Convex Functions: Definitions and Theorems

Fonction Convexes: Définition

• A function f, in a subset I of real numbers, is convex if for any x, y in I and t between 0 and 1, f(tx + (1-t) y) ≤ t f(x) + (1-t) f(y).
• A function f, in a subset I of real numbers, is concave if for any x, y in I and t between 0 and 1, f(tx + (1-t) y) ≥ t f(x) + (1-t) f(y).
• A convex function's chord lies above its graph.

Fonction Convexes: Théorèmes

• For a derivable function f in an open interval I, f is convex if and only if f ' is increasing, and concave if and only if f ' is decreasing.
• For a twice-derivable function f in an open interval I, f is convex if and only if f ''(x) ≥ 0 for all x in I, and concave if and only if f ''(x) ≤ 0 for all x in I.
• For a convex function f in an open interval I, if f '(x₀) = 0, then x₀ is a global minimum of f; if f ''(x) > 0 for all x in I, then x₀ is the unique global minimum of f.
• In Jensen's Inequality, for a convex function f: I → ℝ, where I is an interval, for any x₁, ..., xₙ ∈ I and any λ₁, ..., λₙ ≥ 0, where Σᵢλᵢ = 1, f(Σᵢλᵢxᵢ) ≤ Σᵢλᵢf(xᵢ).
• In the special case of Jensen's Inequality, if λᵢ = 1/n for all i, f (1/n Σᵢxᵢ) ≤ 1/n Σᵢ f(xᵢ).

Analisi Matematica 1 - ING. INFORMAZIONE - 15 Gennaio 2018

• Calculate the limit: lim (n→+∞) (2n² + cos(n)) / (n² + n + 1) = 2
• Study the continuity of the function: f(x) = x² if x ∈ ℚ, 0 if x ∈ ℝ ∖ ℚ = Function is continuous only at x₀=0.
• Calculate the integral: ∫₀¹ x eˣ dx = 1
• Study the convergence of the series: Σ(n=1 to +∞) (n² + 1) / (n³ + n + 1) = Series diverges.

Chapter 2: The Chemistry of Life

2.1 The Elements of Life

• An element cannot be broken down into other substances by chemical reactions.
• A compound contains 2+ elements in a fixed ratio, with emergent properties.
• Essential elements are needed for an organism to maintain health and reproduce, they include carbon, oxygen, hydrogen, and nitrogen, which make up 96% of the body's mass.
• Trace elements are required in very small amounts.

2.2 Atoms and Molecules

• An atom is the smallest unit of matter, and is composed of neutrons, protons, and electrons.
• The atomic number is the number of protons in an atom's nucleus
• The mass number is the sum of protons and neutrons in an atom's nucleus.
• Isotopes are atoms of the same element that have different numbers of neutrons, creating different mass numbers.
• A valence shell is the outermost electron shell
• Chemical bonds are the attractive forces that hold atoms together.

Types of Chemical Bonds

• Covalent bond: Sharing of valence electrons between atoms
• Ionic bond: Transfer of valence electrons between atoms, creating ions
• Hydrogen bond: Weak attraction between a hydrogen atom bound to an electronegative atom and another electronegative atom.
• Van der Waals interactions: Weak attractions between molecules that result from transient local charges.

2.3 Chemical Reactions

• Chemical reactions create change in matter by making and breaking chemical bonds.
• Reactants become the products.
• Chemical equilibrium exists when forward & reverse reaction rates are equal, and reactants and products relative concentration remain constant.

2.4 Water and Life

• Water is essential for life dues to cohesion, adhesion, surface tension, high specific heat, heat of vaporization, evaporative cooling, density, and it being a good solvent.
• An acid increases the hydrogen ion ($H^+$) concentration in a solution.
• A base reduces the hydrogen ion concentration in a solution.
• pH measures the acidity or basicity of a solution from 0-14.
  • pH < 7: Acidic
  • pH = 7: Neutral
  • pH > 7: Basic
• Buffers minimize changes in pH.
• Ocean acidification is the process by which the pH of the ocean decreases due to the absorption of excess carbon dioxide ($CO_2$) from the atmosphere.

