Mineral Electromagnetic Properties

Electromagnetic Properties of Minerals

• Electromagnetic properties determine how minerals interact with electric and magnetic fields, and electromagnetic radiation.
• They are useful for mineral identification and understanding mineral behavior.
• These properties include di- and para-magnetism, ferro-, ferri- and anti-ferromagnetism, piezoelectricity, pyroelectricity, ferroelectricity, electrical conductivity, and dielectric properties.

Magnetism

• Electrons behave as tiny magnets due to their orbital motion and spin.
• Electron spin is the stronger magnetic effect.
• Two electrons in the same orbital have opposite spins, canceling their magnetic moments.
• Atoms with unpaired electrons can have a net magnetic moment.
• Macroscopic magnetic properties depend on how these moments interact.

Diamagnetism

• Diamagnetism results from the repelling effect of a magnetic field on the electron clouds of atoms/ions.
• It is present in all materials, but very weak.
• Diamagnetism is observed only if stronger forms of magnetism are absent.

Paramagnetism

• Paramagnetism occurs in materials with unpaired electrons that do not interact.
• Magnetic moments of unpaired electrons align parallel to an external magnetic field, enhancing it.
• Alignment is lost when the field is removed due to thermal agitation.

Ferromagnetism

• In ferromagnets, unpaired electrons interact, aligning parallel even without an external field within magnetic domains.
• Applying an external field causes domains aligned with the field to grow.
• Some alignment is retained when the field is removed, resulting in a permanent magnet.

Ferrimagnetism

• In ferrimagnetism, magnetic moments align anti-parallel, but are unequal, resulting in a net magnetic moment.

Anti-ferromagnetism

• Anti-ferromagnetism is similar to ferrimagnetism, but magnetic moments are equal and opposite, resulting in no net magnetic moment.

Magnetism Summary

Electrical Properties

Piezoelectricity

• Piezoelectricity is the ability of some minerals to generate an electrical potential when subjected to mechanical stress.
• Ion displacement in the crystal structure creates a dipole moment.
• Applying an electric field causes the crystal to deform, the effect is reversible.
• Examples: Quartz, Tourmaline.

Pyroelectricity

• Pyroelectricity is the ability of some minerals to generate an electrical potential when heated or cooled.
• Temperature change alters atom positions in the crystal structure, creates a dipole moment.
• Pyroelectric minerals have a polar axis and are also piezoelectric.
• Example: Tourmaline.

Ferroelectricity

• Ferroelectricity is the ability of some minerals to exhibit a spontaneous electric polarization.
• This polarization is reversible by applying an external electric field.
• Ferroelectric minerals are polar, pyroelectric, and piezoelectric.
• Example: Barium Titanate.

Electrical Conductivity

• Electrical conductivity is the ability of a material to conduct electricity.
• Conductivity depends on mobile charge carriers and their ease of movement.
  • Metals: high conductivity due to free electrons.
  • Semiconductors: intermediate conductivity, controlled by temperature and impurities.
  • Insulators: low conductivity due to lack of free charge carriers.

Dielectric Properties

• Dielectric properties describe how a material stores electrical energy when subjected to an electric field.
• Material polarization reduces the electric field inside the material.
• The dielectric constant measures how much a material reduces the electric field compared to a vacuum.

Vectors in $\mathbb{R}^n$

• An n-dimensional vector is an ordered n-tuple of real numbers $\mathbf{x} = (x_1, x_2,...,x_n)$, where $x_i \in \mathbb{R}$ for $i = 1, 2,..., n$.
• $\mathbb{R}^n$ is the set of all n-dimensional vectors.

