Autonomic Pharmacology

Autonomic Pharmacology

• Regulates cardiac muscle, smooth muscle, and glands.
• Divided into sympathetic (SNS) and parasympathetic (PNS) nervous systems.
• SNS prepares the body for "fight or flight."
• PNS promotes "rest and digest" functions.

Neurotransmitters

• Acetylcholine (ACh) is the primary neurotransmitter of the PNS.
• Norepinephrine (NE) is the primary neurotransmitter of the SNS.

Receptors

• Recognize and bind specific neurotransmitters.
• Initiate intracellular events to cause physiological responses.

Cholinergic Receptors

• Bind ACh.
• Divided into nicotinic and muscarinic receptors.

Adrenergic Receptors

• Bind NE and epinephrine (Epi).
• Divided into alpha (α) and beta (β) receptors.

Drugs Affecting the ANS

• Can mimic neurotransmitters (agonists) or block them (antagonists).
• Can affect neurotransmitter synthesis, storage, release, or reuptake.

Cholinergic Pharmacology

• Cholinergic agonists mimic ACh effects.
• Also known as parasympathomimetics.

Direct-Acting Agonists

• Bind directly to cholinergic receptors.

Examples

• Bethanechol treats urinary retention and gastrointestinal motility disorders.
• Pilocarpine treats glaucoma and dry mouth.
• Nicotine affects autonomic ganglia and the neuromuscular junction.

Indirect-Acting Agonists

• Increase ACh levels by inhibiting acetylcholinesterase (AChE).
• Also known as cholinesterase inhibitors.

Examples

• Physostigmine treats glaucoma and anticholinergic toxicity.
• Neostigmine treats myasthenia gravis and reverses neuromuscular blocking agents.
• Organophosphates are irreversible cholinesterase inhibitors, used as insecticides and nerve gases.

Cholinergic Antagonists

• Block ACh effects.
• Also known as parasympatholytics.

Muscarinic Antagonists

• Block ACh at muscarinic receptors.
• Also known as anticholinergics.

Examples

• Atropine treats bradycardia, dilates pupils, and reduces secretions.
• Scopolamine prevents motion sickness and reduces secretions.
• Ipratropium treats asthma and COPD.

Nicotinic Antagonists

• Block ACh at nicotinic receptors.

Examples

• Hexamethonium blocks nicotinic receptors in autonomic ganglia.
• Tubocurarine is a non-depolarizing neuromuscular blocking agent, blocking nicotinic receptors at the neuromuscular junction.
• Succinylcholine is a depolarizing neuromuscular blocking agent that initially stimulates nicotinic receptors, then causes paralysis.

Adrenergic Pharmacology

• Adrenergic agonists mimic NE and Epi effects.
• Also known as sympathomimetics.

Direct-Acting Agonists

• Bind directly to adrenergic receptors.

Examples

• Phenylephrine is an α1-adrenergic agonist used to treat nasal congestion and raise blood pressure.
• Clonidine is an α2-adrenergic agonist used to treat hypertension.
• Dobutamine is a β1-adrenergic agonist used to treat heart failure.
• Albuterol is a β2-adrenergic agonist used to treat asthma.
• Epinephrine is non-selective, activating both α and β receptors to treat anaphylaxis and cardiac arrest.
• Norepinephrine is non-selective, activating both α and β1 receptors to treat hypotension.

Indirect-Acting Agonists

• Increase NE levels by promoting release or inhibiting reuptake.

Examples

• Amphetamine promotes NE and dopamine release, used to treat ADHD and narcolepsy.
• Cocaine inhibits NE, dopamine, and serotonin reuptake, used as a local anesthetic and stimulant.

Adrenergic Antagonists

• Block NE and Epi effects.
• Also known as sympatholytics.

Alpha Antagonists

• Block NE and Epi at α receptors.

Examples

• Prazosin is an α1-adrenergic antagonist used to treat hypertension and BPH.
• Phenoxybenzamine is a non-selective α-adrenergic antagonist used to treat pheochromocytoma.

Beta Antagonists

• Block NE and Epi at β receptors.
• Also known as beta-blockers.

Examples

• Propranolol is a non-selective β-adrenergic antagonist used to treat hypertension, angina, and arrhythmias.
• Metoprolol is a selective β1-adrenergic antagonist used to treat hypertension, angina, and heart failure.
• Carvedilol is a non-selective β-adrenergic antagonist with α1-adrenergic blocking activity, used to treat heart failure and hypertension.

