Understanding Harmonic Motion and Sound

• Modular Arithmetic focuses on remainders after division
• Key concepts include congruence, modular inverses, and their applications in cryptography

Congruence

• $a$ is congruent to $b$ modulo $m$ if $m$ divides $a-b$
• This is written as $a \equiv b \mod m$
• $23 \equiv 3 \mod 5$ because $23-3 = 20$, which is divisible by $5$

Properties of Congruence

• If $a \equiv b \mod m$ and $c \equiv d \mod m$, then $a+c \equiv b+d \mod m$ and $ac \equiv bd \mod m$
• If $a \equiv b \mod m$, then $a^k \equiv b^k \mod m$ for any positive integer $k$

Modular Inverses

• The inverse of $a$ modulo $m$ is an integer $x$ such that $ax \equiv 1 \mod m$
• The inverse of $3$ modulo $7$ is $5$ because $3 * 5 = 15 \equiv 1 \mod 7$

Finding Modular Inverses

• The Extended Euclidean Algorithm is used to find modular inverses
• To find the inverse of $5$ modulo $12$:
• Apply the Euclidean Algorithm to find $\gcd(5, 12)$:
• $12 = 2 \cdot 5 + 2$
• $5 = 2 \cdot 2 + 1$
• $2 = 2 \cdot 1 + 0$
• $\gcd(5, 12) = 1$
• Apply the Extended Euclidean Algorithm to express $1$ as a linear combination of $5$ and $12$:
• $1 = 5 - 2 \cdot 2$
• $1 = 5 - 2 \cdot (12 - 2 \cdot 5)$
• $1 = 5 - 2 \cdot 12 + 4 \cdot 5$
• $1 = 5 \cdot 5 - 2 \cdot 12$
• From the equation $1 = 5 \cdot 5 - 2 \cdot 12$, we have $5 \cdot 5 \equiv 1 \mod 12$, so the inverse of $5$ modulo $12$ is $5$

Example 1

• What is $5^{100} \mod 7$?
• $5^1 \equiv 5 \mod 7$
• $5^2 \equiv 25 \equiv 4 \mod 7$
• $5^3 \equiv 5 \cdot 4 \equiv 20 \equiv 6 \mod 7$
• $5^4 \equiv 5 \cdot 6 \equiv 30 \equiv 2 \mod 7$
• $5^5 \equiv 5 \cdot 2 \equiv 10 \equiv 3 \mod 7$
• $5^6 \equiv 5 \cdot 3 \equiv 15 \equiv 1 \mod 7$
• Therefore $5^{100} \equiv 5^{6 \cdot 16 + 4} \equiv (5^6)^{16} \cdot 5^4 \equiv 1^{16} \cdot 2 \equiv 2 \mod 7$

Example 2

• What is the inverse of $11 \mod 26$?
• Using the Extended Euclidean Algorithm:
• $26 = 2 \cdot 11 + 4$
• $11 = 2 \cdot 4 + 3$
• $4 = 1 \cdot 3 + 1$
• $1 = 4 - 1 \cdot 3$
• $1 = 4 - 1 \cdot (11 - 2 \cdot 4)$
• $1 = 4 - 1 \cdot 11 + 2 \cdot 4$
• $1 = 3 \cdot 4 - 1 \cdot 11$
• $1 = 3 \cdot (26 - 2 \cdot 11) - 1 \cdot 11$
• $1 = 3 \cdot 26 - 6 \cdot 11 - 1 \cdot 11$
• $1 = 3 \cdot 26 - 7 \cdot 11$
• Therefore $-7 \cdot 11 \equiv 1 \mod 26$
• $-7 \equiv 19 \mod 26$
• $19 \cdot 11 \equiv 1 \mod 26$, the inverse of $11 \mod 26$ is $19$

Bernoulli's Principle

• Johann Bernoulli discovered it in the 18th century
• States for an inviscid flow, an increase in speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy
• Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation

Assumptions for Application

• Steady flow means the fluid's velocity at a point remains constant over time
• Incompressible flow means the fluid's density is constant along the streamline
• Inviscid flow means friction due to viscosity is negligible