Prérequis pour la conception de systèmes à microprocesseurs

Chapitre 1: Introduction aux systèmes à microprocesseurs

• Von Neumann architecture uses a single address space for instructions and data, consisting of the CPU, ALU, control unit, memory, and I/O.
• Harvard architecture separates storage and signal paths for instructions and data with a program memory and a data memory.
• Microprocessors are complete central processing units (CPUs) produced on a single chip, involving the fetch, decode, execute and store cycle to process instructions.
• Factors influencing microprocessor selection include clock frequency, bus sizes, power consumption, core count, and cost.
• Microprocessor manufacturers include Intel, AMD, ARM, Qualcomm, and Samsung.
• Current trends include increasing core counts, AI integration, reduced power consumption, and enhanced security.

Chapitre 2: Architecture d'un microprocesseur typique

• The CPU comprises the ALU, registers (accumulator, program counter, status register, index register, stack pointer), and control unit.
• There is RAM (SRAM, DRAM) and ROM (PROM, EPROM, EEPROM, Flash Memory).
• The primary I/O components consist of I/O Ports (parallel, serial) and I/O Controllers (UART, SPI, I2C, USB).
• The bus consists of the address bus, the data bus and the control bus
• Common modes of addressing are immediate, direct, indirect, indexed, and relative.
• Types of instructions include data transfer (MOVE, LOAD, STORE), arithmetic (ADD, SUB, MUL, DIV), logical (AND, OR, NOT, XOR) instructions, also control flow (JUMP, BRANCH, CALL, RETURN), shift, etc.
• Interrupts can be hardware or software-generated.

Chapitre 3: Programmation en assembleur

• Assembler language is a low-level programming language which uses mnemonics to represent machin instructions.
• Key building blocks of assembler programs include directives, instructions, and comments.
• The syntax includes labels, mnemonics, operands, and comments.
• The instruction set includes data transfer (MOV, LEA, PUSH, POP), arithmetic (ADD, SUB, MUL, DIV, INC, DEC) Logical Instructions (AND, OR, XOR, NOT), comparison instructions and jump instructions.

Algorithmic Trading: Strategies and Their Characteristics

Trend Following

• This strategy aims to profit from sustained trends.
• Simple to implement, effective in steady markets.
• Common Indicators: Moving averages, trendlines, breakout levels.
• Pros: Potential for high profits in steady markets, easy implementation.
• Cons: Whipsaws in sideways markets, entry & exit delay.

Mean Reversion

• This technique capitalizes on tendency to revert to the average.
• More effective in range-bound markets.
• Common Indicators: Bollinger Bands, RSI, MACD.
• Pros: Profitable in sideways markets, generates frequent opportunities.
• Cons: Risk of losses if prices trend in one direction, requires tuning.

Arbitrage

• Exploits price disparities by simultaneous purchasing and selling of an asset in various forms.
• Spatial arbitrage and statistical arbitrage are the most common types.
• The strategy is low risk with consistent profits.
• Advanced tech & limited opportunities.

High-Frequency Trading (HFT)

• Executes many orders at extremely high speeds to profit from market inefficiencies.
• The most common techniques include market making and latency arbitrage.
• The activity is subject to regulation in a limited market.

Sentiment Analysis

• This tech extracts sentiment or emotion using the aid of NLP algorithms.
• Common indicators: Sentiment scores, sentiment indices.
• Can identify turning points though it is subjective.

Chart Pattern Recognition

• Uses historical data recognition algorithms to identify price patterns to make judgement.
• Common Patterns: Head and Shoulders, Double Tops and Bottoms, Triangles.
• It has a subjective interpretation.

1. Chemical context of life

1.1. A few chemical elements are essential for life

• 96% of life is comprised of carbon, hydrogen, oxygen, and nitrogen.
• The remaining 4% consist of phosphorus, sulfur, potassium, calcium and other elements.
• Trace elements are required by an organism in only minute quantities.

1.2. Atoms combine to form molecules

• Covalent bonds are the sharing of a pair of valence electrons by two atoms.
• If two or more atoms are held together by covalent bonds, they constitute a molecule.
• Electronegativity is the measure of the attraction of a particular atom for the electrons of a covalent bond.
• A nonpolar covalent bond is a covalent bond in which the electrons are shared equally between two atoms
• A polar covalent bond is a covalent bond between atoms that differ in electronegativity.
• Ionic bond: Formed through the attraction between oppositely charged ions (cations and anions).
• Hydrogen bond: The hydrogen atom of a polar covalent bond attracted to the negative atom of another molecule.
• Van der Waals interactions: Occur when transient local partial charges create attractions between molecules.

