Merge Sort Basics

• These notes cover the basics of Merge Sort, mechanical vibrations, chemical kinetics, text extraction with Google Cloud Vision AI, Volumes using Cylindrical Shells, Heat Transfer, Algorithmic Trading, Electronic Configuration and Matrix Operations, as well as the Schrödinger equation.
• All are summaries of the provided texts

Algorithmen Und Datenstrukturen: Sortieren: Merge Sort

• Merge Sort is a sorting algorithm based on the "divide and conquer" paradigm.
• The basic steps involve dividing a list into two halves, recursively sorting each half, and then merging the sorted halves back together.
• For lists with more than one element, it divides the list into two halves.
• The sublists are each sorted recursively.
• The sorted sublists are merged together in order.
• It operates at $O(n \log n)$ complexity in all scenarios of best, average and worst cases.
• It has a space complexity of $O(n)$ due to it's non in-place nature.
• Merge sort is considered a stable sorting algorithm.
• Merge sort is a comparison-based sorting algorithm.
• Suited for large, unsorted datasets and functions well with linked lists.

Unidad 1: Vibraciones Mecánicas

• Mechanical vibration is the oscillatory movement of a mechanical system around an equilibrium position.
• Vibrations are studied for machine and structure design via failure prevention with resononance and fatigue.
• Noise control to reduce the emissions caused by machines and equipment is another reason.
• Condition monitoring to detect failures in rotating equipment is benificial for larger repairs in the future.
• Improved comfort by reducing vibrations in vehicles and buildings is useful for occupants.

Clasificación De Las Vibraciones

• Free vibrations occur under initial conditions without external force.
• Forced vibrations are caused by an external force.
• Undamped vibration lacks energy dissipation.
• Damped vibration involves energy dissipation over time.
• Linear systems obey the superposition principle.
• Non-linear systems do not obey superposition.
• Deterministic vibrations can be predicted in time.
• Random vibrations cannot be predicted in time.

Vibración Libre No Amortiguada

• The equation of motion for a mass-spring system is described by $m\ddot{x} + kx = 0$.
• The natural frequency for the same system is given by $\omega_n = \sqrt{\frac{k}{m}}$.
• The general solution for a mass-spring system is $x(t) = A\cos(\omega_n t) + B\sin(\omega_n t)$, where A and B depend on the initial conditions.

Tipos De Amortiguamiento

• Viscous damping exhibits a damping force proportional to velocity.
• Coulomb damping exhibits a damping force that is constant and opposing movement.
• Hysteretic damping's damping force relies on deformation.

Amortiguamiento Viscoso

• The equation of motion follows the form: $m\ddot{x} + c\dot{x} + kx = 0$
• The damping factor is $\zeta = \frac{c}{2\sqrt{mk}}$
• Different types of damping depend on zeta: <1 sub-damped and oscillating, = 1 critically damped where it returns without oscillation, > 1 over-damped where it returns slowly.
• The amortized frequency is given by $\omega_d = \omega_n\sqrt{1 - \zeta^2}$

Chemical Kinetics

• Reaction rate specifies the rate reactants are converted into products.
• It is expressed as the change in concentration per unit of time.
• Rate = $-\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$ for the reaction $aA + bB \rightarrow cC + dD$
• Rate law: $Rate = k[A]^m[B]^n$ for the reaction $aA + bB \rightarrow cC + dD$
• Where k is the rate constant, A and B are the concentration of reatants and m and n are the reaction orders for the given reactants.

Reaction Order Types

• Zero Order: $Rate = k$[A]^0 = k$, since the reaction order is 0 the rate is independent from the concentration.
• First Order: $Rate = k[A]$: rate is directly proportional to the concentration of reactant.
• Second Order: $Rate = k[A]^2$ rate is proportional to the square of the reactant concentration or the concentration of two factors.

Transcribing Images and Documents With Google Cloud Vision AI

• Goal: Extract text using the Vision AI API from images and documents.
• Cloud Project Creation requires a google cloud account or signup for their free trial.
• Then the Cloud Vision API has to be enabled.
• Set up authentication by creating service account keys, downloaded to your computer in JSON format.
• Install the Google Cloud Client Library with pip.
• Set environment variables to load file path of the JSON file containing your service account key.

Install the Google Cloud client library

• Activate Cloud Shell
• Install the Google Cloud client library using pip.

• pip install google-cloud-vision
  

• Code may be written and uploaded and run via the terminal.

