ln(\frac{k_2}{k_1}) = \frac{E_a}{R}(\frac{1}{T_1} - \frac{1}{T_2})$
• k is the rate constant
• A is the frequency factor
• $E_a$ is the activation energy
• R is the gas constant (8.314 J/mol·K)
• T is the temperature in Kelvin

Reaction Mechanisms

• Series of steps that make up an overall reaction and molecular event.

Catalysts

• Catalysis can speed up reactions without the catalyst being consumed.
• Homogeneous catalysts are in the same phase as reactants.
• Heterogeneous catalysts are NOT in the same phase as reactants.
• Enzymes are biological catalysts.

Analisi Matematica 1 - Ing. Edile-Architettura

Disequazioni polinomiali

• Solving polynomial inequalities.

Disequazioni irrazionali

• Solving irrational inequalities.

Disequazioni con valore assoluto

• Solving inequalities with absolute value

Sistemi di disequazioni

• Solving systems of inequalities.

Algorithmic Game Theory

• Is the study of mathematical models of strategic interactions among rational agents.
• Has applications in computer science, logic, systems science, and social science.

Key Concepts of Game Theory

• Players are the entities who make decisions.
• Strategies are the actions each player can take.
• Payoffs are the rewards received by players based on the strageties chosen
• Rationality means players are acting in their own best interests to maximize profits.

Game Theory Types

• Cooperative vs. Non-cooperative: players agree to binding agreements.
• Symmetric vs. Asymmetric: players have the same or different strategy sets and payoffs.
• Zero-sum vs. Non-zero-sum: One player's gain is a player's loss, or they are unrelated.
• Perfect vs. Imperfect Information: players know all previous actions of other players, or they do not.

Algorithmic-Game Theory

• Algorithmic game theory (AGT) is an area that lies at the intersection of game theory and computer science, and more specifically algorithm design
• AGT designs algorithms, especially for the Internet, taking into account that many agents will strategically react to the algorithm.

Aspects of Algorithmic-Game Theory

• Algorithm Design
• Mechanism Design
• Equilibrium Computation
• Complexity

Selfish Routing Example

• With Selfish Routing, users chooses the route that minimizes their delays, with no thought for overall network performance.
• This can lead to a Nash equilibrium, where the overall network performs worse than it could with coordination.
• The Price of Anarchy, measures the ratio betwen the worst-case Nash equilibrium and optimal social welfare.

Basic Game Theory Concepts

• A normal-form game specifies strategies and payoffs as a matrix.

Payoff Matrix Example

• The suspect's cannot communicate with on another after the police apprehend them.

  	Suspect B Confesses	Suspect B Stays Silent
  Suspect A Confesses	-5, -5	0, -10
  Suspect A Stays Silent	-10, 0	-1, -1

Dominant Strategy

Strategy dominates, if it is optimal for a player regardless of what other players do.

Nash Equilibrium

A stragety such that no player can improve their payoff by just making their strategy alone change, with the conditions that the other players keep their conditions the same.