Dell Hymes-S.P.E.A.K.N.G

Algorithmic Game Theory

• Game theory is the study of mathematical models of strategic interactions among rational agents.
• It applies to social science, logic, systems science, and computer science.
• Originally focused on zero-sum games, it now covers a wide range of behavioral relations.
• It is now an umbrella term for the science of logical decision making in humans, animals, and computers.

Non-cooperative Game Theory

• This deals with situations where players make independent decisions.

Example: Prisoner's Dilemma

• Two suspects are arrested, but police lack sufficient evidence for conviction without a confession.
• If both remain silent, both jailed for 1 year.
• If one confesses, the confessor is freed, and the other gets 3 years.
• If both confess, both are jailed for 2 years.
• Payoff matrix represents the choices:
  • Suspect B Stays Silent: Suspect A Stays Silent (-1,-1), Suspect A Confesses (0,-3)
  • Suspect B Confesses: Suspect A Stays Silent (-3,0), Suspect A Confesses (-2,-2)
• Confessing is the dominant strategy for both, resulting in both being jailed for 2 years
• Both would have been jailed for only 1 year if they remained silent.

Nash Equilibrium

• Set of strategies where no player benefits from unilaterally changing their strategy, given the other players' strategies.
• It is a stable state where no player has an incentive to deviate.
• In Prisoner's Dilemma, both confessing is the Nash equilibrium.

Existence of Nash Equilibrium

• John Nash proved every finite game has at least one Nash equilibrium, possibly in mixed strategies.
• A mixed strategy is a probability distribution over pure strategies
• A player might flip a coin to decide whether to play strategy A or strategy B using probabilities.

Algèbre Linéaire et Analyse Vectorielle (Linear Algebra and Vector Analysis)

Chapitre 1: Vecteurs (Vectors)

1.1 Scalaires et Vecteurs (Scalars and Vectors)

• Scalars are defined completely by their magnitude (e.g., mass, time, temperature).
• Vectors are defined by both magnitude and direction (e.g., displacement, velocity, force).
• Vectors are represented geometrically as directed line segments.
  • Segment length represents magnitude
  • Arrow indicates direction

1.2 Algèbre Vectorielle (Vector Algebra)

• Equality of Vectors: Vectors A and B are equal if they have the same magnitude and direction, regardless of starting points.
• Vector Addition: Sum of A and B (A + B) is found by placing B's start at A's end and connecting A's start to B's end.
  • Parallelogram Law: If A and B are sides of a parallelogram, A + B is the diagonal.
• Vector Subtraction: Difference of A and B (A - B) is found by adding A to the opposite of B, i.e., A - B = A + (-B).
• Scalar Multiplication: Product of vector A by scalar m (mA) is a vector with magnitude |m| times A's magnitude.
  • Direction is the same as A if m > 0, opposite if m < 0.
  • If m = 0, then mA = 0 (zero vector).

1.3 Lois Fondamentales de l'Algèbre Vectorielle (Fundamental Laws of Vector Algebra)

• Let A, B, and C be vectors, and m and n be scalars.
• Commutative Law for Addition: A + B = B + A
• Associative Law for Addition: A + (B + C) = (A + B) + C
• Distributive Law for Scalar Multiplication:
  • m (A + B) = mA + mB
  • (m + n) A = mA + nA
• Associative Law for Scalar Multiplication: (m n) A = m (n A) = n (m A)

1.4 Vecteurs Unitaires (Unit Vectors)

• Unit vector: A vector with a magnitude of 1.
• If A has magnitude |A| ≠ 0, then A/|A| is a unit vector in the same direction as A.
• Any vector A can be described using unit vectors.

1.5 Composantes d'un Vecteur (Vector Components)

• In two dimensions, vector A can be expressed as the sum of its x and y components.
• If $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ are unit vectors in the x and y directions, then $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}}$, where $A_x$ and $A_y$ are the components of A.
• Magnitude of A: $|\mathbf{A}| = \sqrt{A_x^2 + A_y^2}$
• Direction of A relative to the x-axis: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

The Big Bang Theory

Synopsis

• Comedy about four physicists: Sheldon, Leonard, Howard, and Raj with inverse social skills to their intelligence.
• Penny, an attractive woman, moves next door
• Leonard falls for her and seeks life outside the lab.
• Sheldon is content with routine and finds Penny disruptive.
• Howard fancies himself a ladies' man.
• Raj cannot speak to women unless drunk.
• The show follows them navigating science, relationships, and social situations.

Main Characters

• Sheldon Cooper: Theoretical physicist with 187 IQ, socially awkward, difficulty with sarcasm, obsessed with routines.
• Leonard Hofstadter: Experimental physicist and Sheldon's roommate, tries to win Penny's affection.
• Penny: Aspiring actress and waitress, outgoing neighbor, helps the guys in social situtations.
• Howard Wolowitz: Aerospace engineer, lives with his mother, makes inappropriate comments.
• Raj Koothrappali: Astrophysicist unable to speak to women unless drunk, relies on friends.

Recurring Characters

• Bernadette Rostenkowski-Wolowitz: Howard's wife, microbiologist, intelligent and assertive.
• Amy Farrah Fowler: Sheldon's girlfriend, neurobiologist, quirky, based on intellectual connection.
• Stuart Bloom: Comic book store owner, lonely, joins the group providing comic relief.

