Double Integrals and Properties

Double Integrals Over Rectangles

• $\iint_R f(x, y) , dA = \lim_{m, n \to \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^, y_{ij}^) \Delta A$ defines the double integral of a function $f(x, y)$ over a rectangle $R$, where $\Delta A$ represents the area of subrectangles.
• If $f$ is continuous on rectangle $R = [a, b] \times [c, d]$, then $\iint_R f(x, y) , dA = \int_c^d \int_a^b f(x, y) , dx , dy = \int_a^b \int_c^d f(x, y) , dy , dx$.
• Example: $\iint_R (x - 3y^2) , dA$ over $R = [0, 2] \times [1, 2]$ evaluates to -12.

Properties of Double Integrals

• $\iint_R [f(x, y) + g(x, y)] , dA = \iint_R f(x, y) , dA + \iint_R g(x, y) , dA$ (Integral of sum)
• $\iint_R c f(x, y) , dA = c \iint_R f(x, y) , dA$; c is a constant (Constant multiple)
• If $f(x, y) \ge g(x, y)$ in $R$, then $\iint_R f(x, y) , dA \ge \iint_R g(x, y) , dA$ (Inequality)
• If $R = R_1 \cup R_2$ and the regions only share boundaries, $\iint_R f(x, y) , dA = \iint_{R_1} f(x, y) , dA + \iint_{R_2} f(x, y) , dA$ (Additivity)
• If $f(x, y) \ge 0$ in $R$, then $\iint_R f(x, y) , dA \ge 0$ (Non-negativity)

Double Integrals Over General Regions

Type I Region

• $D = {(x, y) | a \le x \le b, g_1(x) \le y \le g_2(x) }$ defines a Type I region with continuous functions $g_1$ and $g_2$.
• $\iint_D f(x, y) , dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) , dy , dx$ is the double integral over a Type I region.

Type II Region

• $D = {(x, y) | c \le y \le d, h_1(y) \le x \le h_2(y) }$ defines a Type II region with continuous functions $h_1$ and $h_2$.

• $\iint_D f(x, y) , dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x, y) , dx , dy$ is the double integral over a Type II region.

• Example: $\iint_D (x + 2y) , dA$ over $D$, bounded by $y = 2x^2$ and $y = 1 + x^2$, evaluates to $\frac{32}{15}$.

Análisis de Resultados

Objetivos

• The machine learning algorithm's ability to predict wine quality effectively will be evaluated.
• Identification of key features (or combinations thereof) instrumental in predictive accuracy.
• A comprehensive understanding of the data and inter-variable relationships will be achieved.

Métricas de Evaluación

• Percentage of accurate predictions out of total predictions determines accuracy: $\text{Accuratezza} = \frac{\text{Numero di previsioni corrette}}{\text{Totale delle previsioni}}$.
• Precisione: $\frac{\text{Vero Positivi}}{\text{Vero Positivi} + \text{Falso Positivi}}$ indicates the model's ability to avoid falsely labeling instances as positive.
• Recall: $\frac{\text{Vero Positivi}}{\text{Vero Positivi} + \text{Falso Negativi}}$ measures the model's ability to identify all positive instances.
• F1-Score: $2 \times \frac{\text{Precisione} \times \text{Recall}}{\text{Precisione} + \text{Recall}}$ is a harmonized mean for balancing precision and recall effectively.
• AUC-ROC gauges class differentiability on a scale where 1 is perfect, and 0.5 indicates randomness

Risultati Ottenuti

• Key features such as fixed and volatile acidity, and alcohol levels significantly impact wine quality prediction.
• Ensemble algorithms, including Random Forest and Gradient Boosting, tend to outperform linear algorithms like logistic regression in performance.
• Hyperparameter optimization can notably enhance a model's performance.

Analisi Dettagliata per Algoritmo

Regressione Logistica

• The accuracy is 0.70, Precision is 0.72, Recall is 0.68, F1-Score is 0.70, and AUC-ROC is 0.75.

Support Vector Machine (SVM)

• The accuracy is 0.75, Precision is 0.77, Recall is 0.73, F1-Score is 0.75, and AUC-ROC is 0.80.

Random Forest

• The accuracy is 0.85, Precision is 0.87, Recall is 0.83, F1-Score is 0.85, and AUC-ROC is 0.90.

Gradient Boosting

• The accuracy is 0.88, Precision is 0.90, Recall is 0.86, F1-Score is 0.88, and AUC-ROC is 0.92.

Importanza delle Features

• Key features include alcohol quantity (0.30), volatile acidity (0.25), density (0.15), fixed acidity (0.10), and residual sugar (0.08).

Chapitre 8: Moments d'une force par rapport à un point

8.1 Définition du moment d'une force par rapport à un point

8.1.1 Force dans le plan

• The moment $\vec{M}_A$ of a force $\vec{F}$ relative to a point A is the cross product of the vector $\vec{r}=\overrightarrow{AB}$ by the force $\vec{F}$: $\vec{M}_A = \vec{r} \times \vec{F}$.
• The module of the moment of the force $\vec{F}$ relative to A is $M_A = r F \sin \theta = F (r \sin \theta) = F d $, where $d$ is the shortest distance between A and the action line of $\vec{F}$.
• In the plane, the moment about A is defined by $\vec{M}_A = \pm F d \vec{k}$.
  • $\vec{M}_A$ is positive with the force $\vec{F}$ rotates around A in a counter-clockwise.
  • $\vec{M}_A$ is negative when $\vec{F}$ rotates around A in a clockwise.

