Cardiovascular Pharmacology: Sodium and Beta-Blockers

Cardiovascular System Pharmacology

Class I: Sodium Channel Blockers

• These drugs block voltage-gated sodium channels.
• They reduce sodium ion influx during action potential depolarization.
• Effects include slowing the rise of the action potential (phase 0), reducing conduction velocity, and prolonging the refractory period.
• Class IA drugs like Quinidine, Procainamide, and Disopyramide moderately slow phase 0 and prolong repolarization.
• Class IB drugs such as Lidocaine, Mexiletine, and Tocainide shorten repolarization and weakly block sodium channels.
• Class IC drugs like Flecainide and Propafenone markedly slow phase 0, but have minimal effect on repolarization.

Class II: Beta-Adrenergic Blockers (Beta-Blockers)

• Action involves blocking catecholamine effects at beta-adrenergic receptors.
• Beta-blockers decrease heart rate and contractility, slow AV nodal conduction, and reduce automaticity.
• Propranolol, Metoprolol, Atenolol, and Esmolol are examples of beta-blockers.
• They are particularly effective for arrhythmias associated with increased sympathetic activity.

Class III: Potassium Channel Blockers

• Potassium channel blockers act by blocking potassium channels.
• They prolong the repolarization phase of the action potential (phase 3).
• These drugs increase the effective refractory period and prolong the QT interval.
• Amiodarone, Sotalol, Dofetilide, and Ibutilide are examples of potassium channel blockers.
• Amiodarone affects sodium, potassium, and calcium channels, and beta-adrenergic receptors.

Class IV: Calcium Channel Blockers

• Involve blocking voltage-gated calcium channels, primarily in the SA and AV nodes.
• Results in slowed SA node automaticity and AV nodal conduction, and reduced contractility.
• Verapamil and Diltiazem are examples of calcium channel blockers.
• Calcium channel blockers are mainly used to treat supraventricular tachycardias.

Other Antiarrhythmic Drugs

• Adenosine activates adenosine receptors in the heart.
• It causes hyperpolarization and decreases AV nodal conduction.
• It terminates paroxysmal supraventricular tachycardia (PSVT).
• Digoxin inhibits the Na+/K+ ATPase pump.
• The result is increased intracellular sodium and calcium and enhances vagal tone
• It slows AV nodal conduction.
• It is used for rate control in atrial fibrillation and flutter.
• Magnesium Sulfate's mechanism is not fully understood.
• Thought to affect ion channels and cellular excitability.
• It treats Torsades de Pointes and digitalis-induced arrhythmias.

Pharmacology of Antianginal Drugs

• Goal: Reduce myocardial oxygen demand or increase myocardial oxygen supply.

Nitrates and Nitrites

• Converted to nitric oxide (NO), which activates guanylate cyclase.
• The increase in intracellular cGMP levels leads to smooth muscle relaxation.
• Effects include vasodilation as well as reduced preload.
• Vasodilation primarily venodilation, reduces preload, dilates coronary arteries, increasing O₂ supply.
• Reduced preload: Decreases ventricular volume/pressure, thus reducing myocardial oxygen demand.
• Examples include Nitroglycerin, Isosorbide Dinitrate, and Isosorbide Mononitrate.
• Headache, flushing, hypotension, and reflex tachycardia are adverse effects.
• Tolerance can develop with chronic use, hence nitrate-free intervals are required.

Beta-Adrenergic Blockers (Beta-Blockers)

• Beta-blockers act by blocking beta-adrenergic receptors
• Reduces catecholamine effects on the heart.
• Decreased heart rate, decreases myocardial oxygen demand.
• Reduced contractility also decreases myocardial oxygen demand.
• Reduced blood pressure, reduced afterload, and reduced myocardial oxygen demand
• Examples include Propranolol, Metoprolol, and Atenolol.
• Reduce the frequency and severity of angina attacks, particularly exertional angina.

Calcium Channel Blockers

• Mechanism involves blocking voltage-gated calcium channels.
• Happens in vascular smooth muscle and cardiac muscle.
• Effects include: reduced afterload, increased coronary blood flow by vasodilation.
• Decreased contractility; more pronounced with verapamil and diltiazem.
• Also reduced heart rate, slowing SA node automaticity; more pronounced with verapamil and diltiazem.
• Examples include Amlodipine, Nifedipine, Verapamil, and Diltiazem.
• Prove useful in treating variant (Prinzmetal's) angina by preventing coronary artery spasm.

