Understanding Functions

What is a Function

• Functions assign each element from a set D to a unique element y in another set R.
• The set D is the domain
• The set of all values of f(x) in R is the range.
• The independent variable is x and the dependent variable is y, in the formula y = f(x)
• The function f(x) = x^2 has a domain of all real numbers and a range of all non-negative real numbers of [0, ∞)

Ways to Represent Functions

• Verbally, described in words
• Numerically, by a table of values
• Visually, by a graph
• Algebraically, by formula
• "The function that takes a number and squares it" is verbally expressing a function
• In a table with defined x and f(x) values is a numerical representation

Types of Functions

• Linear Functions: f(x) = mx + b
• Polynomial Functions: f(x) = a_n x^n + a_{n-1} x^{n-1} + ⋯ + a_1 x + a_0
• Power Functions: f(x) = x^a
• Rational Functions: f(x) = P(x)/Q(x)
• Algebraic Functions use algebraic operations
• Trigonometric Functions include sin(x), cos(x), tan(x)
• Exponential Functions: f(x) = a^x
• Logarithmic Functions: f(x) = log_a(x)

Common Transformations

• Vertical Shifts:
  • y = f(x) + c: Shift upward by c units
  • y = f(x) - c: Shift downward by c units
• Horizontal Shifts:
  • y = f(x - c): Shift to the right by c units
  • y = f(x + c): Shift to the left by c units
• Vertical Stretching/Compression:
  • y = c ⋅ f(x): Stretch vertically if c > 1, compress if 0 < c < 1
• Horizontal Stretching/Compression:
  • y = f(cx): Compress horizontally if c > 1, stretch if 0 < c < 1
• Reflections:
  • y = -f(x): Reflect about the x-axis
  • y = f(-x): Reflect about the y-axis

Arithmetic Combination of Functions

• Sum: (f + g)(x) = f(x) + g(x)
• Difference: (f - g)(x) = f(x) - g(x)
• Product: (f ⋅ g)(x) = f(x) ⋅ g(x)
• Quotient: (f/g)(x) = f(x)/g(x)
• (f ∘ g)(x) = f(g(x)) describes the composition of functions
• The domain of f ∘ g is the set of all x where g(x) is in the first domain of f

Even and Odd Functions

• Even Function: f(-x) = f(x), symmetric to y-axis
• Odd Function: f(-x) = -f(x), symmetric about origin
• f(x) = x^2 is an even function
• f(x) = x^3 is an odd function

Increasing and Decreasing Functions

• Increasing Function: f(x_1) < f(x_2) when x_1 < x_2
• Decreasing Function: f(x_1) > f(x_2) when x_1 < x_2
• Constant Function: f(x_1) = f(x_2) for all x_1 and x_2

Planck's Law

• Describes spectral density of electromagnetic radiation emitted by black body
• Foundational concept in quantum mechanics

Mathematical Expression of Planck's Law

• Formula: B(λ, T) = (2hc^2 / λ^5) ⋅ 1 / (e^(hc / λ k_B T) - 1)
• B(λ, T) = spectral radiance
• λ = wavelength
• T = absolute temperature
• c = speed of light
• h = Planck constant
• k_B = Boltzmann constant

Key Points of Planck's Law

• Quantization of Energy indicates that energy is emitted in quanta.
• Wien's Displacement Law states the wavelength is inversely proportional to temperature, and can be derived from Planck's law.
• Stefan-Boltzmann Law includes integration and defines total energy radiating.
• Planck's law is applied in astrophysics, and thermal engineering.
• An increasing wavelength illustrates Wien's displacement law.

