User:Nomalias

snippets
Hilbert 10th problem Pell's conics elliptic curves parallel

stuff
$$\gcd(m,n)=\prod p_i^{\min(m_i,n_i)}$$

$$\text{Cotangent Bundles}\leftrightarrow \text{Pull-backs}\leftrightarrow \text{Differentials}$$ $$\text{Tangent Bundles}\leftrightarrow \text{Push-forward}\leftrightarrow \text{Tangent Vectors}$$ +

https://en.wikipedia.org/wiki/Highly_composite_number https://en.wikipedia.org/wiki/Van_Eck's_sequence

https://en.wikipedia.org/wiki/Dirichlet_kernel https://en.wikipedia.org/wiki/Weyl_character_formula#The_SU(2)_case https://en.wikipedia.org/wiki/Chern-Simons_theory#HOMFLY_and_Jones_polynomials

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

https://en.wikipedia.org/wiki/Agoh-Giuga_conjecture https://en.wikipedia.org/wiki/Daniel_da_Silva_(mathematician) https://en.wikipedia.org/wiki/Euler's_identity#Generalizations

https://en.wikipedia.org/wiki/Crenel_function https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture

https://en.wikipedia.org/wiki/Cyclotomic_polynomial https://proofwiki.org/wiki/Reciprocals_of_Odd_Numbers_adding_to_1

https://en.wikipedia.org/wiki/Chebyshev_function#The_Riemann_hypothesis https://en.wikipedia.org/wiki/Explicit_formulae_(L-function)#Weil's_Explicit_Formula https://en.wikipedia.org/wiki/Hilbert–Pólya_conjecture https://mathoverflow.net/questions/62816/the-guinand-weil-explicit-formula-without-entire-function-theory?rq=1

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

https://en.wikipedia.org/wiki/Generating_function https://en.wikipedia.org/wiki/Table_of_Newtonian_series

stackexchange
The Selberg trace formula is making $$PSL(2,\Bbb{R})$$ act on $$C^\infty(\Gamma \setminus \Bbb{H})$$, the Frobenius formula is making G act on $$\Bbb{C}[G/H]$$ +

$$\Phi(G):=\det(X_{gh^{-1}})=\prod_{\chi\in\widehat{G}}\left[\sum_{g\in G}\chi(g)X_g\right]=\prod_{\rho~\rm irred}\det\left(\sum_{g\in G}X_g\rho(g)\right)^{\deg\rho}$$ +

diferential representations

 * $$e^{D^2} f(x) = \sum_{k=0}^\infty \frac{D^{2k}}{k!}f(x).$$
 * $${e^D - 1 \over D}f(x)= \sum_{n=0}^\infty {D^n \over (n+1)!}f(x)$$
 * $$e^{D^2}f(x)=\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty f(x-y) e^{-y^2/4}\;dy$$
 * $${e^D - 1 \over D}f(x) = \int_x^{x+1} f(u)\,du$$
 * $$e^{-D^2}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n H_n(x/2)$$
 * $${D \over e^D -1}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n B_n(x)$$

+ + +
 * $$\frac{D e^D}{e^D-1}=\sum_{k=0}^\infty \frac{B_k}{k!} D^k$$

+

Group diagrams
https://commons.wikimedia.org/wiki/Category:Mathematical_diagrams https://commons.wikimedia.org/wiki/Category:Commutative_diagrams https://commons.wikimedia.org/wiki/Category:Group_theory https://commons.wikimedia.org/wiki/File:Projective-representation-lifting.svg +]

groups: https://en.wikipedia.org/wiki/Bézout's_identity Rings: https://en.wikipedia.org/wiki/Chinese_remainder_theorem

stuff with det,tr
https://en.wikipedia.org/wiki/Nahm_equations#Nahm-Hitchin_description_of_monopoles https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice#Kronig-Penney_model

http://mathworld.wolfram.com/Convergent.html

simplex determinant Cayley-Menger_determinant

$$\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\ \|\mathbf{b}\|\cos\theta ,$$ $$\zeta(n+m)=\sqrt{\zeta(2n)\zeta(2m)}\cos\theta ,$$ +

+ - https://en.wikipedia.org/wiki/Spectral_gap_(physics) https://en.wikipedia.org/wiki/Duality_gap + 

https://en.wikipedia.org/wiki/Doubly_periodic_function https://en.wikipedia.org/wiki/Fundamental_pair_of_periods

$$\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} =\operatorname{tr}(\mathbf J(f))$$ + $$\Delta f = \nabla^2 f = \nabla \cdot \nabla f =\operatorname{tr}\big(H(f)\big)$$ + +

https://en.wikipedia.org/wiki/Weierstrass_transform#Generalizations https://en.wikipedia.org/wiki/Shift_operator https://en.wikipedia.org/wiki/Operator_(physics)#The_exponential_map +

$$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\log x}\qquad\qquad(1)$$ + $$\pi_2(x) \sim 1.32 \frac {x}{(\log x)^2}$$ +

$$p_n\approx n\log n + n(\log \log n - 1),$$ $$\sum_{n\le x} \sigma_0(n)\approx x\log x + x(2\gamma-1)$$ $$\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^\gamma,$$ $$\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma$$ + + +

https://en.wikipedia.org/wiki/Chern-Simons_theory#HOMFLY_and_Jones_polynomials https://en.wikipedia.org/wiki/Weyl_character_formula#The_SU(2)_case

