Vector Functions

• Functions that assign a vector to each real number are vector functions of a real variable.
• They describe curves and surfaces in space.

Definition

• A vector function is $\overrightarrow{r}: I \subseteq \mathbb{R} \longrightarrow \mathbb{R}^{n}$
• For each real number $t$ in interval $I$, it assigns a vector $\overrightarrow{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right)$
• Its component functions are $f_{i}(t)$.

Example

• The function $\overrightarrow{r}(t)=(\cos t, \operatorname{sen} t, t)$ illustrates a helix in space.

Vector Function Operations

• Vector functions can be used with scalar functions.

Sums and Differences

• Vector functions are added and subtracted componentwise:
• $(\overrightarrow{r}+\overrightarrow{s})(t) = \overrightarrow{r}(t)+\overrightarrow{s}(t)$
• $(\overrightarrow{r}-\overrightarrow{s})(t) = \overrightarrow{r}(t)-\overrightarrow{s}(t)$

Scalar Product

• Each component of the vector function is multiplied by the scalar function:
• $(f \cdot \overrightarrow{r})(t)=f(t) \cdot \overrightarrow{r}(t)$

Dot Product

• Vector functions go through componentwise multiplication and result summation:
• $(\overrightarrow{r} \cdot \overrightarrow{s})(t)=\overrightarrow{r}(t) \cdot \overrightarrow{s}(t)$

Cross Product

• Vector functions go through the definition of vector cross product::
• $(\overrightarrow{r} \times \overrightarrow{s})(t)=\overrightarrow{r}(t) \times \overrightarrow{s}(t)$

Limits and Continuity

• These are specified using the boundary and continuity associated with their component functions.

Limit

• Vector $\vec{L}$ serves as the boundary of a vector function $\overrightarrow{r}(t)$ when $t$ is near $a$ if the corresponding component of $\vec{L}$ matches $t$'s boundary of each component function of $\overrightarrow{r}(t)$.
• $\lim _{t \rightarrow a} \overrightarrow{r}(t)=\left(\lim {t \rightarrow a} f{1}(t), \lim {t \rightarrow a} f{2}(t), \ldots, \lim {t \rightarrow a} f{n}(t)\right)$

Continuity

• At $t=a$, if the vector function $\overrightarrow{r}(t)$ has existing limit where $t$ approaches $a$, then it equals $\overrightarrow{r}(a)$ resulting in being a continual function.

Derivative

• Refers to boundary of components in particular functions.

Definition

• $\overrightarrow{r}^{\prime}(t)$ is the derivative vector function of $\overrightarrow{r}(t)$.
• The component functions of $\overrightarrow{r}^{\prime}(t)$ serve as the derivatives for the component functions of $\overrightarrow{r}(t)$.
• $\overrightarrow{r}^{\prime}(t)=\left(f_{1}^{\prime}(t), f_{2}^{\prime}(t), \ldots, f_{n}^{\prime}(t)\right)$

Geometric Interpretation

• The function's vector derivative is a tanget to the curve formed by that vector function at a specific point.

Rules for finding derivatives

• $(\overrightarrow{r}+\overrightarrow{s})^{\prime}(t)=\overrightarrow{r}^{\prime}(t)+\overrightarrow{s}^{\prime}(t)$
• $(f \cdot \overrightarrow{r})^{\prime}(t)=f^{\prime}(t) \cdot \overrightarrow{r}(t)+f(t) \cdot \overrightarrow{r}^{\prime}(t)$
• $(\overrightarrow{r} \cdot \overrightarrow{s})^{\prime}(t)=\overrightarrow{r}^{\prime}(t) \cdot \overrightarrow{s}(t)+\overrightarrow{r}(t) \cdot \overrightarrow{s}^{\prime}(t)$
• $(\overrightarrow{r} \times \overrightarrow{s})^{\prime}(t)=\overrightarrow{r}^{\prime}(t) \times \overrightarrow{s}(t)+\overrightarrow{r}(t) \times \overrightarrow{s}^{\prime}(t)$
• $(\overrightarrow{r}(f(t)))^{\prime}=\overrightarrow{r}^{\prime}(f(t)) \cdot f^{\prime}(t)$

Integration

• Defined using integrated components in particular functions.

Definition

• Vector function $\overrightarrow{R}(t)$ is an integral for $\overrightarrow{r}(t)$.
• Its component functions will serve as integrals for $\overrightarrow{r}(t)$.
• $\int \overrightarrow{r}(t) d t=\left(\int f_{1}(t) d t, \int f_{2}(t) d t, \ldots, \int f_{n}(t) d t\right)$

Geometric Interpretation

• Underneath the curve's area are integrals from a specific function.