Ranolazine

• Inhibits the late sodium current (INa) in cardiac myocytes.
• Reduces intracellular sodium resulting and calcium overload.
• Improves myocardial efficiency
• Reduces myocardial oxygen demand without big impact on heart rate or blood pressure.
• Antianginal Effects: Reduces the frequency of angina episodes and improves exercise tolerance.
• May prolong the QT interval and should be carefully used in patients with existing QT prolongation or other QT-prolonging drugs.

Pharmacology of Drugs Used in Heart Failure

Diuretics

• Increases renal excretion of sodium and water, reducing extracellular fluid volume.
• Effects include reduced preload, afterload, and symptom relief.
• Reduced preload decreases ventricular volume and pressure, reducing myocardial oxygen demand.
• Reduced afterload reduces blood volume and blood pressure.
• Symptom relief alleviates pulmonary congestion and peripheral edema.
• Loop Diuretics (Furosemide, Bumetanide, Torsemide) are most potent, act on the loop of Henle.
• Thiazide Diuretics (Hydrochlorothiazide, Chlorthalidone) act on the distal convoluted tubule.
• Potassium-Sparing Diuretics (Spironolactone, Eplerenone) are aldosterone antagonists, act on the collecting tubule.

ACE Inhibitors (Angiotensin-Converting Enzyme Inhibitors)

• They reduce angiotensin II levels by inhibiting the conversion of angiotensin I to angiotensin II.
• Effects include vasodilation, reduced aldosterone secretion, and prevented cardiac remodeling.
• Vasodilation reduces afterload.
• Reduced aldosterone secretion decreases sodium and water retention, reducing preload.
• Preventing cardiac remodeling reduces hypertrophy and fibrosis.
• Examples of ACE Inhibitors are Captopril, Enalapril, Lisinopril, and Ramipril.
• Hypotension, cough, hyperkalemia, and angioedema are adverse effects.

Angiotensin Receptor Blockers (ARBs)

• They act like ACE inhibitors by blocking the binding of angiotensin II to its receptors (AT1 receptors).
• Examples are Losartan, Valsartan, Irbesartan, and Candesartan.
• ARBs are to be used in patients who cannot tolerate ACE inhibitors due to cough or angioedema.

Beta-Adrenergic Blockers (Beta-Blockers)

• Reducing the effects of catecholamines on the heart.
• Blocks beta-adrenergic receptors.
• Decreased heart rate reduces myocardial oxygen demand.
• Reduced contractility reduces myocardial oxygen demand.
• Prevents Cardiac Remodeling: Reduces hypertrophy and fibrosis.
• Examples include Carvedilol, Metoprolol Succinate, and Bisoprolol.
• To be used with caution in patients with decompensated heart failure; start with low doses and titrate slowly.

Aldosterone Antagonists

• Reducing sodium and water retention.
• Block aldosterone receptors in the kidneys.
• Effects include reduced preload and prevented cardiac remodeling
• Reduced preload decreases ventricular volume and pressure.
• Preventing cardiac remodeling reduces fibrosis.
• Examples includes Spironolactone and Eplerenone
• Can cause hyperkalemia, potassium levels should be monitored while using.

Digoxin

• It enhances contractility.
• Slowed atrioventricular nodal conduction, reduces heart rate.
• Primarily used for symptom control; does not improve survival.
• Monitor drug levels due to a narrow therapeutic window.

Hydralazine and Isosorbide Dinitrate

• Vasodilators reduces afterload and preload.
• Hydralazine is a direct arterial vasodilator (reduces afterload).
• Isosorbide dinitrate is a venodilator (reduces preload).
• Reduced Afterload (Hydralazine): Decreases systemic vascular resistance.
• Also, reduced Preload (Isosorbide Dinitrate): Decreases ventricular volume and pressure.
• Especially useful in African American patients with heart failure.

Angiotensin Receptor-Neprilysin Inhibitor (ARNI)

• Combines an ARB (valsartan) and a neprilysin inhibitor (sacubitril).
• Neprilysin degrades natriuretic peptides.
• Effects include vasodilation, increased natriuresis, prevent cardiac remodeling
• Vasodilation: Reduces afterload and preload.
• Increased natriuresis: Promotes sodium and water excretion.
• Prevent cardiac remodeling: Reduces hypertrophy and fibrosis.
• Shown to reduce mortality and hospitalization rates in heart failure
• An example is Sacubitril/Valsartan (Entresto) patients.