Summary Table of Autonomic Drugs

Cholinergic Agonists

• Direct-Acting (Bethanechol, Pilocarpine): Bind to and activate cholinergic receptors.
• Clinical uses are urinary retention, gastrointestinal motility disorders, glaucoma, and dry mouth.
• Indirect-Acting (Physostigmine, Neostigmine): Inhibit acetylcholinesterase.
• Clinical uses are glaucoma, anticholinergic toxicity, myasthenia gravis, and reversal of neuromuscular blocking agents.

Cholinergic Antagonists

• Muscarinic (Atropine, Scopolamine): Block muscarinic receptors, leading to Bradycardia, pupil dilation, reduction of secretions, motion sickness, asthma, and COPD treatments.
• Nicotinic (Hexamethonium, Tubocurarine): Blocks nicotinic receptors, causing hypertension (ganglionic blocker), muscle relaxation (neuromuscular blocker)

Adrenergic Agonists

• Direct-Acting (Phenylephrine, Albuterol): Binds and activates adrenergic receptors, used as a treatment for Nasal congestion, hypotension, asthma, anaphylaxis, and cardiac arrest
• Indirect-Acting (Amphetamine, Cocaine): Increases NE levels in the synapse, used as a treatment ADHD, narcolepsy, and as local anesthesia (cocaine)

Adrenergic Antagonists

• Alpha (Prazosin, Phenoxybenzamine): Blocks alpha receptors, treating hypertension, BPH, and pheochromocytoma.
• Beta (Propranolol, Metoprolol): Blocks beta receptors for Hypertension, angina, arrhythmias, and heart failure treatment.
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Matrizes (Matrices)

• A matrix is a table of numbers arranged in rows and columns.
• Matrices are generally represented by capital letters, denoted as: $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$.
• $a_{ij}$ represents the element in the i-th row and j-th column of matrix A.

Ordem de uma Matriz (Order of a Matrix)

• The order of a matrix is given by the number of rows (m) and columns (n).
• A matrix with m rows and n columns is an $m \times n$ matrix.
• Example: $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ is a $2 \times 3$ matrix.

Tipos de Matrizes (Types of Matrices)

• Matriz Quadrada (Square Matrix): Number of rows equals the number of columns ($m = n$).
• Matriz Linha (Row Matrix): It possesses only one row. ($m = 1$).
• Matriz Coluna (Column Matrix): It possesses a single column. ($n = 1$)
• Matriz Nula (Null Matrix): All elements are zero.
• Matriz Identidade (Identity Matrix): Square matrix with 1s on the main diagonal, 0s elsewhere.
• Matriz Transposta (Transposed Matrix): Denoted as $A^T$, is obtained by swapping rows and columns of A. If $A = [a_{ij}]$, then $A^T = [a_{ji}]$. Example: $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \Rightarrow A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$.

Operações com Matrizes (Matrix Operations)

• Adição e Subtração (Addition and Subtraction): Sum or subtract corresponding elements of matrices of the same order. $A + B = [a_{ij} + b_{ij}]$ and $A - B = [a_{ij} - b_{ij}]$.
• Multiplicação por um Escalar (Multiplication by a Scalar): Matrix A multiplied by a scalar k involves multiplying each element of the matrix by k. $kA = [ka_{ij}]$.
• Multiplicação de Matrizes (Matrix Multiplication): Possible if the number of columns of A equals the number of rows of B.
• If A is $m \times n$ and B is $n \times p$, the product AB is $m \times p$. The element $c_{ij}$ of the product AB is obtained by the sum of the products of the elements of the i-th row of A by the elements of the j-th column of B, as such: $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$.
• Example: Given $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$, then $AB = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$.

Determinante de uma Matriz (Determinant of a Matrix)

• Determinant: Associates a real number to a square matrix, denoted as det(A) or |A|.
• Determinant of $2 \times 2$ Matrix: $ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$ then $\det(A) = ad - bc$.
• Determinant of $ 3 \times 3$ Matrix (Sarrus' Rule): $\det(A) = aei + bfg + cdh - ceg - bdi - afh$.
• Determinant Properties include any row or column comprised of zeores results in a determinite equaling zero. If two rows or columns equal one another, or are porportional to each other, the determinate is also equal to zero. Swapping two rows or coulmns will change the sign. The determinate of the product is equal to the product of the determinate for said matrices: $\det(AB) = \det(A) \cdot \det(B)$.

Inversa de uma Matriz (Inverse of a Matrix)

• A square matrix A is invertible (or non-singular / non-zero) if there exists a matrix B such that $AB = BA = I$, where I is the identity matrix.
• B is called the inverse of A and is denoted $A^{-1}$.

