Stemplots, Histograms and Time Plots

Stemplots

• Stemplots are suited for small data sets and retain original observation values.
• To create a stemplot:
  • Separate data into stems and leaves.
  • List stems vertically, smallest to largest, separated by a line.
  • Write leaves in the stem's row, increasing outward from the stem.
• To address data bunching, stems can be split.
• Stemplots should include titles and legends.

Histograms

• Histograms are suitable for larger datasets
• Steps to create a histogram:
  • Divide data into classes with equal width.
  • Tally individuals per class, representing frequency.
  • Draw histogram bases of bars covering classes, with height representing frequency.
• Relative frequency histograms display percentages instead of raw counts.
• Aim for approximately five classes, but adjust to suit your data.
• Too few classes result in a histogram with a few tall bars
• Too many classes result in a histogram with many short bars

Time Plots

• Time plots chart observations against their measurement times.
• Always label axes in a time plot
• Connecting data points with lines highlights trends.

Analyzing Distributions:

• Assess overall pattern and deviations.
• Overall patterns are described by shape, center, and spread
• Outliers are values outside overall pattern.

Symmetric and Skewed Data

• Symmetry indicates mirror-image sides on a histogram.
• Skewed right distributions have longer tails on the right-side (larger values).
• Skewed left distributions have longer tails on the left-side (smaller values).
• Skewness direction corresponds to tail, not the bulk of data.

Describing Time Plots

• In time plots, monitor for trends like upward or downward movement over time
• Review for cycles for repeating patterns over time

Measuring Center: The Mean

• The mean (x̄) is calculated by summing data values and dividing by the number of observations.
• The Formula for the mean is: x̄ = (x₁ + x₂ + ... + xₙ) / n or x̄ = (1/n) Σxᵢ
• The mean is sensitive to outliers.

Measuring Center: The Median

• The median (M) is the dataset midpoint, dividing data into roughly equal halves.
• Steps to find the median:
  • Arrange data from smallest to largest.
  • The median (M) is the center observation if there are an odd number of values.
  • If there is an even number of values, the median (M) is the mean of the two central data points.
• The Median is resistant to outliers.
• The mean and median coincide in symmetric distributions.
• In skewed distributions, the mean is further along the long tail than the median.

Measuring Spread: The Quartiles

• Steps to calculate quartiles (Q1 and Q3):
  • Arrange data ascendingly and find the median (M).
  • The first quartile Q1 is the median of data points to the left of the overall median.
  • The third quartile Q3 is the median of data points to the right of the overall median.
• Exclude the median when finding quartiles for odd-sized datasets.

The Five-Number Summary and Boxplots

• The five-number summary includes, in ascending order: smallest observation, Q1, M, Q3, and largest observation.
• A boxplot is a graph of a five-number summary:
  • A box spans from Q1 to Q3, bisected by a median line.
  • Whiskers extend to the smallest and largest observations.
• Some boxplots may display outliers as separate data points.

Measuring Spread: Standard Deviation

• Standard deviation (s) measures data point's distance from the mean
• Variance (s²) is the average of squared distances and found using equation: s² = Σ(xᵢ - x̄)² / (n-1)
• Standard Deviation equation: s = √[Σ(xᵢ - x̄)² / (n-1)].
• s measures spread about mean, used when the mean measures centers
• s equals zero only when there is no spread.
• s is susceptible to outliers.
• s shares measurement units with original observations

Numerical Summaries

• All basic numerical summaries can be calculated using statistical software packages
• Calculators often omit quartiles, but some packages include them in output

Selecting Statistics

• For skewed distributions, five-number summaries beat mean & standard deviation
• Use x̄ and s only for symmetric, outlier-free distributions.

Density Curves

• Key Aspects of a Density Curve:
  • It must be on or above the horizontal axis.
  • The area underneath is precisely 1.
• Describes overall distribution patterns. The area corresponds to all observations within a data range.
• No real dataset is perfectly described by density curves, they provide an approximation.

Curves for Measuring Center and Spread

• The median of a curve divides the area under curve evenly
• The mean is the balance point of a curve
• Median and mean are identical for symmetrical density curves
• The mean is pulled away from median toward the long tail on skewed curves
• Standard deviation can't be determined visually on a density curve

Normal Distributions

• Normal distributions are described by symmetrical, single-peaked Normal density curves.
• A Normal curve is defined by its mean (μ) and standard deviation (σ).
  • The mean is located at the symmetrical center of the curve.
  • The standard deviation controls the curve's spread.
• The standard deviation (σ) is the spread measurement for Normal distributions, representing the distance from center to curve's change-of-curvature points
• Normal distribution abbreviated: N(μ, σ).