Estadística

Introducción

• Statistics is the science of collecting, describing, and interpreting data.

Branches of Statistics

• Methods to organize, summarize and present data in an informative way.
• Inferential statistics involves methods to draw conclusions or generalizations about a population based on a sample.

Basic Concepts

• Population is the total set of individuals or objects of interest.
• Sample is the subset of the population.
• Variable is the characteristic measured or observed in each individual or object.

Types of variable

• Describes qualities or categories.
  • i.e. Nominal: without order (ex. eye color).
  • Ordinal: with order (ex. level of satisfaction).
• Describes numerical quantities.
  • Discrete: integer values (ex. number of children).
  • Continuous: decimal values (ex. height).

Estadística descriptiva

Medidas de tendencia central

• $\mu = \frac{\sum_{i=1}^{N} x_i}{N}$ is the average of the data.
• Median is the central value of the ordered data.
• Mode is the data that repeats the most.

Medidas de dispersión

• The difference between the maximum and minimum value indicates the range.
• A variance is the average of the squared differences of each data point with respect to the mean.
• $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$
• The square root of the variance is the standard deviation.
• $\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}$
• The ratio between the standard deviation and the mean represents the coefficient of variation.
• $CV = \frac{\sigma}{\mu}$

Representaciones gráficas

• A bar graph that shows the frequency distribution of a quantitative variable is a histogram.
• The diagram that compares the frequencies of different categories of a qualitative variable is the bar diagram A circular graphic that shows the proportion of each category in relation to the total is the sector diagram.
• The visual representation of the median, quartiles, and outliers of a data set is a box and whisker diagram.

Estadística inferencial

Estimación

• A unique value that is used to estimate a population parameter represents a point estimate.
• An interval estimate represents a range of values ​​used to estimate a population parameter with a certain level of confidence.

Pruebas de hipótesis

• The null hypothesis is the claim that is put to the test.
• The alternative hypothesis is the claim that is accepted if the null hypothesis is rejected.
• The probability of rejecting the null hypothesis when it is true represents the level of significance.
• Probability of obtaining a result as extreme as the one observed, assuming that the null hypothesis is true is the p-value.

Tipos de pruebas

• Comparison of the means of two groups is the t-test.
• Is a comparison of the means of more than two groups represents ANOVA.
• The examination of the association between two categorical variables is through chi-square.
• The strength and direction of the linear relationship between two quantitative variables is measured by the correlation.
• The prediction of the value of a variable is possible depending on another or other variables this is through regression.

Vibration - Chapter 14

Introduction

• Vibration is when oscillatory motion occurs about an equilibrium position.
• Free vibration happens without external force
• Forced vibration is driven by an external force.

Simple Harmonic Motion

• Simple harmonic motion: restoring force proportional to displacement from equilibrium.
• $F = -kx$: formula to express that relationship
• $F$: restoring force; $k$: the spring constant (stiffness); $x$: the displacement from equilibrium.

Characteristics of SHM:

• Amplitude (A): Maximum displacement from equilibrium.
• Period (T): Time for one complete oscillation.
• Frequency (f): Number of oscillations per unit time.
• $f = \frac{1}{T}$: express the relationship between frequency and time

Angular Frequency

• $\omega = 2\pi f = \frac{2\pi}{T}$ expression to calculate angular frequency

Kinematics of SHM

• Displacement (x(t)) at time (t) = $A \cos(\omega t + \phi)$
• $A$: Amplitude, $\omega$: Angular Frequency, $\phi$: Phase constant (initial phase).
• $v(t) = -A\omega \sin(\omega t + \phi)$: Velocity
• Acceleration: $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$

Energy in SHM

• $E = K + U$ : Total mechanical energy.
• $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ (m: Mass, v: Velocity, k: Spring Constant, x: Displacement)
• Total Energy: $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2A^2$

The Simple Pendulum

• A simple pendulum consist of a point mass suspended from a massless sting of length L.
• Period: $T = 2\pi \sqrt{\frac{L}{g}}$ where L: Length. g: Acceleration

Damped Oscillations

• Energy is dissipated from the system, causing the amplitude to decrease overtime.
• Types of Damping:
• Underdamped: decrease amplitude when the system oscillates
• Critically Damped: the system as quickly as possible without oscillating.
• Overdamped: returns to equilibrium slowly without oscillating.

