Static Electricity and Charge

Static Electricity & Charge

• Two types of electric charge: positive and negative.
• Like charges repel, opposite charges attract.
• Charge is conserved.
• Charge is quantized.

Charge is Quantized

• Electric charge ($q$) is an integer multiple of elementary charge ($e$): $q = ne$, where $n = \pm 1, \pm 2, \pm 3,...$
• Elementary charge $e = 1.60219 \times 10^{-19} C$.

Example Calculation

• To achieve a negative charge of $3.2 \times 10^{-17} C$ on a conductor, add 200 electrons ($n = \frac{q}{e} = \frac{-3.2 \times 10^{-17} C}{-1.60219 \times 10^{-19} C} = 200$).

Conductors and Insulators

• Conductors: Materials where electric charges move freely (e.g., copper, aluminum, silver).
• Insulators: Materials where electric charges do not move freely (e.g., glass, rubber, wood).
• Semiconductors: Materials between conductors and insulators (e.g., silicon, germanium).

Charging Objects

• Charging by Conduction: Direct contact transfers electrons, resulting in the same sign of charge.
• Charging by Induction: Proximity of a charged object causes charge redistribution in another object.
• Grounding: Connecting a conductor to the Earth allows it to accept or supply electrons freely.
• Polarization: Insulators experience induced charges on their surfaces in the presence of a charged object.

Coulomb's Law

• Electric force between stationary charged particles: $F = k_e \frac{|q_1||q_2|}{r^2}$.
• Coulomb constant $k_e = \frac{1}{4\pi\epsilon_0} \approx 8.9875 \times 10^9 N \cdot m^2 / C^2$.
• Permittivity of free space $\epsilon_0 = 8.8542 \times 10^{-12} C^2 / N \cdot m^2$.
• Force acts along the line joining the charges, attractive for opposite signs, repulsive for like signs.
• Applies exactly to point charges or particles.

Example Electrostatic Force Calculation

• Given three point charges in a triangle configuration: $q_1 = 6.0 \times 10^{-9} C, q_2 = -2.0 \times 10^{-9} C, q_3 = 5.0 \times 10^{-9} C$. Calculate the resultant electrostatic force on $q_3$.

Superposition Principle

• Resultant force on a charge is the vector sum of forces from other individual charges.
• To find the resultant force on $q_3$, determine the forces $F_{23}$ and $F_{13}$ and add them vectorially.

Force Calculation Example

• $F_{23} = k_e \frac{|q_2||q_3|}{r_{23}^2} = (8.9875 \times 10^9 N \cdot m^2 / C^2) \frac{(2.0 \times 10^{-9} C)(5.0 \times 10^{-9} C)}{(0.10 m)^2} = 9.0 \times 10^{-6} N$. This is attractive and acts to the right.
• $F_{13} = k_e \frac{|q_1||q_3|}{r_{13}^2} = (8.9875 \times 10^9 N \cdot m^2 / C^2) \frac{(6.0 \times 10^{-9} C)(5.0 \times 10^{-9} C)}{(0.30 m)^2} = 3.0 \times 10^{-6} N$. This is repulsive and acts upward.

Resultant Force

• $F = \sqrt{F_{23}^2 + F_{13}^2} = \sqrt{(9.0 \times 10^{-6} N)^2 + (3.0 \times 10^{-6} N)^2} = 9.5 \times 10^{-6} N$.
• $\theta = tan^{-1} (\frac{F_{13}}{F_{23}}) = tan^{-1} (\frac{3.0 \times 10^{-6} N}{9.0 \times 10^{-6} N}) = 18^\circ$.

Electric Field

• An electric field surrounds a charged object (source charge).
• A test charge within the field experiences an electric force.

Electric Field Vector

• $\vec{E} = \frac{\vec{F}}{q_0}$ where $\vec{E}$ is the electric field vector, $\vec{F}$ is the electric force, and $q_0$ is the positive test charge.
• Units: N/C

Electric Field Direction

• If $q_0$ is positive, $\vec{F}$ is in the same direction as $\vec{E}$.
• If $q_0$ is negative, $\vec{F}$ is in the opposite direction as $\vec{E}$.

Dijkstra's Algorithm - Problem Statement

• Given a weighted, directed graph G = (V, E) with non-negative edge weights w: E -> R >= 0 and a start node s ∈ V, the goal is to find the shortest paths from s to all other nodes v ∈ V.

Idea Behind Dijkstra's algorithm

• A greedy algorithm visits nodes in order of distance from the starting node s, updating distances to unvisited nodes.

Dijkstra's Algorithm Steps

• Initialization: Set the distance to all nodes to infinity, except for the start node s, with a distance of 0. Create a set Q with all nodes in V.
• Iteration: While Q is not empty,
  • Select a node u ∈ Q with the minimum distance.
  • Remove u from Q.
  • For each neighbor v of u, calculate the distance from s to v via u: alt = dist[u] + w(u, v).
  • If alt < dist[v], update the distance of v: dist[v] = alt, and set the predecessor of v: prev[v] = u.

