Tumors of prostate

Matrizen (Matrices)

• Matrices are rectangular arrays of numbers
• A matrix $A$ is represented as: $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$
• $a_{ij}$ represents elements which are real ($\mathbb{R}$) or complex ($\mathbb{C}$) numbers

Matrix Dimensions

• $m$ denotes the number of rows in the matrix
• $n$ denotes the number of columns in the matrix
• $A \in \mathbb{R}^{m \times n}$ or $A \in \mathbb{C}^{m \times n}$ describes the dimensions of matrix A
• The element $a_{ij}$ is located in the $i$-th row and $j$-th column of the matrix

Matrix Example

• Example matrix: $A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix} \in \mathbb{R}^{2 \times 3}$
• The elements are:
  • $a_{11} = 1$
  • $a_{12} = 2$
  • $a_{13} = 3$
  • $a_{21} = 4$
  • $a_{22} = 5$
  • $a_{23} = 6$

Special Matrices

• Square Matrix: The number of rows equals the number of columns ($m = n$)
• Zero Matrix: All elements are zero ($a_{ij} = 0$ for all $i, j$)
• Identity Matrix ($I_n$): A square matrix with ones on the main diagonal and zeros elsewhere: $I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{pmatrix}$
• Diagonal Matrix: All non-diagonal elements are zero ($a_{ij} = 0$ for all $i \neq j$)
• Upper Triangular Matrix: All elements below the main diagonal are zero ($a_{ij} = 0$ for all $i > j$)
• Lower Triangular Matrix: All elements above the main diagonal are zero ($a_{ij} = 0$ for all $i < j$)
• Symmetric Matrix: A matrix that is equal to its transpose ($A = A^T$), which means $a_{ij} = a_{ji}$ for all $i, j$
• Antisymmetric Matrix: A matrix that is equal to the negative of its transpose ($A = -A^T$), where $a_{ij} = -a_{ji}$ for all $i, j$

Matrix Transpose

• The transpose of a matrix $A \in \mathbb{R}^{m \times n}$ is a matrix $A^T \in \mathbb{R}^{n \times m}$
• Rows become columns and columns become rows
• If $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ a_{31} & a_{32} \end{pmatrix}$, then $A^T = \begin{pmatrix} a_{11} & a_{21} & a_{31} \ a_{12} & a_{22} & a_{32} \end{pmatrix}$

Matrix Operations

Addition

• Defined for matrices $A, B \in \mathbb{R}^{m \times n}$
• $C = A + B$, where $c_{ij} = a_{ij} + b_{ij}$ for all $i, j$

Scalar Multiplication

• Defined for a matrix $A \in \mathbb{R}^{m \times n}$ and a scalar $\lambda \in \mathbb{R}$
• $C = \lambda A$, where $c_{ij} = \lambda a_{ij}$ for all $i, j$

Matrix Multiplication

• Defined for matrices $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{n \times p}$
• $C = A \cdot B \in \mathbb{R}^{m \times p}$, where $c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$ for all $i, j$
• Matrix multiplication is generally not commutative: $A \cdot B \neq B \cdot A$

Matrix Multiplication Example

• For $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}$:
  • $A \cdot B = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}$
  • $B \cdot A = \begin{pmatrix} 5 \cdot 1 + 6 \cdot 3 & 5 \cdot 2 + 6 \cdot 4 \ 7 \cdot 1 + 8 \cdot 3 & 7 \cdot 2 + 8 \cdot 4 \end{pmatrix} = \begin{pmatrix} 23 & 34 \ 31 & 46 \end{pmatrix}$

Inverse Matrix

• A square matrix $A \in \mathbb{R}^{n \times n}$ is invertible if there exists a matrix $A^{-1} \in \mathbb{R}^{n \times n}$
• Condition for invertibility: $A \cdot A^{-1} = A^{-1} \cdot A = I_n$
• $A^{-1}$ is the inverse matrix of $A$

Inverse Matrix Calculation

• The inverse matrix can be computed using:
  • Gauss-Jordan elimination
  • Cramer's rule

Inverse Matrix Properties

• $(A^{-1})^{-1} = A$
• $(A \cdot B)^{-1} = B^{-1} \cdot A^{-1}$
• $(A^T)^{-1} = (A^{-1})^T$