Forced Oscillations and Resonance

• External frequency drivers an oscillator during forced oscillations
• Resonance occurs when the driving frequency matches the natural frequency of the system, resulting in a large amplitude.
• Resonance Frequency: $\omega_0 = \sqrt{\frac{k}{m}}$ (k: Spring Constant, m: Mass))
• Resonance can be applied to musical instruments and radio receivers.
• And could cause problems like bridge collapse and machine failure.

Lecture 18 - September 28, 2023

Example 1

• Volume calculation of the solid with $y=x^2$ and $y=4$ about the line $y=4$ using Disk method.
• Volume is $V = \int_{-2}^{2} A(x) , dx = \int_{-2}^{2} \pi (4-x^2)^2 , dx = 2\pi \int_{0}^{2} (4-x^2)^2 , dx = \frac{512 \pi}{15}$ Volume calculation of the solid with $y=x^2$ and $y=4$ about the line $y=4$ using Cylindrical Shells.
  
• We have $x = \pm \sqrt{y}$, so the radius of a shell is $4-y$ and the height is $2\sqrt{y}$.
• Volume is $V = \int_{0}^{4} 2\pi (4-y) 2\sqrt{y} , dy = 4\pi \int_{0}^{4} (4-y) \sqrt{y} , dy = \frac{512 \pi}{15}$

Example 2

• Enclosed by the curves $y=x$ and $y=x^2$ Rotate about the line $x=-1$ Using Cylindrical Shells.
• Volume is $V = \int_{0}^{1} 2\pi (x+1)(x-x^2) , dx = 2\pi \int_{0}^{1} x^2 - x^3 + x - x^2 , dx = \frac{\pi}{2}$
  

Example 3

• Region with curves $y=x-x^2$ and $y=0$ Rotate about the line $x=2$ Using Cylindrical Shells.
• Volume is $V= \int_{0}^{1} 2\pi (2-x)(x-x^2) , dx = 2\pi \int_{0}^{1} 2x - 2x^2 - x^2 + x^3 , dx= \frac{\pi}{2}$

Capítulo 1: Álgebra

1.1 Números reales

Clasificación

• The correct sequence is $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
  • $\mathbb{N}$: Naturales (enteros positivos, ${1, 2, 3,...}$)
  • $\mathbb{Z}$: Enteros (positivos, negativos y cero, ${..., -2, -1, 0, 1, 2,...}$)
  • $\mathbb{Q}$: Racionales (pueden expresarse como fracción, ${a/b \mid a, b \in \mathbb{Z}, b \neq 0}$)
  • $\mathbb{I}$: Irracionales (no pueden expresarse como fracción, $\sqrt{2}, \pi, e,...$)
  • $\mathbb{R}$: Reales ($\mathbb{Q} \cup \mathbb{I}$)

Representación gráfica

• Cada número real corresponde a un punto en la recta real.

Valor absoluto

• $|a| = \begin{cases} a, & \text{si } a \geq 0 \ -a, & \text{si } a < 0 \end{cases}$ represents the absolute value of any number

Properties of absolute value

• $|a| \geq 0$
• $|a| = |-a|$
• $|a \cdot b| = |a| \cdot |b|$
• $|a + b| \leq |a| + |b|$

Intervalos

• Subconjuntos de la recta real.
  • These intervals and their representations are listed in the table.

Operaciones

• Suma, resta, multiplicación y división.

