Laplace Transform Properties

The Laplace Transform

• Transforms a function $f(t)$ (defined for $t \geq 0$) into a function $F(s)$.
• Defined by $F(s) = \mathcal{L} {f(t)} = \int_{0}^{\infty} e^{-st} f(t) dt$.
• The parameter $s$ is a complex number frequency, $s = \sigma + j\omega$.

Region of Convergence (ROC)

• Specifies where the Laplace transform integral converges,
• Integral converges when $Re(s) > a$ for some real number $a$.
• The region of convergence (ROC) is $Re(s) > a$.

Properties of the Laplace Transform

• Linearity: $\mathcal{L} {af(t) + bg(t)} = a\mathcal{L} {f(t)} + b\mathcal{L} {g(t)}$.
• Time Scaling: $\mathcal{L} {f(at)} = \frac{1}{|a|} F(\frac{s}{a})$.
• Time Shifting: $\mathcal{L} {f(t - a)u(t - a)} = e^{-as}F(s)$, $u(t)$ is the Heaviside step function.
• Shifting in the s-Domain: $\mathcal{L} {e^{at}f(t)} = F(s - a)$.
• Differentiation in the Time Domain:
  • $\mathcal{L} {\frac{d}{dt}f(t)} = sF(s) - f(0)$.
  • $\mathcal{L} {\frac{d^2}{dt^2}f(t)} = s^2F(s) - sf(0) - f'(0)$.
  • $\mathcal{L} {\frac{d^n}{dt^n}f(t)} = s^nF(s) - s^{n-1}f(0) - s^{n-2}f'(0) -... - f^{(n-1)}(0)$.
• Integration in the Time Domain: $\mathcal{L} {\int_{0}^{t} f(\tau) d\tau} = \frac{F(s)}{s}$.
• Differentiation in the s-Domain:
  • $\mathcal{L} {tf(t)} = -\frac{d}{ds}F(s)$.
  • $\mathcal{L} {t^nf(t)} = (-1)^n \frac{d^n}{ds^n}F(s)$.
• Convolution: $\mathcal{L} {(f * g)(t)} = F(s)G(s)$, where $(f * g)(t) = \int_{0}^{t} f(\tau)g(t - \tau) d\tau$.
• Initial Value Theorem: $\lim_{t \to 0} f(t) = \lim_{s \to \infty} sF(s)$.
• Final Value Theorem: $\lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s)$.

Common Laplace Transforms

• Unit Impulse: $f(t) = \delta(t)$, $F(s) = 1$, ROC: All s.
• Unit Step: $f(t) = u(t)$, $F(s) = \frac{1}{s}$, ROC: $Re(s) > 0$.
• Ramp: $f(t) = t$, $F(s) = \frac{1}{s^2}$, ROC: $Re(s) > 0$.
• Exponential: $f(t) = e^{at}$, $F(s) = \frac{1}{s - a}$, ROC: $Re(s) > Re(a)$.
• Sine: $f(t) = \sin(\omega t)$, $F(s) = \frac{\omega}{s^2 + \omega^2}$, ROC: $Re(s) > 0$.
• Cosine: $f(t) = \cos(\omega t)$, $F(s) = \frac{s}{s^2 + \omega^2}$, ROC: $Re(s) > 0$.
• Hyperbolic Sine: $f(t) = \sinh(at)$, $F(s) = \frac{a}{s^2 - a^2}$, ROC: $Re(s) > |a|$.
• Hyperbolic Cosine: $f(t) = \cosh(at)$, $F(s) = \frac{s}{s^2 - a^2}$, ROC: $Re(s) > |a|$.
• Damped Sine: $f(t) = e^{-at}\sin(\omega t)$, $F(s) = \frac{\omega}{(s + a)^2 + \omega^2}$, ROC: $Re(s) > -a$.
• Damped Cosine: $f(t) = e^{-at}\cos(\omega t)$, $F(s) = \frac{s + a}{(s + a)^2 + \omega^2}$, ROC: $Re(s) > -a$.
• t to the power of n: $f(t) = t^n$, $F(s) = \frac{n!}{s^{n+1}}$, ROC: $Re(s) > 0$.

Vectors in Physics

• Vectors are physical quantities with magnitude, direction, and sense.

Vector Addition

Graphical Method

• Vectors are placed one after another maintaining their original magnitude, direction, and sense.
• The resultant vector connects the origin of the first vector to the end of the last vector.