The 68-95-99.7 Rule

• For Normal distributions, the following applies:
  • 68% of values are within 1σ of the mean μ.
  • 95% of values are within 2σ of the mean μ.
  • 99.7% of values are within 3σ of the mean μ.

Standard Normal Distribution

• The standard Normal distribution is abbreviated N(0, 1) with a mean (μ) of 0 and a standard deviation (σ) of 1.

Standardized Observations

• When the mean is μ, and the standard deviation is σ, the value of $x$ is calculated using: z = (x - μ) / σ
• Standardized values are called z-scores.
• The z scores indicate the distance that the observation values fall based on the mean.
• Observations greater than the mean are positive, while less than the mean are negative.

Standard Normal Table usage

• Table A is an area table underneath the standard Normal curve. The table entry for each value z is the area under the curve to the left of z
• The area is a proportion of the total area 1
• Symmetry Exploitation: For positive z-scores, exploit symmetry for area calculations using the formula.
• Backward Table: Find values for proportions below by looking them up in the Table A body
• Read the value from the left and top margin

Normal Quantile Plots

• Straight line points on normal quantile plots suggest approximately Normally distributed Normal data.
• Deviations suggest a non-Normal distribution
• Outliers will look like points far removed on the overall point
• While not definitively proving Normal distributions, quantile plots can detect moderate non-Normality.

Algorithmen und Datenstrukturen: Komplexität

• Die Landau-Symbole (O-Notation) beschreiben das asymptotische Verhalten einer Funktion.

• f wächst asymptotisch nicht schneller als g

  $f(n) = O(g(n)) \Leftrightarrow \exists c>0, n_0 \in \mathbb{N}: \forall n \geq n_0: f(n) \leq c \cdot g(n)$

• f wächst asymptotisch mindestens so schnell wie g

  $f(n) = \Omega(g(n)) \Leftrightarrow \exists c>0, n_0 \in \mathbb{N}: \forall n \geq n_0: f(n) \geq c \cdot g(n)$

• f wächst asymptotisch gleich schnell wie g

  $f(n) = \Theta(g(n)) \Leftrightarrow \exists c_1, c_2>0, n_0 \in \mathbb{N}: \forall n \geq n_0: c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n)$

• f wächst asymptotisch langsamer als g

  $f(n) = o(g(n)) \Leftrightarrow \forall c>0 \exists n_0 \in \mathbb{N}: \forall n \geq n_0: f(n) < c \cdot g(n)$

• f wächst asymptotisch schneller als g

  $f(n) = \omega(g(n)) \Leftrightarrow \forall c>0 \exists n_0 \in \mathbb{N}: \forall n \geq n_0: f(n) > c \cdot g(n)$

Beispiele für Landau-Symbole

• $n^2 = O(n^3)$
• $n^2 = o(n^3)$
• $n^3 = \Omega(n^2)$
• $n^3 = \omega(n^2)$
• $n^2 \neq O(n)$
• $n^2 = \Theta(n^2)$

Rechenregeln von Landau-Symbole

• Konstanter Faktor $O(c \cdot f(n)) = O(f(n))$
• Summe $O(f(n) + g(n)) = O(max(f(n), g(n)))$
• Produkt $O(f(n)) \cdot O(g(n)) = O(f(n) \cdot g(n))$
• Polynom $O(a_k n^k + a_{k-1} n^{k-1} +... + a_1 n + a_0) = O(n^k)$
• Logarithmus $O(log_a n) = O(log_b n)$

Typische Komplexitätsklassen

• $O(1)$ - konstant
• $O(log(n))$ - logarithmisch
• $O(n)$ - linear
• $O(n \cdot log(n))$ - n log n
• $O(n^2)$ - quadratisch
• $O(n^3)$ - kubisch
• $O(2^n)$ - exponentiell
• $O(n!)$ - faktoriell

Introduction to CFD

• Computational Fluid Dynamics predicts fluid flow, heat, mass transfer, chemical reactions using numerical processes solving mathematical equations.