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

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

https://en.wikipedia.org/wiki/Analytic_torsion https://en.wikipedia.org/wiki/Crooks_fluctuation_theorem


 * $$\langle \hat{A}\rangle={1\over Z_0}\operatorname{Tr}\,[\hat{\rho_0}\hat{A}]={1\over Z_0}\sum_n \langle n | \hat{A} |n \rangle e^{-\beta E_n}$$
 * $$\hat{\rho_0}=e^{-\beta \hat{H}_0}=\sum_n |n \rangle\langle n |e^{-\beta E_n}$$

+ + + + + +


 * $$|\psi(t)\rangle = \exp\left(-\frac{i}{\hbar} \hat H t\right) |q_0\rangle \equiv \exp\left(-\frac{i}{\hbar} \hat H t\right) |0\rangle

$$
 * $$ \left \langle F \bigg| \exp\left( {- {i \over \hbar } \hat H T} \right) \bigg |0 \right \rangle = \int Dq(t) \exp\left[ {i\over \hbar} S \right]$$

+

$$\int\exp(-\frac{(X-\operatorname{E}[X])^2}{2\operatorname{E}[(X-\operatorname{E}[X])^2]})=(\operatorname{det}(\operatorname{E}[(X-\operatorname{E}[X])^2))^{\frac{1}{2}}$$

$$-\frac{F}{k T} = \ln \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big)$$ +

$$\operatorname{Index}(D) = \dim\operatorname{Ker}(D)− \dim\operatorname{Ker}(D*)=\operatorname{tr}(\exp(D*D))-\operatorname{tr}(\exp(DD*))$$ + $$ \frac{1}{Z_{\text{GUE}(n)}} e^{- \frac{n}{2} \mathrm{tr} H^2} $$ +

$$\zeta(1-n,a)=-\frac{B_n(a)}{n} \!$$ for $$ n\geq1 \!$$+ $$\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$$ $$\zeta(-n)=(-1)^n \frac{B_{n+1}}{n+1}$$

Hodge duality->Poincaré duality->Grothendieck local duality->Serre duality

kunneth theorem + + [+

https://en.wikipedia.org/wiki/Dedekind_psi_function https://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis

$$p(x) = \frac{1}{2\pi} \int_{\mathbf{R}} e^{itx} P(t)\, dt = \frac{1}{2\pi} \int_{\mathbf{R}} e^{itx} \overline{\varphi_X(t)}\, dt.$$ +

+ + + $$ \widehat{f}(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \widehat\varphi(t)\psi_h(t) e^{-itx} \, dt               = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{1}{n} \sum_{j=1}^n e^{it(x_j-x)} \psi(ht) \, dt               = \frac{1}{nh} \sum_{j=1}^n \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(ht)\frac{x-x_j}{h}} \psi(ht) \, d(ht) = \frac{1}{nh} \sum_{j=1}^n K\Big(\frac{x-x_j}{h}\Big) $$

$$\varphi_{Z_n}(t)=(\varphi_{Y_1}(\frac{t}{\sqrt{n}}))^n=(e^{-\frac{1}{2}(\frac{t}{\sqrt{n}})^2})^n=e^{-\frac{t^2}{2}}$$

$$ \tilde{K}(p; T)=\tilde{G}_\varepsilon(p)^{T/\varepsilon}=(e^{-\frac{1}{2}(\sqrt{\varepsilon} p)^2})^{T/\varepsilon}= e^{-\frac{T p^2}{2}} $$

$$\psi_t(y) = \int \psi_0(x) K(x - y; t) \,dx = \int \psi_0(x) \int_{x(0) = x}^{x(t) = y} e^{iS} \,Dx,$$

$$K(x, y; T) = \int_{x(0) = x}^{x(T) = y} \prod_t \exp\left(-\tfrac12 \left(\frac{x(t + \varepsilon) - x(t)}{\varepsilon}\right)^2 \varepsilon \right) \,Dx,$$

"Index theory"
$$\sum_i(-1)^i\text{Tr}(\text{Frob},H^i_c(X,\bf{Q}_\ell))=|X({\bf F}_q)|$$ $$\sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q}))=\sum_{x\in\mathrm{Fix}(f)}\mathrm{index} _x f$$ $$\sum_i(-1)^i\dim H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Sing}(v)}\mathrm{index}_xv$$ + + + + +

$$\sum(-1)^\gamma C^\gamma\,=\chi(M)$$ + $$\sum_i \operatorname{index}_{x_i}(v) = \chi(M)\,$$ + $$\textrm{Tr}[(-1)^F e^{-\beta H}]=\sum_{p\in\mathbb{Z}}(-1)^pb_p=\chi(M) \. $$ + +

