You should see gathered listings from this directory below:
Mathematics
Topics Analytic Functions & Cauchy-Riemann Equations Contour Integration & Residue Theorem Laurent Series & Singularities Conformal Mapping Important Theorems Cauchy’s Integral Theorem Cauchy’s Integral Formula Residue Theorem Rouché’s Theorem Maximum Modulus Principle
Topics Set Theory & Boolean Algebra Logic & Proof Techniques Combinatorics & Counting Graph Theory Number Theory (Divisibility, Modular Arithmetic) Recurrence Relations Finite Automata & Formal Languages Discrete Probability
Important Theorems De Morgan’s Laws (Logic & Boolean Algebra) Pigeonhole Principle (Combinatorics) Inclusion-Exclusion Principle (Counting) Euler’s Formula for Graphs ( Handshaking Lemma ( Chinese Remainder Theorem (Number Theory) Fermat’s Little Theorem ( RSA Cryptosystem & Modular Inverses Master Theorem (Recurrence Relations)
All of the site favicons that I use have been generated by contour plots of the complex logarithm and complex exponential functions.
Experiments
HSV | Viridis | Cividis | Inferno | Jet | Magma | Plasma | Rainbow | Turbo
Real
Imaginary
Absolute
HSV | Viridis | Cividis | Inferno | Jet | Magma | Plasma | Rainbow | Turbo
- Linear Algebra
Topics Vector Spaces & Linear Independence Matrix Operations & Determinants Eigenvalues & Eigenvectors Linear Transformations Orthogonality & Inner Products Singular Value Decomposition (SVD) Important Theorems Rank-Nullity Theorem Invertible Matrix Theorem Spectral Theorem (Diagonalization of Symmetric Matrices) Cayley-Hamilton Theorem Gram-Schmidt Process Perron-Frobenius Theorem (Positive Matrices)
Mathematical Background
Definition
(Axiom)
A foundational statement accepted without proof. All other results are built ontop.
let no-one ignorant of geometry enter here
notes
these are some more informal notes that I have made after having taken the rigorous Analysis course.
- $c_{00}$ can be thought of as 'finite vectors'. all $\mathbb{R}^n$ tuples can be written as elements of $c_{00}$ with infinitely many zeros after the $n$th term.
- $c_0$ are all sequences that converge to zero. thus they will include all those in $c_{00}$ and more.
- $\ell^2$ are the vectors that fade fast enough for their energy 𐃏 to stay finite
- $\ell^\infty$ are the infinite vectors that never blow up – their entries are bounded.
\[c_{00} \subset c_0 \subset l^\infty \]
This page pairs well with Probability.
Table of Distributions
| Distribution | mass/density function | $$S_X$$ | $$\mathbb{E}(X)$$ | $$\mathrm{Var}(X)$$ | $$\phi_X(s)$$ |
|---|---|---|---|---|---|
| Bernoulli $$\mathrm{Bern}(\pi)$$ | $$P(X=1)=\pi\\P(X=0)=1-\pi$$ | $$\{0,1\}$$ | $$\pi$$ | $$\pi(1-\pi)$$ | $$(1-\pi)+\pi e^{s}$$ |
| Binomial $$\mathrm{Bin}(n,\pi)$$ | $$p_X(x)=\binom{n}{x}\pi^{x}(1-\pi)^{n-x}$$ | $$\{0,1,\dots,n\}$$ | $$n\pi$$ | $$n\pi(1-\pi)$$ | $$(1-\pi+\pi e^{s})^{n}$$ |
| Geometric $$\mathrm{Geo}(\pi)$$ | $$p_X(x)=\pi(1-\pi)^{x-1}$$ | $$\{1,2,\dots\}$$ | $$\pi^{-1}$$ | $$(1-\pi)\pi^{-2}$$ | $$\frac{\pi}{e^{-s}-1+\pi}$$ |
| Poisson $$\mathcal{P}(\lambda)$$ | $$p_X(x)=e^{-\lambda}\lambda^{x}/x!$$ | $$\{0,1,\dots\}$$ | $$\lambda$$ | $$\lambda$$ | $$\exp\{\lambda(e^{s}-1)\}$$ |
| Uniform $$U[\alpha,\beta]$$ | $$f_X(x)=(\beta-\alpha)^{-1}$$ | $$[\alpha,\beta]$$ | $$\frac{1}{2}(\alpha+\beta)$$ | $$\frac{1}{12}(\beta-\alpha)^2$$ | $$\frac{e^{\beta s}-e^{\alpha s}}{s(\beta-\alpha)}$$ |
| Exponential $$\mathrm{Exp}(\lambda)$$ | $$f_X(x)=\lambda e^{-\lambda x}$$ | $$[0,\infty)$$ | $$\lambda^{-1}$$ | $$\lambda^{-2}$$ | $$\frac{\lambda}{\lambda-s}$$ |
| Gaussian $$\mathcal{N}(\mu,\sigma^{2})$$ | $$f_X(x)=\\\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^{2}}\right\}$$ | $$\mathbb{R}$$ | $$\mu$$ | $$\sigma^{2}$$ | $$e^{\mu s+\frac{1}{2}\sigma^{2}s^{2}}$$ |
| Gamma $$\Gamma(\alpha,\lambda)$$ | $$f_X(x)=\\\frac{1}{\Gamma(\alpha)}\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}$$ | $$[0,\infty)$$ | $$\alpha\lambda^{-1}$$ | $$\alpha\lambda^{-2}$$ | $$\left(\frac{\lambda}{\lambda-s}\right)^{\alpha}$$ |
Statistical Inference
Definition
(Random Sample & Model)
Let \(X=(X_1,\ldots,X_n)\) be i.i.d. from a parametric family \(\{F_\theta:\theta\in\Theta\subset\mathbb{R}^p\}\).
The parameter \(\theta\) is unknown; inference uses the randomness of \(X\) to learn about \(\theta\).
This page pairs well with Statistics.
Elements of Probability Theory
Definition
(Random Experiment, Sample Space, Events)
A random experiment has uncertain outcomes.
The sample space $S$ is the set of all possible outcomes.
An event $E$ is a subset of $S$. The certain event is $S$; the impossible event is $\varnothing$.
I am finding Real Analysis to be more difficult than any other mathematics that I have studied before. I can seem to verify the truth of statements because they seem right; but I am having a difficult time producing rigorous and correct proofs.
It seems that High-School children (on the internet) are able to self-study Fomin with success. Bitterly, we remind ourselves:
"Comparison is the thief of Joy"—Theodore Roosevelt (probably)