Cálculo da Inversa (Inverse Calculation)

1. Calculate the determinant of A. If $\det(A) = 0$, the matrix is not invertible.
2. Find the matrix of cofactors of A.
3. Transpose the matrix of cofactors to obtain the adjoint matrix of A.
4. Divide the adjoint matrix by the determinant of A: $A^{-1} = \frac{1}{\det(A)} \cdot adj(A)$.

• Example: For $A = \begin{bmatrix} 2 & 1 \ 4 & 3 \end{bmatrix}$, $\det(A) = 2 \cdot 3 - 1 \cdot 4 = 6 - 4 = 2$. The inverse of A is $A^{-1} = \frac{1}{2} \begin{bmatrix} 3 & -1 \ -4 & 2 \end{bmatrix} = \begin{bmatrix} \frac{3}{2} & -\frac{1}{2} \ -2 & 1 \end{bmatrix}$.

Aplicações de Matrizes (Applications of Matrices)

• Solves Linear equation systems. Also useful in linear transformations, Graphics Computing or Data Analysis.
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Física (Physics)

Vectores (Vectors)

Suma de Vectores (Sum of Vectors)

Método Analítico (Analytical Method)

• Define Componentes de un vector (Vector components): $A_x = A\cos(\theta)$, $A_y = A\sin(\theta)$.
• Calculate the Angle and Magnitude: $\theta = \arctan(\frac{A_y}{A_x})$, $A = \sqrt{A_x^2 + A_y^2}$.

Suma (Sum)

• $R_x = A_x + B_x +...$
• $R_y = A_y + B_y +...$
• $R = \sqrt{R_x^2 + R_y^2}$
• $\theta_R = \arctan(\frac{R_y}{R_x})$

Producto Escalar (punto) (Scalar Product or Dot Product)

• $\vec{A} \cdot \vec{B} = AB\cos(\theta)$
• $\vec{A} \cdot \vec{B} = (A_x\hat{i} + A_y\hat{j} + A_z\hat{k}) \cdot (B_x\hat{i} + B_y\hat{j} + B_z\hat{k})$
• $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$

Producto Vectorial (cruz) (Vector Product or Cross Product)

• $\vec{A} \times \vec{B} = AB\sin(\theta)\hat{n}$
• $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} $
• $\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$

Cinemática (Kinematics)

MRU (Uniform Rectilinear Motion)

• $v = \frac{x}{t} = \frac{\Delta x}{\Delta t}$

MRUV (Uniformly Varied Rectilinear Motion)

• Define acceleration as $a = \frac{v}{t} = \frac{\Delta v}{\Delta t}$
• State Equations of Motion: $x = x_0 + v_0t + \frac{1}{2}at^2$, $v = v_0 + at$, $v^2 = v_0^2 + 2a\Delta x$.

Caída Libre (Free Fall)

• Acceleration due to gravity: $a = -g = -9.8 m/s^2$
• Equations: $y = y_0 + v_0t - \frac{1}{2}gt^2$, $v = v_0 - gt$, $v^2 = v_0^2 - 2g\Delta y$.

Tiro Vertical (Vertical Shot)

Velocity is 0 at maximum height.

Tiro Oblicuo (Oblique Shot)

• It is an example, where $v_y = 0$ at maximum height.
• $v_x = v_0\cos(\theta)$ remains constant.
• $v_y = v_0\sin(\theta) - gt$ $x = v_0\cos(\theta)t$, $y = v_0\sin(\theta)t - \frac{1}{2}gt^2$.

Alcance Horizontal Máximo (Maximum Horizontal Range)

• $\Delta x = \frac{v_0^2\sin(2\theta)}{g}$

MCU (Uniform Circular Motion)

• Angular velocity: $\omega = \frac{\theta}{t} = \frac{\Delta \theta}{\Delta t}$
• Linear velocity: $v = \omega r$
• Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2r$

Período (Period)

• $T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$

Frecuencia (Frequency)

• $f = \frac{1}{T} = \frac{\omega}{2\pi}$

MCUV (Uniformly Varied Circular Motion)

• Angular acceleration: $\alpha = \frac{\omega}{t} = \frac{\Delta \omega}{\Delta t}$
• $\omega = \omega_0 + \alpha t$
• $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$

Dinámica (Dynamics)

Leyes de Newton (Newton's Laws)

• First Law: If $\sum \vec{F} = 0$ then $\vec{v} = cte$.
• Second Law: If $\sum \vec{F} = m\vec{a}$
• Third Law: $\vec{F}{AB} = -\vec{F}{BA}$

Fuerza de Rozamiento (Frictional Force)

