C++: Data Types, I/O, and Operators

Lab 4: Data Types, Input/Output, and Operators

• This lab focuses on understanding data types in C++, performing input and output operations, using operators, and writing interactive programs.

Resources

• Key resources include Chapters 2-4 of "Starting Out with C++" and the C++ Reference website.

Part 1: Data Types and Variables

• Declare variables for int, double, float, char, bool, and string data types.
• Initialize each declared variable with a relevant value.
• Output the value and data type of each variable using cout.

Part 2: Input/Output Operations

• Prompt the user to enter their name and age using cout.
• Store the user's input using cin.
• Display a personalized message using the stored values such as "Hello, [Name]! You are [Age] years old.".

Part 3: Operators

• Declare two integer variables, num1 and num2, and assign them values.
• Perform addition, subtraction, multiplication, division, and modulus operations on these variables.
• Print the result of each operation.
• Demonstrate increment (++) and decrement (--) operators.
• Show the use of compound assignment operators like +=, -=, *=, and /=.

Part 4: Simple Calculator

• Write a program that takes two numbers and an operator (+, -, *, /) as input from the user.
• Use a switch statement to perform the appropriate operation based on the operator entered.
• Display the result of the calculation.
• Handle the division-by-zero error.

Part 5: Type Conversion

• Declare an integer variable and a double variable, initializing them with appropriate values.
• Convert the integer to a double and vice versa.
• Print the original and converted values to observe the effects of type conversion.

Part 6: Interactive Program

• Develop a simple interactive program that asks the user a question.
• Use cin to get the user's response.
• Based on the response, provide different outputs using cout and if-else statements.
• This exercise aims to integrate the knowledge gained in the previous parts.

Chapter 5: Early Embryonic Development

Fertilization

• The fusion of oocyte and sperm pronuclei marks fertilization.
• Stimulates the oocyte to complete the second meiotic division.
• Results in the formation of the zygote.

Steps

• (1) The sperm penetrates the corona radiata.
• (2) Sperm binds zona pellucida (ZP3).
• (3) Acrosome reaction occurs.
• (4) Sperm penetrates the zona pellucida.
• (5) Fusion of sperm and oocyte plasma membranes then happens.
• (6) Cortical reaction is initiated.
• (7) Oocyte completes the second meiotic division.
• (8) Formation of male and female pronuclei.
• (9) DNA synthesis and replication.
• (10) Pronuclei fuse.

Cleavage

• A series of mitotic divisions of the zygote.
• Results in an increase in cell number with cells becoming smaller (blastomeres).
• Occurs while the zygote travels along the fallopian tube toward the uterus.

Compaction

• After the 8-cell stage, blastomeres undergo compaction
• Cells change shape and tightly align against each other

Morula

• Formed by 12-32 blastomeres.
• Inner cell mass (embryoblast) gives rise to tissues of the embryo.
• Outer cell mass (trophoblast) forms the trophoectoderm, which gives rise to the placenta.

Blastocyst Formation

Blastocyst

• Contains fluid-filled cavity (blastocoel) which forms inside the morula.
• The embryoblast is located at one pole.
• The trophoblast forms the wall of the blastocyst.

Hatching

• Blastocyst hatches from the zona pellucida to allow implantation.

Implantation

• Blastocyst adheres to the endometrial epithelium.
• The Trophoblast differentiates into: Cytotrophoblast, and Syncytiotrophoblast.
• Syncytiotrophoblast invades the endometrial connective tissue.
• The Blastocyst becomes embedded in the endometrium.

Week 2

Day 8

• Cytotrophoblast and syncytiotrophoblast are present.
• The Embryoblast differentiates into Epiblast which forms Amniotic Cavity, and Hypoblast which forms Yolk Sac.

Day 9

• Blastocyst embeds more deeply into the endometrium.
• A coagulation plug closes the penetration defect.
• Lacunae appear in the syncytiotrophoblast.

Days 11-12

• Blastocyst is completely embedded in the endometrium.
• Lacunar networks form via fusion of lacunae.
• Sinusoids erode and lacunar networks fill with maternal blood.
• Extraembryonic mesoderm forms.

Day 13

• Primary Chorionic villi appear.
• Extraembryonic coelom forms
• Yolk sac reduces in size.