Analytical Method

• Rectangular Components:
  • $A_x = A \cos \theta$.
  • $A_y = A \sin \theta$.
• Sum by Components:
  • $R_x = A_x + B_x +...$.
  • $R_y = A_y + B_y +...$.
  • $R = \sqrt{R_x^2 + R_y^2}$.
  • $\theta = \arctan \frac{R_y}{R_x}$.

Vector Multiplication

Scalar (Dot) Product

• $\vec{A} \cdot \vec{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z$.
• Result is a scalar.

Vector (Cross) Product

• $\vec{A} \times \vec{B} = AB \sin \theta \hat{n}$.
• $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$.
• Result is a vector perpendicular to the plane of A and B.
• Direction is determined by the right-hand rule.

Energy Bands in Semiconductors

E vs. k Diagram

• Represents solutions to the Schrödinger equation for electrons in a crystal lattice.
• Illustrates the relationship between energy ($E$) and wave vector ($k$).

Metals

• Fermi level lies within an allowed band, which causes a partially filled band.

Semiconductors

• Fermi level lies within a band gap, separating a full valence band from an empty conduction band.

Insulators

• Fermi level lies within a band gap, which is significantly larger compared to semiconductors.

Intrinsic Semiconductor

Definition

• A pure crystal is an intrinsic semiconductor.
• It contains no impurities or lattice defects.

Materials

• Silicon (Si) and Germanium (Ge) are Group IV elements with a diamond structure.
• Atoms are covalently bonded to four neighbors.

Energy Band Diagram at 0K

• All valence band states are full.
• Conduction band states are empty.

Intrinsic Carrier Concentration at T > 0K

• Thermal excitation creates electron-hole pairs.
• Electron concentration ($n$) equals hole concentration ($p$), denoted as $n = p = n_i$.
• $n_i$ is the intrinsic carrier concentration.

Temperature Dependence

• Intrinsic carrier concentration ($n_i$) increases exponentially with temperature.

Fermi Level in Intrinsic Semiconductor

• The Fermi level ($E_F$) is near the middle of the band gap ($E_g$).

Mathematical Expression for Intrinsic Carrier Concentration

• Formula: $n_i = \sqrt{N_c N_v} e^{-E_g / 2kT}$.
  • $N_c$ = effective density of states in the conduction band.
  • $N_v$ = effective density of states in the valence band.
  • $E_g$ = band gap energy.
  • $k$ = Boltzmann's constant.
  • $T$ = temperature in Kelvin.

Extrinsic Semiconductor

Doping Definition

• Intentional impurities are added into a semiconductor in order to manage its electrical properties.

n-type Semiconductor

• Doped with donor impurities which have more valence electrons.
• Electrons are contributed to the conduction Band.
• Electron concentration ($n > n_i$) is increased.
• $E_F$ is closer to $E_c$.

p-type Semiconductor

• Doped with acceptor impurities with fewer valence electrons.
• Holes are created in the valence band.
• Hole concentration ($p > n_i$) is increased.
• $E_F$ is closer to $E_v$.

Compensation

• Both donor and acceptor impurities are present in the material.
• The conductivity type is determined by the higher concentration impurity.

Carrier Transport Phenomena

Drift

• Motion of charge carriers responding to an electric field.
• Drift current density: $J_{drift} = \sigma E = q(n\mu_n + p\mu_p)E$.
  • $\sigma$ = conductivity.
  • $E$ = electric field.
  • $q$ = elementary charge.
  • $n$ and $p$ = electron and hole concentrations.
  • $\mu_n$ and $\mu_p$ = electron and hole mobilities.

Diffusion

• Movement of charge carriers is from high to low concentration areas.
• Diffusion current density: $J_{diffusion} = qD_n \frac{dn}{dx} - qD_p \frac{dp}{dx}$.
  • $D_n$ and $D_p$ = electron and hole diffusion coefficients.
  • $\frac{dn}{dx}$ and $\frac{dp}{dx}$ = electron and hole concentration gradients.

Einstein Relation

• Relates the diffusion coefficient and mobility of charge carriers:
  • $\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q}$.
  • $k$ = Boltzmann's constant.
  • $T$ = temperature in Kelvin.
  • $q$ = elementary charge.

Algorithmic Complexity: Quantifying Resource Usage

Purpose of Algorithmic Complexity

• Measures resources (time, memory) required by an algorithm.
• Compares the efficiency of algorithms.
• Forecasts algorithm scaling with increasing input size.

