Liam Henderson

I am studying graph theory and I need to understand Dijkstra's algorithm for finding the shortest path from a source vertex to all other vertices in a weighted graph. I know the basic idea: 1. Set distance to source = 0, all others = $\infty$ 2. Mark all vertices as unvisited 3. For the current vertex, consider all unvisited neighbors and update their distances 4. Mark current as visited, select the unvisited vertex with smallest distance 5. Repeat until all visited

I'm studying multivariable calculus and I need to understand the method of Lagrange multipliers for constrained optimization. I understand that to find the extrema of $f(x,y)$ subject to $g(x,y) = 0$, we solve:

I am studying complex analysis and I learned the argument principle: $$\frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz = N - P$$

I am learning discrete mathematics and I need help with propositional logic. I understand the basic connectives: $\land$ (and), $\lor$ (or), $\neg$ (not), $\to$ (implies), $\leftrightarrow$ (iff). My questions: 1. How do I construct a truth table for a compound proposition like $\neg(p \lor q) \leftrightarrow (\neg p \land \neg q)$? 2. What does it mean for two propositions to be logically equivalent? 3. How do I prove De Morgan's laws without truth tables? 4. What is the difference between a tautology, contradiction, and contingency?

I'm studying geometry and I learned that for any convex polyhedron: $$V - E + F = 2$$

In my AP Calculus BC class, we are studying infinite series and I am confused about the distinction between absolute convergence and conditional convergence. I understand that if $\sum |a_n|$ converges then $\sum a_n$ converges absolutely. But what does it really mean for a series to be conditionally convergent? Can someone provide an intuitive explanation with an example like the alternating harmonic series?