The Laplace Transform

The Laplace Transform

• Transforms a function f(t), defined for t ≥ 0, into a function F(s).
• F(s) = ∫₀^∞ e⁻ˢᵗ f(t) dt, where s is a complex frequency parameter (s = σ + jω).

Region of Convergence (ROC)

• Laplace transform exists when the integral converges.
• Convergence occurs when Re(s) > a for some real number a.
• ROC is defined as Re(s) > a.

Properties of Laplace Transforms

• Linearity: The transform of a linear combination is the linear combination of the transforms such that L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}.
• Time Scaling: L{f(at)} = (1/|a|)F(s/a).
• Time Shifting: L{f(t - a)u(t - a)} = e⁻ᵃˢF(s), where u(t) is the Heaviside step function.
• Shifting in the s-Domain: L{eᵃᵗf(t)} = F(s - a).
• Differentiation in the Time Domain: L{d/dt f(t)} = sF(s) - f(0).
• Differentiation in the Time Domain: L{d²/dt² f(t)} = s²F(s) - sf(0) - f'(0).
• Differentiation in the Time Domain: L{dⁿ/dtⁿ f(t)} = sⁿF(s) - sⁿ⁻¹f(0) - sⁿ⁻²f'(0) -... - f⁽ⁿ⁻¹⁾(0).
• Integration in the Time Domain: L{∫₀ᵗ f(τ) dτ} = F(s)/s.
• Differentiation in the s-Domain: L{tf(t)} = -d/ds F(s).
• Differentiation in the s-Domain: L{tⁿf(t)} = (-1)ⁿ dⁿ/dsⁿ F(s).
• Convolution: L{(f * g)(t)} = F(s)G(s), where (f * g)(t) = ∫₀ᵗ f(τ)g(t - τ) dτ.
• Initial Value Theorem: lim t→₀ f(t) = lim s→∞ sF(s).
• Final Value Theorem: lim t→∞ f(t) = lim s→₀ sF(s).

Common Laplace Transforms

• Unit Impulse: f(t) = δ(t), F(s) = 1, ROC: All s.
• Unit Step: f(t) = u(t), F(s) = 1/s, ROC: Re(s) > 0.
• Ramp: f(t) = t, F(s) = 1/s², ROC: Re(s) > 0.
• Exponential: f(t) = eᵃᵗ, F(s) = 1/(s - a), ROC: Re(s) > Re(a).
• Sine: f(t) = sin(ωt), F(s) = ω/(s² + ω²), ROC: Re(s) > 0.
• Cosine: f(t) = cos(ωt), F(s) = s/(s² + ω²), ROC: Re(s) > 0.
• Hyperbolic Sine: f(t) = sinh(at), F(s) = a/(s² - a²), ROC: Re(s) > |a|.
• Hyperbolic Cosine: f(t) = cosh(at), F(s) = s/(s² - a²), ROC: Re(s) > |a|.
• Damped Sine: f(t) = e⁻ᵃᵗsin(ωt), F(s) = ω/((s + a)² + ω²), ROC: Re(s) > -a.
• Damped Cosine: f(t) = e⁻ᵃᵗcos(ωt), F(s) = (s + a)/((s + a)² + ω²), ROC: Re(s) > -a.
• t to the power of n: f(t) = tⁿ, F(s) = n!/sⁿ⁺¹, ROC: Re(s) > 0.

Vectors

• Vectors are fundamental in physics.

Sum of Vectors

Graphical Method

• Vectors A and B are placed sequentially, maintaining their magnitude, direction, and sense.
• The resultant vector R connects the origin of the first vector to the end of the last vector.

Analytical Method

• Rectangular components of a vector:
  • Aₓ = A cos θ.
  • Ay = A sin θ.
• Sum of vectors using components:
  • Rₓ = Aₓ + Bₓ + ...
  • Ry = Ay + By + ...
  • R = √(Rₓ² + *Ry*²).
  • θ = arctan(Ry/Rₓ).