Propiedades

• Conmutativa: $a + b = b + a$, $a \cdot b = b \cdot a$
• Asociativa: $(a + b) + c = a + (b + c)$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Distributiva: $a \cdot (b + c) = a \cdot b + a \cdot c$

1.2 Exponentes y radicales

Exponentes

• $a^n = \underbrace{a \cdot a \cdot... \cdot a}_{n \text{ veces}}$

Propiedades

• $a^m \cdot a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{m \cdot n}$
• $(a \cdot b)^n = a^n \cdot b^n$
• $(\frac{a}{b})^n = \frac{a^n}{b^n}$
• $a^{-n} = \frac{1}{a^n}$
• $a^0 = 1$

Radicales

• $\sqrt[n]{a} = b \Leftrightarrow b^n = a$ formula to expresses

Propiedades

• $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
• $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$
• $\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}$
• $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

Racionalización

• Eliminar radicales del denominador de una fracción is called Rationalization
• Example: $\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

1.3 Expresiones algebraicas

Polinomios

• $P(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0$
  • $a_i:$ coefficients
  • $n:$ degree
  • $a_n:$ principal coeficient
  • $a_0:$ term independent

Operaciones

• Suma, resta, multiplicación y división.

Productos notables

• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$
• $(a + b) \cdot (a - b) = a^2 - b^2$
• $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
• $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

Factorización

• -Escribir un polinomio como producto de factores

Ejemplos:

• Factor común: $ax + ay = a(x + y)$
• Diferencia de cuadrados: $a^2 - b^2 = (a + b)(a - b)$
• Trinomio cuadrado perfecto: $a^2 \pm 2ab + b^2 = (a \pm b)^2$

Fracciones algebraicas

• Cociente de dos polinomios
• Simplificación is dividing Numerador and Denominador per common factor

Operaciones

• Suma, resta, multiplicación y división (similar a las fracciones numéricas).

1.4 Ecuaciones

Lineales

• Solve solving equation $ax + b = 0$.
• $x = -\frac{b}{a} $

Cuadráticas

• $ax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ this is the expression to calculate square roots
• Discriminante: $\Delta = b^2 - 4ac$
  • $\Delta > 0$: dos soluciones reales distintas
  • $\Delta = 0$: una solución real (doble)
  • $\Delta < 0$: dos soluciones complejas conjugadas Systems of equations that represent the conjugation of 2 o more equations
  • Métodos de solución
    • Sustitución
    • Igualación
    • Reducción (eliminación)

Aplicaciones

Modelado y resolución de problemas.

1.5 Desigualdades

Lineales

$ax + b > 0$ (o $ b$ y $c$ es cualquier número, entonces $a + c > b + c$

• Si $a > b$ y $c > 0$, entonces $ac > bc$
• Si $a > b$ y $c < 0$, entonces $ac < bc$ (cambia el sentido)

No lineales

that involve Polinómicas o racionales. Solve following this:

• Llevar a cero
• Factorizar
• Tabla de signos
• Intervalos de solución

Valor absoluto

• $|x| < a \Leftrightarrow -a < x < a$
• $|x| > a \Leftrightarrow x < -a \text{ o } x > a$

Aplicaciones

• Problems of optimización y modelado.

Introduction to Differential Equations

1.1 Definitions and Terminology

Definition

• Includes: Derivatives of one or more dependent variables, in terms of one or more independent variables. $dy/dx = f(x, y)$
  • Classification:
• Ordinary Differential Equation (ODE): Derivatives respect one single independent variable. $ d^2y/dx^2 + dy/dx + y = 0$
• Partial Differential Equation (PDE): Derivatives respect two or more independent variables. $\partial u/\partial x + \partial u/\partial y= 0$
• Order: of its he differential equation depends of its he highest derivative. $d^4y/dx^4 + dy/dx + y = x³$, represents a (4th order of a ODE)
• An nth-order ODE is linear when

$a_n(x) d^ny/dx^n + a_{n-1}(x)d^{n-1}y/dx^{n-1} +......+ a_1(x)dy/dx + y_0(x) + y = g/x$

• Solution:
• It is a function that satisfies the equation
• Explicit: y= f(x)
• Implícit: G(x, y)= 0
• Types of Solutions:
• General: arbitrary constants
• Particular: obtained by assigning specific values to the arbitrary constants.
• Singular : A solution that cannot be obtained from the general solution.

