Fourier Transform Properties

Fourier Transform Properties

• Fourier Transform is a linear operator
• Expressed mathematically: $F{af(t) + bg(t)} = aF(f(t)) + bF(g(t))$

Time Shift

• A shift in the time domain results in a linear phase shift in the frequency domain
• Mathematically: $F{f(t - t_0)} = e^{-jw t_0}F(w)$

Frequency Shift

• A shift in the frequency domain results in a complex exponential multiplication in the time domain
• Represented as: $F{e^{jw_0t}f(t)} = F(w - w_0)$

Scaling

• Scaling in the time domain is inversely proportional to scaling in the frequency domain
• Expressed as: $F{f(at)} = \frac{1}{|a|}F(\frac{w}{a})$

Time Reversal

• Time reversal in the time domain corresponds to time reversal in the frequency domain
• $F{f(-t)} = F(-w)$

Differentiation

• Differentiation in the time domain corresponds to multiplication by $jw$ in the frequency domain
• Mathematically: $F{\frac{df(t)}{dt}} = jwF(w)$

Integration

• Integration in the time domain corresponds to division by $jw$ in the frequency domain plus an impulse at the origin
• $F{\int_{-\infty}^{t}f(\tau)d\tau} = \frac{1}{jw}F(w) + \pi F(0)\delta(w)$

Multiplication

• Multiplication in the time domain corresponds to convolution in the frequency domain
• $F{f(t)g(t)} = \frac{1}{2\pi}[F(w) * G(w)]$

Convolution

• Convolution in the time domain corresponds to multiplication in the frequency domain
• $F{f(t) * g(t)} = F(w)G(w)$

Parseval's Theorem

• Relates the energy of a signal in the time domain to its energy in the frequency domain
• $\int_{-\infty}^{\infty}|f(t)|^2dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|F(w)|^2dw$

Duality

• If $F(w)$ is the Fourier transform of $f(t)$, then the Fourier transform of $F(t)$ is $2\pi f(-w)$
• if $f(t) \leftrightarrow F(w)$, then $F(t) \leftrightarrow 2\pi f(-w)$

Algorithmic Game Theory

• Study of multi-agent decision problems where agents aim to maximize their own utility in cooperative or non-cooperative settings

Goal of Algorithmic Game Theory

• Define solution concepts and predict agent behavior

Computer Science Perspective on Game Theory

• Design rules to achieve goals
• Predict game outcomes, assessing computational difficulty

Economics/Game Theory Perspective on Algorithm Design

• Design algorithms for strategic environments
• Guarantee good performance despite self-interested agents

Example: Selfish Routing

• Network of $n$ agents, each controlling a unit of traffic between a source and destination
• Agent $i$ chooses a path $p_i$ from source $s_i$ to destination $t_i$
• Cost of edge $e$ depends on the total traffic $f_e$ on that edge: $c_e(f_e)$
• Cost to agent $i$ is the sum of the costs of edges on its path: $\sum_{e \in p_i} c_e(f_e)$
• Agents want to minimize their cost

Key Questions of Selfish Routing

• Understanding the equilibrium's nature and inefficiency
• Computational complexity of finding an equilibrium
• Designing mechanisms to improve equilibrium efficiency

Topics in Algorithmic Game Theory

• Congestion Games, Selfish Routing
• Mechanism Design
• Auctions
• Social Choice
• Fair Division
• Matching
• Coalition Formation
• Network Formation
• Learning in Games
• Price of Anarchy
• Complexity of Equilibria
• Algorithmic Mechanism Design
• Online Mechanism Design

Regulation of Gene Expression

• Process by which information from a gene is used in the synthesis of a functional gene product
• These products are often proteins, but can also be functional RNAs (tRNA, snRNA)
• All known life uses gene expression to produce macromolecules
• Regulated to control when, where, and how much of a gene product is produced

Mechanism of Gene Expression Regulation

• Regulation can occur at various steps, from transcription to post-translational modification
• Used by viruses, prokaryotes, and eukaryotes to enhance adaptability by controlling protein synthesis

