Alkanes Nomenclature

Química

Nomenclatura de Alcanos de Cadena Lineal

• Count the number of carbon atoms in the chain.
• Use the prefix that indicates the number of carbons.
• Add the suffix "-ane".
• Examples include methane ($CH_4$), ethane ($C_2H_6$), propane ($C_3H_8$), butane ($C_4H_{10}$), pentane ($C_5H_{12}$), hexane ($C_6H_{14}$), heptane ($C_7H_{16}$), octane ($C_8H_{18}$), nonane ($C_9H_{20}$), decane ($C_{10}H_{22}$), undecane ($C_{11}H_{24}$), dodecane ($C_{12}H_{26}$), tridecane ($C_{13}H_{28}$), eicosane ($C_{20}H_{42}$), and triacontane ($C_{30}H_{62}$).

Nomenclatura de Alcanos Ramificados

• Identify the parent chain, which is the longest continuous chain of carbon atoms.
• Number the parent chain so that the substituents have the lowest possible locator numbers.
• Name the substituents or alkyl groups by changing the "-ane" suffix to "-yl".
• Write the full name of the compound; list substituents in alphabetical order, preceded by their locator number, and use "di-," "tri-," "tetra-," to show multiple identical substituents.

Nomenclatura de Alcanos Cíclicos

• Name them as linear alkanes, but add the prefix "cyclo-".
• If the cycle has substituents, number it so that the substituents have the lowest locator numbers.
• Example: Cyclohexane, 1-Methylcyclohexane, 1,2-Dimethylcyclohexane

Estadística Descriptiva

Definición

• Focuses on summarizing and presenting data in a clear and concise manner.
• Uses measures such as mean, median, standard deviations, coefficients of correlation and mode, and graphs.

Tipos de Estadística Descriptiva

• Univariate: It focuses on the analysis of a single variable.
  • Includes measures of central tendency (mean, median, mode) and dispersion (range, variance, standard deviation), and graphs (histograms, bar charts, box plots).
• Bivariate: It analyzes the relationship between two variables.
  • Includes contingency tables, coefficients of correlation to measure the strength and direction of linear relationships, and scatter plots.
• Multivariate: This examines the relationships between multiple variables.

Aplicaciones de la Estadística Descriptiva

• It allows one to obtain a general vision of the data.
• It helps identify trends and relationships.
• It facilitates the presentation of information to different audiences.

Introduction to Forces

Definition of Force

• A force is a push or pull on an object, causing it to start or stop moving, change direction, or change shape.
• Forces are measured in Newtons (N).

Types of Forces

• Contact Force: Requires contact between objects.
  • Examples include applied force, frictional force, tension force, normal force, air resistance force, and spring force.
• Action-at-a-distance Force: Can exist without objects touching.
  • Examples include gravitational force, electrical force and magnetic force.

Common Forces

• Applied Force ($F_{app}$): A force applied to an object.
• Gravity/Weight ($F_g$): Attracts objects towards massive objects.
  • Calculated as $F_g = mg$, where $g$ is 9.8 m/s².
• Normal Force ($F_N$): Support force exerted by a stable object always perpendicular to the surface.
• Friction Force ($F_{fric}$): Force exerted by a surface as an object moves or tries to move across it. Two types exist static and kinetic friction
• Tension Force ($F_{tens}$): Transmitted through a string, rope, or wire when pulled tight.
• Air Resistance Force ($F_{air}$): Acts upon objects moving through the air.
• Spring Force ($F_{spring}$): Exerted by a compressed or stretched spring.

Net Force

• The vector sum of all forces on an object known as the overall force acting upon the object.
• Forces in the same direction are added, and opposite directions are subtracted.
• Net force is zero when forces are balanced, and non-zero when forces are unbalanced.
• Acceleration occurs only with a net or unbalanced force.

Free Body Diagrams

• A diagram showing all forces acting on an object with arrows starting from the center and proportional to their magnitude.