Initial Value Problem (IVP)

Is an equation and initial condition dy/ dy =f(X,Y), y (Xo) = yo

Examples and Exercises

• To determine the order of the equation $ d^4y/dx^4 + dy/dx + y = x³$, take a look ate the higghest order for example the solution is 4th order
• Verify if y= exp /s a solution if $dy/ dx = y$; The derivate of $y =exp/x$ is $dy/dx= exp/x$ y hence is a solution
• Determine if what indicates the folowing equation is line $1 x^2 d^3y/dx² + x dy/dx = sin /x$:
• Verify $Y = sin(x)$ es a solution sin $(x)$ de $ d^2y/dx² + y = 0$

Lecture 19: Bayesian Inference

Bayesian vs Frequentist

Goal: Estimate parameters $\theta$ with uncertainty given data $\mathcal{D}$ in Bayesian vs Estimate a single "best" $\hat{\theta}$ given data $\mathcal{D}$ in Frequentist. | Parameter(s) | Parameter(s) | | Bayesian | Frequentist | Parameter(s): Random variable(s) with a prior distribution $p(\theta)$ vs Fixed, unknown value(s) Data: Fixed vs Random variable Inference $p(\theta \vert \mathcal{D}) = \frac{p(\mathcal{D} \vert \theta) p(\theta)}{p(\mathcal{D})}$ vs inference Choose an estimator $\hat{\Theta}(\mathcal{D})$ Interpretation$p(\theta = 0.5 \vert \mathcal{D})$ = probability $\theta = 0.5$, given $\mathcal{D}$ vs in Frequentist Confidence intervals: If I repeat this experiment many times, X% of the time Example: "Given that a user clicked on 10 ads, there is a 95% probability that their click-through rate is between 2% and 5%" vs "I am 95% confident that the click-through rate is between 2% and 5%"

| Pros | | Bayesian |

• More interpretable Can incorporate prior knowledge Principled way to quantify uncertainty Can naturally do prediction | |Frequentist | Computationally simpler Can be applied even when prior knowledge is limited Cons |

Need to choose a prior vs Comoutationally intensive Less interpretable / Difficult to quantify uncertainty | | notation review | | $ θ$ parameters of your model vs $ p(D I θ)$: Likelyhood vs Data over model to evaluate |

choosig a prior

if if the posterior with then that prior is a conjugate

Funciones vectoriales de una variable real

Definicion A vectirial function is a fincion that numbers reales mapped to vectors

$\overrightarrow{r}: \mathbb{R} \longrightarrow \mathbb{R}^n $

Dominio

• It is a intersection of dominio $ f1 (t)$

Limite $ 𝑟(𝑡)=〈𝑓1(𝑡),𝑓2(𝑡),…,𝑓𝑛(𝑡)〉 $$ lim𝑡→𝑎𝑟(𝑡)=〈lim𝑡→𝑎𝑓1(𝑡),lim𝑡→𝑎𝑓2(𝑡),…,lim𝑡→𝑎𝑓𝑛(𝑡)〉

Continuous : if vector function is C=A if

$ lim 𝑡→𝑎𝑟(𝑡)=𝑟(𝑎) $

• The derivative of vector function $ r(t)$

$ 𝑟′(𝑡)=limℎ→0𝑟(𝑡+ℎ)−𝑟(𝑡)ℎ $

The intagral with function function

∫𝑏𝑎𝑟(𝑡)𝑑𝑡=〈∫𝑏𝑎𝑓1(𝑡)𝑑𝑡,∫𝑏𝑎𝑓2(𝑡)𝑑𝑡,…,∫𝑏𝑎𝑓𝑛(𝑡)𝑑𝑡〉

Fisíca

Vectores

Suma de vectores

Analityc metog

Ax+ A cos thetra Ay= A sin theta

A = $ A²+ Ay²$ T = Arartan ay / AX

$R = A =B= ( AX= By; AY = By)$

EjerciciO

5 u, 30

AX=5=cos (3O)= 4.33y

Algorithmic Game Theory

Introduction: Presents public project, spectrum auctions, and sponsored search as examples.

Mechanism Design: Outlines components including possible outcomes, agents, types, and valuation functions.

Social Choice Functions: Defines social choice functions (SCFs) and conditions for a mechanism to implement an SCF in dominant strategies.

Revelation Principle: States and proves the revelation principle for dominant strategies, indicating that direct and truthful mechanisms can implement any SCF implemented by a mechanism in dominant strategies.

Implementation with Transfers: Discusses quasi-linear preferences and introduces transfer functions, indicating the existence of a direct truthful mechanism implementing a social choice function with transfers.

Gibbard-Satterthwaite Impossibility Theorem: Establishes that strategy-proof and onto social choice functions are dictatorial, under certain conditions, including that the number of outcomes is at least three.