Regulation During Transcription

• The first control point in gene expression
• Transcription speed varies by gene
• Highly expressed genes are frequently transcribed, this is common in genes coding for proteins/RNAs which are essential to cell function
• Less needed genes are transcribed less frequently

Control Points During Transcription

• Initiation of transcription is most important, mediated through chromatin structure, promoters, enhancers, and silencers
• RNA transcript processing, including splicing, capping, and polyadenylation

Regulation During Translation

• The second major control point
• Rate of mRNA translation into protein is variable, depending on ribosome availability, tRNA availability, and the presence of initiation/elongation factors

Regulation After Translation

• The third major control point
• Post-translational modifications alter protein activity, location, or stability such as chemical groups, removal of amino acids and addition of carbohydrates and lipids

Heat Kernel Defined

• Fundamental solution to the heat equation
• $\frac{\partial}{\partial t} K(x,y;t) = \Delta_x K(x,y;t)$
• Initial condition: $\lim_{t \to 0} K(x,y;t) = \delta(x-y)$
• $\Delta_x$ is the Laplacian with respect to $x$
• $K(x,y;t)$ represents the temperature at point $x$ at time $t$ if an initial unit of heat is placed at point $y$ at time $t=0$.

Heat Kernel Properties

• Symmetry: $K(x,y;t) = K(y,x;t)$
• Positivity: $K(x,y;t) > 0$ for all $x, y$ and $t > 0$
• Normalization: $\int K(x,y;t) dx = 1$ for all $y$ and $t > 0$
• Convolution Property: $K(x,y; t_1 + t_2) = \int K(x,z; t_1) K(z,y; t_2) dz$

Heat Kernel Examples

• Euclidean Space $\mathbb{R}^n$: $K(x,y;t) = \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x-y|^2}{4t}}$
• Circle $S^1$: $K(x,y;t) = \frac{1}{2\pi} \sum_{k=-\infty}^{\infty} e^{-k^2 t} e^{ik(x-y)}$
• Compact Riemannian Manifold $M$: $K(x,y;t) = \sum_{k=0}^{\infty} e^{-\lambda_k t} \phi_k(x) \phi_k(y)$, here $\lambda_k$ are the eigenvalues of the Laplacian and $\phi_k$ are the corresponding eigenfunctions

Heat Kernel Applications

• Solving the Heat Equation: If $u(x,0) = f(x)$, then $u(x,t) = \int K(x,y;t) f(y) dy$
• Spectral Geometry: Analysis of the Laplacian's spectrum using the heat kernel via $\int K(x,x;t) dx$, which has an asymptotic expansion as $t \to 0$, whose coefficients are spectral invariants
• Diffusion Processes: Relation to diffusion processes and Brownian motion

Gas Pressure Defined

• Gas pressure is the force exerted per unit area by gas molecules on the surfaces of objects
• Expressed as: $Pressure = \frac{Force}{Area}$

Units of Pressure

• Pascal (Pa)
• Atmosphere (atm)
• Torr
• mm Hg
• Inches Hg
• PSI (Pounds per square inch)

Pressure Unit Conversions

• 1 atm = 760 mm Hg = 760 torr
• 1 atm = 101,325 Pa
• 1 atm = 14.7 PSI

Boyle's Law

• The volume of a gas is inversely proportional to its pressure
• $P_1V_1 = P_2V_2$

Charles's Law

• The volume of a gas is directly proportional to its temperature
• $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Avogadro's Law

• The volume of a gas is directly proportional to the number of moles
• $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

The Ideal Gas Law

• $PV = nRT$
• $R = 0.0821 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$

Using the Ideal Gas Law to Find Molar Mass

• Molar Mass $(ℳ) = \frac{m}{n} = \frac{mRT}{PV}$
• m = mass (g)
• n = number of moles (mol)

Using the Ideal Gas Law to Find Density

• Density (ρ) $ρ = \frac{m}{V} = \frac{Pℳ}{RT}$

Dalton's Law of Partial Pressures

• The total pressure of a gas mixture is the sum of the partial pressures of the component gases
• $P_T = P_1 + P_2 + P_3 +...$