8.1.2 Force dans l'espace

• The moment of the force $\vec{F}$ relative to point A is defined by $\vec{M}_A = \vec{r} \times \vec{F}$.
• If $\vec{r} = x \vec{i} + y \vec{j} + z \vec{k}$ and $\vec{F} = F_x \vec{i} + F_y \vec{j} + F_z \vec{k}$, then the moment of the force $\vec{F}$ relative to point A is $\vec{M}_A = (yF_z - zF_y)\vec{i} + (zF_x - xF_z)\vec{j} + (xF_y - yF_x)\vec{k} $.

8.1.3 Théorème de Varignon

• The moment relative to a given point A of the resultant of several concurrent forces equals the sum of the moments of the various forces relative to the same point A: $\vec{M}_A = \sum (\vec{r} \times \vec{F}) = \vec{r} \times \sum \vec{F} $.

Algorithmic Game Theory - Summer Term 2023

Organizational Information

• Course lecturer: Prof. Dr. Martin Hoefer; exercises by Julian Würschmidt.
• Course consists of lectures and exercises.
• Credit points are obtained by passing the exam at the end of the course.
• Course content includes: basics of game theory, congestion games, potential games, mechanism design, and auctions.

Game Theory - Basic Definitions

Definition: Strategic Game

• A strategic game is a triple $(N, A, u)$ involving a finite set of players $N$, actions $A_i$ for each player, and a payoff function $u_i$ for each player.

Examples

• The Prisoner's Dilemma exemplifies a strategic game where two players choose to cooperate or defect, affecting each other's payoffs.
• Bach or Stravinsky (BoS) is a coordination game where players choose between two options but have different preferences.
• Matching Pennies exemplifies no Nash equilibrium in pure strategies
• Rock-Paper-Scissors exemplifies cyclical preferences and strategic decision-making.

Partial Differential Equations

Introduction

What are PDEs?

• Involve unknown functions of two or more variables and their partial derivatives.
• They are generally expressed by the form $F(x, y, u, u_x, u_y, u_{xx}, u_{xy}, u_{yy},...) = 0$

Examples of PDEs

• Heat Equation: $u_t = ku_{xx}$
• Wave Equation: $u_{tt} = c^2u_{xx}$
• Laplace's Equation: $u_{xx} + u_{yy} = 0$

Solving PDEs

Analytical and numerical methods are used to solve PDEs.

The Heat Equation

Derivation

• The amount of heat flowing through a cross-section per unit time is given by $\phi(x, t) = -K_0 \frac{\partial u}{\partial x}(x, t)$ (Fourier's Law of Heat Conduction).
• The 1D Heat Equation is given by $u_t = ku_{xx}$.

Initial and Boundary Conditions

• Initial Condition: $u(x, 0) = f(x)$
• Boundary Conditions:
  • Dirichlet BC: $u(0, t) = T_1$, $u(L, t) = T_2$
  • Neumann BC: $u_x(0, t) = 0$, $u_x(L, t) = 0$
  • Robin BC: $u_x(0, t) + h u(0, t) = 0$, $u_x(L, t) + h u(L, t) = 0$

Example: Heat Equation with Dirichlet Boundary Conditions

• The general solution is $u(x, t) = \sum_{n=1}^{\infty} \left[ \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \right] e^{-k\left(\frac{n\pi}{L}\right)^2 t} \sin\left(\frac{n\pi x}{L}\right)$.

Algorithmic Trading

What is Algorithmic Trading?

• Algorithmic trading employs computer programs following defined instructions to place trades based on Time, Price, and Quantity.
• Primary benefits include minimizing emotion, executing at optimal prices, instantaneous order execution, monitoring market conditions, and backtesting.

Algorithmic Trading Strategies

Trend Following Strategies

• Moving Averages, Channel Breakouts, Price Level Movements, Related Assets.

Arbitrage Opportunities

• Index Fund Rebalancing, Mathematical Models, Mean Reversion, Delta-Neutral Strategies.
• Mathematical Model Example: $S = P \cdot e^{rT}$

Implementation

• Requires a computer, internet connection, and access to a trading platform using languages like Python.
• There are useful online courses, like Quantopian and books.

The Future of Algorithmic Trading

• Includes AI and ML.
• Increased accessibility and Market Efficiency is likely.
• Warning: Not a guaranteed path to profits!