Bernoulli's Equation

• $P + \frac{1}{2} \rho v^2 + \rho g h = constant$
• $P$ is the static pressure of the fluid
• $\rho$ is the density of the fluid
• $v$ is the velocity of the fluid
• $g$ is the acceleration due to gravity
• $h$ is the height of the point above a reference plane
• This equation implies that if the speed of a fluid increases, the pressure decreases, and vice versa

Applications of Bernoulli's Principle

• Airplane Wings: Designed so air flows faster over the top, creating lower pressure, which results in lift
• Venturi Tubes: Constricted sections increase speed and decrease pressure used in carburetors
• Atomizers and Sprayers: Fast-moving air creates low pressure, drawing liquid up a tube for a fine spray
• Chimneys: Tall design utilizes wind to reduce pressure, drawing smoke out

Dijkstra's Algorithm

Motivation

• Given a graph $G = (V,E)$ and source node $s \in V$, find the shortest path from s to every other vertex $v \in V$
• BFS finds the shortest path in unweighted graphs
• Solution: Explore vertices in order of their distances from s, while maintaining $dist[v]$ for every vertex
• $dist[v]$ = length of the shortest path seen so far from $s$ to $v$

Algorithm Steps

1. Initialize $dist[s] = 0$ and $dist[v] = \infty$ for all other $v \neq s$
2. Let $S = {}$
3. While $S \neq V$:

• Find the vertex $v \notin S$ with the smallest value of $dist[v]$
• Add $v$ to $S$
• For each neighbor $u$ of $v$:
• $dist[u] = \min(dist[u], dist[v] + w(v,u))$

Example Application

• Running Dijkstra's algorithm on an example graph (A to E) visually outlines the process of iteratively updating distances
• Step-by-step examples trace the changes in the dist array and the set $S$ as the algorithm progresses

Correctness Proof

• A base case, inductive hypothesis, and inductive step formally prove that when a vertex $v$ is added to $S$, dist[v] represents the shortest path from $s$ to $v$

Implementation Considerations

• Two main approaches to implementing the line "Find the vertex $v \notin S$ with the smallest value of $dist[v]$" are:
• Iterate through all vertices not in S:
• Time complexity O(n), leading to a total time complexity of O(n^2)
• Using a priority queue (min-heap):
• Time complexity O(log n), leading to a total time complexity of O(m log n)

Code Snippets

• Presented are Python code implementations using both an iterative search and a priority queue, each with its corresponding time complexity analysis

Negative Edge Weights

• Dijkstra's algorithm does not work correctly when graphs have negative edge weights
• The Bellman-Ford algorithm can be used in these cases

Statistiques descriptives univariées (Univariate Descriptive Statistics)

Définitions (Definitions)

• Population: All individuals or objects of interest
• Échantillon (Sample): A subset of the population
• Variable: A measured or observed characteristic for each individual or object in the population or sample
• Données (Data): The set of observed values for one or more variables

Types de variables (Types of Variables)

1. Variables qualitatives (catégorielles)/Categorical Variables

• Nominales (Nominal): Categories cannot be ordered (e.g., eye color, car brand)
• Ordinales (Ordinal): Categories can be ordered (e.g., satisfaction level, social class)

2. Variables quantitatives (numériques)/Quantitative Variables

• Discrètes (Discrete): Values are integers (e.g., number of children, number of cars)
• Continues (Continuous): Values can be any number within an interval (e.g., height, weight, temperature)

Mesures de tendance centrale (Measures of Central Tendency)

• Moyenne (Mean): Sum of all values divided by the number of values
• Notation: $\qquad \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Médiane (Median): The value that separates the data into two equal parts
• Mode: The value that appears most often in the data

Mesures de dispersion (Measures of Dispersion)

• ÉTendue (Range): Difference between the maximum and minimum values
• Variance: Average of the squares of the deviations from the mean
• Notation: $\qquad s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$
• ÉCart type (Standard Deviation): Square root of the variance
• Notation: $\qquad s = \sqrt{s^2}$
• Coefficient de variation (CV)/Coefficient of Variation: Ratio of the standard deviation to the mean (expressed as a percentage)
• Notation: $\qquad CV = \frac{s}{\bar{x}} \cdot 100$

Mesures de position (Measures of Position)