Quantization of Energy

• Can only take on discrete values
• E = hf = hc/λ is the energy of a photon where h is Planck's constant, f = frequency of the photon, λ = wavelength of the photon, and c is the speed of light
• What is the energy of a photon with a wavelength of 500nm?
• (6.626 x 10^-34 J times s)(3.00 x 10^8 m/s)/(500 x 10^-9 m) = 3.98 x 10^-19 J is the energy of a photon

Work Function

• The minimum energy for removing a surface electron
• Equation E = h f_0 = hc/λ_0, where E equals work function, f0 is the threshold frequency and λ0 is the threshold wavelength
• Sodium's work function is 2.3 eV.
• Threshold frequency of sodium is 5.55 * 10^14 Hz

Kinetic Energy of Photoelectrons

• Maximum kinetic energy equation: K max = hf - E
• K max stands for kinetic energy of ejected electron, and hf for energy of the incident photon (E is the work function)
• Photoelectrons in metal with work function of 3.0 eV, illuminated by light of 400nm wavelength, has energy of 1.64 * 10^-20J

Configuración (Configuration)

• git config --global user.name "nombre" Sets the name that will be attached to your commits.
• git config --global user.email "correo" Sets the email address that will be attached to your commits.
• git config --global core.editor "editor de texto" Sets the default text editor that will be used with Git.
• git config --list Lists all configurations.

Crear (Create)

• git init Initializes a new Git repository in the current directory
• git clone url Clones a Git repository from a remote URL

Cambios (Changes)

• git add archivo Adds the file to the staging area.
• git add . Adds all changes in the current directory to the staging area.
• git commit -m "mensaje" Commits the staged changes with a message.
• git status Shows the status of the working directory and staging area.
• git diff Shows the differences between the working directory and the staging area.

Historial (History)

• git log Shows the commit history.
• git log --oneline Shows the commit history in one line.
• git log --graph Shows the commit history as a graph.
• git show id-commit Shows metadata and changes introduced in the specified commit.

Ramas (Branches)

• git branch Lists all local branches.
• git branch nombre-rama Creates a new branch.
• git checkout nombre-rama Switches to the specified branch.
• git merge nombre-rama Merges the specified branch with the current branch.
• git branch -d nombre-rama Deletes the specified branch.
• git branch -D nombre-rama Deletes the specified branch, even if it hasn’t been merged.

Remoto (Remote)

• git remote add nombre url Adds a remote repository.
• git remote -v Lists all remote repositories.
• git fetch nombre Downloads changes from the specified remote repository.
• git pull nombre rama Downloads changes from the specified remote repository and merges them with the current branch.
• git push nombre rama Uploads changes to the specified remote repository.

Deshacer (Undo)

• git reset archivo Removes the file from the staging area.
• git checkout -- archivo Discards the changes in the specified file.
• git revert id-commit Creates a new commit that undoes the changes introduced in the specified commit.

Espace vectoriel

• A vector space includes $E$ and two operations

Addition:

• $E \times E \rightarrow E$, $(x, y) \mapsto x + y$

Scalar multiplication:

• $K \times E \rightarrow E$, $(\lambda, x) \mapsto \lambda \cdot x$
• K can be $\mathbb{R}$ or $\mathbb{C}$
• $(x + y) + z = x + (y + z)$ for all $x, y, z \in E$
• $x + y = y + x$ for use $x, y \in E$
• An element $0 \in E$ such that $x + 0 = x$ exists
• For all $x \in E$, $\exists -x \in E$, such that $x + (-x) = 0$.
• $\lambda (x + y) = \lambda x + \lambda y$ for use $\lambda \in K$ and $x, y \in E$
• $(\lambda + \mu) x = \lambda x + \mu x$ for all $\lambda, \mu \in K$ and $x \in E$

Sous-espace vectoriel

• A non-empty sub-ensemble $F$ of an array vectoriel $E$
• For all $x, y \in F$, $x + y \in F$
• $\lambda x \in F$

Combinaison linéaire

• A combination of vector $x_1, x_2,..., x_n \in E$ with the form
• $\lambda_1 x_1 + \lambda_2 x_2 +... + \lambda_n x_n$ with $\lambda_1, \lambda_2,..., \lambda_n \in K$

Espace engendré

• Span of vectors $x_1, x_2,..., x_n \in E is noted in the form
• $ Vect(x_1, x_2,..., x_n)$ is a general combination of the list