MTH132 - Section 6.3 - Volumes by Cylindrical Shells

• Instead of slicing perpendicular to the axis of rotation, the cylindrical shell method involves slicing parallel to it.

Formula

• The formula for the volume of a cylindrical shell with respect to x is: $V = 2\pi \int_a^b r(x)h(x) , dx$, where: $r(x)$ is the radius, $h(x)$ is the height, and a and b are the limits of integration.
• The formula for the volume of a cylindrical shell with respect to y is: $V = 2\pi \int_c^d r(y)h(y) , dy$, where: $r(y)$ is the radius, $h(y)$ is the height and c and d are the limits of integration.

Heat Transfer

• There are three modes of heat transfer: conduction, convection, and radiation.
• Heat transfer in a solid or stationary fluid due to a temperature gradient.
• $\dot{Q} \propto A \frac{dT}{dx}$ the amount of heat transfered is proportional to the Area * The temperature gradient.
• $\dot{Q} = -kA \frac{dT}{dx}$: Fourier's Law of Heat Conduction defines the Heat transfer rate = proportionality constant aka k * Area * Temperature Gradient.
• $R = \frac{L}{kA}$: The Thermal resistance to a plane wall follows the equation: Thickness of wall / Thermal conductivity * Area.
• The Thermal resistance to a Cylinder follows the equation: $R = \frac{ln(r_2/r_1)}{2\pi Lk}$
• The Thermal resistance to a Sphere follows the equation: $R = \frac{r_2 - r_1}{4\pi kr_1r_2}$
• Law of Cooling follows the equation: $\dot{Q} = hA(T_s - T_{\infty})$, where h is the convection heat transfer coefficient, $T_s$ is the surface temp and $T_{\infty}$ is the fluid temperature. $R = \frac{1}{hA}$: Thermal resistance to convection is given by 1 / (convection heat transfer coefficient * area)
• Stefan-Boltzmann Law for Radiation transfers through electromagnet waves $Q = \epsilon \sigma A (T_s^4 - T_{surr}^4)$
• Where $\epsilon$ is surface emissivity and sigma is Boltzmann constant.
• Total thermal resistance $R_{total} = R_{conduction} + R_{convection} + R_{radiation}$
• Heat transfer rate $\dot{Q} = \frac{\Delta T}{R_{total}}$

Algorithmic Trading

• Algorithmic trading uses computer programs executing trading orders off of predetermined instructions.
• Strategy development, algorithm design, backtesting and deployment are all steps in algorithmic trading.
• It can increase speed and efficinecy, reduces emotional bias and improve order execution.

Key Components

• Trading Strategy: Logic that defines when to buy or sell assets.
• Execution Algorithm: Code that submits orders to the market.
• Risk Management: Rules to limit potential losses.
• Data Analysis: Tools to analyze market data.
• Infrastructure: Hardware and software to support the trading system.

Types of Algorithms

• Trend Following, mean reversion, arbitrage, and market making are various algorithms to utilize in algorithmic trading. There are many benefits, challenges, components, resources and risks of automated trading.

Quimica

Distribuição Eletrónica

• Electronic distribution or electron configuration describes how electrons are arranged within an atom.
• Electronic distribution is crucial for chemical properties of elements, it influences how they interact.
• The Hund Rule establishes that when filling orbitals with the same amount of energy, electrons are distributed individually with each before pairing.
• Diagram of orbitals represent electron configuration, the level and sub level energy of an atom in notation.
• Valence electrons are the outermost electrons and participate in chemical bonds.

Matrices:

• A matrix is a rectangular array made of numbers, entries or elements all arranged in rows and columns.
• Square matrices, are same row and column count, rows matrix's exist one row and column matrixes have one column.
• Determinants can be found from square matrixes with some special scalar quantity.
• Matrix operations such as addition, subtraction, and multiplication by scalar or other matrixes is possible.

The Schrödinger Equation

• The Schrödinger equation is a mathematical equation that describes the time evolution of a physical system.
• It is a key result in quantum mechanics.
• More specifically, the time-dependent Schrödinger equation describes how a physical system evolves in time.
• The time-independent Schrödinger equation describes the stationary states of a system.
• The Time-Dependent Schrödinger Equation is $i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r}, t)$
• The Time-Independent Schrödinger Equation is $\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})$.
• Hamiltonian operators correspond to the total energy of the system.