Within this definition of the finite-time SEO, the Witten index can be recognized as the sharp trace of the generalized transfer operator. It also links the Witten index to the Lefschetz index,$ \textstyle I_L=\operatorname{Tr} (-1)^{\hat n} M_{t't}^*= \sum_{x\in \operatorname{fix} M_{tt'}} \operatorname{sign} \operatorname{det} (\delta_j^i -\partial M_{tt'}^i(x) /\partial x^j) $, a topological constant that equals the Euler characteristic of the (closed) phase space. Namely, $ \textstyle {\mathcal W} = \operatorname{Tr} (-1)^{\hat n} \langle M_{t't}^* \rangle_\text{noise} = \langle \operatorname{Tr} (-1)^{\hat n} M_{t't}^* \rangle_\text{noise} = I_{L} $ . +

$$\sigma(n)=\sum_{i\in\mathbb{Z}} (-1)^{i+1}\left(\sigma(n{-}\frac12(3i^2{-}i))+\delta(n,\frac12(3i^2{-}i))\,n\right)=\sigma(n{-}1)+\sigma(n{-}2)-\sigma(n{-}5)-\sigma(n{-}7)+\sigma(n{-}12)+\sigma(n{-}15)+ \cdots$$ +

Genus Stuff
Riemann-Roch_theorem: $$\ell (\mathcal K_X - D) = \dim H^0 (X, \omega_X \otimes \mathcal L(D)^\vee),H^0 (X, \omega_X \otimes \mathcal L(D)^\vee)$$ Line bundle-Riemann surface Vector Bundle-Complex manifold Quotient stack sheaf-Orbifold Chain-complex sheaf-Scheme Arithmetic

Degenerancy theory
covering map manifold

Poincaré–Hopf theoremHairy ball theorem

Banach fixed point theorem(existence and uniqueness) Brouwer_fixed-point_theorem(existence) Fixed point degree - + +

Hall's marrriage theorem equivalences

Cayley's theorem equivalencesWagner-preston theorem (Cayley' theorem on inver semigroup)+

group theory
group theory: definitions basics factsnon basic facts

$$|X^{P}|\equiv |X| \mod p\quad \text{(P p-group)}$$ + $$a^p \equiv a \mod p\quad \text{(p prime)}$$

Thompson order formula

+ $$|G|=|G/G_x||G_x|=|G_x \backslash G||G_x|$$+ $$|G|=|G/H||H|=|H \backslash G||H|$$+ $$G/Z(G) \cong Inn(G)$$ $$Aut(G)/Inn(G) \cong Out(G)$$

Centralizer-Normalizer Orbit stabilizer coset-index

https://en.wikipedia.org/wiki/Wedderburn's_little_theorem

Fundamental theorem of abelian groups Fundamental theorem of cyclic groups Fundamental theorem of free groups Jordan–Hölder theorem Finitely generated abelian group

https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Abelian_group#Classification

mobius table
+ +

elliptic stuff
$$\int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega$$ +

primes stuff
$$\sum_{p\leq x}\frac{1}{p}=\int_2^x \frac{1}{t}\,d(\pi(t))$$ + $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$ +

++ $$\frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx$$ + $$\psi_0(x) = \dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds\quad$$ $$\frac{\zeta^\prime(s)}{\zeta(s)} = - s\int_1^\infty \frac{\psi(x)}{x^{s+1}}\,dx$$ +

$$\int_0^\infty x^{s}\ln(1-e^{-x})dx = - \int_0^\infty x^{s-1} \frac{e^{-x}}{1 -e^{-x}} dx$$ $$\int_{0}^{1} | \mathsf{Li}_{s}(e^{2 \pi i x})|^{2} dx = \sum_{k \geq 1} \left| \frac{e^{2 \pi i k x}}{k^{s}} \right|^{2} = \sum_{k \geq 1} \frac{1}{k^{2s}} = \zeta(2s)$$ +

[https://math.stackexchange.com/questions/1484265/mellin-transform-of-a-function-related-to-the-derivative-of-riemann-zeta-functio/2222444 ¿? $$\sum_{p} \frac{\log p}{p^s} = \int_{1}^{\infty} \frac{ d \vartheta(x)}{x^s} = s \int_{1}^{\infty} \frac{ \vartheta(x)}{x^{s+1}} dx$$] $$\sum_{p} \frac{\log p}{p^s-1} = \int_{1}^{\infty} \frac{ d \vartheta(x)}{x^s} = s \int_{1}^{\infty} \frac{ \vartheta(x)}{x^{s+1}} dx$$

$$\quad \psi'(x)=\ln(x)\,\Pi_0'(x)$$  $$\quad T'(x)=\ln(x)\,S'(x)$$

$$\quad S[x]=\sum_{n=1}^{\lfloor x\rfloor}1=\lfloor x\rfloor $$  $$\quad T[x]=\sum_{n=1}^{\lfloor x\rfloor}\log n $$

$$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty (e ^ x - 1)^{-1}x ^ {s-1}\, \mathrm{d}x=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}\mathrm{d}x\quad$$ $$\Gamma(s) = \int_0^\infty e^{-x} \,x^{s-1}\, \mathrm{d}x $$ + $$\zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}\int_0^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x,\quad$$ $$\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}$$ $$\zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}(\frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(ix)-x^{-\frac12}\right)x^{\frac{s}{2}-1}\,\mathrm{d}x + \frac{1}{2}\int_1^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x)$$ +

$$\sum_p\frac{1}{p}= \infty$$+ $$\sum_p\frac{1}{p-1}=1$$+

$$\gamma = \lim_{n\to\infty}\left(\ln n - \sum_{p\le n}\frac{\ln p}{p-1}\right)= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.$$