Vector Operations

• Addition: $\mathbf{x} + \mathbf{y} = (x_1 + y_1, x_2 + y_2,..., x_n + y_n)$
• Scalar Multiplication: $\alpha \mathbf{x} = (\alpha x_1, \alpha x_2,..., \alpha x_n)$, where $\alpha \in \mathbb{R}$

Vector Properties

• For all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^n$ and $\alpha, \beta \in \mathbb{R}$:
  • $\mathbf{x} + \mathbf{y} = \mathbf{y} + \mathbf{x}$ (Commutativity)
  • $(\mathbf{x} + \mathbf{y}) + \mathbf{z} = \mathbf{x} + (\mathbf{y} + \mathbf{z})$ (Associativity)
  • $\exists \mathbf{0} \in \mathbb{R}^n$ such that $\mathbf{x} + \mathbf{0} = \mathbf{x}$ (Additive Identity)
  • $\exists -\mathbf{x} \in \mathbb{R}^n$ such that $\mathbf{x} + (-\mathbf{x}) = \mathbf{0}$ (Additive Inverse)
  • $\alpha (\mathbf{x} + \mathbf{y}) = \alpha \mathbf{x} + \alpha \mathbf{y}$ (Distributivity)
  • $(\alpha + \beta) \mathbf{x} = \alpha \mathbf{x} + \beta \mathbf{x}$ (Distributivity)
  • $\alpha (\beta \mathbf{x}) = (\alpha \beta) \mathbf{x}$ (Associativity)
  • $1 \mathbf{x} = \mathbf{x}$ (Multiplicative Identity)

Dot Product

• For $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, the dot product is defined as: $\mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 +... + x_n y_n$

Dot Product Properties

• $\mathbf{x} \cdot \mathbf{y} = \mathbf{y} \cdot \mathbf{x}$ (Commutativity)
• $\mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z}$ (Distributivity)
• $(\alpha \mathbf{x}) \cdot \mathbf{y} = \alpha (\mathbf{x} \cdot \mathbf{y})$ (Associativity)
• $\mathbf{x} \cdot \mathbf{x} \geq 0$, and $\mathbf{x} \cdot \mathbf{x} = 0$ if and only if $\mathbf{x} = \mathbf{0}$ (Positive Definiteness)

Euclidean Norm

• The Euclidean norm (or magnitude) of a vector $\mathbf{x} \in \mathbb{R}^n$ is defined as: $||\mathbf{x}|| = \sqrt{\mathbf{x} \cdot \mathbf{x}} = \sqrt{\sum_{i=1}^{n} x_i^2}$

Euclidean Norm Properties

• $||\mathbf{x}|| \geq 0$, and $||\mathbf{x}|| = 0$ if and only if $\mathbf{x} = \mathbf{0}$
• $||\alpha \mathbf{x}|| = |\alpha| ||\mathbf{x}||$
• $|\mathbf{x} \cdot \mathbf{y}| \leq ||\mathbf{x}|| \ ||\mathbf{y}||$ (Cauchy-Schwarz Inequality)
• $||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}||$ (Triangle Inequality)

Cross Product (in $\mathbb{R}^3$)

• For $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$, the cross product is defined as: $\mathbf{x} \times \mathbf{y} = (x_2 y_3 - x_3 y_2, x_3 y_1 - x_1 y_3, x_1 y_2 - x_2 y_1)$

Cross Product Properties

• $\mathbf{x} \times \mathbf{y} = - (\mathbf{y} \times \mathbf{x})$ (Anti-commutativity)
• $\mathbf{x} \times (\mathbf{y} + \mathbf{z}) = \mathbf{x} \times \mathbf{y} + \mathbf{x} \times \mathbf{z}$ (Distributivity)
• $(\alpha \mathbf{x}) \times \mathbf{y} = \alpha (\mathbf{x} \times \mathbf{y})$ (Associativity)
• $\mathbf{x} \times \mathbf{x} = \mathbf{0}$
• $||\mathbf{x} \times \mathbf{y}|| = ||\mathbf{x}|| \ ||\mathbf{y}|| \sin \theta$, where $\theta$ is the angle between $\mathbf{x}$ and $\mathbf{y}$.
• $\mathbf{x} \cdot (\mathbf{x} \times \mathbf{y}) = 0$ and $\mathbf{y} \cdot (\mathbf{x} \times \mathbf{y}) = 0$ (Orthogonality)

Lines

• A line in $\mathbb{R}^n$ can be described by: $\mathbf{r}(t) = \mathbf{a} + t\mathbf{v}$
  • $\mathbf{a}$ is a point on the line.
  • $\mathbf{v}$ is the direction vector of the line.
  • $t \in \mathbb{R}$ is a parameter.

