Vectors in R^n: Definition and Operations

Vecteurs dans $\mathbb{R}^n$

• A vector in $\mathbb{R}^n$ is an ordered n-tuple of real numbers $(a_1, a_2,..., a_n)$ where each $a_i$ is a real number.
• The numbers $a_1, a_2,..., a_n$ are the components of the vector.
• Two vectors $\vec{u} = (u_1, u_2,..., u_n)$ and $\vec{v} = (v_1, v_2,..., v_n)$ in $\mathbb{R}^n$ are equal if and only if $u_i = v_i$ for all $i = 1, 2,..., n$.

Opérations Vectorielles

• The sum of two vectors $\vec{u} = (u_1, u_2,..., u_n)$ and $\vec{v} = (v_1, v_2,..., v_n)$ in $\mathbb{R}^n$ is defined as $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2,..., u_n + v_n)$.
• The product of a vector $\vec{u} = (u_1, u_2,..., u_n)$ in $\mathbb{R}^n$ by a scalar $c$ is defined by $c\vec{u} = (cu_1, cu_2,..., cu_n)$.

Propriétés des Opérations Vectorielles

• For vectors $\vec{u}, \vec{v}, \vec{w}$ in $\mathbb{R}^n$ and scalars $c, d$:
  • $\vec{u} + \vec{v} = \vec{v} + \vec{u}$ (Commutativity)
  • $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$ (Associativity)
  • There exists a zero vector $\vec{0} = (0, 0,..., 0)$ such that $\vec{u} + \vec{0} = \vec{u}$.
  • There exists an additive inverse $-\vec{u} = (-u_1, -u_2,..., -u_n)$ such that $\vec{u} + (-\vec{u}) = \vec{0}$.
  • $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$ (Scalar distributivity over vector addition)
  • $(c + d)\vec{u} = c\vec{u} + d\vec{u}$ (Scalar distributivity over scalar addition)
  • $c(d\vec{u}) = (cd)\vec{u}$ (Scalar associativity)
  • $1\vec{u} = \vec{u}$ (Scalar identity)

Combinaisons Linéaires

• A linear combination of vectors $\vec{v_1}, \vec{v_2},..., \vec{v_k}$ in $\mathbb{R}^n$ is a vector of the form $\vec{v} = c_1\vec{v_1} + c_2\vec{v_2} +... + c_k\vec{v_k}$, where $c_1, c_2,..., c_k$ are scalars.
• The scalars $c_1, c_2,..., c_k$ are the coefficients of the linear combination.

Reactions in Aprotic Solvents

• Aprotic solvents do not have acidic protons and cannot donate hydrogen bonds.
• Polar aprotic solvents can enhance the reactivity of nucleophiles.

Common Aprotic Solvents

• Dimethylformamide (DMF): $HCON(CH_3)_2$
• Dimethylacetamide (DMA): $CH_3CON(CH_3)_2$
• Dimethylsulfoxide (DMSO): $(CH_3)_2SO$
• Acetonitrile: $CH_3CN$
• Hexamethylphosphoramide (HMPA): $(N(CH_3)_2)_3PO$

Usefulness of Aprotic Solvents

• Protic solvents solvate nucleophiles strongly via hydrogen bonds, which reduces their reactivity.
• Aprotic solvents solvate nucleophiles less strongly, making them more reactive.
• As an example the reaction rate $CH_3Cl + I^- \rightarrow CH_3I + Cl^-$ has a reaction rate of 1 in methanol but 1,000,000 in DMF.

$\mathbf{S_N1}$ Reactions

• The rates of $\mathbf{S_N1}$ reactions are largely unaffected by the solvent because the rate-determining step is the formation of a carbocation, so solvent effects are small.

Elimination Reactions

• Aprotic solvents are beneficial for elimination reactions as "naked" bases in aprotic solvents are more effective, whereas strongly solvated "caged" bases are less effective.

Définitions

• Given two $\mathbb{K}$-vector spaces $E$ and $F$, a function $f: E \rightarrow F$ is linear if:
  • $\forall x, y \in E, f(x+y) = f(x) + f(y)$
  • $\forall \lambda \in \mathbb{K}, \forall x \in E, f(\lambda x) = \lambda f(x)$

Proposition

• Given two $\mathbb{K}$-vector spaces $E$ and $F$ and a function $f: E \rightarrow F$:
  • $f$ is linear $\Leftrightarrow \forall \lambda, \mu \in \mathbb{K}, \forall x, y \in E, f(\lambda x + \mu y) = \lambda f(x) + \mu f(y)$

Proposition 2

• Given two $\mathbb{K}$-vector spaces $E$ and $F$ and a linear map $f: E \rightarrow F$:
  • $f(0_E) = 0_F$
  • $\forall x \in E, f(-x) = -f(x)$

Operations sur les applications linéaires & Définitions

• Given two $\mathbb{K}$-vector spaces $E, F$, denote the set of linear transformations from $E$ to $F$ as $\mathcal{L}(E, F)$.
• Addition: $\forall f, g \in \mathcal{L}(E, F)$, we define $f+g : E \rightarrow F$ by $(f+g)(x) = f(x) + g(x)$.
• Scalar multiplication: $\forall \lambda \in \mathbb{K}, \forall f \in \mathcal{L}(E, F)$, we define $\lambda \cdot f : E \rightarrow F$ by $(\lambda \cdot f)(x) = \lambda \cdot f(x)$.
• This $(\mathcal{L}(E, F), +, \cdot)$ forms a $\mathbb{K}$-vector space.
• Let $E, F, G$ be three $\mathbb{K}$-vector spaces, $f \in \mathcal{L}(E, F)$ and $g \in \mathcal{L}(F, G)$. The composition of $g$ and $f$, denoted $g \circ f$:
  • $g \circ f : E \rightarrow G$ given by $x \mapsto g(f(x))$.
  • Which is another way of saying $g \circ f(x) = g(f(x))$.