Applications of CFD

• Aerospace: Aircraft design, aerodynamic analysis
• Automotive: Vehicle design, engine simulation, thermal management
• Chemical Processing: Reactor design, mixing, separation
• Civil Engineering: Building aerodynamics, wind loading, bridge design
• Electronics: Heat sink design, thermal management
• Environmental: Pollution dispersion, weather modeling
• HVAC: Heating, ventilation, and air conditioning system design
• Marine: Ship hydrodynamics, propeller design
• Oil & Gas: Pipeline flow, reservoir simulation

CFD Process

• The CFD process involves the following steps:
  • Problem Definition: Define the physical problem and analysis goals of the simulation
  • Geometry and Mesh Generation: Create a geometric model and divide it into small cells or elements.
  • Physics and Fluid Properties: Define the fluid properties, boundary conditions, and physical models
  • Solving: Solve the governing equations numerically using a CFD solver.
  • Post-processing: Analyze and visualize the results.
  • Validation: Compare the CFD results with experimental data.

Governing Equations

• CFD simulations are based on the fundamental governing equations of fluid dynamics:
  • Continuity Equation: Conservation of mass; ∂ρ/∂t + ∇ ⋅ (ρV) = 0
  • Momentum Equation: Conservation of momentum (Navier-Stokes equations); ∂(ρV)/∂t + ∇ ⋅ (ρVV) = −∇p + ∇ ⋅ τ + ρg + F
  • Energy Equation: Conservation of energy; ∂(ρh)/∂t + ∇ ⋅ (ρhV) = ∇ ⋅ (k∇T) + Sh
  • Where:
    • ρ = Density
    • t = Time
    • V = Velocity vector
    • p = Pressure
    • τ = Viscous stress tensor
    • g = Gravitational acceleration
    • F = External body forces
    • h = Enthalpy
    • k = Thermal conductivity
    • T = Temperature
    • Sh = Heat source term

Turbulence Modeling

• Turbulence requires models to approximate effects on mean flow
  • RANS (Reynolds-Averaged Navier-Stokes)
    • k-ε model
    • k-ω model
    • Spalart-Allmaras model
  • LES (Large Eddy Simulation)
  • DES (Detached Eddy Simulation)
  • DNS (Direct Numerical Simulation)

Discretization Methods

• Discretization methods are approximation equations used on the computational mesh
  • Finite Difference Method (FDM)
  • Finite Volume Method (FVM)
  • Finite Element Method (FEM)

CFD Software

• Commercial CFD software
  • ANSYS Fluent
  • ANSYS CFX
  • STAR-CCM+
  • COMSOL Multiphysics
• Open Source CFD software
  • OpenFOAM
  • SU2

Advantages of CFD

• Cost-effective reduced need for physical experiments.
• Detailed Information: Provides detailed flow field information.
• Parametric Studies: modification of design parameters and operating conditions.
• Virtual Prototyping: Testing different designs and scenarios.

Limitations of CFD

• Accuracy based on mesh quality, turbulence model, and accuracy of boundary conditions.
• Computational Cost: Simulations can be computationally expensive.
• Model Validation: Results must be validated against data to ensure accuracy.
• User Expertise: Requires expertise in fluid dynamics, numerical methods, and CFD software.

Static Electricity

• Static electricity is an accumulation of electric charges on an object's surface.
• Charges can be positive or negative, creating an electric field.
• "Static" means charges are not moving, unlike current electricity.

Charging Mechanisms

• Friction (Triboelectric Effect):
  • Dissimilar materials rubbed together transfer electrons; material losing electrons becomes positively charged, and the material gaining electrons becomes negatively charged.
• Conduction:
  • A charged object contacts a neutral object to transfer the charges until they reach the same electric potential and acquire the same charge type as the charged object.
• Induction:
  • A charged object brought near a neutral object causes charge redistribution, without contact; one side becomes charged with the opposite polarity, the other with the same polarity.

Properties of Electric Charge

• Quantization of Electric Charge:
  • Electric charge is quantized in discrete units.
  • This equals the elementary charge (e), the magnitude of charge by a proton or electron, at $1.602 \times 10^{-19}$ coulombs (C).
  • The charge of an object is an integer multiplied by the elementary charge.
    • q = ne
      • q is an electrical object
      • n is an integer
      • e is the elementary charge
• Conservation of Electric Charge:
  • Total electric electrical charge in a closed system remains constant.
  • Charge can transfer from one object to another.
• Additivity of Electric Charge:
  • The algebraic sum is the total charge the object has from the sum of individual charges on the object.
    • qtotal = q1 + q2 + q3 + ... qn
      • q total is the total charge of the object
      • q1, q2, q3...qn are the individual charges present on the object

Coulomb's Law

• Electrostatic force between stationary electrically charged particles.
• Force is proportional to charges and inversely proportional to squared distance.
• Force is attractive for opposite charges and repulsive for like charges.
• F = k |q₁q₂| / r²
  • F is electrostatic force
  • k is Coulomb's constant
  • q₁ and q₂ are magnitudes of charges
  • r distance between charges

Electric Fields

• A region around a charged object where another charged object experiences force.
• It is a vector quantity.
• Created by electric charges and exert forces on other charges.