• Quartiles: Divide the data into four equal parts
• $Q_1$: First quartile (25% of data are less than or equal to this value)
• $Q_2$: Second quartile (median)
• $Q_3$: Third quartile (75% of data are less than or equal to this value)
• Déciles (Deciles): Divide the data into ten equal parts
• Percentiles: Divide the data into one hundred equal parts

Représentations graphiques (Graphical Representations)

• Variables qualitatives (Qualitative Variables)
• Diagramme à barres (Bar chart)
• Diagramme circulaire/camembert (Pie chart)
• Variables quantitatives (Quantitative Variables)
• Histogramme (Histogram)
• Boîte à moustaches (Box plot)
• Nuage de points (Scatter plot)

Tableaux de fréquences (Frequency Tables)

• Fréquence absolue (Absolute Frequency): Number of occurrences of a value
• Fréquence relative (Relative Frequency): Proportion of occurrences of a value relative to the total number of values
• Fréquence cumulée (Cumulative Frequency): Sum of the absolute or relative frequencies up to a given value

Exemple de tableau de fréquences (Example Frequency Table)

Formules utiles (Useful Formulas)

• Moyenne pondérée (Weighted Average)
• Notation: $\qquad \bar{x}w = \frac{\sum{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$
• $w_i$ are the weights associated with the values $x_i$
• Covariance (échantillon)/Sample Covariance
• Notation: $\qquad cov(x, y) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{n-1}$
• Coefficient de corrélation de Pearson/Pearson Correlation Coefficient
• Notation: $\qquad r = \frac{cov(x, y)}{s_x s_y}$
• $s_x$ and $s_y$ are the standard deviations of $x$ and $y$, respectively

Atomic Habits by James Clear

Introduction

• Guides improvement actionably rather than theoretically

My Story

• Detailed account of events in James Clear's life that led to the development of good habits
• This includes baseball career, injury, recovery, academic achievements, and athletic accomplishments through small habits

The Surprising Power of Atomic Habits

Why Small Improvements Matter

• 1% better every day counts in the long run
• Habits are the compound interest of self-improvement
• Success stems from daily habits, not single transformations

Forget About Goals, Focus on Systems Instead

• Defining goals versus systems
• Goals = desired outcome
• System = processes leading to the goal
• Discusses problems that arise when focusing on goals rather than systems, and that the focus be on the system

How Your Habits Shape Your Identity (and Vice Versa)

• The three Layers of behavior change: Outcomes, Processes and Identity
• Focus on who to become, not only what to achieve
• Every action is a vote for the type of person you wish to become
• Two-step process: decide the type of person you aspire to be, prove it with small wins

The Four Laws of Behavior Change

• A loop is composed of Cue, Craving, Response and Reward

The 1st Law (Cue): Make it Obvious

• Implementation intention: Specify WHEN and WHERE the habit will occur
• "I will [BEHAVIOR] at [TIME] in [LOCATION]."
• Habit stacking: Pair a new habit with a current habit
• After [CURRENT HABIT], I will [NEW HABIT].

The 2nd Law (Craving): Make It Attractive

• Temptation bundling: Pair a habit you need to do with a habit you want to do.
• After [HABIT I NEED TO DO], I will [HABIT I WANT TO DO].
• Join a culture where your desired behavior is the normal behavior

The 3rd Law (Response): Make It Easy

• Reduce friction: lower the barrier to entry for good habits.
• Master the Two-Minute Rule: Start new habits in under two minutes.

The 4th Law (Reward): Make It Satisfying

• Use a Habit Tracker: A simple way to measure whether you did a habit.
• Never Miss Twice: Getting back on track as soon as possible.
• Missing one measurement is an accident. Missing two is the start of a new habit.