Determinant

• The determinant of a square matrix $A \in \mathbb{R}^{n \times n}$ is a scalar value, denoted as $\det(A)$ or $|A|$

Determinant Calculation

• $2 \times 2$ matrix: $\det(A) = a_{11} a_{22} - a_{12} a_{21}$
• $3 \times 3$ matrix: Use the Rule of Sarrus
• $n \times n$ matrix: Use the Laplace expansion

Determinant Properties

• $\det(A^T) = \det(A)$
• $\det(A \cdot B) = \det(A) \cdot \det(B)$
• $\det(A^{-1}) = \frac{1}{\det(A)}$
• $\det(I_n) = 1$
• $\det(A) = 0$ if $A$ has linearly dependent rows or columns

Linear Equation Systems (LES)

• A linear equation system is a set of linear equations: $a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n = b_1$ $a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} x_n = b_2$ $\vdots$ $a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n = b_m$
• Where $a_{ij}, b_i \in \mathbb{R}$ or $\mathbb{C}$

Matrix Notation for LES

• The linear equation system can be expressed in matrix form as: $A \cdot x = b$
• Where:
  • $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$
  • $x = \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix}$
  • $b = \begin{pmatrix} b_1 \ b_2 \ \vdots \ b_m \end{pmatrix}$

Methods for Solving LES

• Methods to solve linear equation systems:
  • Gaussian Elimination
  • Cramer's Rule
  • Inverse Matrix Method

Number of Solutions for LES

• A linear equation system can have:
  • No solution
  • A unique solution
  • Infinitely many solutions

Eigenvalues and Eigenvectors

• For a square matrix $A \in \mathbb{R}^{n \times n}$, a non-zero vector $v \in \mathbb{R}^n$ is an eigenvector of $A$
• Condition: There exists a scalar $\lambda \in \mathbb{R}$ such that $A \cdot v = \lambda v$
• $\lambda$ is the eigenvalue of $A$ corresponding to the eigenvector $v$

Eigenvalue Calculation

• The eigenvalues can be found using the characteristic polynomial: $\det(A - \lambda I_n) = 0$

Eigenvector Calculation

• Eigenvectors are found by solving the linear equation system: $(A - \lambda I_n) \cdot v = 0$

Properties of Eigenvalues and Eigenvectors

• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• The sum of the eigenvalues is equal to the trace of the matrix
• The product of the eigenvalues is equal to the determinant of the matrix

Vector Operations

Scalars and Vectors

• Scalar: A quantity characterized by a numerical value.
• Vector: A quantity with both magnitude and direction in space.

Vector Operations

Vector Multiplication by a Scalar

• (\vec{B} = a\vec{A}), where (\vec{A}) is a vector and a is a scalar.
  • Magnitude: (|B| = |aA|)
  • Direction:
    • Same as (\vec{A}) if a is positive.
    • Opposite to (\vec{A}) if a is negative.

Vector Addition

Triangle Rule

• (\vec{R} = \vec{A} + \vec{B})
• "Tip-to-tail" method where the tail of the second vector is placed at the tip of the first vector.

Parallelogram Law

• Method:
  1. Join the tails of (\vec{A}) and (\vec{B}).
  2. Draw a line from the head of each vector parallel to the other vector.
  3. The resultant (\vec{R}) is the diagonal of the parallelogram.

Vector Subtraction

• (\vec{R} = \vec{A} - \vec{B} = \vec{A} + (-\vec{B}))
• Defined as the addition of a negative vector.

Cartesian Vectors

Right-Handed Coordinate System

• Direction: Curl the fingers of the right hand from the x-axis toward the y-axis; the thumb points along the positive z-axis.

Rectangular Components of a Vector

• (\vec{A} = A_x\hat{\imath} + A_y\hat{\jmath} + A_z\hat{k})
  • (A_x), (A_y), (A_z) are the scalar components in the x, y, z directions.
  • (\hat{\imath}), (\hat{\jmath}), (\hat{k}) are the unit vectors in the x, y, z directions.

