Fourier Transform Properties

Fourier Transform Properties

• Linearity in Fourier Transforms is expressed as: $F[af(t) + bg(t)] = aF(f(t)) + bF(g(t)) = aF(\omega) + bG(\omega)$.
• A time shift in the time domain corresponds to a phase shift in the frequency domain: $F[f(t - t_0)] = e^{-j\omega t_0}F(\omega)$.
• Frequency shifting shows modulation in the time domain corresponds to a shift in the frequency domain: $F[e^{j\omega_0 t}f(t)] = F(\omega - \omega_0)$.
• Scaling property illustrates as $F[f(at)] = \frac{1}{|a|}F(\frac{\omega}{a})$.
• Time Differentiation property in frequency domain: $F[\frac{df(t)}{dt}] = j\omega F(\omega)$.
• Higher order as Time Differentiation generalises to $F[\frac{d^n f(t)}{dt^n}] = (j\omega)^n F(\omega)$.
• Time integration: $F[\int_{-\infty}^{t} f(\tau)d\tau] = \frac{1}{j\omega}F(\omega) + \pi F(0)\delta(\omega)$.
• Multiplication in the time domain corresponds to convolution in the frequency domain: $F[f(t)g(t)] = \frac{1}{2\pi}F(\omega)*G(\omega)$.
• Convolution in the time domain corresponds to multiplication in the frequency domain: $F[f(t)*g(t)] = F(\omega)G(\omega)$.
• Duality states that if $f(t) \leftrightarrow F(\omega)$, then $F(t) \leftrightarrow 2\pi f(-\omega)$.

Guía IA Generativa para el Sector Público

• IA generativa is transforming industries and sectors, including the public sector.
• The guide provides public officials a clear understanding for the IA generativa and its potential government applications.
• It also outlines key considerations for the responsible adoption of IA generativa.

¿Qué es la IA Generativa?

• IA generativa creates new and original content including, text, images, audio, and video.
• Unlike traditional AI, IA generativa uses machine learning algorithms to generate resembling data.

Tipos de Modelos de IA Generativa

• Modelos de Lenguaje generate coherent content from indications.
• Modelos de Imagen create realistic or stylised images from descriptions or images.
• Modelos de Audio generate music, voice, or sound effects.
• Modelos de Video create videos from text, images, or videos.

Aplicaciones en el Sector Público

• IA generativa can improve the efficiency, transparency and quality of public services.

Mejora de la Comunicación y el Servicio al Ciudadano

• Chatbots Inteligentes provide quick and accurate answers to citizen questions related to complex tasks.
• Generación de Contenido creares automated press releases, and social media content.
• Traducción Automática facilitates communication with citizens who speak different languages.
• Personalización de Servicios adapts information and services based on individual citizen needs.

Optimización de Procesos Internos

• Automatización de Tareas automates repetitive administrative tasks.
• Generación de Informes creates detailed reports from raw data quickly and efficiently.
• Asistencia en la Toma de Decisiones provides relevant information and analysis to help officials make informed decisions.
• Creación de Material de Capacitación generates tailored training materials for public employees.

Fomento de la Innovación y la Creatividad

• Diseño de Políticas Públicas helps policy makers explore different scenarios and identify innovative solutions.
• Generación de Ideas provides new perspectives and approaches to address public sector challenges.
• Creación de Contenido Educativo generates engaging educational content for students of all ages.
• Visualización de Datos creates data visualisations for easier understanding and decision making.

Consideraciones Clave para la Adopción de la IA Generativa

Ética y Responsabilidad

• Sesgos ensure IA generativa models do not perpetuate any biases.
• Transparencia includes understanding how IA generativa models work and finding conclusions.
• Responsabilidad has clear definitions of who is accountable for the decisions of IA generativa.
• Privacidad protects privacy of citizens when using IA generativa to process data.

