Laplace Transforms: Properties and Applications

The Laplace Transform

• The Laplace transform of a function $f(t)$ is defined as $F(s) = \mathcal{L} {f(t)} = \int_{0}^{\infty} e^{-st} f(t) dt$ for $t \geq 0$.
• The parameter $s$ is a complex number frequency, $s = \sigma + j\omega$.
• The Laplace transform exists when the integral converges, which occurs when $Re(s) > a$ for some real number $a$; this is the region of convergence (ROC).

Properties of Laplace Transforms

• 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: $\mathcal{L} {f(t - a)u(t - a)} = e^{-as}F(s)$, where $u(t)$ 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:
  • $\mathcal{L} {(f * g)(t)} = F(s)G(s)$.
  • $(f * g)(t) = \int_{0}^{t} f(\tau)g(t - \tau) d\tau$.
• 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

Vectors

Vector Addition

Graphical Method:

• Vectors A and B are placed one after the other, maintaining their original magnitude, direction, and sense.
• Resultant vector R is found by connecting the origin of the first vector to the end of the last vector.

Analytical Method:

• Rectangular Components of a Vector:
• $A_x = A \cos \theta$
• $A_y = A \sin \theta$
• Vector Addition via 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 Products

Scalar Product (Dot Product)

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

Vector Product (Cross Product)

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

Energy Bands and Charge Carriers

4.1 Energy Bands

• E vs. k Diagram: solutions of Schrödinger's equation for an electron in a crystal lattice.
• Metals: Fermi level lies within an allowed band, resulting in a partially filled band.
• Semiconductors: Fermi level lies within a band gap separating a completely filled valence band and an empty conduction band.
• Insulators: Fermi level lies within a large band gap.

4.2 Intrinsic Semiconductor

• Intrinsic Semiconductor: perfect crystal with no impurities or lattice defects.
• Silicon (Si) and Germanium (Ge) are Group IV elements with each atom covalently bonded to four neighbors.
• Energy Band Diagram at $T=0K$: all valence band states are full, conduction band states are empty.
• Intrinsic Carrier Concentration at $T > 0K$: Some electrons are thermally excited creating electron-hole pairs.
• $n = p = n_i$, where $n_i$ is the intrinsic carrier concentration.
• Temperature Dependence: $n_i$ increases exponentially with temperature.
• Fermi Level ($E_F$) in an intrinsic semiconductor lies near the middle of the band gap ($E_g$).
• Mathematical Expression: $n_i = \sqrt{N_c N_v} e^{-E_g / 2kT}$, where $N_c$ and $N_v$ are the effective density of states, $E_g$ is the band gap energy, k is Boltzmann's constant, and T is temperature.

4.3 Extrinsic Semiconductor

• Doping: Intentionally adding impurities to control electrical properties.
• n-type Semiconductor: doped with donor impurities (e.g., Phosphorus in Silicon).
• Donors contribute electrons to the conduction band, $n > n_i$.
• $E_F$ is closer to $E_c$
• p-type Semiconductor: Doped with acceptor impurities (e.g., Boron in Silicon).
• Acceptors create holes in the valence band, $p > n_i$.
• $E_F$ is closer to $E_v$
• Compensation: Both donor and acceptor impurities are present, conductivity type determined by the impurity with the higher concentration.

4.4 Carrier Transport Phenomena

• Drift: Motion of charge carriers in response to an electric field.
• $J_{drift} = \sigma E = q(n\mu_n + p\mu_p)E$,
• $\sigma$ is the conductivity, $E$ is the electric field, $q$ is the elementary charge.
• $n$ and $p$ are the electron and hole concentrations, $\mu_n$ and $\mu_p$ are the electron and hole mobilities.
• Diffusion: Movement of charge carriers from high to low concentration.
• $J_{diffusion} = qD_n \frac{dn}{dx} - qD_p \frac{dp}{dx}$,
• $D_n$ and $D_p$ are diffusion coefficients, $\frac{dn}{dx}$ and $\frac{dp}{dx}$ are concentration gradients.
• Einstein Relation: Relates diffusion coefficient and mobility.
• $\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q}$,
• $k$ is Boltzmann's constant, $T$ is the temperature (Kelvin), $q$ is the elementary charge.

Algorithmic Complexity

• Algorithmic complexity quantifies resource needs like time or memory.
• It's used to compare algorithm efficiency and predict scaling with input size, $n$.
• This helps in algorithm selection, code optimization, and understanding computational limits.

