User:Nomalias


 * CLT-Bayesian CLT-Arithmetic CLT-Ensemble CLT-Flow CLT
 * Gibbs entropy-LK entropy-arithmetic entropy?-maximun entropy-openprob entropy
 * AEP+LLN+CLT=LAN
 * $S=k_\text{B} \ln \Omega_{\rm mic} = k_\text{B} (\ln Z_{\rm can} + \beta \bar E) = k_\text{B} (\ln \mathcal{Z}_{\rm gr} + \beta (\bar E - \mu \bar N)) $
 * Helmholtz decomposition-poloidal decomposition-clebsch decomposition-FlRW decomposition-cauchy-pompieu decomposition
 * T^{3}=~chart SO(3)(euler angles),RP^{3}=~SO(3)(quotient Spin(3))(Rotations),S^{3}=~spin(3) (quaternions)//(charts on SO(3), gimbal lock, plate trick)
 * $\Delta \subset \mathbf{C} \subset \widehat{\mathbf{C}}$(complex disk,plane,sphere)
 * algebraic classification 2-manifolds: sum connected toris,sum connected projective plane, sphere, geometric classification, topological classification, combinatoric classification
 * obstruction theory: whitehead torsion /analytic(reideimester)-Hauptvermutung

Renormalon(Borel istanton)
 * soliton (localised in space)(fixed points of the flow?)
 * breather (periodic soliton), oscilon (standing periodic soliton), kink(steady state periodic soliton)
 * ADHM instanton construction
 * Instanton(localised in time soliton)(1d:tunneling,2d:vortices,3d:higgs field-TP monopole, 4d gauge bundle)
 * https://en.wikipedia.org/wiki/Chern_class#Via_the_Chern–Weil_theory (characteristic class of a four form with unitary group symmetry)
 * https://en.wikipedia.org/wiki/Pontryagin_class#Pontryagin_classes_and_curvature (characteristic class of a four form with orthogonal group symmetry)
 * https://en.wikipedia.org/wiki/Minakshisundaram%E2%80%93Pleijel_zeta_function#Applications

atiyah-singer proof techniques: pseudodifferential operator, cobordism, k theory, heat operator https://en.wikipedia.org/wiki/Signature_operator

https://en.wikipedia.org/wiki/Manifold_decomposition https://en.wikipedia.org/wiki/Borel_summation (best summation) https://en.wikipedia.org/wiki/Pad%C3%A9_approximant (best rational approximant) https://en.wikipedia.org/wiki/Chowla–Selberg_formula https://en.wikipedia.org/wiki/Spinor

https://en.wikipedia.org/wiki/Supersymmetry_nonrenormalization_theorems

https://en.wikipedia.org/wiki/List_of_small_groups
 * https://en.wikipedia.org/wiki/Category:Mathematics-related_lists
 * https://en.wikipedia.org/wiki/List_of_cohomology_theories
 * https://en.wikipedia.org/wiki/List_of_dualities
 * https://en.wikipedia.org/wiki/List_of_finite_simple_groups
 * https://en.wikipedia.org/wiki/List_of_prime_knots (https://en.wikipedia.org/wiki/Knot_tabulation)
 * https://en.wikipedia.org/wiki/List_of_mathematical_knots_and_links
 * https://en.wikipedia.org/wiki/List_of_manifolds


 * complexity classes lists: wikipedia+complexity zoo

The Berger classification
$\det(\exp(A))=\exp(\mathrm{tr}(A))$

$\ln(\det(A))=\mathrm{tr}(\ln(A))$

$\mathrm{\hat H}\Psi=E\Psi$

$R(M,x):=\frac{x^{*}Mx}{x^{*}x}$

$\varepsilon\left[\Psi\right]=\frac{\left\langle\Psi|\hat{H}|\Psi\right\rangle}{\left\langle\Psi\mid \Psi\right\rangle}$


 * $$\begin{align}

\frac{\langle{y,Ly}\rangle}{\langle{y,y}\rangle} &=\frac{\int_a^b y(x)\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + q(x)y(x)\right)dx}{\int_a^b{w(x)y(x)^2}dx}\\ &= \frac{ \left \{ \int_a^b y(x)\left(-\frac{d}{dx}\left[p(x)y'(x)\right]\right) dx \right \}+ \left \{\int_a^b{q(x)y(x)^2} \, dx \right \}}{\int_a^b{w(x)y(x)^2} \, dx} \\ &= \frac{ \left \{\left. -y(x)\left[p(x)y'(x)\right] \right |_a^b \right \} + \left \{\int_a^b y'(x)\left[p(x)y'(x)\right] \, dx \right \} + \left \{\int_a^b{q(x)y(x)^2} \, dx \right \}}{\int_a^b w(x)y(x)^2 \, dx}\\ &= \frac{ \left \{ \left. -p(x)y(x)y'(x) \right |_a^b \right \} + \left \{ \int_a^b \left [p(x)y'(x)^2 + q(x)y(x)^2 \right] \, dx \right \} } {\int_a^b{w(x)y(x)^2} \, dx}. \end{align}$$

Ten Physical Applications of Spectral Zeta Functions https://en.wikipedia.org/wiki/Edward_Witten https://en.wikipedia.org/wiki/Srinivasa_Ramanujan https://en.wikipedia.org/wiki/Barry_Simon https://en.wikipedia.org/wiki/Leonhard_Euler https://en.wikipedia.org/wiki/Serge_Lang https://www.quora.com/Where-can-I-find-problems-notes-and-lectures-of-Math-55-and-Statistics-class https://www.quora.com/What-is-it-like-to-take-Harvards-Math-55-purported-the-most-difficult-undergraduate-math-class-in-the-country-teaching-four-years-of-math-in-two-semesters