Mining, Minerals and Elements

Mining

• Process of extracting ore from the ground

Environmental Impact

• Deforestation occurs
• Bulldozers and excavators are used
• Contamination is a risk
• Butterfly Effect can occur

Reasons

• Mining is a money-making business for companies
• Mining supports governments through revenues
• Workers receive income and benefits

Civilisation

• Civilisation occurs

Minerals

• Rocks are made of minerals and minerals are made of elements
• Granite is an example of a rock
• Quartz, feldspar, and biotite are minerals found in granite
• Consists of silicon and oxygen
• Rocks consist of solidified magma composed of two or more minerals
• Minerals are naturally occurring, inorganic substances in the Earth's crust

Elements

• Elements are pure substances

Compounds

• Compounds are mixed substances and must make profit

Ores

• Ores are minerals containing sufficient elements to extract
• Hematite is an ore containing iron oxide

Common Elements & Their Ore Minerals

• Aluminum ore is Bauxite
• Copper ore is Cuprite
• Iron ore is Hematite
• Lead ore is Magnetite
• Tin ore is Galena
• Mercury ore is Cinnabar

Rehabilitation

• Underground mines need to be rehabilitated
• To rehabilitate, the top soils are removed to create the open-cut mine are replaced, soils are then graded to ensure the landscape drains well, the area is then fertilised and replanted with plants native to area
• The biggest concern when rehabilitating a mine site are tailings

Tailings

• Tailings are waste materials from the mine or mineral processing plant
• They are often toxic and contain acids or poisonous chemicals
• Gold mine tailings usually contain the poison cyanide

Australia Mining Regulations

• Laws require mining companies to comply with strict procedures.
• Environmental scientists design and monitor pollution control measures.
• Water pollution is controlled by chemically treating water and leaving it to settle out in tailing dams.
• Clean run-off can then be discharged into surrounding creeks & rivers.

Physical Properties

• Used to identify minerals
• Hardness uses Moh's hardness by scratching
• Luster denotes how shiny a mineral is
• Color and shape describe a mineral's features
• Streak color denotes a mineral's color

Exploration

• Many methods can be employed to explore for minerals, some of the most common include, aerial methods (magnetic gravity & electromagnetic surveys), on site methods (seismic & geochemical surveys), Satellite images

Aerial Method Types

• Aerial methods of surveying are done from an aircraft or helicopters.
• Aerial surveying advantage is covering vast areas fairly quickly

Magnetic Surveys

• The Earth has a magnetic field running between the north and south poles
• Minerals such as iron, nickel or cobalt change the magnetic field around them.
• Magnetometers are sensitive instruments that detect any change, and show the effect
• Magnetic data from the magnetometer is fed into a computer and a colored map can then be constructed.

Gravity Surveys

• Small differences in the gravitational pull of the earth can be detected by very sensitive instruments called gravimeters
• Dense rocks and minerals affect the gravitational pull on the aircraft
• Variations in the gravitational pull are then shown on a map using color to show the differences

Electromagnetic Surveys

• Electromagnetic surveys detect ore bodies that are good electrical conductors.
• A magnetic field is sent from a transmitter on an aircraft of helicopter.
• The magnetic field creates tiny electrical currents called eddy currents in minerals in the earth.
• These currents create their own magnetic field called secondary field.
• This method is good for detecting metal sulfides such as copper sulfide, lead sulfide and zinc sulfide, and also for detecting gold and silver

Mining Methods, Affecting Factors

• The cost of how much the mineral is worth.
• Type of mineral the miner is searching for
• The Geology of an area

Mining Methods

• Surface & Underground Mining
• Leach & Dredging Mining
• Petroleum Wells

Open Pit Mining

• Ore body is near the surface
• Open-cut pit is made with roads
• Ores are removed using giant drills & large machines and shovels
• Minerals mined using this method include Gold & Copper
• Fimiston & Boddington are example mines

Leach

• Materials are mined by dissolving them in fluid that's injected in ores
• Involves drilling holes in ore deposit
• Is used in mining of uranium
• Minerals include Uranium