Gastrulation

Week 3

• Formation of the three germ layers, which are: Ectoderm, Mesoderm and Endoderm.

Primitive Streak

• Begins as a thickened linear band of epiblast.
• Forms at the caudal end of the embryo.
• Primitive node forms at the cephalic end of the primitive streak.
• The Primitive groove develops in the primitive streak.
• Cells of the epiblast migrate through the primitive streak and node to form the mesoderm and endoderm.

Germ Layers

• Ectoderm forms the epidermis, central and peripheral nervous systems, and retina.
• Mesoderm forms the muscle, bone, blood, and urogenital system.
• Endoderm forms the epithelial lining of the respiratory and digestive tracts.

Notochord

• Develops from the primitive node.
• Induces the formation of the neural tube.
• Defines the longitudinal axis of the embryo.

Neurulation

• Formation of the neural tube from the ectoderm.
• The Neural plate forms from the ectoderm.
• The Neural folds elevate and fuse to form the neural tube.
• Neural crest cells migrate from the neural folds.

Radiative Processes

3.1 The Einstein Coefficients

• Considers two energy levels: a lower level 1 and an upper level 2.
• $E_1$ and $E_2$ are their respective energies.
• $h\nu = E_2 - E_1$, relates frequency ($\nu$) of emitted/absorbed radiation.

3.1.1 Absorption

• Absorption is proportional to the atoms in the lower level ($N_1$) and energy density ($u_{\nu}$).
• $B_{12}$ is the Einstein coefficient for absorption.
• $R_{12} = B_{12} N_1 u_{\nu}$ (3.1) gives the absorption rate.

3.1.2 Spontaneous Emission

• Atoms in the upper level spontaneously decay, emitting a photon.
• Rate of emission is proportional to atoms in the upper level ($N_2$).
• $A_{21}$ is the Einstein coefficient of of spontaneous emission.
• $R_{21} = A_{21} N_2$ (3.2) gives the spontaneous emissions rate.

3.1.3 Stimulated Emission

• Stimulated emission is proportional to $N_2$ and the energy density ($u_{\nu}$).
• $B_{21}$ is the Einstein coefficient for stimulated emission.
• Rate is expressed at $R_{21} = B_{21} N_2 u_{\nu}$ (3.3).

3.1.4 Thermal Equilibrium

• Equilibrium achieved when absorption and emission rates balance.
• Described by equation $N_1 B_{12} u_{\nu} = N_2 A_{21} + N_2 B_{21} u_{\nu}$ (3.4).
• The energy density can then be expressed as $u_{\nu} = \frac{A_{21}}{B_{12} \frac{N_1}{N_2} - B_{21}}$ (3.5).
• The Boltzmann distribution describes the population ratio: $\frac{N_1}{N_2} = \frac{g_1}{g_2} e^{\frac{h\nu}{kT}}$ (3.6).
• Here, $g_1$ and $g_2$ are level statistical weights.
• In thermal equilibrium: $u_{\nu} = \frac{A_{21}}{B_{12} \frac{g_1}{g_2} e^{\frac{h\nu}{kT}} - B_{21}}$ (3.7).

3.1.5 Einstein Relations

• Compare 3.7 with the Planck blackbody spectrum: $B_{\nu}(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{kT}} - 1}$ (3.8)
• The Einstein relations can be described as: $g_1 B_{12} = g_2 B_{21}$ (3.9), and $A_{21} = \frac{2h\nu^3}{c^2} B_{21}$ (3.10).

Chapitre 1 Logique et raisonnement

1.1. Propositions et opérateurs logiques

Définition 1.1.1. Proposition

• A proposition is a declarative sentence that is either true or false, but not both.
• Exemple 1.1.1.
• “5 is a prime number” is a true proposition.
• “Canada is the richest country in the world” is a false proposition.
• “It is raining” is a proposition: truth depends on time and place.
• “What time is it?” is not a proposition.
• “Close the door” is not a proposition.
• “x + 5 = 3” is not a proposition: truth depends on the value of x.

Opérateurs logiques

• These can form new propositions from existing ones: negation, conjunction, disjunction etc.

Négation

• The negation of p, written ¬p, is true when p is false, and false when p is true. | p | ¬p | | :---- | :---- | | T | F | | F | T |
• Exemple 1.1.2.
• If p is “5 is a prime number,” ¬p is “5 is not a prime number.”