Linear Algebra

• Algebraic equations that deal with linear relations

Table of Contents

• Linear applications and matrices
• Linear systems
• Vector spaces of dimension fin
• Matrix Calculation
• Determinants
• Endomorphism reduction
• Scalar product and Euclidian spaces
• Affine geometry

Machine Learning Algorithms

• Machine learning is a field of AI (Artificial Intelligence)

Supervised learning

• The utilization of tagged data.
• Helps train for data projection.

Types

• Classification: Categorizes a specific category depending on the information provided (spam or medical etc)
• Common Algorithms
  • Logistic regression
  • Support Vector machines (SVM)
  • Decision trees
  • Random forests

Regression

• Algorithm utilized to project a value, it can be in sales or prices
• Common Algorithm
  • Linear regression
  • Polynomial regression
  • Regression trees

Unsupervised Learning

• The utility of a naked data sample
• Its goal is to expose veiled relationships

Clustering

• A Group of similar data sets
• Common Algorithm
  • K- Means
  • Hierarchical clustering
  • DBSCAN

Dimensional Reduction

• It lessens variables whereas the information is still intact
• It can be used on data representations
• Common Algorithm
  • Principal component analysis (ACP)
  • T-distributed Stochastic Neighbor Embedding (t-SNE)

Reinforcement- Learning

• An agent is prepared to make actions
• It improves in an enviornment to improve in it
• Games or even robots can follow this pattern
• Common Algorithms
  • Q- learning
  • Deep Q-Network(DQN)
  • Actor-Critic

Specific Algorithms

Linear Regression

• Linear Model utilzied to have a focus on projection of continous target value
• Formulaic
  • $h_\theta(x) = \theta_0 + \theta_1x_1 +... + \theta_nx_n$
• Cost Function
  • $J(\theta) = \frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x^{(i)}) - y^{(i)})^2$
• Gradient Descent
  • $\theta_j := \theta_j - \alpha \frac{1}{m} \sum_{i=1}^{m}(h_\theta(x^{(i)}) - y^{(i)})x_j^{(i)}$

Logistic Regression

• It has a focus on binary focus projection
• Sigmoid Fxn
  • $g(z) = \frac{1}{1 + e^{-z}}$
• Hypothesis
  • $h_\theta(x) = g(\theta^T x)$
• cost function-
  • $J(\theta) = -\frac{1}{m}\sum_{i=1}^{m}[y^{(i)}log(h_\theta(x^{(i)})) + (1-y^{(i)})log(1-h_\theta(x^{(i)}))]$

K-Means

• K clusters is how it divies the data
• Algorithm
  1. Pick cluster area
  2. Assign a point to a cluster area
  3. Retotal the clusters
  4. Till the algorithim converges, reapeat #2/3

Descision Trees

• Classification and regression of data through a tree structure
• How its made
  • Divide data depending on high infomration or less impurity
• Focus the direction based on information sets

Random Forest

A collection of desicion trees

• How its made
  • By the means of a subset and training the info with various desicion trees
• Projection
  • The selection or a certain class based on frequency or individual trees

Support vector machines (SVM)

• Seperating data by the means of hyperplanes
• Target
  • To enhance the margins within given classes
• Kernel Funx
  • Data is linear when using core funx like RBF

Physics Essentials

• Sheet for constant acceleration

Constant Acceleration

• $v=v_{0}+a t$
• $x=x_{0}+v_{0} t+\frac{1}{2} a t^{2}$
• $v^{2}=v_{0}^{2}+2 a\left(x-x_{0}\right)$
• $x-x_{0}=\frac{1}{2}\left(v_{0}+v\right) t$
• $x-x_{0}=v t-\frac{1}{2} a t^{2}$

Free Fall

• $a_{y}=-g=-9.8 \mathrm{~m} / \mathrm{s}^{2}$

Projectile Motion

• $v_{x}=v_{0 x}=$ constant
• $x=x_{0}+v_{0 x} t$
• $v_{y}=v_{0 y}-g t$
• $y=y_{0}+v_{0 y} t-\frac{1}{2} g t^{2}$

Force

• $\vec{F}=m \vec{a}$
• $F_{x}=m a_{x}, F_{y}=m a_{y}, F_{z}=m a_{z}$
• $F_{\text {gravity }}=m g$
• $F_{\text {spring }}=-k\left(x-x_{0}\right)$

Frixion

• $f_{\text {static}} \leq \mu_{s} F_{\text {normal }}$
• $f_{\text {kinetic }}=\mu_{k} F_{\text {normal }}$

