Understanding the Wave Equation

The Wave Equation

• The wave equation has physical contexts involving vibrations and waves in strings, bars, air, and electromagnetism.

Wave Equation in 1-D

• The wave equation in one dimension is represented as $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$.
• $c$ can be calculated as $c = \sqrt{\frac{T}{\rho}}$, $c = \sqrt{\frac{E}{\rho}}$, or $c = \sqrt{\frac{B}{\rho}}$ based on the medium.

General Solution

• The general solution of the wave equation is $u(x,t) = f(x+ct) + g(x-ct)$.
• Here, $f$ and $g$ are arbitrary functions.

Example 1

• Given $\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}$ with $u(x,0) = \sin(x)$ and $\frac{\partial u}{\partial t}(x,0) = 0$, the solution is $u(x,t) = \sin(x) \cos(2t)$.

Example 2

• Given $\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}$ with $u(x,0) = 0$ and $\frac{\partial u}{\partial t}(x,0) = \sin(x)$, the solution is $u(x,t) = \frac{1}{2} \sin(x) \sin(2t)$.

Example 3

• Given $\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}$ with $u(x,0) = \sin(x)$ and $\frac{\partial u}{\partial t}(x,0) = \sin(x)$, the solution is $u(x,t) = \sin(x) \cos(2t) + \frac{1}{2} \sin(x) \sin(2t)$, which can be simplified to $u(x,t) = \sin(x) \left( \cos(2t) + \frac{1}{2} \sin(2t) \right)$.

Evaluating Set Theory Expressions

• $(\mathbb{Z} \times \mathbb{Q}) \cap (\mathbb{Q} \times \mathbb{Z}) = \mathbb{Z} \times \mathbb{Z}$ because the intersection requires both elements to be integers.
• $(\mathbb{R} \times \mathbb{N}) - (\mathbb{Q} \times \mathbb{N}) = (\mathbb{R} - \mathbb{Q}) \times \mathbb{N}$ as it involves irrational numbers in the real number set.
• $(\mathbb{Z} \times \mathbb{R}) \cup (\mathbb{R} \times \mathbb{Z})$ represents all points in the plane where at least one coordinate is an integer.

Set Identity Proof

• $(A \cap B) \cup C = A \cap (B \cup C)$ holds if and only if $C \subseteq A$.
• This is proven by showing implication in both directions: if $(A \cap B) \cup C = A \cap (B \cup C)$, then $C \subseteq A$ and vice versa.

Function Properties

• For any subset $C$ of $B$, $f(f^{-1}(C)) \subseteq C$.
• If $f$ is surjective, then for any subset $C$ of $B$, $C \subseteq f(f^{-1}(C))$.
• If $f$ is surjective, then $f(f^{-1}(C)) = C$ for any subset $C$ of $B$.

Function Compositions

• $f(A \cap B) = f(A) \cap f(B)$ is false; a counterexample demonstrates this.
• $f(A \cup B) = f(A) \cup f(B)$ is true as demonstrated by showing that $f(A \cup B) \subseteq f(A) \cup f(B)$ and $f(A) \cup f(B) \subseteq f(A \cup B)$.

Definition of Logic

• Logic is the study of methods and principles used to evaluate correct vs. incorrect reasoning.

Proposition Definition

• A proposition is a declarative statement that is either true or false, but not both.

Examples of Propositions

• "Paris is the capital of France" is a true proposition.
• "1 + 1 = 2" is a true proposition.
• "The Moon is made of cheese" is a false proposition.

Non-Propositions Examples

• Exclamations like "Hello!"
• Questions like "How are you?"
• Imperative sentences like "Close the door."

Logical Operators

• Negation ($\neg$) means "Not".
• Conjunction ($\land$) means "And".
• Disjunction ($\lor$) means "Or".
• Conditional ($\to$) means "If... then".
• Biconditional ($\leftrightarrow$) means "If and only if".

Truth Tables - Negation

• If $p$ is true, $\neg p$ is false; if $p$ is false, $\neg p$ is true.

Truth Tables - Conjunction

• $p \land q$ is true only when both $p$ and $q$ are true.

Truth Tables - Disjunction

• $p \lor q$ is true when either $p$ or $q$ or both are true.

Truth Tables - Conditional

• $p \to q$ is false only when $p$ is true and $q$ is false; otherwise, it is true.

Truth Tables - Biconditional

• $p \leftrightarrow q$ is true when both $p$ and $q$ have the same truth value (both true or both false).

Logical Equivalencies

• Two propositions are logically equivalent if they have the same truth table.

Example Equivalencies

• $p \to q \equiv \neg p \lor q$
• DeMorgan's Law: $\neg (p \land q) \equiv \neg p \lor \neg q$
• DeMorgan's Law: $\neg (p \lor q) \equiv \neg p \land \neg q$

Logical Implication

• Proposition $p$ logically implies proposition $q$ if whenever $p$ is true, $q$ is also true.

Arguments

• An argument is a sequence of propositions where the last one is the conclusion and the others are premises.

Argument Validity

• An argument is valid if the conclusion is a logical consequence of the premises.

Introduction to the Néperien Logarithm Function

• The study notes focus on the Néperien Logarithm Function.
• Information about definitions, algebraic properties, analysis, derivatives, variations, limits and graphic representation are reviewed.
• Assumes prior knowledge of algebra, differentiation, and exponential functions.