$$\frac {\zeta^\prime(s)}{\zeta(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}=-\sum_{p\in\mathcal{P}}\frac{\log(p)}{p^{s}-1}=P_{p-1}'(s)$$ +

$$L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx\quad A(x)=\sum_{n\le x} \chi(n)$$ $$\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx$$+ $$\zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} dx$$ + +

$$\zeta'(s) = -\sum_{n \mathop = 2}^\infty \frac{\ln \left({n}\right)}{n^s}$$ + $$\left(\frac{\zeta'(s)}{\zeta(s)}\right)^2 = \sum_{n=1}^\infty \sum_{d|n} \frac{\Lambda(d) \Lambda(n/d)}{ n^{s}}$$ + $$\frac{d}{ds}\left(\frac{\zeta^{(k)}(s)}{\zeta(s)}\right)=\frac{\zeta^{(k+1)}(s)}{\zeta(s)}-\frac{\zeta'(s)}{\zeta(s)}\frac{\zeta^{(k)}(s)}{\zeta(s)}$$ +


 * $$\beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.$$

+
 * $$\omega^p = (x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p},$$

+

$$\zeta(s) =\prod_{p \text{ prime}} (1-p^{-s})^{-1}=\sum_{n=1}^\infty\frac{1}{n^s}=(\sum_{n=1}^\infty \frac{\mu(n)}{n^s})^{-1}$$

$$f(q)=\prod_{n=1}^\infty (1-q^n) = \sum_{n=0}^\infty a_nq^n=(\sum_{n=0}^\infty p(n) q^n)^{-1}$$

$$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^{n})=(\sum_{n>0}\tau(n)q^n)^{-24}=(\sum_{n=0}^{\infty}p(n)q^{n-\frac{1}{24}})^{-1}$$ + +

$$\left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}=j(p)-j(q)$$ +

$$e^x = \prod_{n \geq 1}(1-x^n)^{-\mu(n)/n}$$ + + + + + + $$x = \sum_{n \geq 1}\frac{\mu(n)}{n}\ln((1-x^n)^{-1})$$

$$\frac{1}{1-x}=\prod_{n\geq 0} (1+x^{2^{n}})$$ + + $$\prod_{n\geq 0}\frac{1}{1-x^{2n+1}}=\prod_{n\geq 0} (1+x^{n})$$+

Rodrigues's formula Li's criterion

Szegő_limit_theorems Jensen's formula five value theorem

ELSV formula

https://en.wikipedia.org/wiki/Fredholm's theorem Fredholm alternative Farkas_lemma Hyperplane separation theorem Hanh Banach separation theorem Positive-definite matrix Positive-definite kernel Positive definiteness

homology-homotopy dictionary+ number field-function field dictionary Kapranov-Reznikov-Mazur dictionary/arithmetic topology + + + + + + Diophantine dictionary/Arithmetic dynamics algebraic geometry dictionary + wu-yang dictionary elliptic-parabolic dictionary feynman-intersection number dictionary-like - petterson-weil volume+ witten's volume orbifolds volume orbifold euler characteristic+

varieties: Grassman Segre veronese

global-local homology global-local homotopy +

nowhere differentiable: everywhere continuos, nowhere continuos

$\max( |A+A|, |A \cdot A| ) \geq c \cdot |A|^{1+\varepsilon} $ $\max(|a|,|b|,|c|) \geq C \cdot rad(abc)^{1+\varepsilon} $+ $\max\{\deg(a),\deg(b),\deg(c)\} \le \deg(\operatorname{rad}(abc))-1.$

mle-entropy euler-lagrange-gauss-principle

(Elliptic function-Elliptic curve) (Modular form-Modular curve)

Arithmetic geometry Fermat's squares theorem Minkowski's theorem + Gauss circle problem Dirichlet's divisor problem

Class field theory Class number Class number formula List of number fields with class number one Lists of discriminants of class number 1 Stark-Heegner_theorem Kronecker-Weber_theorem Kummer theory Fundamental discriminant + +

+ +

+ + + + +

+

+ +

Perron's formulaShimura correspondence

$$\zeta(s)=\frac{\int}_0^\infty{{v^{\frac{s}{2}}}\left({\frac{2}}\right)\frac{v}}$$+ $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$ $$\zeta(s)=\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$$ $$ \lim_{s\to 1}(s-1)\zeta(s)=1$$

$$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ $${L_D}(s)\!\!:=\mathop {\prod}\limits_{p\,{\rm{prime}}} {\frac{1}} $$ $$\frac{L_D}(1)=\frac{2}$$ $$Q(x, y)=Ax^2+Bxy+Cy^2,D=B^{2}-4AC$$ $$Q(x, y)=Q(ax+by,cx+dy),ad-bc=1$$

$${L_E}(s) = \frac{\int}_0^\infty {{v^s}{f_E}(iv)\frac{v}}$$ $${L_E}(s)\!\!:=\mathop{\prod}\limits_{p\,{\rm{prime}}}{\frac{1}}$$ $${\rm{li}}{{\rm{m}}_{s \to 1}}{(s-1)^{-{r_E}}}{L_E}(s)<\infty$$ $$\mathop \limits_{s \to 1}\frac{1}\frac = {c_E}\left|{{\text{Ш}}(E)}\right|$$ $${y^2}={x^3}+Ax+B,4A^3+27B^2\neq0$$ $${(Ncz + d)^{ - 2}}{f_E}\left({\frac}\right)={f_E}(z),ad-Nbc=1$$

