Economic

Static Equilibrium

• Static equilibrium occurs when an object is at rest with no net force or net torque acting on it.
• Conditions for static equilibrium:
  • The vector sum of forces must equal 0: ∑🡒F = 0, ensuring no translational acceleration.
    • In component form: ∑Fx = 0, ∑Fy = 0, ∑Fz = 0.
  • The vector sum of torques must equal 0: ∑🡒τ = 0, ensuring no angular acceleration.
  • ∑τ = 0
• Torque is the tendency of a force to rotate an object, defined as τ = rFsinθ.
  • r: distance from the axis of rotation to the force's application point.
  • F: magnitude of the force.
  • θ: angle between the force vector and the lever arm.

Experiment Procedures

• Equilibrium of a Particle
  • Apparatus include a force table, weight hangers, strings, slotted weights, protractor, and ruler Experimental Setup:
  • Strings connect the center ring to weight hangers crossing a pulley
  • Add weight and adjust force magnitude and direction until the central ring is centered Data Collection:
  • The values recorded from the magnitudes and directions
  • Force's magnitude equals weight of hanger & weights; its direction is the angle from the 0° line. Data Analysis:
  • Calculate forces, focusing on $x$ and $y$ components.
  • Assess the vector sum's proximity to zero, indicating equilibrium.
  • Find the percentage difference to evaluate accuracy.
• Equilibrium of a Rigid Body
  • Apparatus includes a meter stick, its fulcrum string, weight hangers with slotted weights, and a triple beam balance Experimental Setup:
  • Determine the center of gravity of the meter stick
  • The meter stick will sit on the fulcrum and will be the base for all objects being weighed Data Collection:
  • Magnitude and position of each weight being recorded
  • The values are weight of hanger & weights, and position relative to fulcrum. Data Analysis:
  • Torque for each weight is calculated relative to fulcrum
  • Sum all torque values
  • If this nears zero, the meter stick is in equilibrium
  • The experiment's accuracy will be determined by percentage difference

Algorithmic Trading

• Involves computer programs that execute trades according to a predefined rule set
• These rules can be based on price, quantity, time, or any other variable Key Advantages:
  • Executes faster than humans
  • Monitors market conditions and trades 24/7
  • Removes emotional decision-making.
  • Can be tested against historical data to see how well it would have performed Key Disadvantages:
  • Susceptible to technical problems like software bugs or hardware failures.
  • Can be over-optimized to perform well on historical data, but not on new data
  • May perform poorly during periods of high market volatility.
  • Might make errors if not properly monitored Common Strategies:
  • Trend Following aims to identify & follow trends in the market.
  • Mean Reversion assumes prices will revert to their average level.
  • Arbitrage profits from price differences in different markets.
  • Market Making gives liquidity to the market by creating buy and sell orders

Newton's Method

• Root Finding Problem
  • Given a function, f(x), determine the points where the function's value is zero
• Key Facts
  • An iterative method: Starts with an initial guess (x0) and refines it.
  • The improved approx formula is, $x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$
• Quadratic convergence: near the root, the method typically converges quadratically.
  • Requires knowledge of f'(x): need to know derivative of the function.
  • Initial guess needs to be close enough to the root to ensure convergence.
  • Newton's method can exhibit chaotic behavior and may not converge.
• Example of $f(x) = x^3 - x$: converges quickly when the initial guess is close to a root Newton's method may diverge or oscillate if the initial guess isn't close enough to a root
• The derivative ($f'(x)$) plays an important role in how well and fast Newton's method converges.

Mechanism Design

• Standard Setup for Mechanism Design
  • n agents/players within a set N = {1, ..., n} and O is the set of possible outcomes.
  • Each agent i has valuation function vi : O → ℝ with agent i's (private) type being vi
• Mechanism defined as algorithim
  • Takes types v = (v1, ..., vn) as an input, computes allocation/outcome o ∈ O, and computes payments p = (p1, ..., pn) where pi is the payment for agent i Goal:
  • To design a mechanism that achieves soem desirable objective Examples of objectives:
  • Maximizing social welfare, making good decisions, and maximizing revenue
• Important Factors:
  • Strategyproofness which means agents need to be incentivised to reveal their type
  • Computationally efficient which means the algorithm can be used in polynomial time
  • Information is very important as mechanism has to know the type probability distribution

