We began by reviewing concepts I had already encountered in Discrete Mathematics, focusing on:
One key observation from class is that seemingly “big” or intimidating names often describe straightforward ideas. For instance, De Morgan’s Laws are intuitive statements about complements of unions and intersections:
\[ (A \cup B)^c = A^c \cap B^c \] \[ (A \cap B)^c = A^c \cup B^c \]
Although these laws may look formal, they reflect simple set relationships.
Two sets \(A\) and \(B\) are disjoint (or mutually exclusive) if:
\[ A \cap B = \varnothing. \]
We also looked at examples that involve basic counting principles, laying the groundwork for permutations and combinations. These are powerful tools for counting the number of ways to arrange or choose items from a collection, which we will explore in more depth in future sessions.
Again for this we mostly did review of concepts from previous classes that would prove key in our future sessions. I'll do a writeup for Multivariable Calculus and ODEs next week.
In linear algebra, an inner product on a (real) vector space \(V\) is a function \(\langle \,\cdot\,, \,\cdot\, \rangle: V \times V \to \mathbb{R}\) that generalizes the geometric notion of the dot product. It must satisfy the following properties for all \(f, g, h \in V\) and all real scalars \(c\):
When \(V = \mathbb{R}^n\), the inner product is given by the dot product. For two vectors \(\mathbf{u} = (u_1, u_2, \dots, u_n)\) and \(\mathbf{v} = (v_1, v_2, \dots, v_n)\), the dot product is:
\[ \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_i \, v_i. \]
This definition satisfies the five properties above, making it a valid inner product on \(\mathbb{R}^n\).
For function spaces, one common way to define an inner product is via an integral. For example, if \(f\) and \(g\) are real-valued functions on an interval \([a, b]\), one often sets:
\[ \langle f, g \rangle = \int_a^b f(x)\,g(x)\,dx. \]
This, too, can be shown to satisfy the five properties, providing a powerful way to measure “angles” and “lengths” of functions in various spaces.
Determinism is the philosophical concept that every event or state of affairs is the inevitable result of preceding events and the laws of nature. Historically, many thinkers believed that the universe operates like a clockwork machine, following precise and predictable laws.
A famous articulation of determinism came from Pierre-Simon Laplace (1749–1827). He suggested a hypothetical “demon” that, given perfect knowledge of all particles’ positions and velocities at a given moment, could predict the entire future (and retrodict the entire past) of the universe. This notion underlies classical determinism: if we know all initial conditions precisely, the future is entirely predictable.
By the late 19th and early 20th centuries, classical physics seemed to confirm determinism for large-scale objects. Physical laws like Newton’s laws worked exceedingly well to predict the motions of planets and projectiles.
However, at the atomic and subatomic scales, quantum mechanics (QM) introduced inherent probabilities:
While quantum mechanics challenged determinism at small scales, chaos theory showed that even classical deterministic systems can exhibit unpredictable behavior if they are sensitive to initial conditions.
Chaos theory does not negate determinism in a strict sense—these systems still follow deterministic laws—but it highlights our practical inability to predict outcomes unless we know the initial conditions with extreme (often unachievable) precision.
Note: It is often stated that chaos is impossible if the degrees of freedom are fewer than three. The double pendulum, having four, makes a wonderful physical example of chaotic motion.
This session reminded us that:
By and large, what we covered in the Nonlinear Dynamics and Chaos class was most interesting. I'll from today refer to it as Chaos Theory for short. I'll also add DSP to next week's post. I'm figuring out how to neatly render math using html, so you may see the formatting as a bit off. Line spacing where math symbols appear and mathjax is used is a visibly messed up.