Planes

• A plane in $\mathbb{R}^3$ can be described by: $\mathbf{n} \cdot (\mathbf{x} - \mathbf{a}) = 0$
  • $\mathbf{n}$ is the normal vector to the plane.
  • $\mathbf{a}$ is a point on the plane.
  • $\mathbf{x}$ is any point on the plane.
• Alternatively, the equation of a plane can be written as: $ax + by + cz = d$ where $(a, b, c)$ is the normal vector to the plane.

Choosing the Right Chart

• Different chart types serve different use cases.
• Bar/Column charts compare numerical values across categories.
• Line charts show trends and changes in data over time.
• Pie charts display proportions of a whole.
• Scatter plots illustrate relationships between two variables.
• Histograms show the distribution of a single variable.
• Area charts highlight the magnitude of change over time.
• Bubble charts display relationships between three or more variables.
• Box plots summarize data distributions with outliers.
• Heat maps visualize the magnitude of a phenomenon as color.
• Word clouds represent the frequency of words in a text.
• Treemaps display hierarchical data as nested rectangles.
• Network graphs illustrate relationships between entities.
• Choropleth maps visualize data distributions across geographic regions.
• Radar charts compare multiple quantitative variables.
• Funnel charts illustrate the stages in a process.
• Candlestick charts describe price movements over time.
• Gantt charts illustrate project schedules and timelines.

Data Visualization Principles

• Clarity: Ensure visualizations are easy to understand and simple.
• Accuracy: Ensure that data is represented truthfully without distortion.
• Efficiency: Convey information effectively and with minimal clutter.
• Aesthetics: Create visually appealing and engaging graphics.

Common Mistakes to Avoid

• Misleading scales can distort data, start axes at zero unless there's a reason to do otherwise.
• Too much information can clutter visualizations, include only necessary details.
• Poor color choices can confuse viewers, use color intentionally and be mindful of color blindness.
• Inconsistent design can create a jarring effect, using a constant look and feel across visualizations.

Tools for Data Visualization

• List of tools: Tableau, Python (Matplotlib, Seaborn, Plotly), R (ggplot2), D3.js, Google Charts, Excel

Visualisation elements

• Title - a clear, descriptive title for the chart
• Axes labels - to explain the x and y axis
• Data labels - on important data points
• Legend - explains the symbols and colours used
• Gridlines - to aid reading values

Color Palettes

• Sequential: Progresses from low to high
• Diverging: Data has a mid-point
• Categorical: Distinct categories with order

Data Preprocessing Steps

• Data Collection: Gather data from various sources.
• Data Cleaning: Handle missing values, outliers, and inconsistencies.
• Data Transformation: Scale, normalize, or aggregate data as needed.
• Data Reduction: Reduce the size of the dataset using techniques such as PCA.

Advanced Visualization Techniques

• Interactive Dashboards: Create dynamic dashboards that allow users to explore data.
• Animation: Use animation to show changes in data over time.
• Geospatial Visualization: Overlay data on maps to reveal spatial patterns.
• Network Analysis: Visualize relationships and connections between entities.

Data Storytelling

• Narrative: Craft a compelling narrative around your data.
• Context: Provide relevant context to help your audience understand the data.
• Visual Cues: Use visual cues to guide the audience's attention.
• Engagement: Engage your audience with interactive elements and storytelling techniques.

Fundamental Theorem of Calculus Part 1

• If $g(x) = \int_{a}^{x} f(t) dt$, then $g'(x) = f(x)$.
• The derivative of an integral is the original function.