Why Use It?

• Select appropriate algorithms.
• Optimize code.
• Understand computational limitations.

How It's Determined

1. Select a computation model.
2. Define input Size $n$.
3. Count operations as a function of $n$.
4. Express complexity with Big-O notation.

Models of Computation

• Turing Machine: theoretical model using symbols on tape to manipulate symbols according to rules.
• Random Access Machine (RAM): Practical model that allows random memory access.
• Word RAM: RAM model with memory divided into fixed-size words.
• Real RAM: permits real numbers stored in memory.

Big-O Notation

• Definition: $f(n) = O(g(n))$ if positive constants $c$ and $n_0$ exist such that $f(n) \le cg(n)$ for all $n \ge n_0$.
• Intuition: $f(n)$ grows no faster than $g(n)$ as $n$ approaches infinity.
• Examples:
  • $n^2 + n = O(n^2)$.
  • $100n = O(n)$.
  • $log(n) = O(n)$.

Common Complexities

• Constant: $O(1)$, accesses an array element.
• Logarithmic: $O(log n)$, binary search.
• Linear: $O(n)$, loops through an array.
• Log-Linear: $O(n log n)$, merge sort.
• Quadratic: $O(n^2)$, nested loops.
• Cubic: $O(n^3)$, matrix multiplication.
• Exponential: $O(2^n)$, traveling salesman problem.

Simplifying Big-O Expressions

• Drop lower-order terms: e.g., $n^2 + n \rightarrow n^2$.
• Ignore constant factors: e.g., $100n \rightarrow n$.
• Use simplest possible expression.
• Logarithms to different bases are equivalent.

Examples

Searching Algorithms

• Linear Search: $O(n)$.
• Binary Search: $O(log n)$.

Sorting Algorithms

• Bubble Sort: $O(n^2)$.
• Merge Sort: $O(n log n)$.

Caveats

• Big-O describes asymptotic behavior.
• Constant factors matter in practice.
• Big-O doesn't give actual running time.
• Consider worst-case vs average-case complexity.

Algorithmic Game Theory

Definition of Game Theory

• Studies mathematical models of strategic interactions between rational agents.
• Has applications in social science, logic, systems science, and computer science.

Cooperative Games

• Focus on groups to form coalitions.

Non-Cooperative Games

• Focus on individual players and their strategies.

Normal-Form Games

Defining Games via Payoff Matrices

• A tuple $(N, A, u)$, where:
  • $N$ is a finite set of n players.
  • $A$ is a a set of actions available to players, $a$ is an action profile.
  • $u$ is the utility function; payoff when action profile is $a$.

Prisoner's Dilemma

• Two suspects are arrested.
• Insufficient evidence without a confession.
• Suspect confesses and other doesn't: confessor freed, other jailed for 3 years.
• Both confess: both jailed for 2 years.
• Neither confesses: both jailed for 1 year.

Prisoner's Dilemma Game Representation

• Players: $N = {1, 2}$.
• Actions: $A_i = {\text{Cooperate}, \text{Defect}}$.
• Utility matrix:

Game Strategies

Pure Strategy

• An particular action $a_i \in A_i$ for player $i$.

Mixed Strategy

• A probability distribution covering $A_i$, the strategies available for player $i$.
• $S_i$: set of player $i$'s mixed strategies.
• $s_i(a_i)$: probability of playing $a_i$ under mixed strategy $s_i$.

Strategy Profile

• Tuple $s = (s_1, \dots, s_n)$, where $s_i \in S_i$.
• $S = S_1 \times \dots \times S_n$: set of all strategy profiles.

Expected Utility

• An player's expected utility under the strategy profile $s$:
  • $u_i(s) = \sum_{a \in A} u_i(a) \prod_{j \in N} s_j(a_j)$.

Nash Equilibrium

Definition

• Strategy profile $s^* \in S$ is where player $i$ and strategy $s_i$ are at equilibrium.

$u_i(s_i^, s_{-i}^) \ge u_i(s_i, s_{-i}^*)$

• Where $s_{-i}^*$ denotes the strategies of all players except $i$.

Interpretation

• No player benefits from changing unilaterally.

Prisoner's Dilemma Example

• (Defect, Defect) is the Nash Equilibrium for Prisoners Dilemma.

Existence Theorem

• Every finite normal-form game has at least one Nash Equilibrium.
• Possibly in mixed strategies.