Fun Facts

• Theme song is performed by Barenaked Ladies.
• Jim Parsons (Sheldon) won four Primetime Emmy Awards.
• Series finale aired May 16, 2019, after 12 seasons and 279 episodes.
• Spin-off "Young Sheldon" follows Sheldon's childhood.
• References real scientific concepts, making it both educational and entertaining.

Episode Guide

• A table displays season, episodes, and originally aired dates for seasons 1-12.

Quotes

• "Bazinga!" - Sheldon Cooper.
• "I'm not crazy. My mother had me tested." - Sheldon Cooper.
• "That's my spot." - Sheldon Cooper.
• "Oh, gravity, thou art a heartless bitch." - Sheldon Cooper.
• "We're physicists! We have tenure! We can do whatever we want!" - Leonard Hofstadter.

Awards

• A table displays the year awards won, the award earned category, and whether they won.
• 2010, Primetime Emmy Award, Outstanding Lead Actor, Won
• 2011, People's Choice Award, Favorite TV Comedy, Won
• 2013, Critics' Choice Television Award, Best Comedy Series, Won
• 2014, Primetime Emmy Award, Outstanding Lead Actor, Won.
• 2017, People's Choice Award, Favorite Network TV Comedy, Won.

Matrizenmultiplikation (Matrix Multiplication)

Definition

• For $A = (a_{ij}){m \times n}$ and $B = (b{ij}){n \times p}$ matrices, the product $C = A \cdot B$ is defined as the $m \times p$ matrix $C = (c{ij})$ with $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} +... + a_{in}b_{nj}$.
• Example:
  • $\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}$

Eigenschaften (Properties)

• Associativity: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
• Distributivity: $A \cdot (B + C) = A \cdot B + A \cdot C$ and $(A + B) \cdot C = A \cdot C + B \cdot C$
• Non-commutative: $A \cdot B \neq B \cdot A$ (in general)

Einheitsmatrix (Identity Matrix)

• The identity matrix $I_n$ is an $n \times n$ matrix with ones on the main diagonal and zeros elsewhere.
  • $I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{pmatrix}$
• It holds: $A \cdot I_n = A$ and $I_m \cdot A = A$

Transponierte Matrix (Transposed Matrix)

• The transpose $A^T$ of an $m \times n$ matrix $A$ is an $n \times m$ matrix in which rows and columns are swapped.
  • $(A^T){ij} = A{ji}$
• Example: $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 \end{pmatrix} \Rightarrow A^T = \begin{pmatrix} 1 & 3 & 5 \ 2 & 4 & 6 \end{pmatrix}$

Rechenregeln für Transponierte (Rules for Transposes)

• $(A + B)^T = A^T + B^T$
• $(\lambda A)^T = \lambda A^T$
• $(A \cdot B)^T = B^T \cdot A^T$
• $(A^T)^T = A$

Inverse Matrix (Inverse Matrix)

• An $n \times n$ matrix $A$ is called invertible if there is a matrix $A^{-1}$ such that $A \cdot A^{-1} = A^{-1} \cdot A = I_n$ holds.
• $A^{-1}$ is called the inverse matrix of $A$.
• Not every matrix is invertible.

Algèbre Linéaire et Géométrie Analytique I (Linear Algebra and Analytical Geometry I)

Chapitre 1 : Systèmes d'équations linéaires (Systems of Linear Equations)

1.1 Introduction

• A system of linear equations is a set of equations in the form:
  • $a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1$
  • $a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2$...
  • $a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m$
  • where $x_1, x_2,..., x_n$ are unknowns, $a_{ij}$ are coefficients, and $b_i$ are constants.
• Example:
  • $2x_1 + 3x_2 = 5$
  • $x_1 - x_2 = 1$

1.2 Résolution des systèmes d'équations linéaires (Solving Systems of Linear Equations)

• Methods to solve linear equations:
  • Substitution
  • Elimination (Gauss)
  • Matrices (Gauss-Jordan)

1.3 Matrices et opérations matricielles (Matrices and Matrix Operations)

• A matrix is a table of numbers used to represent and operate on linear systems.
• Definition: An $m \times n$ matrix A is a table of numbers with m rows and n columns.
  • $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \ a_{21} & a_{22} &... & a_{2n} \... &... &... &... \ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$
• Matrix Operations:
  • Addition: $A + B$
  • Subtraction: $A - B$
  • Scalar Multiplication: $cA$
  • Matrix Multiplication: $AB$

1.4 Forme échelonnée et forme échelonnée réduite (Row Echelon Form and Reduced Row Echelon Form)

• Row Echelon Form:
  • All non-zero rows are above zero rows.
  • The leading coefficient (pivot) of each non-zero row is to the right of the leading coefficient of the previous row.
  • All elements in the column below a leading coefficient are zero.
• Reduced Row Echelon Form:
  • Meets Row Echelon Form conditions.
  • Leading coefficient of each non-zero row is 1.
  • All elements in the column above a leading coefficient are zero.
• Example:
  • Row Echelon Form: $\begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix}$
  • Reduced Row Echelon Form: $\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$

1.5 Théorème de Rouché-Fontené (Rouché-Capelli Theorem)

• Gives a necessary and sufficient condition for a system of linear equations to have solutions.
• Theorem: A system of linear equations has solutions if and only if the rank of the coefficient matrix equals the rank of the augmented matrix.
  • $rang(A) = rang([A|b])$
  • If $rang(A) = rang([A|b]) = n$, then the system has a unique solution.
  • If $rang(A) = rang([A|b]) < n$, then the system has infinite solutions.
  • If $rang(A) < rang([A|b])$, then the system has no solution.