Partial Pressure and Mole Fraction

• $P_1 = X_1P_T$
• $X_1$ = mole fraction of component 1 = $\frac{n_1}{n_T}$
• $n_1$ = number of moles of component 1
• $n_T$ = total number of moles in the mixture

Kinetic Molecular Theory

• A gas is composed of particles that are in constant, random straight-line motion
• The particles collide elastically with each other and with the walls of the container
• The particles are so small and the distances between them are so large that the volume of the individual particles is negligible
• There are no attractive or repulsive forces between the particles
• The average kinetic energy of the particles is proportional to the absolute temperature

Average Kinetic Energy

• $KE_{avg} = \frac{3}{2}RT$

Root Mean Square Velocity

• $v_{rms} = \sqrt{\frac{3RT}{ℳ}}$
• R = 8.314 J/(mol⋅K)
• ℳ = molar mass (kg/mol)

Graham's Law of Effusion

• The rate of effusion of a gas is inversely proportional to the square root of its molar mass
• $\frac{Rate_A}{Rate_B} = \sqrt{\frac{ℳ_B}{ℳ_A}}$

Van der Waals Equation

• Real Gases: Deviations from Ideal Behavior
• $(P + a(\frac{n}{V})^2)(V - nb) = nRT$
• a and b are van der Waals constants
• a: reflects the attractive forces between gas molecules
• b: reflects the finite volume of the gas molecules

Algorithmic Game Theory

• Branch of theoretical computer science that uses game theory to model the behaviour of interacting computing systems

Algorithmic Game Theory Perspective

• Game Theory: Study of strategic interactions among rational, selfish agents
• Algorithm Design: How to design efficient algorithms for computational problems
• Design games so that selfish behavior of agents leads to desirable outcomes
• Design algorithms that perform well when inputs are controlled by selfish agents

Selfish Routing Model

• Network of $n$ nodes and $m$ edges
• Each edge $e$ has a cost function (or latency function) $l_e(x)$ which is the cost incurred per unit of traffic when the edge carries $x$ units
• Non-atomic routing games involve many players, each controlling a small amount of traffic
• There are $k$ source-destination pairs $(s_i, t_i)$ and traffic demand $r_i$ for each pair

Objective of Selfish Routing

• For each traffic to minimize cost
• Evaluate the quality of the equilibrium when selfish routing has reached an equilibrium

Selfish Routing Definitions

• Flow $f$ specifying how traffic is routed
• $f_e$: amount of flow on edge $e$
• A flow $f$ is feasible if it satisfies the traffic demand for each source-destination pair
• The cost of a path $P$ is $\sum_{e \in P} l_e(f_e)$
• The cost of a flow $f$ is $\sum_{e \in E} f_e \cdot l_e(f_e)$

Wardrop Equilibrium Defined

• A flow $f$ at Wardrop equilibrium if, for any source-destination pair $(s, t)$, all flow is routed along paths of minimum cost

System Optimum Defined

• A flow $f$ at system optimum if it minimizes the total cost $\sum_{e \in E} f_e \cdot l_e(f_e)$

Price of Anarchy Explained

• Price of anarchy (PoA) is the ratio between the cost of the worst-case equilibrium and the cost of the social optimum

Price of Anarchy Equation

• $PoA = \frac{\text{Cost of Worst-Case Equilibrium}}{\text{Cost of Social Optimum}}$

Braess's Paradox Demonstrated

• Shows adding an edge to the network can worsen the performance at equilibrium

Bounding the Price of Anarchy Assumptions

• All latency functions are linear: $l_e(x) = a_e x + b_e$ with $a_e, b_e \geq 0$

Theorem on Linear Latency Functions

• With linear latency functions, the price of anarchy is at most $\frac{4}{3}$

Tightness Demonstrated

• Bound is tight, demonstrated through example: two nodes $v_1, v_2$, source $v_1$, destination $v_2$, 1 unit of traffic, two parallel edges
• $l_1(x) = x$
• All traffic goes through edge 1 costing 1 for system optimum
• All traffic goes through edge 2 costing 1 too for Wardrop Equilibrium
• $l_2(x) = 1$
• With $PoA = \frac{4}{3}$