Seguridad y Protección de Datos

• Ciberseguridad protects AI generative models against cyberattacks and guarantees integrity of data.
• Protección de Datos follows data protection laws and regulations when using the IA generativa to process information.
• Control de Acceso limits access to AI generative models along with training data to personnel.
• Monitoreo oversees the use AI generative models to detect potential misuse.

Capacitación y Desarrollo de Habilidades

• Formación provides public officials necessary training to understand and use AI generativa effectively.
• Desarrollo de Habilidades encourages development of skills in data science, machine learning, and the ethics of AI.
• Colaboración promotes to ensure that IA generativa solutions are useful.
• Conciencia raises awareness about benefits and risks of AI generativa between public officials and citizens.

Infraestructura y Recursos

• Hardware involves investing in hardware infrastructure for the efficient execution of AI generative models.
• Software uses software and open source tools to reduce costs and foster innovation.
• Datos accesses high quality relevant data to train AI generative models.
• Financiamiento allocates funds to support the implementation of AI generative models within the public sector.

Conclusión

• AI generativa transforms public sector by improving efficiency, transparency and quality of service.
• However, ethical, safety, and training must be address to guarantee effective and responsible use.
• A strategic and collaborative approach means optimal benefits from AI generativa.

Algorithmic Game Theory

• Game theory is a mathematical framework for analyzing strategic interactions between individuals or entities.
• The outcome of one's choices depends on the choices of others.
• It provides tools and concepts to model and predict behavior in competitive situations.

Key Concepts

• Players are the decision-makers involved in the game.
• Strategies are the possible actions or plans available to each player.
• Payoffs are the outcomes or rewards that players receive based on the combination of strategies chosen.
• Rationality is the assumption that players act in their own best interest to maximize their payoffs.

Types of Games

• Cooperative vs. Non-cooperative define if players can form binding agreements.
• Zero-sum vs. Non-zero-sum define if one player's gain is another player's loss.
• Complete vs. Incomplete Information define players knowledge of the game's structure and payoffs.
• Simultaneous vs. Sequential define if players make decisions simultaneously or in a specific order.

What is Algorithmic Game Theory?

• Algorithmic Game Theory (AGT) combines game theory and computer science to address computational issues in strategic settings.
• It focuses on designing efficient algorithms and mechanisms for games and analyzing their computational complexity.

Key Aspects

• Mechanism Design involves designing rules and incentives to achieve desired outcomes in strategic environments.
• Computational Complexity involves analyzing the computational resources required to find optimal strategies or outcomes.
• Approximation Algorithms means developing algorithms that provide near-optimal solutions when finding exact solutions is computationally hard.
• Equilibrium Computation involves finding stable states in games where no player has an incentive to deviate.

Selfish Routing

• Selfish routing is a classic example in algorithmic game theory.
• Multiple players (e.g., drivers) choose routes in a network to minimize their individual travel time.
• Each player acts selfishly, without considering the impact of their choices on others.

Braess's Paradox

• Adding a new road to a network can actually increase the travel time for all players.
• The new road can change the equilibrium routing pattern, leading to a less efficient outcome.

Price of Anarchy

• The price of anarchy (PoA) is a measure of the inefficiency caused by selfish behavior in a system.
• It compares the total cost of the worst-case Nash equilibrium to the total cost of the optimal (socially efficient) outcome.
• Formula: $$ PoA = \frac{\text{Total cost of the worst-case Nash equilibrium}}{\text{Total cost of the optimal outcome}} $$
• A high PoA indicates that selfish behavior leads to significant degradation in overall system performance.

Example: Pigou's Network

• A network containing two parallel links connecting a starting node to a destination node.
  • $c_1(x) = 1$
  • $c_2(x) = x$
• Total cost function calculation as:
• $C(x) = x \cdot c_1(x) + x \cdot c_2(x) = x \cdot 1 + x \cdot x = x + x^2$
• PoA in Pigou's network as: $$ PoA = \frac{1}{0.75} = 1.33 $$

Chapitre 2 : Algèbre relationnelle

• L'algèbre relationnelle is a formal query language for relational databases.
• It constitutes the theoretical foundations of relational DBMS.
• It manipulates relations to obtain new relations via mathematical operators.