$$\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}$$ $$\zeta_K (s) = \prod_{P \subseteq \mathcal{O}_K} \frac{1}{1-(N_{K/\mathbf{Q}}(P))^{-s}},\text{ for Re}(s)>1.$$ $$ \lim_{s \to 1} (s-1) \zeta_K(s) = \frac{2^{r_1} \cdot(2\pi)^{r_2} \cdot \operatorname{Reg}_K \cdot h_K}{w_K \cdot \sqrt{|D_K|}}$$

Winding number-Eisenbud Levine Khimshiashvili signature formula

Roth's_theorem-Duffin-Schaeffer conjecture

Kutsenov trace formula-Gutzwiller trace formula

Min-Max theoremMax-min_inequality

Kolmogorov equation-Fokker Planck equation Koopman operator-Perron-Frobenius operator

not recursive function not computable function not ZFC-dependent function bound+

+

Euler product + +

Feller-Tornier constant + Pólya conjecture Chebyshev's bias + + Goldfeld conjecture + Parity_problem

$\pi_{n,a}(n) \sim \frac{\pi(n)}{\varphi(n)}$ $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$ + Artin's conjecture +

$$ \Pi^*(x;q,a) = \sum^*_{p\le x, p\equiv a\pmod q} 1 + \sum^*_{p^2\le x, p^2\equiv a\pmod q} \tfrac12 + \sum^*_{p^3\le x, p^3\equiv a\pmod q} \tfrac13 + \cdots = \pi^*(x;q,a) + \tfrac12 \sum_{b\pmod q, b^2\equiv a\pmod q} \pi^*(x^{1/2};q,b) + \tfrac13 \sum_{c\pmod q, c^3\equiv a\pmod q} \pi^*(x^{1/3};q,c) + \cdots $$ +

Bateman-Horn conjecture square free distribution +

algebraic topology
$$\langle a,b,c,d\ |\ abcd=1\rangle$$ $$\langle a,b,c\ |\ [a,b]c=1\rangle$$ +

https://en.wikipedia.org/wiki/Homology_(mathematics)
 * NOTES:
 * For a non-orientable surface, a hole is equivalent to two cross-caps.
 * Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere $$S^2$$, g = c = 0.

$$\mathbb{S}^2,\mathbb{E}^2,\mathbb{H}^2,SO(3),ISO(\mathbb{R}^2)^+,SL(2,\mathbb{R})=SO(1,3)^+,\pi_1(\mathbb{S}^2)=0,\pi_1(T \cong \R^2/\Z^2)=0$$

RPn CPn HPn

???????????????


 * https://en.wikipedia.org/wiki/Modular_elliptic_curve
 * https://en.wikipedia.org/wiki/Weierstrass's_elliptic_functions#General_theory
 * https://en.wikipedia.org/wiki/Period_mapping


 * https://en.wikipedia.org/wiki/Poincaré–Bendixson_theorem
 * https://en.wikipedia.org/wiki/Nielsen-Thurston_classification


 * https://en.wikipedia.org/wiki/Wirtinger_derivatives
 * https://en.wikipedia.org/wiki/Creation_and_annihilation_operators

Real n-Cauchy-Riemann:$Df^TDf = (\det(Df))^{2/n}I$


 * https://en.wikipedia.org/wiki/Darboux's_theorem
 * https://en.wikipedia.org/wiki/Period_mapping

Volume form + Connection_form torsion form curvature form + spin connection khäler form + Solder form

Poincaré series Igusa zeta function

Character theory Weyl character formula Kirillov_character_formula

Representation theory Representation of Lie algebra Representation of Lie group Representation of finite groups ℓ-adic representations

Hyperbolic geometryHyperbolic manifold++

{5,5}-tilling-Poincaré Sphere+

Minimal program model +

Abel's theorem converse Abel's theorem

Associahedron Pemutohedron

https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
 * invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)
 * primary decomposition + companion matrix yields primary rational canonical form
 * primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)

zeta table
+ + + + + + + +

$$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$ + + $$S=\operatorname{tr}_g \operatorname{Ric}$$ + $$R_{k \overline{l}}=\partial_{k} \partial_{\overline{l}} \ln (\operatorname{det}(g)),\ \text{Ricci-Chern form}$$ + + + $$\frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi};$$$$exp( - \zeta'(0)) / Vol(S)$$ + +

$$n!=\exp(\ln(n!))=\exp(\sum_n\ln n)=\exp(\sum_n\frac{\ln n}{n^s}|_{s=0})=\exp(-\zeta'(0)))$$ $$p\#=\exp(\ln(p\#))=\exp(\sum_p\ln p)=\exp(\sum_p\frac{\ln p}{p^s}|_{s=0})=\exp(-P'(0)))$$

$$\zeta(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}dx$$ $$P(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{p>0}e^{-px}dx$$