Magnitude of Cartesian Vector

• (A = \sqrt{A_x^2 + A_y^2 + A_z^2})

Direction of a Cartesian Vector

• Defined by the coordinate direction angles (\alpha), (\beta), (\gamma).
  • (\cos{\alpha} = \frac{A_x}{A}), (\cos{\beta} = \frac{A_y}{A}), (\cos{\gamma} = \frac{A_z}{A})
  • (\cos{\alpha}), (\cos{\beta}), (\cos{\gamma}) are the direction cosines of (\vec{A}).
  • Relationship: (\cos^2{\alpha} + \cos^2{\beta} + \cos^2{\gamma} = 1)

Unit Vector

• Unit vector in the direction of (\vec{A}):
  • (\hat{u}_A = \frac{\vec{A}}{A} = \frac{A_x}{A}\hat{\imath} + \frac{A_y}{A}\hat{\jmath} + \frac{A_z}{A}\hat{k} = \cos{\alpha}\hat{\imath} + \cos{\beta}\hat{\jmath} + \cos{\gamma}\hat{k})

Addition of Cartesian Vectors

• (\vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath} + (A_z + B_z)\hat{k})

Dot Product (Scalar Product)

• (\vec{A} \cdot \vec{B} = A B \cos{\theta})
  • (\theta) is the angle between vectors (\vec{A}) and (\vec{B}).
  • The dot product results in a scalar value.

Laws of Operation

• Commutative law: (\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A})
• Multiplication by a scalar: (a (\vec{A} \cdot \vec{B}) = (\alpha \vec{A}) \cdot \vec{B} = \vec{A} \cdot (\alpha \vec{B}))
• Distributive law: (\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C})

Cartesian Vector Formulation

• (\vec{A} \cdot \vec{B} = (A_x\hat{\imath} + A_y\hat{\jmath} + A_z\hat{k}) \cdot (B_x\hat{\imath} + B_y\hat{\jmath} + B_z\hat{k}))
• (\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z)

Applications

Angle Between Two Vectors

• (\cos{\theta} = \frac{\vec{A} \cdot \vec{B}}{AB} = \frac{A_x B_x + A_y B_y + A_z B_z}{AB})

Projection of a Vector

• (A' = A \cos{\theta} = \vec{A} \cdot \hat{u}_B), where (A') is the projection of (\vec{A}) onto (\vec{B}).

Cross Product (Vector Product)

• (\vec{C} = \vec{A} \times \vec{B})
  • Magnitude: (C = AB \sin{\theta})
  • Direction: (\vec{C}) is perpendicular to the plane containing (\vec{A}) and (\vec{B}) (right-hand rule).
  • The cross product results in a vector.

Laws of Operation

• Commutative law: (\vec{A} \times \vec{B} = - \vec{B} \times \vec{A})
• Multiplication by a scalar: (a (\vec{A} \times \vec{B}) = (\alpha \vec{A}) \times \vec{B} = \vec{A} \times (\alpha \vec{B}))
• Distributive law: (\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C})

Cartesian Vector Formulation

• (\vec{A} \times \vec{B} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{\imath} - (A_x B_z - A_z B_x)\hat{\jmath} + (A_x B_y - A_y B_x)\hat{k})

Algorithmic Game Theory

• Algorithmic Game Theory examines the connections between game theory and computer science.

Scope

• Algorithmic issues in game theory focus on the design and analysis of algorithms for solving game-theoretic problems like computing Nash equilibria or finding optimal auctions.
• Game-theoretic issues in algorithms concentrate on designing systems with strategic agents, such as mechanism design or creating networks robust to selfish actions.

Routing Games Example

• Each agent $i$ chooses a path $P_i$ from $s_i$ to $t_i$
• Each edge $e$ has a cost or latency $l_e(x)$, which is a function of the number of agents $x$ using $e$.
• The cost to agent $i$ is: $c_i = \sum_{e \in P_i} l_e(x_e)$ where $x_e$ is number of agents using edge $e$

Nash Equilibrium

• A state where no agent benefits by unilaterally changing their strategy, assuming other agents' strategies remain constant.

Braess Paradox Example

• A network with two paths from s to t, each having two edges: one with latency x and one with latency 1
• Adding a zero-latency edge from u to v (middle nodes on each path) causes all agents to switch to the path s->u->v->t
• The total cost for each agent increases, even though network capacity has increased.