2.1.1 Opérateurs de l'algèbre relationnelle

• Opérateurs ensemblistes include Union, Intersection, Difference and Cartesian Product.
• Opérateurs spécifiques. are Selection, Projection, Join, Division, Renaming.

2.2 Opérateurs ensemblistes

2.2.1 Union ($\cup$)

• The operator for two relations $R_1$ and $R_2$ which combines all tuples which can be found in either or bothe relations.
• $R_1$ and $R_2$ must have the same attribute types.
• Result a relation scheme with the same attribute that of $R_1$ and $R_2$.

2.2.2 Intersection ($\cap$)

• The operator for two relations $R_1$ and $R_2$ which returns all tuples that are part of BOTH relations.
• $R_1$ and $R_2$ must have the same attribute types.
• Result in a relation with an identical attribute schema for $R_1$ and $R_2$.

2.2.3 Différence (–)

• The operator for two relations $R_1$ and $R_2$ which returns only the attributes that are found within $R_1$, that are not found in $R_2$.
• $R_1$ and $R_2$ must have the same attribute types.
• Result a relation scheme with the same attributes that of only $R_1$.

2.2.4 Produit cartésien ($\times$)

• The combination tuples of two relations $R_1$ and $R_2$, each tuble in $R_1$ is combined with everything in $R_2$.
• No types restrictions.
• Result is all attributes in $R_1$ and all attributes in $R_2$.
• if $R_1$ has $n_1$ tuples and $R_2$ has $n_2$ tuples, then $R_1 \times R_2$ has $n_1 \times n_2$ tuples.

2.3 Opérateurs spécifiques

2.3.1 Sélection ($\sigma$)

• An operator to selects specific tubles from a relation based on a specified condition.
• $\sigma_{\text{condition}}(R)$
• Condition is a boolean that uses logical comparison.

2.3.2 Projection ($\pi$)

• An operator to selects only column/ attributes from a relation.
• The results deletes duplicate entries.
• Syntax : $\pi_{\text{attribut1, attribut2,...}}(R)$

2.3.3 Jointure ($\Join$)

• An operator that combines tuples from two relations based on a combination condition.
• $\text{Employés} \Join_{\text{Employés.dept_id = Départements.dept_id}} \text{Départements}$ (combines employees with their departments).

2.3.4 Division ($\div$)

• The operator returns from a certain relation those that are associated to every touple in another relation.
• Must specify which relations the operator affects.

2.3.5 Renommage ($\rho$)

• The operator changes the name of a relation or to its attributes.

2.5 Conclusion

• Relation algebra is a powerful tool to manipulate data.
• A way to formally request data and SQL.

Chemical Thermodynamics

• Thermodynamics handles energy changes in both physical and chemical processes.
• It informs about the feasibility of a process under conditions.
• Limitations; It informs only of feasibiltiy and says nothing about he rate of process.

Types of Systems

• Open System: Exchanges both matter and energy.
• Closed System: Exchanges energy only.
• Isolated System: Exchanges neither matter or energy with surround environment.

Thermodynamic Processes

• Isothermal Process: Constant temperature.
• Adiabatic Process: No heat exchange.
• Isobaric Process: Constant pressure.
• Isochoric Process: Constant volume.
• Reversible Process: A process where the system can return to initial conditions without affecting the environment.
• Irreversible Process: A change to the system and environment.

Thermodynamic Functions

• State Functions count on end states NOT the path taken.
• Examples: Internal energy ($U$), Enthalpy ($H$), Entropy ($S$), Gibbs free energy ($G$).

