FOURIER FIELD
f^(ω)=∫−∞∞f(x)e−iωxdxf(x)=2π1∫f^(ω)eiωxdωPROJECTION
P=QQ⊤r=b−Pb⊥col(Q)GREEN LOOP
∮∂S−ydx+xdy=2A∬S(∇×F)⋅dSNORM GEOMETRY
∥v∥S2=v⊤Sv∥A∥2=∥x∥2=1max∥Ax∥2RANK NULLITY
rank(A)+dimkerA=nAx=0LAPLACE MODE
L{f}(s)=∫0∞f(t)e−stdtL{f′}=sF(s)−f(0)STOKES RIM
∬S(∇×F)⋅dS=∮∂SF⋅dr∂S→SLAURENT BAND
f(z)=n=−∞∑∞an(z−z0)nan=2πi1∮(z−z0)n+1f(z)dzTENSOR TRACE
(AB)ij=k∑AikBkjtr(A)=i∑AiiTIKHONOV
xmin∥Ax−b∥22+λ∥x∥22xλ=(A⊤A+λI)−1A⊤bSHERMAN MORRISON
(A+uv⊤)−1=A−1−1+v⊤A−1uA−1uv⊤A−1ΔA=uv⊤SPARSE CODE
x=Ds,∥s∥0≪nsmin∥y−Ds∥22CONDITION NUMBER
κ2(A)=σminσmax∥x∥∥δx∥≲κ(A)∥b∥∥δb∥ORTHOGONAL MAP
Q⊤Q=I∥Qx∥2=∥x∥2FINITE DIFFERENCE
h2ui+1−2ui+ui−1≈u′′(xi)∂xu≈2hui+1−ui−1SVD SKETCH
X≈UrΣrVr⊤∥X−Xr∥2=σr+1BAYES UPDATE
P(A∣B)=P(B)P(B∣A)P(A)P(B)=i∑P(B∣Ai)P(Ai)HEAT TRACE
∂tu=αΔuu(x,t)=k∑cke−αλktϕk(x)EIGEN FLOW
A=SΛS−1etA=SetΛS−1KRYLOV SPACE
Km(A,r0)=span{r0,Ar0,…}xm∈x0+KmHOUSEHOLDER
H=I−2vv⊤H⊤H=IMARKOV STEP
pk+1=Ppk1⊤P=1⊤WAVE MODE
utt=c2Δuu^tt=−c2∥k∥2u^GAUSS FLUX
∭V∇⋅FdV=∬∂VF⋅ndSΦ=F⋅nRESIDUE
∮γf(z)dz=2πik∑Res(f,zk)a−1=Res(f,z0)DE MOIVRE
(cosθ+isinθ)n=cosnθ+isinnθzn=rneinθHESSIAN
Hij=∂xi∂xj∂2fδf≈∇f⊤δx+21δx⊤HδxKALMAN GAIN
K=PH⊤(HPH⊤+R)−1x^+=x^−+KrRLS UPDATE
Pk=λ−1(Pk−1−Kkxk⊤Pk−1)Kk=λ+xk⊤Pk−1xkPk−1xkVANDERMONDE
Vij=λji−1x(0)=VcCHOLESKY
S=LL⊤x⊤Sx>0GRAM SCHMIDT
qk=∥⋅∥2ak−∑j<k(qj⊤ak)qjspan{qj}=span{aj}RK4 STEP
xn+1=xn+6h(k1+2k2+2k3+k4)k1=f(tn,xn)LEAST SQUARES
xmin∥Ax−b∥22x⋆=(A⊤A)†A⊤bPDE FLOW
∇⋅u=0∂tu+∇p=νΔuQUADRATIC FORM
q(x)=x⊤Sx∇xq=(S+S⊤)xCOMPRESSED SENSING
xmin∥x∥1s.t.Φx=yQR FACTOR
A=QRQ⊤Q=IPCA AXIS
C=m1XX⊤Cuk=λkukDFT GRID
Xk=n=0∑N−1xne−i2πkn/NF−1=N1FPOISSON FIELD
−Δu=f∇⋅∇u=ΔuCAUCHY RIEMANN
ux=vy,uy=−vxf′(z)=ux+ivxEULER SPIRAL
eiθ=cosθ+isinθz=reiθJACOBIAN
Jij=∂xj∂Fidy=JdxGRADIENT STEP
xk+1=xk−η∇f(xk)∇f=0WOODBURY
(A+UCV)−1=A−1−A−1UM−1VA−1M=C−1+VA−1UINCOHERENCE
μ=ni,jmax∣⟨ϕi,ψj⟩∣y=ΦΨsLU PIVOT
PA=LUdetA=detPi∏UiiPSEUDOINVERSE
A†=VΣ†U⊤AA†b=projcol(A)bDETERMINANT VOLUME
Vol=∣detA∣det(AB)=detAdetB

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