Determining Complexity

1. Model Selection: Choose a model of computation (e.g., Turing Machine, RAM).
2. Input Size: Define the input size $n$.
3. Operation Count: Count operations as a function of $n$.
4. Big-O Notation: Express complexity using Big-O notation.

Computational Models

• Turing Machine: theoretical model of computation that manipulates symbols on a strip of tape according to a table of rules.
• Random Access Machine (RAM): A more practical model of computation that allows random access to memory.
• Word RAM: A RAM model where memory is divided into words of a fixed size.
• Real RAM: A RAM model that allows real numbers to be stored in memory.

Big-O Notation

• Definition: $f(n) = O(g(n))$ if there exist positive constants $c$ and $n_0$ 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

Simplifying Big-O Notation

• Drop lower order terms (e.g., $n^2 + n \rightarrow n^2$).
• Ignore constant factors (e.g., $100n \rightarrow n$).
• Use the simplest possible expression (e.g., $O(n^2 + n) \rightarrow O(n^2)$).
• Logarithms to different bases are equivalent ($O(log_a n) = O(log_b n)$).

Example: Searching

• Linear Search: Complexity is $O(n)$
• Binary Search: Complexity is $O(log n)$

Example: Sorting

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

Caveats

• Big-O describes asymptotic behavior.
• Constant factors can matter in practice.
• Big-O doesn't provide actual running time.
• There's a distinction between worst-case and average-case complexity.

Algorithmic Game Theory

What is Game Theory?

• Game theory is the study of mathematical models of strategic interactions among rational agents, applicable in social science, logic, systems science, and computer science.
• Cooperative: Focuses on groups of players forming coalitions.
• Non-Cooperative: Focuses on individual players and their strategies.

Normal-Form Games

• Games described via a payoff matrix.

Definition:

• A normal-form game is a tuple $(N, A, u)$, where:
• $N$ is a finite set of $n$ players, indexed by $i$.
• $A = A_1 \times \dots \times A_n$, where $A_i$ is a finite set of actions available to player $i$. Each $a = (a_1, \dots, a_n) \in A$ is an action profile.
• $u = (u_1, \dots, u_n)$, where $u_i : A \mapsto \mathbb{R}$ is a utility function for player $i$. $u_i(a)$ gives the payoff to player $i$ when the action profile is $a$.

Example: Prisoner's Dilemma

• Two suspects are arrested; police lack evidence unless one confesses.
• If one confesses and the other doesn't, the confessor is freed, and the other gets 3 years.
• If both confess, they both get 2 years; if neither confesses, they both get 1 year.

Game Representation:

• $N = {1, 2}$
• $A_i = {\text{Cooperate}, \text{Defect}}$, for $i \in N$
• $u_i$ is defined by the table:

Strategies

Pure Strategy:

• A pure strategy for player $i$ is simply an action $a_i \in A_i$.

Mixed Strategy:

• A mixed strategy for player $i$ is a probability distribution over $A_i$.
• $S_i$ denotes the set of all possible mixed strategies for player $i$.
• For $s_i \in S_i$, $s_i(a_i)$ is the probability of playing action $a_i \in A_i$ under mixed strategy $s_i$.

Strategy Profile:

• A strategy profile is a tuple $s = (s_1, \dots, s_n)$, where $s_i \in S_i$ is a mixed strategy for player $i$.
• $S = S_1 \times \dots \times S_n$ denotes the set of all strategy profiles.

Expected Utility:

• The expected utility of player $i$ under strategy profile $s$ is:
• $u_i(s) = \sum_{a \in A} u_i(a) \prod_{j \in N} s_j(a_j)$

Nash Equilibrium

Definition:

• A strategy profile $s^* \in S$ is a Nash Equilibrium if, for every player $i \in N$ and every strategy $s_i \in S_i$:
• $u_i(s_i^, s_{-i}^) \ge u_i(s_i, s_{-i}^*)$
• where $s_{-i}^*$ denotes the strategies of all players except $i$.

Interpretation:

• No player can unilaterally improve their payoff by deviating from their strategy in a Nash Equilibrium.

Example: Prisoner's Dilemma: The only Nash Equilibrium is (Defect, Defect).

Existence of Nash Equilibrium

Theorem (Nash, 1950):

• Every normal-form game with a finite number of players and actions has at least one Nash Equilibrium (possibly in mixed strategies).