Dredging

• Is used where the ore is located near a plentiful water source or a large pit, dug & filled with water
• A small lake and dredging boats floats on surface, with dredge that moves slowly through water & draws up material
• Minerals Mined Titanium from Cooljarloo

Underground

• Is used when the ore is deep, fairly concentrated & confined to small space
• A decline path is used which enables trucks to drive down & pick up crushed ores
• Minerals Mined; Gold, Nickel, Copper, Uranium, Diamond & Silver
• Example mines are Birla & Slopes

Petroleum Wells

• Petroleum occurs as liquid
• Hole is drilled deep to Earth until oil or gas are found
• Natural pressure of Earth forces oil & gas up through holes to large storage tanks
• Used to extract petroleum

Process of Getting Minerals

• 3 Steps include Mining, Processing & Rehabilitation.
• Mining - resource is removed from the ground
• Processing - all unwanted materials are removed, and the wanted material is extracted
• Rehabilitation - after mining & processing is completed, the mining site is repaired so that it returns to something similar to the original environment

Figure 8.5.5 shows:

• Underground mining using a decline enables trucks to drive down to pick up crushed ore & carry it to surface

Processing Steps

• Step 1: Crushing is done over several stages to produce rocks of similar size. It increases surface area of the rock, making it easier to treat further
• Step 2: Minerals that are magnetic are removed by magnets, while dense materials are allowed to drop out from rest of ore. The remainder of ore continues through further processing
• Step 3: Minerals are enriched to concentrate them & improve their quality. Enrichment can be done through many different processes, depending on their chemical & physical properties
• Step 4: The desired metal is extracted. Extraction is the chemical process of separating metals from compounds in which they occur. For example, iron, lead & zinc are commonly extracted from ores by heating up in a blast furnace from themselves or with limestone

Common Elements & Their Ore Minerals

• Aluminum: Bauxite
• Copper: Cuprite
• Iron: Hematite
• Lead: Magnetite
• Tin: Galena
• Mercury: Cassiterite
• Mercury; Cinnabar

Algorithmic Game Theory

• Study of strategic interactions between rational agents: Game Theory
• How to design efficient algorithms: Algorithm Design

Basic Solution Concepts

• Nash Equilibrium: Set of strategies (one for each player) where no player can improve their payoff by unilaterally changing their strategy.
• Pareto Optimality: Outcome where impossible to make one player better off without making another player worse off.
• Social Welfare: Sum of all players' payoffs.

Price of Anarchy (PoA)

• POA = (Social Welfare of the Optimal Outcome) / (Social Welfare of the Worst Nash Equilibrium)
• Measures how much social welfare is lost due to selfish behavior of agents in a system.
• Quantifies inefficiency of Nash Equilibria compared to a centrally enforced optimal solution.

Congestion Games

• Models situations where delay (or cost) incurred by players depends on number of players using same resource.
• N players & E edges within a network
• Each player chooses path from source to destination.
• Cost Function: cₑ(x) is cost of edge e if x players use it.
• Player Cost: sum of costs of edges in chosen path.
• Social Cost: sum of all players' costs.

Braess's Paradox

• Adding a new road to a network can increase average travel time for all players.
• This arises players selfishly choose paths individually optimal, but collectively lead to a worse outcome.

Bounding the Price of Anarchy

• Theorem: In a congestion game with linear cost functions, the Price of Anarchy is at most 4/3.
• Proof Idea: Compare cost of Nash Equilibrium flow to cost of optimal flow.
• Even in systems where users act selfishly, the resulting social cost is not much worse than the optimal social cost, this provides a guarantee on the efficiency of selfish behavior in certain types of systems.

Mechanism Design

• Goal: Implement a social choice function to align incentives of players with the desired outcome.
• Example: Selling an item in an auction.

Vickrey-Clarke-Groves (VCG) Mechanism

• General mechanism for implementing efficient outcomes in settings with externalities.
• Properties:
  • Efficiency: Outcome maximizes social welfare
  • Strategy-proofness: Truthful bidding is a dominant strategy for each player.
• Payment Rule: Each player pays the negative externality they impose on other players Payment of player i is equal to the harm that player i's presence causes to the other players.