Vector Fields

• A vector field on $\mathbb{R}^2$ assigns a two-dimensional vector $\mathbf{F}(x, y) = \lang P(x, y), Q(x, y) \rang$ to each point $(x, y)$, where $P$ and $Q$ are scalar functions.
• A vector field on $\mathbb{R}^3$ assigns a three-dimensional vector $\mathbf{F}(x, y, z) = \lang P(x, y, z), Q(x, y, z), R(x, y, z) \rang$ to each point $(x, y, z)$, where $P$, $Q$, and $R$ are scalar functions.
• Examples include gravitational and electric force fields.

Gradient Vector Field

• A gradient vector field in two dimensions is defined as $\mathbf{F}(x, y) = \nabla f(x, y) = \lang f_x(x, y), f_y(x, y) \rang$ for a scalar function $f(x, y)$.
• A gradient vector field in three dimensions is defined as $\mathbf{F}(x, y, z) = \nabla f(x, y, z) = \lang f_x(x, y, z), f_y(x, y, z), f_z(x, y, z) \rang$ for a scalar function $f(x, y, z)$.
• An example of finding the gradient vector field of $f(x, y) = x^2y^3$ is $\nabla f(x, y) = \lang 2xy^3, 3x^2y^2 \rang$.

Conservative Vector Fields

• A vector field $\mathbf{F}$ is conservative if there exists a scalar function $f$ that results in $\mathbf{F} = \nabla f$.
• The scalar function $f$ is its "potential function".

Algorithmic Game Theory

Definition of Game Theory

• Game theory studies mathematical models of strategic interactions among rational agents.
• The agents act to maximize its own utility, which depends on what other agents do.

Price of Anarchy

• PoA = (social cost of the worst-case Nash equilibrium) / (optimal social cost)
• Social cost is the sum of all players' costs.

Theorem in selfish routing game

The price of anarchy in the parallel link routing game is 1.

Algèbre linéaire

Vecteurs

• A vector is defined by a direction, a sense on this direction, and a length.
• One can represent it by an arrow, a couple of points, or components in a base.

Matrices

• A matrix is a table of numbers, generally noted with a capital letter.
• The sum of two matrices depends on the number of columns of A and the number of rows of B.

Applications linéaires

• A linear application is a function $f: E \rightarrow F$ between two vectorial spaces $E$ and $F$.
• It can be represented by a matrix.

Matplotlib Tutorial

Overview

• Matplotlib is a plotting library for Python used for data visualization.
• Pyplot is a module within Matplotlib that offers an interface similar to MATLAB, making plot creation easier.

Plotting

• Line Plot: Connects individual data points with lines.
• Scatter Plot: Displays values for two variables in a dataset.
• Bar Chart: It compares different sets of data among different groups.
• Histogram: It represents the distribution of numerical data.
• Pie Chart: It showcases numerical proportions in a circular graphical format.

Thermodynamics

Energy Transfers

• Heat (Q): Transfer of energy between objects due to temperature difference, measured in Joules (J). $Q = mc\Delta T$
• Work (W): Transfer of energy when a force causes displacement, measured in Joules (J). $W = P\Delta V$

The First Law

• The change in internal energy of a system equals added heat minus work done. Formula: $\Delta U = Q - W$

Thermodynamic Processes

• Isobaric: Constant pressure, $W = P\Delta V$
• Isochoric (Isovolumetric): Constant volume, $W = 0, \Delta U = Q$
• Isothermal: Constant temperature, $\Delta U = 0, Q = W$
• Adiabatic: No heat transfer, $Q = 0, \Delta U = -W$

Heat Engines

• It converts thermal energy to mechanical work.
• $e = \frac{W_{net}}{Q_H} = 1 - \frac{Q_C}{Q_H}$

Refrigerators and Heat Pumps

• A device that transfers heat from a cold reservoir to a hot reservoir

Second Law of Thermodynamics

• Heat can't spontaneously flow from cold to hot.
• Entropy always increases.