Conjonction

• The conjunction of p and q, written p ∧ q, is true only when both are true | p | q | p ∧ q | | :---- | :---- | :-------- | | T | T | T | | T | F | F | | F | T | F | | F | F | F |
• Exemple 1.1.3.
• If p is “5 is a prime number” and q is “Ottawa is the capital of Canada,” then p ∧ q is true.

Disjonction

• The disjunction of p and q, written p ∨ q, is true if at least one is true. | p | q | p ∨ q | | :---- | :---- | :-------- | | T | T | T | | T | F | T | | F | T | T | | F | F | F |
• Exemple 1.1.4.
• If p is “5 is a prime number” and q is “Quebec is the capital of Canada,” then p ∨ q is true.

Disjonction exclusive

• The exclusive disjunction of p and q, written p ⊕ q, is true when exactly one is true. | p | q | p ⊕ q | | :---- | :---- | :-------- | | T | T | F | | T | F | T | | F | T | T | | F | F | F |
• Exemple 1.1.5.
• If p is “5 is a prime number” and q is “Quebec is the capital of Canada,” then p ⊕ q is true.

Implication

• The implication of p and q, written p → q, is false only when p is true and q is false. | p | q | p → q | | :---- | :---- | :-------- | | T | T | T | | T | F | F | | F | T | T | | F | F | T |
• p is the hypothesis and q is the conclusion.
• Exemple 1.1.6.
• If p is “it is raining” and q is “the ground is wet,” then p → q is “If it is raining, then the ground is wet."
• Remarque 1.1.1.
• p → q is true even when p is false. (“If 2 + 2 = 5, then the Earth is flat” is true).

Équivalence

• The equivalence of p and q, written p ↔ q, is true when p and q have the same truth value. | p | q | p ↔ q | | :---- | :---- | :-------- | | T | T | T | | T | F | F | | F | T | F | | F | F | T |
• Exemple 1.1.7.
• If p is “5 is an even number” and q is “2 is a prime number,” then p ↔ q is false, since p is false and q is true.

Priorité des opérateurs logiques

• Operations have an order of priority:
• Negation (¬)
• Conjunction (∧)
• Disjunction (∨)
• Implication (→)
• Equivalence (↔)
• Exemple 1.1.8.
• $p ∧ ¬q ∨ r → s$ is equivalent to $((p ∧ (¬q)) ∨ r) → s$.

1.2. Tables de vérité et équivalences logiques

Table de vérité

• A truth table indicates a compound proposition’s truth value for all combinations of truth values of its components.
• Exemple 1.2.1.
• The truth table of $(p → q) ↔ (¬q → ¬p)$ yields all true values.

Équivalence logique

• Propositions p and q are logically equivalent if they have the same truth value for all possible combinations of truth values. Written $p ≡ q$.
• Exemple 1.2.2.
• $(p → q) ≡ (¬q → ¬p)$ shown in Example 1.2.1.

Some important logical equivalencies:

• Loi de De Morgan :
• $¬(p ∧ q) ≡ ¬p ∨ ¬q$
• $¬(p ∨ q) ≡ ¬p ∧ ¬q$
• Loi de la double négation : $¬(¬p) ≡ p$
• Loi de l'implication : $p → q ≡ ¬p ∨ q$
• Loi de la contraposée : $p → q ≡ ¬q → ¬p$
• Loi de la commutativité :
• $p ∧ q ≡ q ∧ p$
• $p ∨ q ≡ q ∨ p$
• Loi de l'associativité :
• $(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)$
• $(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)$
• Loi de la distributivité :
• $p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)$
• $p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)$
• Loi de l'idempotence :
• $p ∧ p ≡ p$
• $p ∨ p ≡ p$
• Loi de l'identité :
• $p ∧ V ≡ p$
• $p ∨ F ≡ p$
• Loi de domination :
• $p ∨ V ≡ V$
• $p ∧ F ≡ F$
• Loi de complémentarité :
• $p ∨ ¬p ≡ V$
• $p ∧ ¬p ≡ F$

1.3. Prédicats et quantificateurs

Prédicat

• A predicate is an expression containing one or more variables; it becomes a proposition when values are substituted for the variables.
• Exemple 1.3.1.
• Let $P(x)$ be “$x > 5$”.
• $P(3)$ is “$3 > 5$”, which is false.
• $P(10)$ is “$10 > 5$”, which is true.