Dijkstra's Pseudocode

• Function Dijkstra(Graph G, Startnode s): initializes distances to infinity, iterating to relax distances and record shortest paths to other nodes.

Example of Dijkstra's algorithm

• Consider a sample graph with nodes A to F and corresponding edge weights to illustrate the algorithm's execution.
• The table displays the step-by-step execution of the algorithm, showing the updated distances for each node.

Dijkstra's Algorithm properties

• Finds shortest paths from a start node to all other nodes in a weighted graph with non-negative edge weights.
• Runtime depends on the implementation of the priority queue Q. Simple implementation is O(|V|^2), Heap-based implementation is O((|V| + |E|) log |V|).

Limitations of Dijkstra's Algorithm

• Only works for graphs with non-negative edge weights.
• The Bellman-Ford algorithm must be used for graphs with negative edge weights.

Matrix Sum

• For matrices A and B of the same dimension mxn, the sum S = A + B is another matrix of the same dimension, with elements $s_{ij} = a_{ij} + b_{ij}$.
• Provides an example matrix sum to clarify.

Product of a Scalar & Matrix

• The product of a scalar k and a matrix A yields matrix B = kA of the same dimension where each element is $b_{ij} = k \cdot a_{ij}$.
• Provides example multiplying a matrix by scalar.

Properties of Matrix Sum & Scalar Product

• Commutative: A + B = B + A
• Associative: A + (B + C) = (A + B) + C
• Identity Element: Exists a matrix 0 where A + 0 = A (Zero Matrix).
• Inverse Element: Exists a matrix -A for each matrix A where A + (-A) = 0; If A = (aij), then -A = (-aij).
• Distributive 1: λ(A + B) = λA + λB
• Distributive 2: (λ + μ)A = λA + μA
• Pseudo-associative: λ(μA) = (λμ)A
• Unit Element of Product: 1 * A = A

Product of Matrices

• The product A·B is possible only if the number of columns in A equals the number of rows in B; if A is mxn and B is nxp, then A·B is mxp. Each element is calculated by $c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} +... + a_{in}b_{nj}$.
• Provides example of matrices product.

Properties of Matrix Product

• Associative: A(BC)=(AB)C
• Identity Element: There is matrix I, s.t., AI = IA =A
• Distributive 1: A ∙(B+C)=A∙B+A∙C
• Distributive 2: (A+B)∙C=A∙C+B∙C
• Not commutative. AB != BA
• General simplification: Invalid. AB = AC does not mean B= C
• Annullation: invalid. With matrices A*B=0, it does not mean A or B = 0 matrix
• Product to scalar: k(A ∙B)=(kA)∙B=A∙(kB)

Algorithmic Trading, Introduction

• Algorithmic trading (AT) involves using computer programs to automate trading decisions.
• Quantitative strategies use mathematical and statistical techniques to exploit market inefficiencies.

Algorithmic Trading Explained

• AT executes pre-programmed instructions, handling data faster and removing emotions.
• It's also known as automated trading, black-box trading, or systematic trading.

Historical Developments in Algorithmic Trading

• History intertwined with tech advancements in computing and telecommunications.
• Early forms of AT started already in the 1970s.
• 1975: US SEC eliminated fixed brokerage commissions.
• Computer use in trading rooms grew dramatically in the 1980s.
• 1986: LSE introduced the Stock Exchange Automated Quotations (SEAQ) system.
• Late 1980s: Introduction of program trading.
• Quantitative finance research grew significantly in the 1990s.
• 1998: US equity markets decimalized.
• 2001: Nasdaq introduced SuperMontage.
• 2000s: High-frequency trading (HFT) emerged.
• AT accounted for over 60% of US equity trading volume by 2009.

Categories of Algorithms

• Execution Algorithms: Aim to execute large orders at the best possible price (Examples: VWAP, TWAP, IS).
• Market Making Algorithms: Provide liquidity by placing bid and offer quotes.
• Statistical Arbitrage Algorithms: Exploit temporary price discrepancies. (Examples: pairs trading, index arbitrage, triangular arbitrage).
• Trend Following Algorithms: Capitalize on trends in asset prices. (Examples: moving averages, breakout strategies).

Quantitative Strategies Definition

• Involves using mathematical and statistical models to identify and exploit market inefficiencies.

Process of implementing Quantitative strategies

• Idea Generation: Identify potential sources of alpha.
• Data Collection and Preparation: Gather and clean historical data.
• Model Development: Build a mathematical model.
• Backtesting: Test the model on historical data.
• Risk Management: Identify and mitigate risks.
• Implementation: Deploy the strategy live.
• Monitoring and Refinement: Continuously monitor and refine the model.

Metrics of Performance

• Alpha: Excess return compared to a benchmark.
• Beta: Sensitivity to market movements.
• Sharpe Ratio: Risk-adjusted return, calculated as $\frac{R_p - R_f}{\sigma_p}$.

Pitfalls of Algorithms

• Overfitting: Poor performance due to too closely matching the training data sets.
• Data Mining Bias: Spurious patterns can cause problems.
• Transaction Costs: Neglecting the impact of trading expenses
• Market Impact: A trader’s order's effect on asset values may be detrimental.
• Model Risk: Overlooking the possible model misspecification or invalidity which causes errors.