Product of Vectors

Scalar Product (Dot Product)

• Formula: A · B = AB cos θ = AₓBₓ + AyBy + AzBz.
• The result is a scalar value.

Vector Product (Cross Product)

• Formula: A × B = AB sin θ n̂.
• Cross product represented as a determinant: result = (AyBz - AzBy)î + (AzBₓ - AₓBz)ĵ + (AₓBy - AyBₓ)k̂.
• Result is a vector perpendicular to the plane formed by A and B.
• Direction determined by the right-hand rule.

Energy Bands and Charge Carriers in Semiconductors

Energy Bands

E vs. k Diagram

• Solutions to Schrödinger's equation for an electron in a crystal lattice can be expressed in this diagram.

Metals

• Fermi level lies within an allowed band, causing it to be partially filled.

Semiconductors

• The Fermi level is within the band gap, separating the filled valence band and the empty conduction band.

Insulators

• The Fermi level lies within a band gap, much larger than that of semiconductors.

Intrinsic Semiconductor

Definition

• A perfect crystal structure without impurities or lattice defects.

Materials

• Silicon (Si) and Germanium (Ge) are Group IV elements.
• Each atom is bonded covalently with four neighbors in a diamond crystal structure.

Energy Band Diagram at T = 0K

• All valence band states are full, and all conduction band states are empty.

Intrinsic Carrier Concentration (T > 0K)

• Thermally excited electrons jump from valence to conduction band, forming electron-hole pairs.
• n = p = ni, where ni is the intrinsic carrier concentration.

Temperature Dependence

• The intrinsic carrier concentration (ni) increases exponentially with temperature.

Fermi Level

• In an intrinsic semiconductor, the Fermi level (EF) lies near the middle of the band gap (Eg).

Intrinsic Carrier Concentration - Expression

• ni = √(NcNv) e^(-Eg/2kT).
  • Nc is the effective density of states in the conduction band.
  • Nv is the effective density of states in the valence band.
  • Eg is the band gap energy.
  • k is Boltzmann's constant.
  • T is in Kelvin.

Extrinsic Semiconductor

Doping

• Intentional addition of impurities to control electrical properties.

n-type Semiconductor

• Doped with donor impurities (e.g., Phosphorus in Silicon).
• Donors contribute electrons, increasing electron concentration (n > ni).
• EF moves closer to Ec.

p-type Semiconductor

• Doped with acceptor impurities (e.g., Boron in Silicon).
• Acceptors create holes, increasing hole concentration (p > ni).
• EF moves closer to Ev.

Compensation

• Occurs when both donor and acceptor impurities are present.
• Determined by impurity concentration to define conductivity type.

Carrier Transport Phenomena

Drift

• Charge carrier movement due to an electric field.
• Jdrift = σE = q(nμn + pμp)E.
  • σ is the conductivity.
  • E is the electric field.
  • q is the elementary charge.
  • n and p are electron and hole concentrations.
  • μn and μp are electron and hole mobilities.

Diffusion

• Movement from high to low concentration.
• Jdiffusion = qDn dn/dx - qDp dp/dx.
  • Dn and Dp are electron and hole diffusion coefficients.
  • dn/dx and dp/dx are the electron and hole concentration gradients.

Einstein Relation

• Relates diffusion coefficient and mobility.
  • Dn/ μn = Dp/ μp = kT/ q.
  • k is Boltzmann's constant.
  • T is temperature in Kelvin.
  • q is the elementary charge.

Algorithmic Complexity

Definition and Purpose

• Quantification the amount of resources, such as time or memory needed by an algorithm.
• Enables the comparison of efficiencies between different algorithms that solve the same problem.
• Prediction of how well an algorithm will scale with increasing input size is possible.