Work And Energy

• $W=F_{x} \Delta x+F_{y} \Delta y+F_{z} \Delta z=\vec{F} \cdot \Delta \vec{r}$
• $K E=\frac{1}{2} m v^{2}$
• $U_{\text {gravity }}=m g y$
• $U_{\text {spring }}=\frac{1}{2} k\left(x-x_{0}\right)^{2}$
• $E=K E+U$
• $W_{\text {nc }}=\Delta E$
• $P=\frac{W}{\Delta t}=F v(\text { if } \vec{F} | \vec{v})$

Momentum

• $\vec{p}=m \vec{v}$
• $\vec{F}_{\text {ext }}=\frac{d \vec{p}}{d t}$
• $\vec{p}{i}=\vec{p}{f}$
• $m_{1} v_{1 i}+m_{2} v_{2 i}=m_{1} v_{1 f}+m_{2} v_{2 f}$
• $v_{1 f}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) v_{1 i}+\left(\frac{2 m_{2}}{m_{1}+m_{2}}\right) v_{2 i}$
• $v_{2 f}=\left(\frac{2 m_{1}}{m_{1}+m_{2}}\right) v_{1 i}+\left(\frac{m_{2}-m_{1}}{m_{1}+m_{2}}\right) v_{2 i}$
• $\left(v_{1 i}-v_{2 i}\right)=-\left(v_{1 f}-v_{2 f}\right) \quad$ (elastic collision)

Trigonometry

• $\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}$
• $\cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}$
• $\tan \theta=\frac{\text { opposite }}{\text { adjacent }}$

Trigonometric Functions

Definition

• Any function that uses trigonometry as its means of definition.

EX

• $f(x) = 5 \sin(4x) - 3$
• $g(x) = -2 \cos(0.5x + 1) + 4$
• $h(x) = 7 \tan(2x) - 6$

Diagrams

• Shows the basics for each Trigonometric function

Sine Function

• Amplitude : 1
• Domain : $\mathbb{R}$
• Image : $[-1, 1]$
• Period : $2\pi$

Cosine Function

• Amplitude : 1
• Domain : $\mathbb{R}$
• Image : $[-1, 1]$
• Period : $2\pi$

Tangent Function

• Amplitude : Undefined
• Domain : $\mathbb{R} \setminus {\frac{\pi}{2} + k\pi \mid k \in \mathbb{Z}}$
• Image : $\mathbb{R}$
• Period: $\pi$

Parametric

• Sine and Cosine functions are changed by parameters

Parametric equations

• $f(x) = a \sin(b(x - h)) + k$
• $f(x) = a \cos(b(x - h)) + k$
• Where the following are
  • a: The reflection of x axis
  • b: $\frac{2\pi}{|b|}$- Period
  • h: Horizontonal Direction
  • k: Vertical direction

Tangent Equations

• $f(x) = a \tan(b(x - h)) + k$
• a: Vertical Direction (Scale) and reflection
• b: $\frac{\pi}{|b|}$ - Period
• h: Horizontonal Direction
• k: Vertical direction

Transformations

• Refers ot the way the paremters make the transformations

Cardiovascular System

Refers to the transfer of nutrients throughout the bodies circulatory system.

Blood Vessels

Part of the Circulatory System

Arteries

• To have a system of blood flow away from the heart
• They are elastic/ thick
• It has walls that hold high pressured force
• There are different types
  • Elastic arteries: Hold the capacity of volume and are the most voluminous of their type
  • Muscular arteries: Have muscle tissue within them (media)
  • Arterioles: Smallest in size and the regulator of blood flow.
• Functionaity ensures constant blood transport

Capillaries

• Exchange between blood and tissue

• Singular tissue of endothelium

  • Continuous capillaries: Most universal, sealed junctions

  • Fenestrated capillaries: enhanced porous

  • Sinusoid capillaries: Gaps

• Functionality is to faciliate gas transfers

Veins

• Sends blood to the heart
• Very fluid , less dense, contains valves
  • Venules: Receives the blood flow
• Medium-Vein: Valves function to stop backflow
• Large-Vein: To restore flow, directly restore flow -Reservoir- The function of this vein is to transport blood and to act as a blood resovoir

Blood Pressure

Force applied of blood - It is to be applied on veins and wall tissue

• Measurement system -
  • Blood pressure during a heartbeat (systolic)
• Blood pressure during ventricle relax (diastolic)
• 120/80 MMHG is the norm
• Regulation: - Via hormones, ADh epinephrine

Cardiac Cycle

• Ventricles Push blood - Valves open
• atria pushes blood throughout the vertricile
• phases are separated

1. Atrial Contraction 2) System that Pushes out the blood 3) the ventricle relaxes

Heart sounds

• Dub or lub - How the valves close
• S1 or S2 - the valves for each

Theorems

• Fermats law
• "p" = prime integer (number)
• "$a^p$ - a" always can be multpled by P

Formula

$a^p \equiv a (mod \ p)$

• Not being divisible means it is = $a^{p-1} \equiv 1 (mod \ p)$

Proof

Set {1,2...p-1}

$f: A \rightarrow A$ defined by $x \mapsto ax \ mod \ p$.