1.3. Chemical reactions make and break chemical bonds

• Chemical reactions involve the making and breaking of chemical bonds, leading to changes in the composition of matter.
• A reactant is a starting material in a chemical reaction, while a product is a resulting material.
• Chemical equilibrium: Rate of the forward reaction equals the rate of the reverse reaction.

Algorithmique Complexité

Introduction

• A measure of ressources to execute systematic procedures to resolve problems.

Modèles de Calcul

• Turing Machine
• Random Access Machine

Analyse de la Complexité

• Complexité Temporelle
• Complexité Spatiale
• Notations de Landau: O(f(n)), Ω(f(n)), Θ(f(n))

Classes de Complexité

• Classe P: Problèmes résolubles en temps polynomial.
• Classe NP: Problèmes dont la solution peut être vérifiée en temps polynomial.
• Problèmes NP-Complets
• Problèmes NP-Difficiles

Techniques de Conception d'Algorithmes

• Diviser pour Régner
• Programmation Dynamique
• Algorithmes Gloutons

Exemples d'Algorithmes et Complexité

• Tri: Tri à bulles O(n²), Tri par insertion O(n²), Tri par sélection O(n²), Tri fusion O(n log n), Tri rapide O(n log n).
• Recherche: Recherche linéaire O(n), Recherche binaire O(log n).
• Parcours de Graphes: Largeur d'abord O(V + E), Profondeur d'abord O(V + E).

TAREA 1

Ejercicio 1 - Lenguaje Natural

• (a): Existe un número real x tal que su cuadrado es igual a 2.
• (b): Para todo número real x, su cuadrado es mayor o igual a 0.
• (c): Existe un número natural x tal que para todo número natural y, x es menor o igual que y.
• (d): Para todo número real x, existe un número real y tal que la suma de x y y es igual a 0.
• (e): Para todo número real x, si x es mayor que 0, entonces existe un número real y tal que y es mayor que 0 y el cuadrado de y es igual a x.

Ejercicio 2 - Formalización

• (a): No existe un número natural cuyo cuadrado sea 2 ≡ ∼∃x∈ℕ:x²=2
• (b): Para todo número real existe un número natural mayor que él ≡ ∀x∈ℝ:∃n∈ℕ:n>x
• (c): Existen dos números naturales distintos cuya suma es 5 ≡ ∃x∈ℕ:∃y∈ℕ:(x≠y)^(x+y=5)
• (d): Para todo número natural existe otro número natural mayor que él ≡ ∀x∈ℕ:∃y∈ℕ:y>x
• (e): Existen dos números reales cuyo cuadrado suman cero, entonces ambos números son cero ≡ ∀x∈ℝ:∀y∈ℝ:(x²+y²=0)⇒(x=0^y=0)

Chapter 14: Oscillation

14.1 Simple Harmonic Motion

• Periodic motion repeats itself in regular intervals, while oscillation is repeating motion in time.
• Simple Harmonic Motion (SHM) features a restoring force proportional to displacement: F=-kx.
• Key characteristics of SHM include amplitude (A), period (T), frequency (f = 1/T), and angular frequency (ω = 2πf).
• Equation of Motion: x(t)=Acos(ωt+ϕ).
• Velocity: v(t)=−Aωsin(ωt+ϕ); v_max = Aω.
• Acceleration: a(t)=−ω^2x(t) , a_max = Aω^2.
• Energy in SHM: Potential U=12kx²=12kA²cos²(ωt+ϕ), Kinetic K=12mv²=12mA²ω²sin²(ωt+ϕ), Total E=12kA²=12mA²ω².

14.2 The Simple Pendulum

• The Angular Frequency is ω=squareroot(g/L) and the Period is T=2πsquareroot(L/g).

14.3 Damped Oscillations

• Occur with a frictional force that opposes the motion reducing amplitude: Fd=−bv, as well as: x(t)=Ae^(−bt/2m)cos(ω′t+ϕ).
• Various types of damping can occur, depending on the nature of friction (underdamped, critically damped, overdamped).

14.4 Forced Oscillations and Resonance

• Forced oscillations occur when a periodic force is applied, expressed by: F(t)=F0cos(ωdt).
• Resonance occurs when driving frequency is close to the natural frequency, resulting in the largets amplitude for these oscillations.