Fundamental Theorem of Calculus Part 2

• $\int_{a}^{b} F'(x) dx = F(b) - F(a)$
• The integral of a derivative is the original function evaluated at the bounds.
• To evaluate a definite integral: Find the antiderivative, plug in the upper bound, plug in the lower bound, and subtract.

Derivatives of Integrals - Examples

• Find the derivative of $g(x) = \int_{1}^{x} (t^2 + 1) dt$
  • $g'(x) = x^2 + 1$
• Find the derivative of $g(x) = \int_{1}^{x^3} (t^2 + 1) dt$
  • $g'(x) = ((x^3)^2 + 1) \cdot 3x^2 = (x^6 + 1) \cdot 3x^2 = 3x^8 + 3x^2$
• Find the derivative of $g(x) = \int_{x^2}^{x^3} (t^2 + 1) dt$
  • Use properties of integrals to write the integral as $g(x) = \int_{x^2}^{0} (t^2 + 1) dt + \int_{0}^{x^3} (t^2 + 1) dt$ .
  • Rewrite as $g(x) = -\int_{0}^{x^2} (t^2 + 1) dt + \int_{0}^{x^3} (t^2 + 1) dt$.
  • Then $g'(x) = -(x^4 + 1) \cdot 2x + (x^6 + 1) \cdot 3x^2 = -2x^5 - 2x + 3x^8 + 3x^2 = 3x^8 - 2x^5 + 3x^2 - 2x$

Algorithmic Game Theory

Game Theory Topics

• Routing: Price of Anarchy is $\leq \frac{4}{3}$
• Mechanism Design without money

Todays Game Theory Topic

• Mechanism Design with money (Auctions)

Single Item Auctions

• One seller, one item
• $n$ bidders. Bidder $i$ has valuation $v_i$
• Sealed bid auctions:
  • Each bidder submits bid $b_i$ to auctioneer
  • Based on bids $(b_1, \dots, b_n)$, auctioneer determines:
    • Allocation rule: Which bidder gets the item?
      • $x_i(b_1, \dots, b_n) \in {0, 1}$
      • $\sum_i x_i(b_1, \dots, b_n) \leq 1$
    • Payment rule: How much does each bidder pay?
      • $p_i(b_1, \dots, b_n) \geq 0$

Auction Goals

• Efficiency: Allocate item to bidder with highest valuation.
  • Maximize $\sum_i v_i x_i(b_1, \dots, b_n)$
• Revenue Maximization: Seller wants to maximize revenue
  • Maximize $\sum_i p_i(b_1, \dots, b_n)$
• Truthfulness: Bidders should bid their true valuation
  • Bidding truthfully should maximize bidder's utility

Auction Truthfulness

• Quasilinear utility: $v_i x_i - p_i$
• A mechanism is truthful or incentive compatible if for every bidder $i$ and every possible set of bids by other bidders $b_{-i}$, bidding truthfully $b_i = v_i$ maximizes bidder i's utility.
  • $v_i x_i(v_i, b_{-i}) - p_i(v_i, b_{-i}) \geq v_i x_i(b_i, b_{-i}) - p_i(b_i, b_{-i}) ; \forall b_i$

List of Common Actions

• First Price Auction:
  • Allocation rule: Item goes to highest bidder
  • Payment rule: Winner pays their bid
• Second Price Auction:
  • Allocation rule: Item goes to highest bidder
  • Payment rule: Winner pays second highest bid
• All Pay Auction:
  • Allocation rule: Item goes to highest bidder
  • Payment rule: Everyone pays their bid

Which Actions are Truthful?

• First Price Auction: Not truthful. Bidding true valuation is not a Nash Equilibrium.
• Second Price Auction: Truthful!
• All Pay Auction: Not Truthful.