Electric Field Strength

• Force (F) on a small positive charge (q₀) at a point, divided by the charge's magnitude.
  • E = F / q₀
• Direction matches the force on the positive test charge.

Electric Field of Point Charge

• E = k |q| / r²
  • E electric field strength
  • k Coulomb's constant
  • q magnitude of the point charge
  • r distance from the point charge

Charging by Induction

• Bring a Charged Object Near:
  • Bring a charged object near a neutral one object proximity must be close but no contact
• Redistribution of Charges:
  • The presence of either a positive or negative charge causes electrons to redistribute from one side to the other
• Ground the System:
  • Grounding wire connected provides ground for electron flow
    • If the charged object is negative, excess electrons will flow from the conductor to the ground through the grounding wire
    • If the charged object is positive, electrons will flow from the ground to the conductor through the grounding wire
• Removal after Redistributing :
  • Remove the ground leaving the redistribution
• The Charged Object:
  • Once the charged object is removed, a uniform redistribution of charge occurs

Chemical Kinetics: Reaction Rate

• The Reaction Rate represents the rate of reactant to product change/unit time .
• Rate Expression: For a Reaction: aA + bB → cC + dD $$ Rate = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} $$
  • [A], [B], [C], [D] are the concentrations of reactants and products
  • a, b, c, d are the stoichiometric coefficients Rate Law: expresses a reaction's rate in the function of reactant quantities and a consistent k.

$$ Rate = k[A]^m[B]^n $$

Where:

- k is the rate constant.
- m and n are the reaction orders with respect to reactants A and B, respectively.
- The overall reaction order is m + n.

Factors Affecting Reaction Rate

- Concentration of Reactants: Increasing the concentration increases the rate
- Temperature: Generally temperature increases reaction rate
- Surface Area: Solid reaction surface area increase
- Catalysts: Catalysts increases rate without being consumed
- Pressure: increasing pressure increases reaction rate in reactions with gases

Rate Constant

Arrhenius Equation

• The equation describes temperature dependence using the following equation
• k = A * e^-(Ea/RT)
  • k = the rate constant
  • A pre-exponential/frequency factor
  • Ea = Activation energy
  • R is 8.314 J/mol * K
  • T is Temperature

Determining Activation Energy

• The activation energy (E a) can be determined from the Arrhenius equation by plotting ln(k) versus 1/T.

$$ \ln(k) = \ln(A) - \frac{E_a}{R}\left(\frac{1}{T}\right) $$

• The slope of the plot is Ea / R, from which Ea can be calculated.

Reaction Order

Zero Order Reaction

• Rate does not depend on concentration amounts
• Rate law = K
• Integrated rate law [A]t = -Kt + [A]0
• Half-life t1/2 = [A]/2K

First Order Reaction

• Rate depends on concentration amounts
• Rate Law - K[A]
• Integrated Rate law ln[A]T -Kt + ln[A]0
• Half-Life t1/2 = 0.693/K

Second Order Reactions

• This proportional from concentration rates between two or two compounds
• Rate Law K[A]^2 or K[A][b]
• Integrated ln rate law is ln(1/[A]t = Kt +1/[A]0
• Half Life is 1/K[A]0

Mechanisms of Reaction

Elementary Step

• Steps which are are single in a general mechanism

Rate Determining Step

• Rates that are slowest in the mechanisms

Molecularity

• Molecule number in elementary steps

Catalyst

• Substances that increase reaction rate
• Homogenous - Occurs same phase
• Heterogenous - Occurs different phase from reactant and gaseous state

Enzymes

• Proteins which catalyse biological reactions
• Michalels-Mentea mechanism type

Algorithmic Game Theory

• Modeling Strategic Interactions with Rational Agents Using Mathematical Models
• Games are formally described with strategic situations
  • Players- Who or how many are participating
  • States - Types of possible options
  • Intel - Available knowledge on those options
  • Payoffs - What is the outcome for those involved

Selfish Routing

• Network: $G = (V,E)$.
• Players: Users of the network.
• Strategies: Paths from source to destination.
• Payoffs: Travel time (cost).
• Solution Concept: Wardrop equilibrium.