Definition of Néperien Logarithm

• Defined on $]0;+\infty[$ and is the inverse of the exponential function.
• $y=ln(x) \Leftrightarrow x=e^{y}$
• $ln(e)=1$, $ln(1)=0$
• $e^{ln(x)}=x$ for $x>0$ and $ln(e^{x})=x$

Algebraic Properties of Logarithms

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

Function Analysis: Derivative of ln

• The derivative of $ln(x)$ is $ln'(x)=\frac{1}{x}$.

Variations

• $ln$ is strictly increasing on $]0;+\infty[$.

Limits of ln

• Basic limits include: $\lim_{x \to 0}{ln(x)}=-\infty$, $\lim_{x \to +\infty}{ln(x)}=+\infty$
• $\lim_{x \to +\infty}{\frac{ln(x)}{x}}=0$ and $\lim_{x \to 0}{\frac{ln(1+x)}{x}}=1$.

Multiple Choice Questions

• Questions regarding solubility, vapor pressure, boiling points, and colligative properties are covered.
• Molarity, molality, reaction orders, and reaction mechanisms are also reviewed.

Short Answer Questions

• Saturated, unsaturated, and supersaturated solutions need defining.
• Concepts regarding salt melting ice, heating curves for water, and determining reaction orders are reviewed.
• Rate laws and constants are also covered.

Zeroth Law of Thermodynamics

• States that if two systems are each in thermal equilibrium with a third, then they are also in thermal equilibrium with each other.
• This law allows to define temperature as a state function.

First Law of Thermodynamics

• States that the change in internal energy of a system equals the heat added to the system minus the work done by the system: $\Delta U = Q - W$.
• Is a restatement of the law of energy conservation.

Second Law of Thermodynamics

• The total entropy of an isolated system can only increase over time in spontaneous processes or remain constant in reversible processes, mathematically expressed as $\Delta S \geq 0$.
• Entropy ($S$) indicates the disorder in a system.

Third Law of Thermodynamics

• As the temperature of a system approaches absolute zero, the entropy approaches a minimum or zero value.
• Absolute zero cannot be reached in a finite number of steps.

Thermodynamic Processes: Isobaric

• Process occurring at constant pressure, with work $W = P\Delta V$ and heat transfer $Q = nC_p\Delta T$.

Thermodynamic Processes: Isochoric (Isovolumetric)

• Process occurring at constant volume, work $W = 0$, heat $Q = nC_v\Delta T$, and first law relation $\Delta U = Q$.

Thermodynamic Processes: Isothermal

• Process at constant temperature, with work $W = nRT \ln\left(\frac{V_2}{V_1}\right)$, heat transfer $Q = W$, and $\Delta U = 0$.

Thermodynamic Processes: Adiabatic

• Process with no heat transfer ($Q = 0$), work $W = -\Delta U$, and relation $PV^\gamma = \text{constant}$, where $\gamma = \frac{C_p}{C_v}$.

State Functions

• Definition: Properties that depend only on the state the system is in, not how that state was reached.
• Changes in state functions only depend on initial and final states.

Path Functions

• Definition: Properties dependent on the path taken in reaching its final state.
• Changes in path functions depend on the process.

Heat Engine Efficiency

• Equal to $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$.

Refrigerator Performance

• Described by the Coefficient of Performance, $\text{COP} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$.

Carnot Cycle

• Theoretical maximum efficiency is given by $\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$.

Phase Transitions

• Involve changes in the physical state of a substance.
• Includes melting, boiling, condensation, and freezing.

Latent Heat

• Amount of heat absorbed or released during phase transition at constant temperature: $Q = mL$.

Boltzmann Distribution

• Gives the probability of a system being in a certain state which depends on the energy of that state, described by $P(E) = \frac{e^{-E/(kT)}}{Z}$.

Entropy and Microstates

• Equation $S = k \ln(\Omega)$ relates entropy to the number of microstates $\Omega$.

Algorithmic Game Theory

Second-Price Auction

• The task requires calculating payments in the VCG mechanism, arguing about Nash equilibrium, and computing payments in the GSP mechanism.

Directed Graph Game

• Players choose strategies corresponding to nodes and utilities are defined based on cycles.
• It is required to specify a Nash equilibrium and determine if the game always has one.

Set Cover

• Requires proving that an $\alpha$-approximate set cover algorithm can be used to construct an $\alpha$-approximate mechanism for the problem.

Single-Item Auction

• Need to prove or disprove the truthfulness of the proposed single-item auction mechanism.

Introduction to Game Theory

• Game theory is a mathematical framework for analyzing strategic interactions among players.

Key Concepts in Game Theory

• Game: Strategic interaction of multiple players.
• Player: Decision-maker in a game.
• Strategy: Action plan for a player.
• Payoff: Player's gain or loss.
• Equilibrium: Strategy set where no player benefits from unilateral change.

Game Types

• Cooperative vs. non-cooperative.
• Zero-sum vs. non-zero-sum.
• Single vs. repeated.

Game Representations: Normal Form

• A normal form represents players, strategies, and payoffs in a table.

Game Representations: Extensive Form

• The extensive form represents a game's flow as a tree.

Dominant Strategy

• Is the best option regardless of other players' choices.

Dominated Strategy

• Gives worse payoff than another, regardless of others' actions.

Nash Equilibrium Definition

• In a Nash equilibrium, no player can increase their payoff by unilaterally changing their strategy.

Nash Equilibrium Example

• The Prisoner's Dilemma illustrates Nash equilibrium where rational prisoners both confess.

Game Theory Applications

• Economics: Pricing and negotiations.
• Political Science: Voting behavior and international policy.
• Biology: Analyzing evolutionary strategies.
• Computer Science: AI and algorithm design.