Why Second Price Auction is Truthful

• Vickrey Auction
• Theorem: Second Price Auction is truthful
  • Proof:
    • Fix $i$ and $b_{-i}$. Let $b' = \max_{j \neq i} b_j$.
    • Case 1: $v_i > b'$
      • If $b_i > b'$, then $i$ wins and pays $b'$. Utility $= v_i - b' > 0$
      • If $b_i < b'$, then $i$ loses and pays 0. Utility $= 0$
      • So bidding $b_i > b'$ is optimal
    • Case 2: $v_i < b'$
      • If $b_i > b'$, then $i$ wins and pays $b'$. Utility $= v_i - b' < 0$
      • If $b_i < b'$, then $i$ loses and pays 0. Utility $= 0$
      • So bidding $b_i < b'$ is optimal
    • Case 3: $v_i = b'$
      • $i$ is indifferent between winning and losing
    • So bidding $b_i = v_i$ is optimal!

Auctions Theorems

• Theorem: There is no truthful auction that maximizes revenue.
  • Proof:
    • Consider single bidder. Want to sell item to bidder if $v_i \geq x$.
    • If auction is truthful, then bidder with valuation $x$ must have utility 0.
    • So they must pay $x$ if they win.
    • But then seller gets 0 revenue if $v_i < x$.

Auction - VCG

• Vickrey-Clarke-Groves (VCG) Auction
  • Mechanism for general setting where we want to maximize social welfare.
    • $n$ bidders
    • Set of possible outcomes $O$.
    • Bidder $i$ has valuation $v_i(o)$ for outcome $o \in O$
    • Mechanism asks bidders to submit bids $b_i(o)$ for each outcome $o \in O$
    • Allocation Rule: Choose outcome $o^*$ that maximizes $\sum_i b_i(o)$
      • $o^* = \arg \max_{o \in O} \sum_i b_i(o)$
    • Payment Rule: Bidder $i$ pays
      • $p_i = \max_{o \in O} \sum_{j \neq i} b_j(o) - \sum_{j \neq i} b_j(o^*)$

VCG

• Bidder $i$'s utility is
  • $v_i(o^) - p_i = v_i(o^) + \sum_{j \neq i} b_j(o^*) - \max_{o \in O} \sum_{j \neq i} b_j(o)$
• Bidding truthfully means $b_i(o) = v_i(o)$. So bidder $i$'s utility is:
  • $v_i(o^) + \sum_{j \neq i} v_j(o^) - \max_{o \in O} \sum_{j \neq i} v_j(o)$
• But $o^*$ maximizes $\sum_i v_i(o) = v_i(o) + \sum_{j \neq i} v_j(o)$.
• So bidder $i$ cannot increase their utility by misreporting their valuation!

VCG Examples

• Single item auction:
  • Outcome is who gets the item
  • $v_i(o) = v_i$ if $i$ gets the item, 0 otherwise
  • VCG auction is second price auction!
• Combinatorial Auction:
  • $m$ items for sale
  • Bidder $i$ wants a subset $S_i$ of items
  • $v_i(o) = v_i$ if $i$ gets $S_i$, 0 otherwise
  • VCG auction is computationally hard!

VCG Limitations

• Not always revenue maximizing
• Not budget balanced:
  • $\sum_i p_i$ can be negative!
  • Auctioneer has to pay bidders!
• Susceptible to collusion:
  • Bidders can collude to lower their payments

Mechanism Design without Complete Knowledge

• Bayesian Mechanism Design
  • Each bidder has a prior distribution over their possible valuations
  • Mechanism designer knows these prior distributions
  • Goal: Design a mechanism that maximizes expected revenue
• Prior-Independent Mechanism Design
  • Mechanism designer does not know the prior distributions
  • Goal: Design a mechanism that performs well regardless of the prior distributions
• Learning-Based Mechanism Design
  • Mechanism designer learns the prior distributions from data
  • Goal: Design a mechanism that maximizes expected revenue based on the learned prior distributions

Linear Functions (Chapitre 9)

General Information

• Definition 1: A function $f: E \rightarrow F$ is linear if:
  • $\forall x, y \in E, f(x+y) = f(x)+f(y)$
  • $\forall \lambda \in \mathbb{K}, \forall x \in E, f(\lambda x) = \lambda f(x)$

Linear Function terms

• Morphism of vector spaces = linear Function
• Endomorphism = If $E=F$,
  • Note $\mathscr{L}(E)$ is the set of endomorphisms of $E$.
• Isomorphism = If $f$ is bijective.
• Automorphism = Endomorphism is bijective.
  • The set of automorphisms of $E$ is noted $GL(E)$.
• Proposition 1: $f$ is linear $\Longleftrightarrow \forall \lambda, \mu \in \mathbb{K}, \forall x, y \in E, f(\lambda x + \mu y) = \lambda f(x) + \mu f(y)$.