Wardrop Equilibrium

When flows can reach equilibrium where change is not able to increase performance

Braess's Paradox

As links are added, the potential travel time increase increases for the user

Mechanism Design

Settings

• Possible amount
• Set value
• Center must make a choice

Mechanism

• Agents report the values
• Computes the outcome based on said values

Examples

• Auctions values
• Medical exchanges to set allocations
• Advertising for sponsored based ad slots and value for maximizing income

Algoritmo de Euclides

Introducción:

• El algoritmo de Euclides es para calcular números enteros

Explicación:

• Dados dos números el MCD es el número entero mayor

Pasos:

• Datos dos numeros enteros a y b
• Si b es 0 , entonces el mcd es a y termina
• Resto del resultado a y obtenga r como resultado
• Mueva a con b y b con r en las siguientes
• Vuelva a probar los resultados cuando r no sea 0

Ejemplo

1.Datos a = 48 y b = 18

2.48 / 18 = 2 r = 12

3.Con a = 18 y b = 12 pruebalo

4.18 / 12 = 1 r = 6

5.Con a = 12 y b = 6 pruebalo

6. 12 / 6 = 2 r = 0

7.Con a = 6 pruebalo

8.El mcd es 6

• Con este ejemplo el mcd de 48 y 18 es 6

Pseudocódigo

función MCD(a, b)
mientras b ≠ 0 hacer
r = a mod b
a = b
b = r
fin mientras
regresar a
fin función

Implementación en Python

def mcd(a, b):
while b:
a, b = b, a % b
return a

Conclusión

El algoritmo de Euclides es muy útil en la teoría de números así como tiene aplicaciones en encriptación y también informática

• Matriisien peruslaskutoimitukset: Yhteenlasku ja vähennyslasku:
• Matriiseilla oltava samat mitat.
• Summa A+B saadaan laskemalla matriisien A ja B vastinealkiot.
• Erotus A-B saadaan vähentämällä matriisien A ja B vastinealkiot.

Skalaarikertolasku:

• Matriisia A voidaan kertoa skalaarilla c.
• Tulo cA saadaan kertomalla A:n jokainen alkio c:llä

Matriisikertolasku:

• Tulon AB määrittely onnistuu, jos A:n sarakkeiden lkm = B:n rivien lkm.
• Olkoot A m x n -matriisi ja B n x p -matriisi.
• Tulo AB on m x p -matriisi, jonka tulojen summa alkiosta i ja matriisista B:
• (AB)ij = mnk=1 AikBkj = Ai1B1j + Ai2B2j + ··· + AinBnj

Chapitre 5: Espaces Vectoriels

• Ce chapitre generalise les idées de l'algèbre linéaire dans Rn à des espaces plus abstraits.

Définition:

• Un espace vectoriel est un ensemble non vide V d'objets, appelés vecteurs, sur lequel sont définies deux opérations: addition et multiplicatioun scalaire.

Pour tout u à V

• u + v à V
• cu a V
• u + v = v + u
• (u + v) + w = u + (v + w)
• Il existe un élément 0 de base avec u + 0 = u pour tout u
• chaque u u + (-u) = 0
• c(u + v) = cu + cv
• (c + d)u = cu + du
• c(du) = (cd)u
• 1u = u

Exemples

• Rn: vecteurs colonnes
• Pn: polynômes de degré au plus n
• Mmxn: matrices Mmxn
• F(R)

Sous Espaces

• Un sous-ensemble H d'un espace vectoriel V est un sous-espace de V si H est lui-même un espace vectoriel sous les opérations définies sur V.

Théorème

• Si H est un sous-ensemble non vide de V, alors H est un sous-espace de V si et seulement si les deux conditions suivantes sont satisfaites:
• Si u et v sont dans H, alors u + v est dans H.
• Si u est dans H et c est un scalaire, alors cu est dans H.

Bernoulli's Principle

• States that increase in fluid speed occurs with decrease in pressure or potential energy

How Wings Generate List

• Airfoil Design - Curved air travel vs flat area creating

Bernoulli's Principle in Action

• Increase air speed - Decrease in Pressure
• Bottom vs Top - Bottom is Slower and Higher

Lift Generation

• Creates force to lift wing

Key Concepts

• Fluid dynamics

Applications

• Cars Aerodynamics ,Aircraft Wings, Venturi Masks, Spray Atomizers