Quantificateurs

• A quantifier expresses the quantity of elements satisfying a predicate. Most common are the universal and existential quantifiers.

Quantificateur universel

• The universal quantifier, written ∀, means “for all.”
• $∀xP(x)$ is true if $P(x)$ is true for all values of $x$ in the domain of discourse.
• Exemple 1.3.2.
• Let $P(x)$ be "$x^2 ≥ 0$". In real numbers, $∀xP(x)$ is true because every real number squared is non-negative.

Quantificateur existentiel

• The existential quantifier, written ∃, means “there exists.”
• ∃xP(x) is true if P(x) is true for at least one value of x in the domain of discourse.
• Exemple 1.3.3.
• Let P(x) be “x > 5”. If the domain is real numbers, ∃xP(x) is true because there's a real number greater than 5.

Négation des quantificateurs

• $¬∀xP(x) ≡ ∃x¬P(x)$
• $¬∃xP(x) ≡ ∀x¬P(x)$
• Exemple 1.3.4.
• The negation of "All students in this course are nice" is "There is at least one student in this course who is not nice."
• The negation of "There is a student over 30 in this course" is "All students in this course are at most 30 years old."

Chemical Kinetics

Reaction Rate

• The reaction rate defines how fast reactants convert into products.
• Reaction rate is affected by reactant concentration, temperature, surface area, and catalysts.

Rate Law

• Expresses rate as a function of reactant concentrations.
• $aA + bB \rightarrow cC + dD$.
• $Rate = k[A]^m[B]^n$, where:
• k is the rate constant.
• m and n are the reaction orders with respect to A and B.
• m + n is the overall reaction order.

Determining Reaction Order

• Method of Initial Rates measures initial rate with varied initial concentrations.
• Rates are then compared to determine the order with respect to each reactant.
• Integrated Rate Laws relate the concentration to time.
• Zero order: $[A]_t = -kt + [A]_0$
• First order: $ln[A]_t = -kt + ln[A]_0$
• Second order: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

Activation Energy

• The definition of activation energy is the minimum energy required for a reaction to occur.
• With the Arrhenius Equation: $k = Ae^{\frac{-E_a}{RT}}$.
• Where $E_a$ is the activation energy.
• R is (8.314 J/mol·K).
• A is the pre-exponential factor.
• T is the Kelvin temperature.

Catalysis

• Catalyst: Speeds up the reaction without being consumed.
• Homogeneous: Same phase as reactants.
• Heterogeneous: Different phase.
• Mechanism: Catalysts lower the energy by providing alternate pathway.

Reaction Mechanisms

• Elementary Steps: Simple steps summing to the overall reaction.
• The rate-determining step is the slowest step and thus determines the overall rate.
• Intermediates: Substances produced and then consumed

Validation of a Mechanism

• The elementary steps must be balanced.
• The predicted rate law must match experimental one.

Chemical Kinetics

Definition of Reaction Rate

• The change in concentration of reactants or products per unit time.

Definition of Rate Law

• The Rate Law is the relationship between reaction rate and reactant concentrations, where: $aA + bB \rightarrow cC + dD$.
• Rate = $k[A]^m[B]^n$.
• k is the rate constant, which is related to the Order of reactions
• m and n are reaction orders of A and B.

Definition of the Order of Reaction

• This is the sum of exponents in the rate law.
• Measured as : Overall order = $m + n$

Factors Affecting Reaction Rate

Temperature

• Increasing temperature usually increases reaction rate.
• Has relationships to Arrhenius Equation

Concentration

• By increasing concentration, increasing reaction rate.

Catalyst

• Speeds up reaction without being consumed.

Surface Area

• Increasing surface area of a solid means usually increasing reaction rate.

Definition of Rate Constant

Rate Constant

• Is a proportional constant and relates the reaction rate to a concentration of certain reactants.

Arrhenius Equation

• This defines the temperature dependence of rate constant, described by:
• $k = Ae^{-\frac{E_a}{RT}}$ where:
• A is Pre-Exponential Factor
• Ea is Activation Energy
• R is GAs Constant (8.314 J/mol * K) T is (absolute temperature in Kelvin)

Activation Energy

• Definition
  • The minimum energy needing to occur

Transition State Theory

• Meaning it expresses rate of reactions that are based on an activated complex of transition state.