Examples of Linear Functions

• The null function $f: E \rightarrow F$ defined by $\forall x \in E, f(x) = 0_{F}$ is linear.
• Soit $E$ un $\mathbb{K}$-ev. The function $I d_{E}: E \rightarrow E$ defined by $\forall x \in E, I d_{E}(x) = x$ is linear.
• Soit $a \in \mathbb{K}$. The function $f: E \rightarrow E$ defined by $\forall x \in E, f(x) = ax$ is linear.
• Soit $E = \mathbb{K}[X]$ et $F = \mathbb{K}[X]$. The function $f: E \rightarrow F$ defined by $f(P) = P'$ is linear.
• Soit $E = \mathscr{C}^{1}([a,b], \mathbb{R})$ et $F = \mathscr{C}^{0}([a,b], \mathbb{R})$. The function $f: E \rightarrow F$ defined by $f(g) = g'$ is linear.
• Soit $E = \mathscr{C}^{0}([a,b], \mathbb{R})$ et $F = \mathbb{R}$. The function $f: E \rightarrow \mathbb{R}$ defined by $f(g) = \int_{a}^{b} g(t) dt$ is linear.
• Soit $E$ un $\mathbb{K}$-ev et $u \in E$. The function $f: \mathbb{K} \rightarrow E$ defined by $f(\lambda) = \lambda u$ is linear.

Operations on functions

• Proposition 2: Soient $E, F$ deux $\mathbb{K}$-ev et $f, g \in \mathscr{L}(E,F)$. Alors :
  • $\forall \lambda \in \mathbb{K}, f+g \in \mathscr{L}(E,F)$
  • $\lambda f \in \mathscr{L}(E,F)$
• $(\mathscr{L}(E,F), +,.)$ is a $\mathbb{K}$-ev.
• Proposition 3: Soient $E, F, G$ trois $\mathbb{K}$-ev et $f \in \mathscr{L}(E,F), g \in \mathscr{L}(F,G)$. Alors $g \circ f \in \mathscr{L}(E,G)$.

Image and Kernel

• Definition 2: Soit $f \in \mathscr{L}(E,F)$
  • The image of $f$ is the set $\operatorname{Im}(f) = {y \in F / \exists x \in E, f(x) = y } = { f(x) / x \in E } = f(E)$.
  • The kernel of $f$ is the set $\operatorname{Ker}(f) = { x \in E / f(x) = 0_{F} } = f^{-1}({0_{F}})$.
• Proposition 4: Soit $f \in \mathscr{L}(E,F)$.
  • $\operatorname{Im}(f)$ is a sev of $F$.
  • $\operatorname{Ker}(f)$ is a sev of $E$.

Propositions around Linear Functions

• Proposition 5: Soit $f \in \mathscr{L}(E,F)$.
  • $f$ injective $\Leftrightarrow \operatorname{Ker}(f) = {0_{E}}$.
• Proposition 6: Soit $f \in \mathscr{L}(E,F)$.
  • Si $E$ is of limited dimension, then $\operatorname{Im}(f)$ is of limited dimension and $\operatorname{dim}(\operatorname{Im}(f)) \leq \operatorname{dim}(E)$.
  • Si $F$ is of limited dimension, then $\operatorname{Ker}(f)$ is of limited dimension and $\operatorname{dim}(\operatorname{Ker}(f)) \leq \operatorname{dim}(F)$.