Internal Energy ($U$)

• Total energy of system (kinetic and potential).
• Absolute value cannot be calculated.
• Change in internal energy: $\Delta U = U_{final} - U_{initial}$.

First Law of Thermodynamics

• Energy cannot be destroyed nor can it be created; it is conserved.
• Mathematical expression: $\Delta U = q + w$, $q$ is heat and $w$ is work.

Enthalpy ($H$)

• $H = U + PV$, $P$ is pressure and $V$ is volume.
• Change in enthalpy: $\Delta H = \Delta U + P\Delta V$ (at constant pressure).
• $\Delta H = q_p$, $q_p$ is the heat at constant pressure.

Heat Capacity

• Definition is amount of heat to raise temperature of substance by $1^\circ C$ or $1 K$.
  • Molar Heat Capacity = Heat capacity for one mole of a substance.
  • Specific Heat Capacity, heat capacity for one gram of a substance.

Relationships

• Relation using heat capacity (when $C$ represents mass); $q = mc\Delta T$, $m$ is mass, $c$ is specific heat, and $\Delta T$ is the temperature change.
• Relationship using molar heat capacity with number moles, $q = nC_m\Delta T$, $n$ is the number of moles and $C_m$ is the molar heat capacity.

Enthalpy Changes in Chemical Reactions

• Standard Enthalpy of Formation ($\Delta H_f^\circ$), enthalpy change to make amount of compound from elements, the compound and elements must be the standard states.
• Standard Enthalpy of Combustion ($\Delta H_c^\circ$), amount of energy required to burn one mole of substance.
• Enthalpy of Atomization ($\Delta H_a$), amount of heat to isolate gaseous atom.
• Bond Enthalpy, change to individual break molecular bond for gas molecules.
• Enthalpy of Solution, solution, dissolve one mole in solvent, specified amounts.
• Enthalpy of Hydration, liquid.

Hess's Law

• Change is identical regardlesss of multistep or single step.
• $\Delta H_{reaction} = \sum \Delta H_f^\circ (products) - \sum \Delta H_f^\circ (reactants)$.

Second Law of Thermodynamics

• Entropy ($S$), measure the degree of randomness or disorder.
• Entropy = $\Delta S = \frac{q_{rev}}{T}$, $q_{rev}$ is the heat absorbed in a reversible process.
• Spontaneous process will occur if there is some form influence.

Third Law of Thermodynamics

• Entropy of a perfectly crystalline solid at absolute zero (0 K) is zero.

Gibbs Free Energy ($G$)

• Amount of energy available to work with constant temperature and pressure.
• Definition. $G = H - TS$.

Change in Gibbs Free Energy

• When $\Delta G = \Delta H - T\Delta S$ is less than 0, the process is spontaneous or feasible.
• Is $\Delta G$ is more than zero the process can occur.
• At equillibrium the total energy is zero as $\Delta G = 0$.

Gibbs Energy and Equilibrium Constant

• Constant. $\Delta G^\circ = -RT\ln K$, $R$ is the gas constant, $T$ is the temperature, and $K$ is the equilibrium constant.

Reguläre Ausdrücke (regex)

• Ein regulärer Ausdruck (regex) ist eine Zeichenkette, die ein Suchmuster definiert.
• Reguläre Ausdrücke werden oft verwendet, um Text zu suchen oder zu validieren.

Anwendungsfälle for regular expressions are;

• Suchen: Finde alle E-Mail-Adressen in einem Text.
• Ersetzen: Ersetze alle Vorkommen eines Wortes durch ein anderes.
• Validieren: Stelle sicher, dass eine Eingabe einem bestimmten Format entspricht (z.B. eine Postleitzahl).

Syntax of a expression is

• Reguläre Ausdrücke bestehen aus normalen Zeichen und Sonderzeichen (Metazeichen).