Auctions

• First-Price Auction: Highest bidder wins and pays their bid.
• Second-Price Auction: Highest bidder wins and pays second-highest bid.
• Revenue Equivalence Theorem: Under certain conditions, different auction formats will yield same expected revenue for the seller.

Sponsored Search Auctions

Challenges:

• Clickthrough Rates: Taking into account the probability that a user will click on an ad.
• Position Effects:
  • Higher positions on the search results page tend to get more clicks.
  • Algorithmic Game Theory provides tools for analyzing and designing systems where strategic interactions are important.
  • It combines concepts from a variety of domains.

Álgebra

Expresiones algebraicas

• Combinación de letras y números ligados por operaciones aritméticas
  • Ejemplos
    • $\sqrt{x} + \frac{y}{z}$
    • $4a^2b - 7ab^2 + b^3$

Polinomios

• Expresión algebraica formada por suma finita de monomios
  • Ejemplos:
    • P(x) = $3x^2$ - 5x + 2
    • Q(x) = $x^3$ + 2x - 1
• El grado de un polinomio es el mayor de los grados de los monomios
• Valor numérico de un polinomio P(x), en x=a, es el número que se obtiene al sustituir x por a

Operaciones con polinomios

• Suma y Resta: Sumar o restar términos semejantes.
• Multiplicación: Multiplicar cada término de un polinomio por cada término del otro y sumar términos semejantes.

Identidades Notables

• Cuadrado de Suma: (a + b)² = $a^2$ + 2ab + $b^2$
• Cuadrado de Diferencia: (a - b)² = $a^2$ - 2ab + $b^2$
• Suma por Diferencia: (a + b)(a - b) = $a^2$ - $b^2$

División de Polinomios

• Dados dos polinomios P(x) (dividendo) y Q(x) (divisor), existen dos únicos polinomios C(x) (cociente) y R(x) (resto), tal que grado R(x) < grado Q(x)
• Regla de Ruffini: Método para dividir un polinomio P(x) entre (x - a).
• Teorema del resto: El resto de la división de un polinomio P(x) entre (x - a) es igual al valor numérico del polinomio para x = a, es decir, P(a).

Factorización de Polinomios

• Factorizar un polinomio es expresarlo como producto de polinomios de menor grado.
• Métodos:
  • Factor común
  • Identidades notables
  • Regla de Ruffini
  • Resolver ecuaciones de segundo orden

Cálculo vectorial

Campos escalares

• Es una función que asocia un valor escalar a cada punto en el espacio.
  • Ejemplo: La función de temperatura T(x, y, z)

Campos Vectoriales

• Es una función que asocia un vector a cada punto en el espacio.
  • Ejemplo: La función de velocidad V(x, y, z) de un fluido

Derivada direccional

Derivada direccional de un campo escalar f(x,y,z) un vector $\hat{u}$ ($D{\hat{u}} f = \nabla f \cdot \hat{u}$)

Gradiente $\nabla f$

• $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$
• Es perpendicular a las curvas de nivel de f
  • Propiedades:
    • Operador Lineal
    • Regla del producto
    • Regla de la cadena

Divergencia

$\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$ - La divergencia es un escalar - Propiedades: - Es un operador lineal - Regla del producto

Rotacional

$\nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)$ - El rotacional es un vector - Propiedades: - Es un operador lineal - Regla del producto

Laplaciano

$\nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$

Laplaciano

• $\nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$

Identidades Vectoriales

• $\nabla \times (\nabla f) = \vec{0}$
• $\nabla \cdot (\nabla \times \vec{F}) = 0$
• $\nabla \times (\nabla \times \vec{F}) = \nabla (\nabla \cdot \vec{F}) - \nabla^2 \vec{F}$

Fourier Transform

• A signal is transformed from time domain to frequency domain using the Fourier Transform,