Linear Functions Theorem

• Soit $f \in \mathscr{L}(E,F)$ with $E$ of limited dimension.
• Then $\operatorname{dim}(E) = \operatorname{dim}(\operatorname{Ker}(f)) + \operatorname{dim}(\operatorname{Im}(f))$

Projections and symetries

Projector definition

• Soit $p \in \mathscr{L}(E)$. On dit que $p$ est un projecteur si $p \circ p = p$.
  • (i.e. $p^{2} = p$).
• Soit $p \in \mathscr{L}(E)$ tel que $p^{2} = p$. Alors :
  • $E = \operatorname{Im}(p) \oplus \operatorname{Ker}(p)$
  • $p$ est la projection sur $\operatorname{Im}(p)$ parallèlement à $\operatorname{Ker}(p)$.

Symetries definition

• Soit $s \in \mathscr{L}(E)$. On dit que $s$ est une symétrie si $s \circ s = I d_{E}$.
  • (i.e. $s^{2} = I d_{E}$).
  • Soit $s \in \mathscr{L}(E)$ tel que $s^{2} = I d_{E}$. Alors :
  • $E = \operatorname{Ker}(s - I d_{E}) \oplus \operatorname{Ker}(s + I d_{E})$
• $s$ est la symétrie par rapport à $\operatorname{Ker}(s - I d_{E})$ parallèlement à $\operatorname{Ker}(s + I d_{E})$.

Linear Form

• On appelle forme linéaire sur $E$ toute application linéaire de $E$ dans $\mathbb{K}$.
  • The set of linear forms of $E$ is noted $E^{*}$ and called dual of $E$.
• $E^{*}$ is a $\mathbb{K}$-ev.
• Soit $E$ un $\mathbb{K}$-ev of limited dimension.
  • Then $E$ et $E^{}$ sont isomorphes et $\operatorname{dim}(E) = \operatorname{dim}(E^{})$.
• Si $E$ is of illimited dimension, $E$ and $E^{*}$ ne are not isomorphes

Hyperplanes

• Soit $E$ un $\mathbb{K}$-ev.
  • On appelle hyperplan de $E$ tout sev de $E$ of dimension $dim(E) - 1$.
• Soit $E$ un $\mathbb{K}$-ev of limited dimension and $H$ un sev de $E$.
  • $H$ est un hyperplan de $E \Longleftrightarrow \exists f \in \mathscr{L}(E, \mathbb{K})$ telle que $H = \operatorname{Ker}(f)$.
• $H$ est le noyau d'une forme linéaire.
• Soit $H$ un hyperplan de $E$. Alors $H$ est un sev maximal de $E$, c'est-à-dire que si $H \subset F \subset E$ avec $F$ un sev de $E$, alors $F = H$ ou $F = E$.

Convolutional Neural Networks (CNNs)

Introduction

• CNNs are effective for computer vision tasks like image classification, object detection, and facial recognition.

Convolutional Layers

• Convolutional layers are the core of CNNs, applying filters to the input image.
• Each filter learns weights during training and slides over the input, performing a convolution operation.
• The convolution operation calculates the dot product between filter weights and the input region.
• The output is a feature map that emphasizes important features.

Pooling Layers

• Pooling layers reduce the dimensionality of feature maps.
• It reduces the number of parameters and avoids overfitting.
• It makes the network robust to variations in object position and size.
• Max pooling selects the maximum value in each region of the feature map.
• Average pooling calculates the average value in each region.

Fully Connected Layers

• Fully connected layers perform the final classification.
• They take the output of convolutional and pooling layers and feed it to a traditional neural network.
• The neural network learns to assign the extracted features to different classes.

CNN Architecture Example

• Input Image -> Convolutional Layer -> Pooling Layer -> Convolutional Layer -> Pooling Layer -> Fully Connected Layer -> Output
• CNNs transform the image into feature maps that highlight important patterns.

Advantages of CNNs

• They learn complex features from input images.
• They are robust to variations in object position and size.
• They are efficient in terms of the number of parameters.

Disadvantages of CNNs

• They can be computationally expensive to train.
• They can be prone to overfitting if not trained correctly.