• It is only injective if 1 = 1 ; P is a prime numer, thus is always will be. Always will show the result

$a^{p-1} \equiv 1 \mod \ p$

Corollary

• For every zero or greater integer $a^n \equiv a^{n \mod (p-1)} \mod \ p$.

Wilson theory

Integer Theory

• if $p>1$ prime then $(p-1)! \equiv -1 \mod \ p$

proof

• A series of sets will need to be calculated for both the right and left
• If not the prime factors $gcd((p-1)! + 1, p) = 1$

Euler

Theory

• For intergers of $n \geq 1$, let $\phi(n)$ denote the number of integers in ${1, 2, ... , n}$ $\phi(p) = p-1$ in prime cases

Theorem

if $gcd(m, n) = 1$, then $\phi(mn) = \phi(m) \phi(n)$. `

Algorithmic Trading/Order Execution

Is the way humans delegate to machines the ability run a trading decision

Algorithm-

• Its computer programs trading actions
• Trade depending on factors
• Improves transation/trading
• Employed by hedge/mutual funds
• Used to sell or acquire bondds contracts and currencies and shares

SEC

• Has regulations like SEC 15c2-3

Electronis oreder book

Offers and prices

Limit Order

Buy 10 shares at some USD, is to put a limit/ specific price

Dark Pool

• Dark pools don’t post any informaiton publicly
• Utilized by orginizations to deal transactions anonymously

Participants

• Btokers in the markets
• Exchange companies
• Comms net works
• Internal

Greatest Exec

• Is the legal obligation of the worker to buy /sell at best rate for profit
  • Must factor in speed , volume and trade when analyzing the optimal route

Hydrogen Atoms

• Definition
• Coordinate sys $(r, \theta \phi)$ - spherical pole
• Has formulas such a Hamiltonian $$ \hat{H} = \frac{-\hbar^2}{2\mu}\nabla^2 - \frac{e^2}{r} $$ $\mu$: mass $m_e$: mass of electron M:nucleus mass

Schrodinger formula

$$ \hat{H}\Psi(r, \theta, \phi) = E\Psi(r, \theta, \phi) $$

Variables

$\Psi(r, \theta, \phi) = R(r)Y(\theta, \phi)$ $$ [-\frac{\hbar^2}{2\mu}\frac{1}{r^2} \frac{d}{dr}(r^2\frac{dR(r)}{dr}) + \frac{l(l+1)\hbar^2}{2\mu r^2} - \frac{e^2}{r}]R(r) = ER(r) $$

Equations

$$ \rho = \frac{r}{a_0} $$ $$ a_0 = \frac{\hbar^2}{\mu e^2} = 0.529 Å $$ $$ [-\frac{1}{2}\frac{d^2}{d\rho^2} + \frac{l(l+1)}{2\rho^2} - \frac{1}{\rho}]P(\rho) = \epsilon P(\rho) $$ `

Solutions

• Formula $$ P_{n,l}(\rho) = N_{n,l}\rho^{l+1}e^{-\rho}L_{n+l}^{2l+1}(2\rho) $$

• $n=1, 2, 3,...$ $l = 0, 1, 2,..., n-1$

Energies

• Formula $$ E_n = -\frac{E_H}{2n^2} $$

Radial Funct

• Formula

$$ R_{n,l}(r) = \frac{P_{n,l}(r)}{r} $$

####Atomic Orbitals/ Shell

• Formula $$ \Psi_{n,l,m}(r, \theta, \phi) = R_{n,l}(r)Y_{l,m}(\theta, \phi) $$

Thermodynamics

Theory

• Therm equilibrium is to exist if two systems are equal in some other element

Equation

$\qquad \Delta U = Q - W$

• If its conserved then its equal to energy

Entropy

• Equation $\qquad \Delta S \ge 0$
• Can’t be reduced in closed Sys

Gibbs Free Energy

• Equation $\qquad G = H - TS = U + PV - TS$ Spontaneity is defined as: $\Delta G < 0$.

Laws

• All them are very fundamental

Process

• Isothermal process: constant temperature.
• Adiabatic process: transfer. heat being free. Isobaric process: A process at constant pressure. -Isochoric process: constant volume.

HEAT engines

• Equation $\qquad \eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$
• Engine effectiveness rating
• carnot enginge is the highest efficiency there is

Refrigerators

• The coefficient of performance (COP) $\qquad COP_{refrigerator} = \frac{Q_C}{W}$
• A heat pump is the device to switch between hot and cold