$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$

Where - $f(t)$ is the signal in the time domain - $F(\omega)$is the signal in the frequency domain - $\omega$ is the frequency in radians per second - t is the time in seconds - $j = \sqrt{-1}$ - Inverse Fourier Transform $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega$

Fourier Transform Properties

• Linearity: The Fourier Transform is a linear operator: $F{af(t) + bg(t)} = aF(\omega) + bG(\omega)$ Shifting a signal in time corresponds to a phase shift in frequency domain
• Time Shifting: $F{f(t - t_0)} = e^{-j\omega t_0} F(\omega)$
• Time Scaling: Scaling a signal in time corresponds to an inverse scaling in the frequency domain: $F{f(at)} = \frac{1}{|a|} F(\frac{\omega}{a})$
• Duality: If $F(\omega)$ is the Fourier Transform of $f(t)$, then the Fourier Transform of $F(t)$ is $2\pi f(-\omega)$
• Convolution: Fourier Transform of convolution of two signals is product of their Fourier Transforms: $F{f(t) * g(t)} = F(\omega)G(\omega)$

Common Fourier Transform Pairs

• $\delta(t)$ becomes 1
• $1$ becomes $2\pi\delta(\omega)$
• $e^{j\omega_0 t}$ becomes $2\pi\delta(\omega - \omega_0)$
• $\cos(\omega_0 t)$ becomes $\pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$
• $\sin(\omega_0 t)$ becomes $j\pi[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)]$
• $rect(t)$ becomes $sinc(\omega/2)$

Application of the Fourier Transform

• Signal & System analysis
• Image Processing
• Data Compression
• Solving differential equations

Example

• If $f(t) = e^{-at}u(t)$, where $a > 0$ and $u(t)$ is the unit step function, then
• $F(\omega) = \frac{1}{a + j\omega}$.

Information Retrieval Systems

• Systems facilitate the efficient discovery of documents relevant to specific queries within extensive collections.

Basic IR System Architecture

• Document Collection: the collection containing the documents to be indexed and searched.
• Indexer: A module that processes documents to create an index.
• Index: A data structure that enables efficient searching (e.g., inverted index). Query Processor: Transforms user queries into a form suitable for searching the index. Ranking Algorithm: Ranks documents based on their relevance to the query.

Bag of Words Model

• Model represents text as bag (multiset) of its words, disregarding word order.

Example

Doc 1 = "The cat sat on the mat." Doc 2 = "The dog sat on the log."

Bag of Words

Doc 1: the(2), cat(1), sat(1), on(1), mat(1) Doc 2: the(1), dog(1), sat(1), on(1), log(1)

Term Weighting

• Assigns weight to each term in a document, reflecting importance within a collection. TF-IDF (Term Frequency-Inverse Document Frequency)
• TF-IDF combines two measures:

Term Frequency (TF)

• Measures the frequency of a term in a document. Tf$(t, d) = $(Number of times term t appears in document d) / (Total number of terms in document d)

Inverse Document Frequency (IDF)

• Measures how rare a term is across collection IDF$(t, D) = \log$(Total number of documents in collection D)/(Number of documents containing term t)

Computing TF-IDF

• Tf-idf(t, d, D) = tf(t, d) * idf(t, D)

Example Calculation

Assume you have: D = 1000 documents and document, d: Information retrieval is important

1. Term Frequency Calculation for "information"
   • term appears 1 time, with total of 4 terms in d tf("information", d) = ¼ = 0.25
2. Inverse Document Frequency Calculation for "information"
   • Assume term appears in 50 of the 1000 documents. idf("information", D) = log(1000/50) = log(20) = ~1.301
3. TF-IDF Calculation

tfidf("information", d, D) = 0.25 * 1.301 = ~0.325

Laws of Thermodynamics

• Zeroth Law
• If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
• First Law
• The change in internal energy equals heat added minus work done: ΔU = Q - W
• Second Law
• Entropy of an isolated system tends to increase towards a maximum at equilibrium.
• Third Law
• As temperature approaches absolute zero, the entropy approaches a minimum (or zero).