[math] fix hot mess pt. 2

Author: ikrima

.markdownlint.json

@@ -1,6 +1,7 @@
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+  "MD045": false,
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content/math/cliff-notes.md

@@ -4,29 +4,38 @@ Condensed cheat sheet/mnemonics of stuff I forget
 
 ## Differential Geometry
 
-- Inner Product: Angle
-- Norm: Length
-- Metric: Distance
-- Measure: Size
-- $L_{p}$ norm: Max() component
-- $L_{0}$ norm: Counting norm
-- Gradient: derivative
-- Jacobian: how a differential patch area is skewed under a (linear only?) transformation
-- Divergence: sink vs source aka volume density of outward flux
-- Laplacian: average of neighborhood
-- Manifold: fancy name of a curved space
-  - Reimannian manifold: manifold with geodesic metric (Reimannian metric)
-- Group requirements: closed under multiplication, commutative, identity function, inverse
-  - Lie Group: curved space with a group structure i.e. a group that is a manifold where multiplication is smooth/infinitely differentiable
-  - Lie Algebra: tangent space of Lie group
-  - Tangent space: is a linear approximation of a curved space
-  - Non-abelian group: non-commutative group i.e. $a*b \neq b*a$ (e.g. SO(3) rotation group)
-  - Dual number is convenient for computation of Lie algebra
-- Banach Space (norm+completeness) ⊇ Hilbert Space (inner-product norm) ⊇ Sobolev Space ("nice" derivatives up to order S)
-- Functionals: functions that take functions as inputs (derivative/integral operators)
+- **Inner Product**: Angle
+- **Norm**:          Length
+- **Metric**:        Distance
+- **Measure**:       Size
+- **$L_{p}$ norm**:  Max() component
+- **$L_{0}$ norm**:  Counting norm
+- **Gradient**:      derivative
+- **Jacobian**:      how a differential patch area is skewed under a (linear only?) transformation
+- **Divergence**:    sink vs source aka volume density of outward flux
+- **Laplacian**:     average of neighborhood
+- **Manifold**:      fancy name of a curved space
+  - **Reimannian manifold**: manifold with geodesic metric (Reimannian metric)
+- **Group**:         closed under multiplication, commutative, identity function, inverse
+  - **Lie Group**:   curved space with a group structure i.e. a group that is a manifold where multiplication is smooth/infinitely differentiable
+  - **Lie Algebra**: tangent space of Lie group
+  - **Tangent        space**: linear approximation of a curved space
+  - **Non-abelian    group**: non-commutative group i.e. $a*b \neq b*a$ (e.g. SO(3) rotation group)
+  - **Dual number**: convenient for computation of Lie algebra
+- **Banach Space** (*norm+completeness*) ⊇ **Hilbert Space** (*inner-product norm*) ⊇ **Sobolev Space** (*"nice" derivatives up to order S*)
+- **Functionals**: functions that take functions as inputs (derivative/integral operators)
+- **Laplacian**: Avg of neighbors at a point - point value
+  - Maximal smoothness/mean curvature is zero
+  - *Poisson equation*: $\Delta u$ = 0
+    - Think of boundary condition being a wire and a soap film covering the wire
+    - That's a $\Delta u(x,y) = 0$
+  - Another interpretation is equilibrium state. Think of temperature
+  - Another interpretation is that there are no bumps or local minimas in that surface
 
 ## Spectral Theory
 
+### Legendre polynomial
+
 The nth Legendre polynomial, $\boldsymbol{L}_{n}$, is orthogonal to every polynomial with degrees less than n i.e.
 
 - $\boldsymbol{L}_{n} \perp \boldsymbol{P}_{i}, \ \forall i\in [0..n-1]$
@@ -36,7 +45,7 @@ $\boldsymbol{L}_{n}$ has n real roots and they are all $\in [-1,1]$
 
 Harmonic functions => $\Delta u(x) = 0$
 
-Homogenous function => $f : \reals^{n} \rightarrow \reals^{n}, \ f(\lambda \mathbf{v})=\lambda^{k} f(\mathbf{v})$ where $k,\lambda \in \reals$
+Homogenous function => $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}, \ f(\lambda \mathbf{v})=\lambda^{k} f(\mathbf{v})$ where $k,\lambda \in \mathbb{R}$
 
 General form of Newton's divided-difference polynomial interpolation:
 
@@ -53,53 +62,43 @@ f_{n}(x)=\sum_{i=0}^{n} L_{i}(x) f\left(x_{i}\right)\newline
 L_{i}(x)=\prod_{j=0 \atop j \neq i}^{n} \frac{x-x_{j}}{x_{i}-x_{j}}
 $$
 