Price of Anarchy (PoA)

• A measure of inefficiency in a Nash equilibrium
• $PoA = \frac{\text{cost of the worstcase Nash equilibrium}}{\text{optimal social cost}}$
• Social cost is the sum of all agents' costs
• Optimal social cost is the minimum social cost, assuming no selfish behavior
• In the Braess paradox where $l_e(x) = a_e x + b_e$ is less than or equal to 4/3

Torsional Vibration

Causes

• Diesel engines, reciprocating compressors, generators etc cause Torsional Vibration

Effects

• Torsional Vibration results in Noise, gear tooth wear, and coupling/shaft failure

Methods to Reduce

• Torsional Vibration Damper and Tuned Absorber can reduce vibration

Types of Torsional Vibration Damper

• Viscous Damper, Pendulum Damper, and Elastomeric Damper

Viscous Damper

• This consists of a housing, viscous fluid, and inertia ring. Vibration causes the inertia ring to lag, providing damping.

Viscous Damper Advantages

• Simple, reliable and effective over a wide range of frequencies. Also relatively inexpensive.

Viscous Damper Disadvantages

• Viscous Dampers can be bulky/heavy and affected by temperature

Pendulum Damper

• This consists of a pendulum attached to the rotating shaft and the out-of-phase oscillation provides damping.

Pendulum Damper Advantages

• Effective at reducing vibration at a specific frequency and is relatively small/lightweight.

Pendulum Damper Disadvantages

• Complicated and expensive. Performance is sensitive to changes in frequency.

Elastomeric Damper

• This consists of an elastomeric material bonded to the shaft and an inertia ring. Deformation of the elastomeric material provides damping.

Elastomeric Damper Advantages

• Simple and relatively inexpensive; effective over a wide range of frequencies

Elastomeric Damper Disadvantages

• Performance affected by temperature/frequency and can be bulky.

Tuned Absorber

• This reduces torsional vibration at specific frequencies.

Types of Tuned Absorber

• Harmonic Absorber and Lanchester Damper are types of Tuned Absorber

Harmonic Absorber

• Harmonic Absorbers consist of a mass and spring system tuned to the vibration frequency and provides damping.

Harmonic Absorber Advantages

• Effective at reducing vibration at a specific frequency and is relatively small/lightweight.

Harmonic Absorber Disadvantages

• Performance is sensitive to changes in frequency and can be complex/expensive.

Lanchester Damper

• This consists of two discs with friction material in between and the friction provides damping.

Lanchester Damper Advantages

• Simple and relatively inexpensive and effective over a wide range of frequencies.

Lanchester Damper Disadvantages

• Performance is affected by the friction material and can be noisy.

The Euclidean Algorithm

• Theorem: If $a, b \in \mathbb{Z}$, then $\exists x, y \in \mathbb{Z}$ such that $ax + by = gcd(a, b)$.

Method

• Given $a > b > 0$ as integers
• $a = q_1b + r_1$, $0 \leq r_1 < b$
• $b = q_2r_1 + r_2$, $0 \leq r_2 < r_1$
• $r_1 = q_3r_2 + r_3$, $0 \leq r_3 < r_2$
• $r_{n-2} = q_nr_{n-1} + r_n$, $0 \leq r_n < r_{n-1}$
• $r_{n-1} = q_{n+1}r_n + 0$
• The greatest common divisor of $a$ and $b$ is then $gcd(a, b) = r_n$.
• The Euclidean Algorithm provides $x, y$ such that $ax + by = gcd(a, b)$

Example

• Find the GCD (greatest common divisor) of 1001 and 330

• $1001 = 3 \cdot 330 + 11$

• $330 = 30 \cdot 11 + 0$

• The GCD is therefore $gcd(1001, 330) = 11$

Example

• Express 11 as the result of a linear combination of 1001 and 330

• $11 = 1001 - 3 \cdot 330 = 1001(1) + 330(-3)$

• So $x = 1$, $y = -3$.

The Fundamental Theorem of Arithmetic

• Every integer $n > 1$ has a unique prime factorization.
• A number $n > 1$, that is not prime can be written as $n = ab$ where $1 < a, b < n$.
• The factors ,$a$ and $b$ may be further factored. The final result will be a product of primes.