Reaction Mechanisms

• Definition
  • A step by step process of by which overall reaction changes occur.

Elementary Step

• a single step

Rate-Determining Step

• The slowest step that determines overall rate

Catalysis

• Homogenous is catalyst in same phase:
• Heterogenous is different

Summary Table

• Reaction Rate: Change in concentration.
• Rate Law: Reaction Rate and reactant concentratinos.
• Rate Constant: K in rate Law
• Order of Reaction: sum of Exponents
• Activation Energy: Minimum Energy Need
• Reaction Mechanism: Step by Step Elementary reactions
• Rate-Determining Step: Slowest Step.
• Catalyst: Lowers Activation Energy

Algèbre Linéaire et Géométrie Analytique I

Chapitre 1: Systèmes d'Équations Linéaires

1.1 Introduction

• Systems of linear equations are explored as a set of linear equations sharing the same variables.
• Solving involves finding values satisfying all equations.

1.2 Définitions

Définition 1.2.1 (Équation Linéaire)

• A linear equation in n variables (x₁, x₂, ..., xₙ) follows the form: a₁x₁ + a₂x₂ + ... + aₙxₙ = b
• a₁, a₂, ..., aₙ and b are real constants, the "coefficients"
• b is the equation's "constant"

Définition 1.2.2 (Système d'Équations Linéaires)

• A system of m linear equations in n variables (x₁, x₂, ..., xₙ):
• a set of m linear equations:
• a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
• a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
• ⁞
• aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ
• aᵢⱼ & bᵢ are real constants.

Définition 1.2.3 (Solution d'un Système d'Équations Linéaires)

• Solution: values for variables x₁, x₂, ..., xₙ that satisfy all system equations.

Définition 1.2.4 (Ensemble des Solutions d'un Système d'Équations Linéaires)

• Set of Solutions: the complete set of solutions that exist

Définition 1.2.5 (Système d'Équations Linéaires Compatible)

• Compatible = 1+ solutions

Définition 1.2.6 (Système d'Équations Linéaires Incompatible)

• Incompatible = no solution

1.3 Exemples

Exemple 1.3.1

• x + y & = 2 and x - y = 0 x = 1 && y = 1 (so compatible)
• Solutions = {(1, 1)}.

Exemple 1.3.2

• x + y = 2 AND x + y = 0
• No Solution (so incompatible), solution set = empty

Chapter 14: Government

Section 1: The Role of Government in the Economy

• Economic goals of government include:
• Economic growth = more output per person long term
• Efficiency = resources used to produce maximum amount of goods
• Equity = fair distribution of income and wealth
• Economic security = protecting people from various risks.
• Economic freedom = ability to make economic choices absent interference
• Government promotes goals by
• Providing public goods and services (national defense, infrastructure, etc)
• They are generally non exclusive to people or companies
• redistribution via welfare and social security.
• Regulating activity via environment, minimum wage and antitrust laws
• Promoting Stability with fiscal and monetary policy

Section 2: Taxation

• Tax Types include

• Progressive = bigger slice from higher earners (income tax)

• Regressive = bigger from lower earners (sales tax)

• Proportional = Same % from everyone (property)

• Tax incidence: who bears the burden depends on supply elasticity

• Demand more elastic than supply = producer pays more

• Supply more elastic than demand = consumer

• Effects of taxes

• Can distort economic activity

• Reduce goods producing/consuming

• Deadweight loss = efficiency lost when not in ideal equilibrium

Section 3: Government Spending

• Types
• Mandatory = by law, such as Soc Sec etc.
• Discretionary = Not required (defense, education)
• Gov Budget
• budget = spending plan
• Surplus = less spending
• Deficit = more spending
• Nat Debt = total owed money

Section 4 : Government Regulation

• Types

• Economic = pricing controls in industries

• Social = health, product safety

• Effects = can increase costs but also provide benefits like environmental protections

Section 5 Market Failure

• Markets fail (bad re allocation)
• Due to such examples such as externalities and information
• Government can intervene on it with interventions such as subsides, taxes and other regulations.
• Intervention can help but be bad