Planck's Constant: Quantum Mechanics

Planck's Constant

• Planck's constant is the quantum of action in quantum mechanics.
• It carries the dimensions of energy multiplied by time, also known as action.
• Planck's constant is denoted by h.
• The constant's value is $6.62607015 \times 10^{-34} J.s$.

Formula

• E = hν is the formula for calculating energy using Planck's constant.
• E represents the energy of the photon.
• h is Planck's constant.
• ν represents the frequency of the radiation.

Significance

• Planck's constant relates the energy of a photon to its frequency.
• It explains the quantization of energy and matter.
• The constant is fundamental in understanding atomic and subatomic processes.

Applications

• Planck's constant is used in calculations involving quantum phenomena.
• It is essential in fields like quantum computing, microscopy, and cryptography.
• Planck's constant is key to technologies like lasers, semiconductors, and medical imaging.

Algorithmic Game Theory

• Game theory studies mathematical models of strategic interactions among rational agents.
• It applies to social science, logic, systems science, and computer science.

Selfish Routing Model

• Network: G=(V, E)
  • V is the set of vertices.
  • E is the set of edges.
• Rate of traffic: $r_{s,t}$ between each pair of nodes $s, t \in V$.
• Set of players: $\bigcup_{s, t \in V} r_{s,t}$
• Strategy: Path from $s$ to $t$.
• Cost: Latency on the path.

Nash Equilibrium

• A Nash Equilibrium is a state where no player can unilaterally change strategy and decrease cost.

Social Cost

• The social cost of a Nash Equilibrium is the sum of all players' latencies.

Question

• How inefficient is a Nash Equilibrium?

Price of Anarchy (PoA)

• The price of anarchy is the ratio between the worst Nash Equilibrium's social cost and the social optimum.
• $PoA = \frac{\text{Social Cost of Worst Nash Equilibrium}}{\text{Social Optimum}}$.

Braess's Paradox Example

• Assume 1 unit of traffic going from s to t.

Without the dashed edge:

• The Nash Equilibrium is that all players take the path $s \rightarrow u \rightarrow t$.
• The social cost is $1 * (1 + 1) = 2$.
• $2$ is also the social optimum.

With the dashed edge:

• The Nash Equilibrium is that all players take the path $s \rightarrow v \rightarrow u \rightarrow t$.
• The social cost is $1 * (1 + 1) = 3$.
• The social optimum is 1/2 of the players take the path $s \rightarrow u \rightarrow t$ and 1/2 of the players take the path $s \rightarrow v \rightarrow t$.
• The social cost is $1/2 * (1 + 1/2) + 1/2 * (1/2 + 1) = 3/2$.
• Therefore, in this example the $PoA = \frac{2}{3} = 4/3$

Algèbre linéaire (Linear Algebra)

Définitions (Definitions)

• Scalaire (Scalar): A scalar is a real or complex number.
• Vecteur (Vector): A vector is an ordered list of scalars.
• Matrice (Matrix): A matrix is a rectangular array of scalars.
• Espace vectoriel (Vector space): A vector space is a set of vectors equipped with two operations: addition and multiplication by a scalar, which satisfy certain properties.

Opérations sur les vecteurs (Operations on Vectors)

• Addition (Addition): The addition of two vectors is done component by component.
• Multiplication par un scalaire (Multiplication by a scalar): The multiplication of a vector by a scalar is done by multiplying each component of the vector by the scalar.
• Produit scalaire (Dot product): The dot product of two vectors is a scalar. It can be calculated in different ways, for example:
  • $ \vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos(\theta)$ where $ \theta $ is the angle between $ \vec{u} $ and $ \vec{v} $.
  • $\vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i$ where $u_i$ and $v_i$ are the components of $ \vec{u} $ and $ \vec{v} $ respectively.
• Produit vectoriel (Cross product): The cross product of two vectors is a vector orthogonal to the two original vectors. It is only defined in three-dimensional space.

Opérations sur les matrices (Operations on Matrices)

• Addition (Addition): The addition of two matrices is done element by element.
• Multiplication par un scalaire (Multiplication by a scalar): The multiplication of a matrix by a scalar is done by multiplying each element of the matrix by the scalar.
• Multiplication de matrices (Matrix multiplication): The multiplication of two matrices $A$ and $B$ is only possible if the number of columns of $A$ is equal to the number of rows of $B$. The result is a matrix whose number of rows is equal to the number of rows of $A$ and the number of columns is equal to the number of columns of $B$.
• Transposition (Transposition): The transposition of a matrix consists of exchanging the rows and columns.
• Inverse (Inverse): The inverse of a matrix $A$, noted $A^{-1}$, is a matrix such that $A \cdot A^{-1} = A^{-1} \cdot A = I$, where $I$ is the identity matrix.

Concepts importants (Important Concepts)

• Indépendance linéaire (Linear independence): Vectors are linearly independent if none of them can be written as a linear combination of the others.
• Base (Basis): A basis of a vector space is a set of linearly independent vectors that span the vector space.
• Dimension (Dimension): The dimension of a vector space is the number of vectors in a basis of that vector space.
• Valeurs propres et vecteurs propres (Eigenvalues and eigenvectors): An eigenvector of a matrix $A$ is a vector $\vec{v}$ such that $A\vec{v} = \lambda \vec{v}$, where $ \lambda $ is a scalar called eigenvalue.
• Déterminant (Determinant): The determinant of a matrix is a scalar that can be calculated from the elements of the matrix. It gives information about the properties of the matrix, for example whether it is invertible or not.
• Rang (Rank): The rank of a matrix is the number of linearly independent columns (or rows) of the matrix.
• Systèmes d'équations linéaires (Systems of linear equations): A system of linear equations is a set of linear equations. It can be solved using different methods, for example the Gauss-Jordan method or Cramer's rule.
• Espaces vectoriels (Vector spaces): A vector space is a set of vectors equipped with two operations: addition and multiplication by a scalar, which satisfy certain properties.

Applications

Linear algebra is used in many fields, for example:

• Computer graphics
• Signal processing
• Machine learning
• Physics
• Economics