Number Representations, Decimal

• Decimal notation (base ten) utilizes ten symbols: 0 to 9.
• Symbol's position determine the value it represents.

Binary Notation

• Binary (base two) notation uses two symbols: 0 and 1 (bits).
• The placement in binary determine the integer value.

Binary to Decimal Conversion

• Multiply each bit by its place value (power of 2) and sum the results.
• Example: $101101_2 = 45_{10}$.

Decimal to Binary Conversion

• Repeatedly divide decimal number by 2 and list remainders.
• Example: $25_{10} = 11001_2$.

Hexadecimal Notation

• Hexadecimal (base sixteen) notation uses sixteen symbols: 0-9 and A-F.
• Symbol's value from right to left is the digit multiplied by 16 to the power of it's location (starting at zero).

Binary to Hexadecimal Conversion

• Group binary into 4-bit sets (from right) translate to hexadecimal digit.
• Example: $1011010111110010_2 = B5F2_{16}$.

Hexadecimal to Binary Conversion

• Convert each hexadecimal digit to 4-bit binary set.
• Example: $3F9_{16} = 001111111001_2$.

Representing Text - ASCII

• ASCII (American Standard Code for Information Interchange).
• Each symbol is represented by a 7-bit pattern, allowing 128 symbols.

Unicode, Text Represented With Different Languages

• Represents different languages with up to 4 bytes.
• Supports over 1 million symbols.
• Includes ASCII as a subset.
• Popular encodings: UTF-8 (1-4 bytes), UTF-16 (2 bytes).

Representing Images

• Bitmap techniques storing pixel values
• Resolution: number of pixels (e.g., 640x480)
• Bit depth is the number of bits/pixel
  • monochrome: 1 bit; grayscale: 8 bits; color: 24 bits (8 bits for R, G, B)

Image Vector Techniques

• Uses lines & curves to represent images, scales properly.
• Use geometric shapes is and lines/curves to form images.
• Commonly used for fonts, logos, etc.

Sound Representation, Sampling

• Sound sampled at intervals, stored as numerical values.
• Sampling rate: samples taken per second (e.g., 44,100 Hz).
• Bit depth: Number of bits per sample.

Sound, MIDI

• MIDI (Musical Instrument Digital Interface).
• Stores instructions for sound recreation rather than the record of sound waves.
• Describes instrument, note, intensity, duration.

Algorithmic Game Theory: Introduction

• Game Theory Goal: to model rational behavior
• Rationaly: players must be trying to optimize and maximize gains
• Mechanism Design Goal: design the rules to achieve desired outcome
• Example = Auctions

Selfish Routing-Definition

• Model directed graph with cost function, source-destination pair, rate for each pair
• Strategy path travels from source -> destination
• Goal: Each of the units flow need to travel through it.
• social cost: the cost of all the edges

Formal definition of a Nash equilibrum

• A flow is under Nash equilibrum if no flow can be made using one unilateral way to decreased its cost
• Price of Anarchy (PoA)
• = social cost of worse Nash equilibrum/optimal social cost

Braess's Paradox

• Adding capacity to a network makes social cost worse at NE.
• Example= 1 unit of flow source -> sink
• Network-Edga a will have cost, b will happen c , etc
• Initial network half the flow goes, social cost 3.
• New network all flow goes , social cost 4

Calculus: Function Definition

• A function f is defined as a rule that binds unique value f(x) , in set E ,to each element x in a set D
• -The name D means its the domain of the function
• The range of variable must be the one with all the possible value of f(x).
• Arbitrary symbol-domain of the functions in the independant variable
• symbol-range of functions is where independant function happens

Types of Function Represenations

• (1) Descriptions with words
• (2)Numerical charts of points
• (3)Visual and graphical
• (4)Algebraic and formulaic

function in temperature

• The relation of temp versus duration is an example

Vertical Line Test

• A funtion's function and graph works if and only if there are intersection points
• comparison of complex numbers

Complex Numbers: Comparison

• Given $z=a + bi$ and $z' = a'+b'i$, where a, a', b, b' ∈ real#s, then z= z' if and only if the reals-real and the b numbers are = to b.
• Notes, comparison has no effect, only when its real comparison makes sense

Conjugate of a complex numbers

• $z= a + bi$, is the conjugate denoted as = −= a- bi
• Example, If z = 3 +2i, then $=3-21

Properties of conjugate

Example

• $z + z$= 2e, where real a+bl
• Where =−=0
• If $ z \neq $0 find reciprocal

Properties of Modules

Example

• Module, where z = aibi , a and be are reals
• Geometry dist origin of affixe z in complexe plans

Physics of Vectores

• addition chart show directions with corresponding charts
• addition analytically shows steps of calculation
• scalar product is defined as AB=AB*cos
• Vecotial product AB = AB*sine
• Kinematics , MRV, trayectoria , velocidad , arelation =0
• MRVU trayectoria aceleretaion consrant trajectory. -equations of trajectory
• vertical tiro libre defined accelerations and what equations of calculations are
• tires oblique trajectory, acceleration vertical calculations

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