Linear Algebra and Matrix Analysis

Chapter 1: Introduction

• Vecteurs (Vectors)
• A one-dimensional array of numbers
• Column Vector example: $\mathbf{x} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$
• Row Vector example: $\mathbf{x} = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
• Matrices (Matrices)
• A two-dimensional array of numbers
• Example: $\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$
• Opérations de base (Basic Operations)
• Addition de vecteurs (Vector Addition)
• If $\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ \vdots \ x_n \end{bmatrix}$ and $\mathbf{y} = \begin{bmatrix} y_1 \ y_2 \ \vdots \ y_n \end{bmatrix}$, then $\mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \ \vdots \ x_n + y_n \end{bmatrix}$
• Multiplication scalaire (Scalar Multiplication)
• If $\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ \vdots \ x_n \end{bmatrix}$ and $c$ is a scalar, then $c\mathbf{x} = \begin{bmatrix} cx_1 \ cx_2 \ \vdots \ cx_n \end{bmatrix}$
• Multiplication de matrices (Matrix Multiplication)
• If $\mathbf{A}$ is a $m \times n$ matrix and $\mathbf{B}$ is a $n \times p$ matrix, then $\mathbf{C} = \mathbf{AB}$ is a $m \times p$ matrix, where $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$
• Transposition (Transposition)
• The transpose of a matrix $\mathbf{A}$, denoted $\mathbf{A}^T$, is obtained by swapping the rows and columns of $\mathbf{A}$
• If $\mathbf{A} = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $\mathbf{A}^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$

Chapter 2: Systèmes d'Équations Linéaires (Systems of Linear Equations)

• Représentation matricielle (Matrix Representation)
• A system of linear equations can be represented in matrix form as $\mathbf{Ax} = \mathbf{b}$, where $\mathbf{A}$ is the matrix of coefficients, $\mathbf{x}$ is the vector of variables, and $\mathbf{b}$ is the vector of constants
• Méthodes de résolution (Solution Methods)
• Élimination de Gauss (Gaussian Elimination)
• A method to solve systems of linear equations by transforming the augmented matrix $[\mathbf{A} | \mathbf{b}]$ into a reduced echelon form
• Règle de Cramer (Cramer's Rule)
• Another method to solve systems of linear equations using determinants

Chapter 3: Espaces Vectoriels (Vector Spaces)

• Définition (Definition)
• A non-empty set $V$ of objects, called vectors, on which two operations are defined: addition and scalar multiplication
• Sous-espaces vectoriels (Vector Subspaces)
• A subset of a vector space that is itself a vector space
• Base et dimension (Basis and Dimension)
• A basis of a vector space is a set of linearly independent vectors that span the vector space
• The dimension of a vector space is the number of vectors in a basis

Chapter 4: Valeurs Propres et Vecteurs Propres (Eigenvalues and Eigenvectors)

• Définition (Definition)
• An eigenvector of a matrix $\mathbf{A}$ is a non-zero vector $\mathbf{v}$ such that $\mathbf{Av} = \lambda\mathbf{v}$, where $\lambda$ is the eigenvalue of $\mathbf{A}$
• Calcul des valeurs propres (Calculating Eigenvalues)
• The eigenvalues of a matrix $\mathbf{A}$ are the solutions of the characteristic equation $\det(\mathbf{A} - \lambda\mathbf{I}) = 0$, where $\mathbf{I}$ is the identity matrix
• Diagonalisation (Diagonalization)
• A matrix $\mathbf{A}$ is diagonalizable if there exists an invertible matrix $\mathbf{P}$ such that $\mathbf{P}^{-1}\mathbf{AP}$ is a diagonal matrix

Formules Importantes (Important Formulas)

• Produit scalaire (Dot Product): $\mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^{n} x_i y_i$
• Norme d'un vecteur (Vector Norm): $||\mathbf{x}|| = \sqrt{\sum_{i=1}^{n} x_i^2}$
• Déterminant d'une matrice 2x2 (Determinant of a 2x2 Matrix): $\det(\begin{bmatrix} a & b \ c & d \end{bmatrix}) = ad - bc$

Regulation of Gene Expression

Introduction

• Gene expression converts a gene's information into a functional gene product, like proteins or functional RNA
• Expression is precisely controlled to adjust to changing conditions while saving energy and resources
• Precise levels of gene products are produced in a spatial-temporal context
• Regulation can occur at transcription, RNA processing, RNA transport, translation, and protein modification
• Transcription regulation, deciding if a gene is transcribed, is a common control mechanism

Regulation of Transcription

Prokaryotes

• Prokaryotes regulate transcription to respond to environmental changes
• Operons are a key mechanism, consisting of:
• Promoter: Where RNA polymerase binds
• Operator: Controls RNA polymerase access
• Genes: DNA stretch transcribed into a single mRNA
• Activators switch on some operons
• Repressors switch off other operons, preventing transcription by:
  • binding the operator
  • blocking RNA polymerase
• Repressor form depends on a corepressor/inducer binding