• $F_r = \mu N$
• $\mu_e$: static
• $\mu_d$: dynamic

Trabajo (Work)

$W = \vec{F} \cdot \Delta \vec{x} = F\Delta x\cos(\theta)$

Energía (Energy)

• Energía Cinética (Kinetic Energy): $K = \frac{1}{2}mv^2$
• Energía Potential Gravitatoria (Gravitational Potential Energy): $U_g = mgy$
• Energía Potential Elástica (Elastic Potential Energy): $U_e = \frac{1}{2}kx^2$

Teorema Trabajo-Energía (Work-Energy Theorem)

$W_{neto} = \Delta K$

Conservación de la Energía Mecánica (Conservation of Mechanical Energy)

$K_i + U_i = K_f + U_f$

Potencia (Power)

$P = \frac{W}{t} = \vec{F} \cdot \vec{v}$

Impulso (Impulse)

$\vec{I} = \vec{F}\Delta t = \Delta \vec{p}$

Cantidad de Movimiento (Momentum)

$\vec{p} = m\vec{v}$

Conservación de la Cantidad de Movimiento (Conservation of Momentum)

• $\vec{p}_i = \vec{p}_f$

Choques (Collisions)

• Elásticos (Elastic): Kinetic energy is conserved.
• Inelastic (No elasticity): Kinetic energy is not conserved

Estática (Statics)

Torque

$\qquad \tau = \vec{r} \times \vec{F} = rF\sin(\theta)$

Equilibrio (Balance)

$\qquad \sum \vec{F} = 0$ $\qquad \sum \vec{\tau} = 0$

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Algorithmic Game Theory (AGT)

• Combines game theory with algorithm design and analysis.
• Deals with the computational aspects of games, such as: Computing equilibria and the design of Mechanisms, as well as the Complexity of solving problems related to Game Theory.

Basic Concepts

• Rationality implies that players act in their own best interests to maximize their payoffs.
• Games are characterized by Players, who have a defined sets of Strategies available to them so as to achieve Payoffs.

Types of Games

• Cooperative games have binding agreements and collaboration. Non-cooperative games emphasize individual strategies.

Zero-sum Game

• One player's gain is another's loss. The counterpart is the Non-zero-sum Game that allows for mutually beneficial outcomes.
• An environment described as Complete Information refers to all players knowing the strategies and payoffs of others. The opposite would be Incomplete Information, with uncertainty.

Key Topics in AGT

Nash Equilibrium

• A set of strategies where no player can improve their payoff by unilaterally changing their strategy.
• In Pure Nash Equilibrium players choose a single strategy. Conversely, in Mixed the Nash Equilibrium players chooses a probability distribution over applicable strategies.
• Computing problems related to finding said Nash Equilibria can be computationally hard (PPAD-complete).

Mechanism Design

• It designs the "rules of the game" to achieve a specific outcome.
• Deals with Auctions to maximize revenue or achieve efficient allocation.
• It has special case of Vickrey Auction.

Price of Anarchy

• It measures the loss of efficiency due to selfish behavior in a system.

Applications

• Internet and Network Economics
• E-commerce
• Social science
• Political sciences
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Lecture 24: Intro. to General Relativity

Principle of Equivalence

• Describes Weak Equivalence Principle: All objects accelerate at the same rate in a given gravitational field, regardless of their mass or composition.

Eötvös Experiment

• Demonstrates the Weak Equivalence Principle.
• Formula for deviation: $\eta = 2 \frac{|a_1 - a_2|}{|a_1 + a_2|}$, where $a_1$ and $a_2$ are the accelerations of two objects.

Einstein's Elevator

• Is a tool and a concept illustrating the equivalence of gravity and acceleration.

Local Inertial Frame

• The laws of physics are the same as in special relativity.
• Freely falling frame is a local inertial frame.

Strong Equivalence Principle

• The laws of physics are the same in all local inertial frames, regardless of their state of motion or gravitational field.

Consequences of Equivalence

• The laws of physics are the same in all local inertial frames, regardless of their state of motion or gravitational field.
• The outcome of any local physical experiment in a gravitational field is independent of the velocity of the freely falling reference frame, and its location in spacetime.

Gravitational Time Dilation

• Time passes slower in stronger gravitational fields.

Gravitational Redshift

• Light emitted from a strong gravitational field is redshifted.

Bending of Light

• Light is deflected by gravity.