-Gaussian quadrature
+### Gaussian quadrature
 
 - allows for accurately approximating functions where $f(x) \in P_{2n-1}$ with only n coefficients
 
-Regression Schemes (Linear or nonlinear)
-
-- Curves do not necessarily go through sample points so error at said points might be large
-- Round-off error becomes pronounced for higher order versions and ill-conditioned matrices are a problem
-- Orthogonal polynomials do not necessarily suffer from this
-
-Interpolation Schemes (splines, lagriangina/newtonian, etc)
+### Approximation Schemes
 
-- Curves must go through sample points so error at said points is small
-- Not ill conditioned
+- **Regression Schemes:** (Linear or nonlinear)
+  - Curves do not necessarily go through sample points so error at said points might be large
+  - Round-off error becomes pronounced for higher order versions and ill-conditioned matrices are a problem
+  - Orthogonal polynomials do not necessarily suffer from this
+- **Interpolation Schemes:** (splines, lagrangian/newtonian, etc)
+  - Curves must go through sample points so error at said points is small
+  - Not ill conditioned
 
-Thin plate splines
+#### Thin plate splines
 
-- construction is based on choosing a function that minimizes anintegral that represents the bending energy of a surface
-- the idea of thin-plate splines is to choose a functionf(x) that exactly interpolates the datapoints (xi,yi), say,yi=f(xi), and that minimizes the bending energy
+- construction is based on choosing a function that minimizes an integral that represents the bending energy of a surface
+- the idea of thin-plate splines is to choose a function f(x) that exactly interpolates the datapoints (xi,yi), say,yi=f(xi), and that minimizes the bending energy
   $E[f]=\int_{\mathbf{R}^{n}}\left|D^{2} f\right|^{2} d X$
 - Can also choose function that doesn't exactly interpolate all control points by using smoothing parameter for regularization
   $E[f]=\sum_{i=1}^{m}\left|f\left(\mathbf{x}_{i}\right)-y_{i}\right|^{2}+\lambda \int_{\mathbb{R}^{n}}\left|D^{2} f\right|^{2} d X$
 
-Spherical Basis Spliens:
-
-- Gross reduction summary: bsplines with slerp instead of lerp between control points
-
-Laplacian => Avg of neighbors at a point - point value
+#### Spherical Basis Splines
 
-- Maximal smoothness/mean curvature is zero
-- Poisson equation = $\Delta u$ = 0
-- Think of boundary condition being a wire and a soap film covering the wire.
-  That's a $\Delta u(x,y) = 0$
-- Another interpretation is equilibriam state. Think of temperature
-- Another interpretation is that there are no bumps or local minimas in that surface
+- Gross reduction summary: b-splines with slerp instead of lerp between control points
 
-### RBF
+#### RBF
 
-- [Integration By Rbf Over The Sphere](https://www.math.unipd.it/~marcov/pdf/AMR05_17.pdf)
+- [Integration By RBF Over The Sphere](https://www.math.unipd.it/~marcov/pdf/AMR05_17.pdf)
 - [RBF for Scientific computing](https://math.boisestate.edu/~wright/montestigliano/RBFsForScientificComputingPartOne.pdf)
 - [Interpolation and Best Approximation for Spherical Radial Basis Function Networks](https://www.hindawi.com/journals/aaa/2013/206265)
-- Spherical Radial Basis Functions, Theory and Applications (SpringerBriefs in Mathematics)
+- Spherical Radial Basis Functions, Theory and Applications (Springer Briefs in Mathematics)
 - [Transport schemes on a sphere using radial basis functions](https://www.math.utah.edu/~wright/misc/msFinal_Grady.pdf)
-- [On choosing a radial basis function and a shape parameterwhen solving a convective PDE on a sphere](https://amath.colorado.edu/faculty/fornberg/Docs/Fornberg_Piret_2.pdf)
-- [A Fast Algorithm For Spherical Basisapproximation](https://www.math.uni-luebeck.de/mitarbeiter/prestin/ps/sharma.pdf)
+- [On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere](https://amath.colorado.edu/faculty/fornberg/Docs/Fornberg_Piret_2.pdf)
+- [A Fast Algorithm For Spherical Basis approximation](https://www.math.uni-luebeck.de/mitarbeiter/prestin/ps/sharma.pdf)
 
-### Spherical Splines
+#### Spherical Splines
 
 - [Spline Representations of Functions on a Sphere for Geopotential Modeling](https://kb.osu.edu/bitstream/handle/1811/78653/1/SES_GeodeticScience_Report_475.pdf)
 - [Fitting scattered data on sphere-like surfaces using spherical splines](https://math.vanderbilt.edu/schumake/ans4.pdf)