Metazeichen list

• . is a Beliebiges Zeichen (ausser Zeilenumbruch).
• ^is a Zeilenanfang.
• $ is a Zeilenende.
• *is a 0 oder mehr Vorkommen des vorherigen Zeichens.
• + is a 1 oder mehr Vorkommen des vorherigen Zeichens.
• ? is a 0 oder 1 Vorkommen des vorherigen Zeichens.
• {n} is Exactly n Vorkommen des vorherigen Zeichens.
• {n,} is n oder mehr Vorkommen des vorherigen Zeichens.
• {n,m} Between n and m Vorkommen des vorherigen Zeichens.
• [] contains a Zeichenklasse: Eines der Zeichen innerhalb der Klammern.
• [^] contains a Negierte Zeichenklasse: Keines der Zeichen innerhalb der Klammern.
• \|means Oder: Entweder der Ausdruck vor oder nach dem |.
• () contains a Gruppierung: Fasst Ausdrücke zusammen.
• \ is the Escape-Zeichen: Wandelt die Sonderbedeutung eines Metazeichens ab.

Zeichenklassen

• \dis a Ziffer (0-9).
• \Dis a Kein Ziffer.
• \w is a Wortzeichen (Buchstaben, Ziffern, Unterstrich).
• \W is a Kein Wortzeichen.
• \s is a Whitespace (Leerzeichen, Tab, Zeilenumbruch).
• \Sis a Kein Whitespace.

Beispiele

• \.is a Punkt (da. ein Metazeichen ist, muss es mit \ "escaped" werden).
• [a-z]+contains Eine oder mehrere Kleinbuchstaben.
• [0-9]{4} contains Genau 4 Ziffern.
• ^Hallo means Die Zeile beginnt mit "Hallo".
• Welt$ means Die Zeile endet mit "Welt".

Verwendung in Programmiersprachen

• Die meisten Programmiersprachen bieten Funktionen zur Verwendung regulärer Ausdrücke an.

Wichtige Hinweise

• Reguläre Ausdrücke können sehr komplex werden.
• Es ist wichtig, reguläre Ausdrücke sorgfältig zu testen, um sicherzustellen, dass sie das gewünschte Ergebnis liefern.

Resúmenes de álgebra lineal

Matrices

• A matrix is an array of expressions,numbers, or symbols found in rows and columns.
• E.g. $\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$.
  • Matriz cuadrada: A square matrix has equal rows and columns.
  • Matriz identidad: Matriz cuadrada with 1s on the main diagonal and 0s.
  • Matriz transpuesta: Interchange filas, rows, and columns.
• Operaciones
• Suma/Resta: Sum/Subtraction are performed element per element if the matrics have the same dimension.
• Multiplicación: The multiplitacion has certain properites.
• Multiplicación por un escalar: The mulitplication matrix and scalar.

Vectores

• A vector is a singular matrix with either one file or column.

Operaciones con vectores

• Suma/Resta: The Sum/ subraction is performed element to element when the dimension is identical.
• Producto escalar (punto): The scalar equation $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)$.
• Producto vectorial (cruz): Only valid in $\mathbb{R}^3$ giving back a vector where the equation equals zero.

Sistemas de ecuaciones lineales

• Un sistema de ecuaciones lineales se puede representar como $Ax = b$, donde $A$ es la matriz de coeficientes, $x$ es el vector de incógnitas y $b$ es el vector de términos independientes.

Representación matricial

• Toda transformación lineal se puede representar mediante una matriz.

Núcleo e imagen

• Núcleo (Kernel): Conjunto de vectores que se transforman en el vector cero.
• Imagen (Rango): Conjunto de vectores que son resultado de la transformación.

Valores y vectores propios

• Given a operator linear $T: V \rightarrow V$, un vector propio $v$ that has a zero equation.

Cálculo

1. Solve for all equations and check if equation value contains zero.
2. For each values the values are set and calculated to find a vector for a certain proprety.