The Ising Model

• Magnetic materials are made of atoms with intrinsic magnetic dipole moments which tend to align.
• Modelled with the Ising Model:
  • A "spin" $s_i = \pm 1$ is placed on each site of a lattice.
  • Spins tend to align with their neighbors.
  • There is also an external magnetic field $H$.
  • The energy is: $$ E = -J \sum_{\langle i, j \rangle} s_i s_j - H \sum_i s_i $$
    • $J$ is the exchange energy, with $J > 0$ favoring alignment.
    • $\langle i, j \rangle$ denotes nearest neighbors on the lattice.
• The partition function is: $$ Z = \sum_{{s_i}} e^{-\beta E} $$

Behavior of the System to Understand

• The behavior of the system, especially the magnetization given by: $$ M = \sum_i s_i $$

1-D Ising Model

• Consider a 1-D chain of $N$ spins with periodic boundary conditions.
• With the 1-D chain the energy is: $$ E = -J \sum_i s_i s_{i+1} - H \sum_i s_i $$
• In turn implying that the partition function is: $$ Z = \sum_{{s_i}} e^{\beta J \sum_i s_i s_{i+1} + \beta H \sum_i s_i} $$

Transfer Matrix Method

• Use the transfer matrix method: $$ Z = \sum_{{s_i}} \prod_i T(s_i, s_{i+1}) $$
  • Where $$ T(s_i, s_{i+1}) = e^{\beta J s_i s_{i+1} + \frac{\beta H}{2} (s_i + s_{i+1})} $$
  • Then in matrix form: $$ T = \begin{bmatrix} e^{\beta (J + H)} & e^{-\beta J} \ e^{-\beta J} & e^{\beta (J - H)} \end{bmatrix} $$
  • Then: $$ Z = \text{Tr}(T^N) = \lambda_1^N + \lambda_2^N $$
    • With $\lambda_1, \lambda_2$ the eigenvalues of $T$.

Overview

• As a quick over view regarding $N$ being so much larger than $1, Z \approx \lambda_{max}^N$ $$ \lambda_{max} = e^{\beta J} \cosh(\beta H) + \sqrt{e^{2\beta J} \sinh^2(\beta H) + e^{-2\beta J}} $$
• The free energy is: $$ F = -k_B T \ln Z \approx -k_B T N \ln \lambda_{max} $$
• The magnetization is: $$ M = -\frac{\partial F}{\partial H} = N k_B T \frac{\partial \ln \lambda_{max}}{\partial H} $$ $$ \langle M \rangle = N \frac{e^{\beta J} \sinh(\beta H)}{\sqrt{e^{2\beta J} \sinh^2(\beta H) + e^{-2\beta J}}} $$
  • Notice that $\langle M \rangle \rightarrow 0$ as $H \rightarrow 0$.
  • There is no spontaneous magnetization in the 1-D Ising model.

Correlation Function

• Consider two spins $s_i$ and $s_j$ separated by a distance $|i - j| = r$.
• The correlation function is defined as: $$ G(r) = \langle s_i s_{i+r} \rangle - \langle s_i \rangle \langle s_{i+r} \rangle $$
  • In the absence of an external field, $\langle s_i \rangle = 0$, so $G(r) = \langle s_i s_{i+r} \rangle$.
  • With $$ G(r) = \tanh^r(\beta J) = e^{-r/\xi} $$
  • Where $\xi = -\frac{1}{\ln(\tanh(\beta J))}$ is the correlation length.
    • As $T \rightarrow 0, \xi \rightarrow \infty$
    • As $T \rightarrow \infty, \xi \rightarrow 0$

Atomic Structure and Chemical Bonds

• Atoms bond to achieve the lowest possible energy state.
• Interactions involve valence electrons.
• The Octet Rule says that Atoms "want" to aim for 8 valence electrons (except H, Be, B).
• Types of Bonds:

Types of Bonds

• Ionic in that it transfers of electrons.
• Covalent where it shares electrons.
• Metalic with a "Sea" of electrons.

Ionic Bonding

• With Metals + Nonmetals: Electrons transfer to form ions.
• As Ions are Attracted: Because Positive and negative charges attract.

Properties of Ionic Compounds

- High Melting Points: Due to having strong electrostatic forces.
- Brittle: Shifting in ions leads to repulsion.
- Conduct Electricity: Done when dissolved in water.

The Heat Equation

• Consider a metal rod of length $L$.
• Let:
• $u(x,t)$ = the temperature at position $x$ and time $t$
• $u(x,0)$ = initial temperature distribution
• $u(0,t)$ = temperature at the left end
• $u(L,t)$ = temperature at the right end
• Assumptions:
• The rod is perfectly insulated, so no heat is lost through the sides.
• The rod is thin enough to ensure the temperature is constant across the cross-section.
• The heat equation is: $$ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} $$
  • where $k$ is the thermal diffusivity

Boundary Conditions Definition

• Dirichlet Boundary Conditions (fixed temperature): $$ u(0,t) = T_1 $$

$$ u(L,t) = T_2 $$

• Neumann Boundary Conditions (insulated): $$ \frac{\partial u}{\partial x}(0,t) = 0 $$

$$ \frac{\partial u}{\partial x}(L,t) = 0 $$

• Initial Condition: $$ u(x,0) = f(x) $$