Mechanism Design Problems

• Payment use can incentivize agents to reveal their true type
• Mechanism Design without Money
  • In design process, payments are not allowed
  • To incentivize agents to reveal the true type need to rely on a restricted domain
  • Also restrict on possible set of outcomes

Example: Facility Location

• Problem involves the placement of a facility on a line to minimize the sum of distances to the agents at N = {1, ..., n}
• Strategyproof Mechanism follows a social choice function
• O = R for the location of the facility with the valuations defined as vi(o) = -|o - xi|
• Social Welfare defined as SW(o; x) = ∑in=1 vi(o) = -∑in=1 |o - xi|
• To be strategyproof agents need to be incentivized to report true location

Regulation of Gene Expression

• Crucial for controlling timing and amount of gene products
• Primarily regulated at transcription (DNA → RNA) and translation (RNA → Protein) levels
• Prokaryotes utilize operons with clustered genes transcribed from a single promoter; the lac Operon of E. coli controls lactose metabolism via:
  • Promoter, Operator, lacZ, lacY, and lacA.
• In the absence of lactose: a protein binds and prevents transcription.
• In the presence of lactose: It converts to bind with the repressor, releasing from the operator + cAMP improves transcription.
• Eukaryotic regulation entails several factors:
  • Transcription Factors; enhancing and silencing DNA sequences.
  • Chromatin packaging affects gene expression, Heterochromatin being inactive, and Euchromatin active.
  • Epigenetics alter expression without changing the DNA sequence via DNA and modification of Histones, thus leading to mRNA degradation or blocked translation.

Basic Propositions of Probability

• Experiment: A process that yields one of several possible outcomes.
• Sample Space (S): The set of all possible outcomes of a random experiment.
• Event (E): A subset of the sample space.
• Probability of Success: Probability measures the likelihood that an vent will occur; where $P(E)$ is the probability of the event E Axioms of Probability:
  • 1. $0 \le P(E) \le 1$
  • 2. $P(S) = 1$
  • 3. For mutually exclusive events $E_1, E_2, E_3,...$
  • $P(E_1 \cup E_2 \cup E_3 \cup...) = \sum_{i=1}^{\infty} P(E_i)$ Basic Propositions:
  • 1. $P(\emptyset) = 0$
  • 2. $P(E^c) = 1 - P(E)$ with P(E) being the event of success and P(Ec) being the complement event of failiure
  • 3. $P(E \cup F) = P(E) + P(F) - P(E \cap F)$
  • 4. $E \subset F$, then $P(E) \le P(F)$

Poisson Distribution

• A discrete probability distribution: measures the chances of a certain number of events happening within a specific timeframe or space, Events should:
  • Occur at a known, constant average rate.
  • Occur independent of last event's time.
• Probability Mass Function (PMF): the chance of observing k events occurring is calculated as:
  • $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$,
• Key:* k = number of events & λ (Lambda) = average rate of events Mean and Variance : For a Poisson random variable $X$ with parameter $\lambda$, then $E[X] = \lambda$ and $Var(X) = \lambda$

Function Composition and Cardinality

• $f: X \rightarrow Y$ is 1-1 (injective) if $f(x_1) = f(x_2) \implies x_1 = x_2$ $f: X \rightarrow Y$ is onto (surjective) if for all $y \in Y$, there exists $x \in X$ such that $f(x) = y$
• $f: X \rightarrow Y$ is bijective if it is 1-1 and onto; meaning there would then exist $f^{-1}: Y \rightarrow X$ Composition of functions: Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$. Then, $(g \circ f)(x) = g(f(x))$. where $g \circ f: X \rightarrow Z$ shows the combination of them Theorem: If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ as bijective, then:
• ) $g \circ f$ is bijective.
• ) $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.
• Cardinality: Two sets $A$ and $B$ have the same cardinality ,(i.e., $|A| = |B|$) if there exists a bijective function $f: A \rightarrow B$. and as such define set B as countable if $|A| = |\mathbb{N}|$ or if $A$ is finite while being uncountable otherwise

Linear Equations Intro

• Linear systems often involve multiple equations with multiple variables ($m$ equations, $n$ variables).
• Represented in matrix form as $Ax = b$, where:
• $A$ is the coefficient matrix.
• $x$ is the variable vector.
• $b$ is the constant vector. Key aspects of dealing with these equations are:
• Each linear equation corresponds to a geometric shapes, and the solution represents where those shapes intersect