$$\zeta(s)=\exp(-\sum_p\ln(1-p^{-s}))=\exp(\sum_{p,n}\frac{p^{-ns}}{n})$$ +

$$\phi(q)=\exp(-\sum_k\ln(1-q^k))=\exp(\sum_{k,n}\frac{q^{-nk}}{n})=\exp(\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{q^n-1})$$ +

$$\phi(q)=\exp(-\sum_k\ln(1-q^k))=\exp(\sum_{k,n}\frac{q^{-nk}}{n})=\exp(\sum_{k|m,m}\frac{kq^{-m}}{m})=\exp(\sum_{m}\frac{q^{-m}}{m}\sum_{k|m}k)=\exp(\sum_{m}\frac{q^{-m}}{m}\sigma(m))$$ $$\log Z(X, T) =\sum_{x \in X}-\log \left(1-T^{\operatorname{deg}(x)}\right)=\sum_{x \in X} \sum_{n=1}^{\infty} \frac{T^{\operatorname{deg}(x) \cdot n}}{n}=\sum_{m=1}^{\infty}\left(\sum_{\operatorname{deg}(x) | m} \operatorname{deg}(x)\right) \frac{T^{m}}{m}=\sum_{m=1}^{\infty}\left|X\left(\mathbb{F}_{q^{m}}\right)\right| \frac{T^{m}}{m}$$ + +

$$Tr_V q^{L_0} = \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}$$ +

$$\wp(z;\Lambda)= -\zeta'(z;\Lambda)=\ln''(\sigma(z;\Lambda)), \mbox{ for any } z \in \Complex $$ + $$\frac{1}{2 \pi i} \frac{d}{d z} \log \Delta(z)=1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n z}}{1-e^{2 \pi i n z}}=1-24 \sum_{m=1}^{\infty} \sigma_{1}(m) e^{2 \pi i m z}=1-24\sum_{n>0}\sigma_1(n)q^n=E_{2}(z)$$ + +

$$ \sum_{n \leq x} \Lambda(n) =\dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds\quad= x - \sum_{\rho} \frac{x^{\rho}}{\rho}- \ln 2\pi - \tfrac{1}{2} \ln (1-x^{-2})=x - \sum_{\rho} \frac{x^{\rho}}{\rho}- \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2}\sum_{k=1}^{\infty} \frac{x^{-2k}}{-2k}$$

$$\ln \mathcal{L}(\mu,\Sigma) = -{n \over 2} \ln \det(\Sigma) -{1 \over 2} \operatorname{tr} \left[ \Sigma^{-1} \sum_{i=1}^n (x_i-\mu) (x_i-\mu)^\mathrm{T} \right]. $$

$$ \sum_\rho F(\rho) = \operatorname{Tr}(F(\widehat T )).\!$$ + ++++++

convergence stuff
as=>p=>d+

+

function field analogy+ + +

Selberg class Abstract analytic number theory + + One can view Selberg’s theorem as a sort of Fourier-analytic variant of the Erdös-Kac theorem. PNT scaled models+ + + mertens=/=PNT+ +

$\Delta \cong \mathbf{H} \subset \mathbf{C} \subset \widehat{\mathbf{C}}$(complex disk,plane,sphere) + +

n>0:TOP=PL=DIFF n>3:TOP=\=PL=DIFF n>6:TOP=\=PL=\=DIFF

Analogies: Quine McCluskey algorithmBuchberger's algorithm

Correspondences: algebraic sets & Ideals Field subextension & Galois Subgroups Space covering & Fundemantal group + + + Modular forms & Elliptic curves Automorphic forms & Algebraic curves Elliptic modular forms & Group representations Geometric langlands Kobayashi–Hitchin correspondence Simpson correspondence Riemann-Hilbert correspondence Robinson-Schensted correspondence Shimura correspondence-Theta correspondence

Enumerative invariants: Sympletic category: Donaldson invariants Seiberg–Witten invariants Gromov-Witten invariants FJRW theory Algebraic category: Donaldson–Thomas +

characterization: Gamma function+ Determinant exponential function

Obstructions characteristic classes: + Pontryagin_class (orthogonal group)+++
 * homotopy=simple homotopy: Whitehead torsion+
 * homeorphism=PL homeomorphism: Kirby-Siebenmann_class+
 * homeomorphims=homotopic homeomorphism:Reidemeister torsion
 * homeomosphism=homological homeomorphism?:Euler class
 * vector bundle morphism=vector bundle isomorphism: Stiefel-Whitney_class
 * complex vector bundle morphism=complex vector bundle isomorphism:Chern-class (unitary group)
 * Todd classreciprocal characteristic class
 * Segre class-inverse chern class
 * Hasse invariant
 * Manin obstruction

Generalizations: (Hodge conjecture-Arithmetic Hodge Conjecture) (Milnor conjecture-Thom conjecture) (Witten conjecture-Virasoro conjecture) (K theory-L theory) (Pontryagin_duality-Tannaka-Krein duality) (Maximun principle-Hopf's Maximum principle) (Padé series-Laurent series-Puiseux series) (Weierstrass factorization-Mittag-Leffer's factorization) (Stone-Weierstrass theorem-Arakelyan's_theorem) (Cantor's_paradox-Ordinal Cantor's paradox(Introduction to Mathematical Logic Mendhelson page 2))(Stone's theorem-Stone-von Neumann theorem) (Morse theory-Picard-Lefschetz theory) (Invariant theory-Geometric invariant theory) (https://en.wikipedia.org/wiki/Differential_Galois_theory)