Reasons to study algorithmic complexity

• Aiding decision making regarding which algorithm best fits the situation.
• Enhancing code performance by optimization.
• Develop an understanding of computation limitations.

Methodology for assessing complexity

• Modeling the computation performed.
• Defining the input size "n".
• Counting operations as "n" changes.
• Expressing complexity using Big-O notation.

Models of Computation

• Turing Machine: Symbols manipulated on tape using rules table.
• Random Access Machine (RAM): Random access to memory.
• Word RAM: RAM with fixed size memory words.
• Real RAM: RAM that stores real numbers internally.

Big-O Notation

Definition

• "f(n) = O(g(n))" means there are constants c > 0 and n₀ > 0 where f(n) ≤ cg(n) for all n ≥ n₀.

Intuition

• "f(n)" grows no faster than "g(n)" as "n" approaches infinity.

Examples

• n² + n = O(n²*)*.
• 100n = O(n).
• log(n) = O(n).

Common complexities

Simplifying Big-O expressions

• Drop lower order terms: n² + n becomes n².
• Ignore constant factors, 100n becomes n.
• Always use the simplest expression: O(n² + n) becomes O(n²).
• Logarithms to different bases are equivalent, O(logₐ n) = O(logb n).

Common Searching algorithm complexities

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

Common Sorting algorithm complexities

• Bubble Sort: O(n²).
• Merge Sort: O(n log n).

Caveats

• Deals with the asymptotic behavior of algorithms.
• Ignores constant factors, which still matters in practice.
• Gives no information about actual run times.
• Differentiates between "worst" against "average" case run times.

Algorithmic Game Theory

• Game theory studies mathematical models of strategic interaction among rational agents playing games.
• It is applied across social science, logic, system science and computer science.

Cooperative game theory

• Focuses on groups of players forming coalitions.

Non-cooperative game theory

• Focuses on individual players and their strategies.

Normal-Form Games

• Games described through payoff matrices.

Definition

A normal-form game is defined by the tuple (N, A, u) where:

• N is the number of n players.
• A is the finite set of actions available to all players with "a" being action profiles.
• u specifies expected payoff (utility) within strategy profiles to individual agents in money or equivalent value.

Illustrative Game: Prisoner's Dilemma

• Scenario is two suspects arrested lacking sufficient evidence, suspects encouraged toward betrayal of each other, but offered limited incentive.

Prisoner's Dilemma representation

• N = {1, 2}.
• A ᵢ = {Cooperate, Defect}, for i ∈ N.

Prisoner's Dilemma payoff matrix

               | Cooperate | Defect    |
- --------------|-----------|-----------|
- *Cooperate**  | -1, -1    | -3, 0     |
- *Defect**     | 0, -3     | -2, -2    |

Strategies

Pure Strategy

• Taking action "ai" from "Ai".

Mixed Strategy

• Probability distribution across "Ai".
• Si denotes set of all possible mixed strategies.
• For si in Si, let si(ai) be the probability of playing action ai in Ai with strategy si

Strategy Profiles

• A tuple s = {s₁, … sn} where each player "i" selects mixed strategy in set "Si".
• Let S = S₁ × … × Sn denote the set of all strategy profiles.

Expected Utility

• Utility of player "i" under strategy profile "s": ui(s) = ∑(a in A) ui(a) ∏(j in N) sj(aj).

Nash Equilibrium

Definition

• The best strategy for a single player depends on what strategies other players are using.
• Profile s* is NE if, for all players i in N and all s ᵢ in Sᵢ, uᵢ(sᵢ*, s*-ᵢ*) >= uᵢ(sᵢ, s*-ᵢ), where s*-ᵢ are strategies apart from i.

Interpretation

• Given others' strategies, no one gains by unilaterally changing strategy.

Example within Prisoner's Dilemma

• Only NE is (Defect, Defect).

Theoretical Underpinning

• Every normal-game model with a finite number of players/actions has NE(s) in its mixed strategizing.