Famille libre

• $x_1, x_2,..., x_n \in E$ implies that $\lambda_1 x_1 + \lambda_2 x_2 +... + \lambda_n x_n = 0$
• $\lambda_1 = \lambda_2 =... = \lambda_n =0$

Famille génératrice

• Generating vectors $x_1, x_2,..., x_n \in E Vect(x_1, x_2,..., x_n) = E$
• Base includes $E$ a family of general vectors

Dimension

• $dim(E)$ implies the # of vectors to a base
• $f : E \rightarrow F$ can be expressed with two vector spaces $E$ and $F$ with the field $K$
• $f(x + y) = f(x) + f(y) $
• $f(\lambda \cdot x) = \lambda f(x)$

Noyau et image

• Can show $Ker(f)$ = kernel of f - $Ker(f) ={x \in E mid f(x) = 0}$ Le rang de $f Im(f) = {y \in F mid \exists x \in E, f(x)=y }$

rang

The range $rg(f)$, equals $dim of (Imf)$. The theorem is an application $f : E \rightarrow F$ given by E. A dimension range.

Matricies

$m \times n$ is the size of a matrix of matrix.

Matrix Operations

• Addition: $A+B$ is given when $A$ and $B$ have the same size.
• Scalar Multiplication: $=\frac{1}{\sqrt{2\pi}\sigma} \sum {\lambda \cdot A}$ Multiplication: is given when columns of $A$ aligns w/ vector columns of $B-

Matrix Inverse

• A matrix is a square when it is inverse ( A * B = B * A = 1) where $I$ equal the identity matrixs in $A$

Value

• The value of an image to scaler operations that matrix with linear transformations

Vector Eigen

• A non zero of A $ A \times 2*2
• Identity Matrices
• $A \times 2 * 2/ is A \plus 1 equals $A /matrix matrix with numbers and all entry matrix matrix numbers matrix matrix Matrix matrix matrix matrix . A A A

A is the matrix

• Matrix can be described as such $m \times n = a, m,n/ number
• A matrix can be added for any multiplication operations as long as you know the #$

Matrix

• Rectangular arrays of numbers
• Numbers of described in arrays by rows, then columns

Matrix Arithmetic

• Dimensions that equal can be added or subtracted:
• Scalar can be multiplied
• Product $AB$ = matrix that must equal A rows and B must equal their columns

special metric .

Identity magic Square of the Matrix matrix

• numbers

Algorithmic Trading and Quantitative Strategies

• Algorithmic trading uses computer programs to automate trading decisions depending on price, timing, etc
• (Automated trading vs. blackbox, ect)
• Quantitative trading subset: Quant trading/systemic, data backtesting

Who use Algo’s ?

• Buy-side. Firms that manage capital for clients, such as hedge funds, mutual funds, pension funds, family offices, and insurance firms.
• Side-Sell Bank investments, Market Makers ect

Why use algorithms rather than doing it manually?

• Increased Efficiency: Automated execution reduces human error, Faster execution improves things such as, ability to monitor markets 24/7
• Reduced Costs: Reduced transaction costs with manpower, through automation.
• Improved Decision Making, based on quantitative analysis without emotion
• Greater. Transparency audit trails

Types of Algo Trading Strategies:

• Trend Following: Identify assets by betting up or trends, as price increases
• Mean Reversion strategies Identify assets as they bet, as they leave from assets.
• Arbitrage: Exploit price differences in different markets for the same assets; by buying /selling
• Execution Algorithms.:
• Maket making buy and sell to place the spread

Example of Moving Average

• **Trend Following: =**Identify assets by betting up or trends, as price increases

Key points

• SMA =

** SMA long =

Challenges

• Change in market, overfit, execution/technology, and/regulatory of an act

resources

• Look at various journals, etc

Risk.

• Use clear loss-stop in order to adapt and increase positions as you see necessary

Testing vs Paper trading

• Simulated for analysis and testing, where a journal online is used.