Math Language

4-Vectors

• Vectors in spacetime, with three spatial components and one time component: $x^{\mu} = (ct, x, y, z)$, where $\mu = 0, 1, 2, 3$

Metric Tensor

• Is a description for the geometry of spacetime: $ds^2 = g_{\mu\nu} dx^{\mu} dx^{\nu}$
• In special relativity, the metric tensor is the Minkowski metric:

$\eta_{\mu\nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$

Gravity in General Relativity

• Gravity is not a force, rather caused by the curvature of spacetime as a result of mass and energy. The Einstein field equations relate the curvature of spacetime to the distribution of mass and energy.

$G_{\mu\nu} = 8\pi G T_{\mu\nu}$

• $G_{\mu\nu}$: the Einstein tensor, which describes the curvature of spacetime
• $T_{\mu\nu}$: the stress-energy tensor, which describes the distribution of mass and energy
• $G$: the gravitational constant
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Statics

• Force Vectors

Scalars and Vectors

• Scalars:
• quantity characterized by a positive or negative number
• examples: length, area, volume, mass, time
• Vectors:
• quantity with magnitude, direction, and sense
• represented graphically by an arrow (defines direction and magnitude)
• examples: position, force, moment

Multiplication/Division of Vectors by Scalars

• Increased magnitude when multiplied by a positive scalar; changed direction if multiplied by negative scalar.

Vector Addition

• Parallelogram Law: Two vectors are added to form a resultant vector R (R is the diagonal of the parallelogram described by P and Q). Can find R through trigonometry.
• Triangle Rule: End-to-end addition of vectors. Vector addition is commutative ($\mathbf{R} = \mathbf{P} + \mathbf{Q} = \mathbf{Q} + \mathbf{P}$) and associative ($\mathbf{R} = (\mathbf{P} + \mathbf{Q}) + \mathbf{S} = \mathbf{P} + (\mathbf{Q} + \mathbf{S})$).
• Vector Subtraction: Defined as addition of a negative vector ($\mathbf{R} = \mathbf{P} - \mathbf{Q} = \mathbf{P} + (-\mathbf{Q})$).

Vector Addition of forces

• Finding a Resultant Force.

Procedure

• Parallelogram Law: Sketch out vectors to be added according to law.
• Calculate Magnitude: Find the magnitude of the resultant vector via the law of cosines or the law of sines.
• Determine and apply Direction: Determine the direction and apply it in a certain way to two vectors to which it is relative.

Law of Cosines

$R = \sqrt{P^2 + Q^2 -2PQ\cos(\theta)}$

Law of Sines

$\frac{P}{\sin(a)} = \frac{Q}{\sin(b)} = \frac{R}{\sin(\theta)}$

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  Calculus I

2. 8 Related Rate

Strategy

1. Read the problem: Read the task at hand and extract the problem carefully.
2. Diagramming possible: Diagram as you move forward in reading and gathering variables/info.
3. Nomenclature: Assign variables to the problem so that they are functions of time.
4. Derivatives: State how the required information and derivatives interact.
5. Use Geometry (subsidiary Equation) to relate quantities: Manipulate these relationships that are various (variables) to get rid of the extraneous variable through the subsidiary equation.
6. Chain Rule: Differentiate as such w/respect to t.
7. Substitution: Fill in given info. into Eqns to get the answer.

Example Air Volume Problem

• Air is being pumped in at $ 100 cm^3/s$.
• How is radius increasing when diameter is 50cm?
• Given: : $dV/dt = 100 cm^3/s$. Find, at $p=50 cm, dr/dt$ $V = \frac{4}{3}\pi r^3$
• Differentiate:

$$\frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt} = \frac{d}{dr}(\frac{4}{3}\pi r^3) \frac{dr}{dt} = 4\pi r^2 \frac{dr}{dt}$$

• Apply $$\frac{dr}{dt}=\frac{1}{4\pi r^2}\frac{dV}{dt}$$
• For r=25:

$$\frac{dr}{dt}=\frac{1}{4\pi (25)^2}100 = \frac{1}{25\pi}$$

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Algorithmic Trading

• Algorithmic traders are using computer programs set to run in an automated way, to trade. Automated instructions accounts on:

• Price

• Timing

• Volume

• Algorithmic trading widely used by institutional investors.

High vs Low

• HFT (High Frequency) is algorithmic, but it has it' own characteristics through:
• High Speed Computing
• Co Location
• Short Term Investments
• And used by prop trading firms:

Pros:

• Reduced transaction costs.
• Improved order execution.
• Increased trading capacity.

Cons:

• Model malfunction (inaccurate parameters.)
• "Fat finger" errors (incorrect input that leads to errors.)
• Data feed/cybersecurity.
• Regulatory risks

Market Impact

• . Liquid
• . Narrow B/A
• . Spreads
• . Volatility and potential . market manipulation.