(Farkas's lemma-Fredholm alternative) (sdf matrix-sdf kernel)

Martingale CLT Functional integralPath integral Hammersley–Clifford theorem
 * CLT-frequentist 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)) $

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

Vector bundles on algebraic curves Birkhoff–Grothendieck theoremAtiyah-Birkhoff–Grothendieck theorem

Bott periodicity&Hurwitz's_theorem Bott_periodicity_theorem Hurwitz theorem Frobenius theorem

Vector bundles on algebraic curves Birkhoff–Grothendieck theoremAtiyah-Birkhoff–Grothendieck theorem

Hesisenberg group transform Special linear group transform +?

Special_unitary_group (spin group) + +2 +3 +4 +5

String group + +2 anomaly conformal invariants + +2


 * soliton (localised in space)(fixed points of the flow?)
 * breather (periodic soliton), oscilon (standing periodic soliton), kink(steady state periodic soliton)
 * ADHM instanton
 * Instanton(localised in time soliton)(1d:tunneling,2d:vortices,3d:higgs field-TP monopole, 4d: gauge bundle)
 * Renormalon(Borel istanton)
 * Cotangent localization

Helmholtz decomposition FLRW decomposition Ricci tensor decomposition Riemann tensor decomposition Electromagnetic tensor decomposition+

cauchy-pompieu decomposition poloidal decomposition clebsch decomposition

Sokhotski Plemelj theorem +

Covariance matrix Covariance process +

Var(x)=2Dt + + +

Cayley-Hamilton theorem Fredholm determinant +

variational quotients
Dirichlet eigenvalue Rayleigh quotient Variational method Fisher's linear discriminant Poincaré constant Cheeger constant

Poincaré inequality Bauer Fike theorem

Min-max theorem Minimax theorem Max-min inequality Duality gap Lagrange multiplier

Geometric_phase Fubini-Study_metric Fisher_information_metric

Coarea formula Smooth coarea formula coarea+ coarea+

Distance preserving Angle preserving Area preserving Volume preserving + Sympletic preserving + +

Knot group Link group Braid group

Knot theory Link theory Braid theory Ribbon theory +

Homology group Homotopy group Holonomy group

Cohomology group Cohomotopy group

Homological algebra Homotopical algebra

stable unitary group stable orthogonal group J-homomorphism

Riemann zeta function Hurwitz zeta function Polylogarithm

Eta invariant Dirichlet_eta_function  1+2+3+4+... Riemann+Euler-Maclaurin Darboux's formula

Analytic torsion Heat kernel signature

Signature operator Atiyah Singer index theorem+

Witten index + +2 +3 +4 +5 +6

Supersymmetric atiyah Singer index theorem + +2 +3 +?

Equivalences:Abel–Plana_formula<=>Euler–Maclaurin_formula<=>Poisson summation formula

Singular Kernel Regularization

Riemman, Ricci, scalar curvature +

isothermal conformal uniformization [https://en.wikipedia.org/wiki/Ricci_flow#Relation_to_diffusion isothermal/conformal? Ricci flow]

Inclusion-exclusion principle Isomorphism theorems

Manifold decomposition
 * atiyah-singer proof techniques: pseudodifferential operator, cobordism, k theory, heat operator
 * https://en.wikipedia.org/wiki/Borel_summation (best summation)
 * https://en.wikipedia.org/wiki/Padé_approximant (best rational approximant)
 * https://en.wikipedia.org/wiki/Category:Mathematics-related_lists
 * https://en.wikipedia.org/wiki/List_of_cohomology_theories +
 * https://en.wikipedia.org/wiki/Vanishing_theorem
 * https://en.wikipedia.org/wiki/Fixed-point_theorem#List_of_fixed-point_theorems
 * https://en.wikipedia.org/wiki/List_of_zeta_functions ++
 * https://en.wikipedia.org/wiki/Trace_formula
 * https://en.wikipedia.org/wiki/List_of_dualities
 * https://en.wikipedia.org/wiki/Category:Duality_theories
 * https://en.wikipedia.org/wiki/List_of_complex_and_algebraic_surfaces
 * https://en.wikipedia.org/wiki/List_of_small_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

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

+ + +
 * https://en.wikipedia.org/wiki/Classification_theorem
 * https://en.wikipedia.org/wiki/Complete_set_of_invariants
 * https://en.wikipedia.org/wiki/Invariant_(mathematics)
 * (https://en.wikipedia.org/wiki/Invariant_theory)
 * https://en.wikipedia.org/wiki/Enriques-Kodaira_classification#Invariants_of_surfaces
 * https://en.wikipedia.org/wiki/Decomposition_(disambiguation)
 * $\Delta \subset \mathbf{C} \subset \widehat{\mathbf{C}}$(complex disk,plane,sphere)
 * algebraic classification 2-manifolds: sum connected toris,sum connected projective plane, sphere, geometric classification, topological classification, combinatoric classification
 * https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification
 * https://en.wikipedia.org/wiki/Wigner's_classification
 * https://en.wikipedia.org/wiki/Bianchi_classification
 * https://en.wikipedia.org/wiki/Petrov_classification
 * https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
 * https://en.wikipedia.org/wiki/Enriques–Kodaira_classification
 * https://en.wikipedia.org/wiki/Bernstein–Zelevinsky_classification
 * https://en.wikipedia.org/wiki/Langlands_classification
 * https://en.wikipedia.org/wiki/Segre_classification
 * https://en.wikipedia.org/wiki/Crystal_system#Classification_of_lattices
 * https://en.wikipedia.org/wiki/Painlevé_transcendents
 * https://en.wikipedia.org/wiki/List_of_finite_simple_groups