Matrices

Définition

• Rectangular array of real numbers.
• A matrix with $m$ rows and $n$ columns is a $m \times n$ matrix.
• Coefficients are elements of the matrix, located at ($i$,$j$)

Vocabulaire

• A matrix where $m = n$ is said to be square.
• A matrix with a single column is a column matrix.
• A matrix with a single row is a row matrix.
• The main diagonal of a square matrix is the diagonal formed from the coefficients
• A matrix diagonal is a square matrix whose elements are all zeros, except those of the main diagonal
• The matrix identity is a matrix diagonal of order n whose all coefficient of its main diagonal values.
• A triangular superior is a matrix square where matrix square whose elements located under the main diagonal are zero.
• A inferior triangular is a matrix square whose elements located over the main diagonal are zero.
• The transposed of a matrix $A$ is the matrix written $A^T$ obtained by interchanging the rows and columns of $A$ .

Opérations sur les matrices

• Addition: $A + B = (a_{ij} + b_{ij})$.
• Multiplication by scalar: $kA = (ka_{ij})$.
• Multiplication of matrix: $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj}$.

Propriétés

• Addition is commutative and associative.
• Multiplication by a scalar is distributive and matrix multiplication is associative.
• Matrix multiplication distributes over addition, but is generally non-commutative
• $AI = IA = A$

Lecture 41: The Endocrine System

Communication

• The nervous system and endocrine systems facilitate communication.

Nervous System

• Fast, electrical signals that are short-lived.

Endocrine System

• Slower, chemical signals (hormones) that last longer.

Endocrine Glands

• Pituitary gland, thyroid gland, parathyroid gland, adrenal glands, pineal gland, thymus, pancreas, ovaries, and testes.

Hormones

• Amino acid-based hormones: water-soluble, unable to cross plasma membrane, using a second messenger system.
• Steroid hormones: lipid-soluble, able to cross plasma membrane, binding to intracellular receptors.

Control of Hormone Release

• Regulated by humoral, neural, and hormonal stimuli. Hormone levels are maintained via negative feedback.

Pituitary Gland

• Located in the brain with anterior (glandular) and posterior (nervous tissue) lobes.

Pituitary Hormones

• Anterior pituitary produces growth hormone (GH), thyroid-stimulating hormone (TSH), adrenocorticotropic hormone (ACTH), follicle-stimulating hormone (FSH), luteinizing hormone (LH), Prolactin (PRL), and melanocyte-stimulating hormone (MSH).
• Oxytocin Posterior pituitary releases oxytocin and antidiuretic hormone (ADH).

Thyroid Gland

• Located in the neck. Produces tetraiodothyronine (T4), triiodothyronine (T3) and thyroid hormones, which control metabolism, and lowering blood calcium with lowering bloods levels, calitonin.

Parathyroid Glands

• Located on thyroid gland.
• Produces parathyroid hormone to increase (PTH) blood calcium levels.

Adrenal Glands

• Located above kidneys.
• Consists of corticosteroids outer (aldosterone-mineralocorticoid, cortisol-glucocorticoid, androgens-gonadocorticoid) region with a production, and inner adrenal medulla region (epinephrine and norepinephrine).

Pancreas

• Located in abdomen
• Secret Insulin (decrease of ) and glucose levels with glucagon in endocrine part.

Pineal Gland

• Located in brain and regulates melatonin.

Thymus

• Located in the chest.
• Regultes thymosin which T cell stimulated for the devolpoments.

Ovaries & Testes

• Located in the pelvic
• Regultes female Ovaries which devlop estrogen and female secondary and menses characteristics with progesterone
• Regultes males Testes and produce testosterone which develops characteristics for it and their secondary..

Statistiques

Nature de la statistique

• Descriptive: Describe, present, summarize information.
• Inféréntielle: Generalize, anticipate, make decisions.

Vocabulaire de base

• Population: All of the individuals or objects of interest.
• Échantillon: All of the under assembly of the population.
• Variable: Characteristics measured either observed on every individual or in the population.
• Données : Data set

Types de variables

• Qualitative:
  • Nominale: No order.
  • Ordinale: With order .
• Quantitative:
  • Discrète: Integer
  • Continue: decimal

Représentation des données

• Tabular*

• Qualitative: With number frequency and percentage Quantitative discrète: with number frequency and percentage

• Graphiques*

• Diagrammes- Diagramme à secteurs( pies), diagramme à bâtons

• Quantitative - Histogramme*

Mesures statistiques descriptives

• Tendance centrale - Mean
• Dispertion- Range
• Position-Quantiles
• Form- Cooefficient of skew

Regulation

Principles

• The objectives of financial regulation include financial stability, consumer protection, market integrity, and competition.
• Regulation can treat different customers differently, with retail clients receiving more protection than professional ones.
• Regulation can be principles-based, relying on judgment, or rules-based, focusing on detailed compliance.

UK Regulatory Bodies

Financial Policy Committee (FPC)

• The Financial Policy Committeeis one key component of Bank of Enghland to ensure finacial stability, tools are such setting Loan to Value* macro-prudential regulation. Loan to Value limits on mortgages.*

Prudential Regulation Authority (PRA)

• The Prudential Regulation Authority is a part of Bank of Enghland, whose objective is to increase the stability of financial stability where they setting standards for those firms.

Financial Conduct Authority (FCA)

• The Financial Conduct Authority is ensuring that objectives are reaching to customers (protection and intergity).