• Una matriz, Matrix, M and A Matrix, is diagonalizable, and there exists M = P^-1AP, an identity matrix..
• • Condición is diagonalized if and only if the linear space is dependant.

Analisi Matematica 1 - Ing. Edile-Architettura

• The text provides problems for students to test their understanding and skills in calculus.

Esercizio 1

• The problem involves calculation of limit with function.

Esercizio 2

• Study the given function: $f(x) = \log(4-x^2)$.
  • This involves deriving function, domain, symmetry, intervals and graph.

Eserecizio 3

• This problem requires to calculate the integration from zero until one for a given calculation.

Eserecizio 4

Study the convergence series - $\qquad \sum_{n=1}^{+\infty} \frac{n^2 + 3}{n^3 + 1}x^n$ al variare di $x \in \mathbb{R}$.

Eserecizio 5

• Determine the answer to a Caulschy problem solution.

Clinical Guidelines on Diagnosing and Treating Low Testosterone

• Testosterone produces testicals through the blood to the body.

What does it do?

• The hormone formation to different organs during featus.
• male characterist during puberty.
• sperm production.
• bone density & muscle strength.
• production of red blood cells.

How is the measure of low testosterone?

• Varies depending on the lab.
• The measure is around 300 - 100 ng/dL.

What is Low Testosterone?

• The level has to be confirmed by two blood test.
• The condition also known as hypogonadism or testosterone deficiency.

Symptoms of Low Testosterone include.

• reduced sex drive, fatique.
• hair loss, infertility.
• depression, small testies.

How is Low Testosterone diagnosed?

• Medical history and blood tests are performed.
  • Total testosterone(drawn between 8-10 a.m.)
  • Free testosterone
  • Luteinizing hormone (LH)
  • Follicle-stimulating hormone (FSH)
  • Prolactin
  • Estradiol
  • Complete blood count (CBC)
  • Prostate-specific antigen (PSA)

Causes of Low Testosterone

• Primary hypogonadism Genetic conditions (e.g., Klinefelter syndrome) Undescended testicles Testicular injury or infection Cancer treatment (chemotherapy or radiation)
• Secondary hypogonadism (problem in the pituitary gland or hypothalamus): Pituitary tumors Inflammatory diseases (e.g., sarcoidosis) Medications (e.g., opioids, steroids) Obesity Aging Combination of primary and secondary hypogonadism

- How is Low Testosterone treated?

• Treatments include hormone therapy and other medications.

Benefits of TRT

• increased sex drive, concentration and bone density.

Risks of TRT

• acne, increase of heart attack.
• decreased sperm and depression.

Monitoring during TRT

• CBC
• Blood and hormone level
• Lipids
• Liver function

Who should NOT receive TRT?

• Men with prostate cancer
• Men with breast cancer
• Men with uncontrolled heart failure
• Men with sleep apnea
• Men who desire fertility

Lecture 24: The Poisson Process

• A way of event occruing on random time.
• Example includes customer arriving and emitting particle.

Definition of prossion.

• This mean the number of evens in the intervl are $P(N(s+t) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots$.

Properties

• The number of events in any interval of length. The interarrival times are exponentially distributed with mean $1/\lambda$.
• Arrival times have a Gamma distribution with parameters .

Example

• Example is $\lambda = 10$ customers per hour. What is the probability that exactly 5 customers arrive in the first hour? What is the probability that at least 2 customers arrive in the first hour? What is the probability that no customers arrive in the first 10 minutes?
• The probability that exactly 5 customers is $P(N(1) = 5) = e^{-10} \frac{10^5}{5!} \approx 0.0378.$ The probability that at least 2 customers is $ P(N(1) \geq 2) = 1 - P(N(1) = 0) - P(N(1) = 1) = 1 - e^{-10} - 10e^{-10} \approx 0.9995$.
• The probaility that no customer as first is $P(N(1/6) = 0) = e^{-10/6} \approx 0.1889.$