 * complexity classes lists: list complexity classes+complexity zoo

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

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

[https://en.wikipedia.org/wiki/Schrödinger_equation#Time_indep endent $$\mathrm{\hat H}\Psi=E\Psi$$]

Twelvefold way +

Plane partition number partition Multiplicative_partition (unordered factorization)



Numerical methods: Garlekin Homotopy analysis Finite difference Finite element Finite volume ≈

Physics
https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)

https://physics.stackexchange.com/questions/295714/whats-the-relation-between-path-integral-and-dyson-series http://bolvan.ph.utexas.edu/~vadim/Classes/2011f/dyson.pdf

$$\Delta H \equiv \Delta H_\text{L} + \Delta H_\text{T}$$ $$\mathcal{H}=\mathcal{H}_{\rm Coulomb}+\mathcal{H}_{\mathrm{kinetic}}+\mathcal{H}_{\mathrm{SO}}+H_{\mathrm{Darwin}}\!$$ + + +

https://en.wikipedia.org/wiki/Quantum_invariant https://en.wikipedia.org/wiki/Periodic_table_of_topological_invariants

$$e^{-\frac{F}{k T}} = \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big)$$ $$-\frac{F}{k T} = \ln \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big)$$

https://en.wikipedia.org/wiki/Bose_gas https://en.wikipedia.org/wiki/Gas_in_a_box https://en.wikipedia.org/wiki/Gas_in_a_harmonic_trap

thermal fluctuations + +

Dilaton Perelman's renormalization Perleman's entropy Perelman's fluctuation Quantropy Jacobson's entropic field Verlinde's entropic force 

velocity limit: light velocity entropy limit: bekenstein bound

Unruh radiation Hawking radiation +

+ Locality breaker: EPR paradox Unitarity breaker: BHI paradox causality breaker: relativity

Spacetime symmetries

Sagnac effect

Crystallographic periodic table crystal structure #

YB equation BMW algebra +

ΛCDM common observations: SNIa CMB H(z) BAO LSS Planck spacecraft + standard candle standard ruler

Hamilton Jacobi_equation Circulation Biot-Savart

Kutta Joukowski theorem Magnus effect Vorticity equation

Kelvin's circulation theorem tao's post Euler fluid equations Hamiltonian fluid mechanics Madelung equations

https://en.wikipedia.org/wiki/Category:Scattering

https://en.wikipedia.org/wiki/Moiré_pattern https://en.wikipedia.org/wiki/Bragg's_law https://en.wikipedia.org/wiki/Compton_scattering

https://en.wikipedia.org/wiki/List_of_quasiparticles https://en.wikipedia.org/wiki/List_of_particles

Quark lepton_complementarityEigthfold way

Phonon-polaron-magnon spinon-orbiton-chargon https://en.wikipedia.org/wiki/Bose-Einstein_condensation_of_quasiparticles

Molecular partition function + + +

volumetric entropy surface entropy gas entropy + + +

+ thermodynamics +

Deconfinement

https://en.wikipedia.org/wiki/Scale-free_ideal_gas


 * electromagnetism
 * relativistic charge distribution
 * relativistic charge point
 * +

https://en.wikipedia.org/wiki/Bragg's_law

wave: string, springs, bars


 * https://en.wikipedia.org/wiki/Chapman-Enskog_theory(https://en.wikipedia.org/wiki/Navier-Stokes_equations https://en.wikipedia.org/wiki/Boltzmann_equation)

--- ++

Group contractionQuantization commutes with reduction

D'Alembert->KdV Hierarchy/Dirac operator->BBGKY

Hamiltonian fluid mechanics Madelung equation De_Broglie–Bohm_theory Convection diffusion equation Langevin dynamics

Lagenvin motion + +

Dyson Series + Functional determinant

Electromagnetism Gravitoelectromagnetism + Kaluza-klein theory Yang-Mills theory

Precession +

Bohr-Sommerfeld EBK-Bohr-sommerfeld cyclotron Bohr-Sommerfeld

Aharonov-Bohm effect Landau quantization

Wilson loop t'hooft loop

T*C->Poison bracket T*M_g->ADM bracket +


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

Black hole: Black hole radiation dark matter - Cosmic Backgrounds: Relics: Photon(CMB) Neutrino Gravitation DEBRA: Infrared X-ray Extragalactic light Radio Gamma-ray CMB

Angles: Weinberg Peccei–Quinn GIM CKM PMKS +

Planck temperature Hagedorn temperature +

Noether's Theorem extensions Wald entropy formula++

stringification=categorization

Types of radioactive decay