Eukaryotes

• Eukaryotic expression is more complex
• Chromatin structure's role:
• DNA packaged with proteins into chromatin
• Heterochromatin is tightly condensed, not usually transcribed
• Euchromatin is loosely packed, able to be transcribed
• Histone acetylation and DNA methylation affect chromatin
• Transcription factors are essential for regulation:
• General factors are needed for all protein-coding transcription
• Specific factors bind control elements, influence transcription rates
• Enhancers are distant DNA sequences
• Activators bind enhancers and interact with other proteins to initiate transcription
• Combinatorial control: needs multiple transcription factors

RNA Processing

• RNA processing includes:
• RNA splicing: Intron removal, exon joining
• Alternative splicing: Different mRNAs from one pre-mRNA by treating segments differently

Translation and Protein Processing

• Translation can be regulated by:
• Initiation factors: Required to bind mRNA to ribosomes
• Regulatory proteins: Bind mRNA, prevent translation
• Protein processing:
• Cleavage: Cutting the polypeptide chain
• Chemical modifications: Adding phosphates/sugars
• Protein degradation: Proteins are marked by ubiquitin and degraded by proteasomes

Non-coding RNAs

• Non-coding RNAs (ncRNAs) regulate expression
• MicroRNAs (miRNAs) bind/block/degrade mRNA
• Small interfering RNAs (siRNAs) are similar to miRNAs
• Piwi-associated RNAs (piRNAs) play a role in heterochromatin formation and block transposons

Cancer

• Cancer may result from mutated genes that regulate growth
• Oncogenes promote growth
• Tumor suppressor genes inhibit growth
• Mutated genes can lead to uncontrolled growth and cancer

Chemistry Thermodynamics

Spontaneous Processes

• Spontaneous processes occur without external intervention, like a ball rolling downhill or rust forming on iron

Entropy (S)

• Entropy measures the dispersal of energy in a system
• Measured in J/K or J/(mol⋅K)
• It is a state function: $\Delta S = S_{final} - S_{initial}$

Changes That Increase Entropy ($\Delta S > 0$)

1. Phase Changes: Order goes from $S_{solid} < S_{liquid} < S_{gas}$ with melting, vaporization and sublimation increasing entropy
2. Increase in Temperature: More kinetic energy exists at higher temperatures
3. Increase in Volume: More space allows more dispersal of energy
4. Increase in the Number of Gas Molecules: More molecules mean more ways to distribute energy

• For example, $2H_2O_2(g) \rightarrow 2H_2O(g) + O_2(g)$ increase entropy

Entropy Changes in Chemical Reactions

• For the reaction $aA + bB \rightarrow cC + dD$:
  • $\Delta S_{rxn}^\circ = \Sigma n \Delta S^\circ (products) - \Sigma n \Delta S^\circ (reactants)$
• Here, n is the stoichiometric coefficient

Example

• Calculating the change in entropy for the reaction $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$ gives:
• $\Delta S_{rxn}^\circ = [2 \cdot S^\circ (NH_3(g))] - [S^\circ (N_2(g)) + 3 \cdot S^\circ (H_2(g))]$
• Using values of:
• $S^\circ (N_2(g)) = 191.5 J/(mol \cdot K)$
• $S^\circ (H_2(g)) = 130.6 J/(mol \cdot K)$
• $S^\circ (NH_3(g)) = 192.3 J/(mol \cdot K)$
• The result is:
• $\Delta S_{rxn}^\circ = -198.7 J/K$

Entropy and the Second Law of Thermodynamics

• Second Law: the entropy of the universe increases during any spontaneous process.
• $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$
• For a reversible process, $\Delta S_{universe} = 0$

Entropy of Surroundings

• $\Delta S_{surroundings} = \frac{- \Delta H_{system}}{T}$
• Exothermic process: $\Delta H_{system} < 0$, so $\Delta S_{surroundings} > 0$
• Endothermic process: $\Delta H_{system} > 0$, so $\Delta S_{surroundings} < 0$