4 Relevant Results

$$ \newcommand{\LetThereBe}[2]{\newcommand{#1}{#2}} \newcommand{\letThereBe}[3]{\newcommand{#1}[#2]{#3}} \newcommand{\ForceToBe}[2]{\renewcommand{#1}{#2}} \newcommand{\forceToBe}[3]{\renewcommand{#1}[#2]{#3}} \newcommand{\MayThereBe}[2]{\newcommand{#1}{#2}} \newcommand{\mayThereBe}[3]{\newcommand{#1}[#2]{#3}} % Declare mathematics (so they can be overwritten for PDF) \newcommand{\declareMathematics}[2]{\DeclareMathOperator{#1}{#2}} \newcommand{\declareMathematicsStar}[2]{\DeclareMathOperator*{#1}{#2}} % striked integral \newcommand{\avint}{\mathop{\mathchoice{\,\rlap{-}\!\!\int} {\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int} {\rlap{\raise.09em{\scriptscriptstyle -}}\!\int} {\rlap{-}\!\int}}\nolimits} % \d does not work well for PDFs \LetThereBe{\d}{\differential} \LetThereBe{\Im}{\IM} \LetThereBe{\Re}{\RE} \letThereBe{\linefrac}{2}{#1/#2} \LetThereBe{\ExtProd}{\mathsf{\Lambda}} \letThereBe{\unicodeInt}{1}{\mathop{\vcenter{\mathchoice{\huge\unicode{#1}}{\unicode{#1}}{\unicode{#1}}{\unicode{#1}}}}\nolimits} \letThereBe{\Oiint}{1}{\underset{ #1 \;}{ {\rlap{\mspace{1mu} \boldsymbol{\bigcirc}}{\rlap{\int}{\;\int}}} }} \letThereBe{\sOiint}{1}{\unicodeInt{x222F}_{#1}} $$ $$ % Simply for testing \LetThereBe{\foo}{\textrm{FIXME: this is a test!}} % Font styles \letThereBe{\mcal}{1}{\mathcal{#1}} \letThereBe{\chem}{1}{\mathrm{#1}} % Sets \LetThereBe{\C}{\mathbb{C}} \LetThereBe{\R}{\mathbb{R}} \LetThereBe{\Z}{\mathbb{Z}} \LetThereBe{\N}{\mathbb{N}} \LetThereBe{\K}{\mathbb{K}} \LetThereBe{\im}{\mathrm{i}} % Sets from PDEs \LetThereBe{\boundaryOf}{\partial} \letThereBe{\closureOf}{1}{\overline{#1}} \letThereBe{\Contf}{1}{\mcal C^{#1}} \letThereBe{\contf}{2}{\Contf{#2}(#1)} \letThereBe{\compactContf}{2}{\mcal C_c^{#2}(#1)} \letThereBe{\ball}{2}{B\brackets{#1, #2}} \letThereBe{\closedBall}{2}{B\parentheses{#1, #2}} \LetThereBe{\compactEmbed}{\subset\subset} \letThereBe{\inside}{1}{#1^o} \LetThereBe{\neighborhood}{\mcal O} \letThereBe{\neigh}{1}{\neighborhood \brackets{#1}} % Basic notation - vectors and random variables \letThereBe{\vi}{1}{\boldsymbol{#1}} %vector or matrix \letThereBe{\dvi}{1}{\vi{\dot{#1}}} %differentiated vector or matrix \letThereBe{\vii}{1}{\mathbf{#1}} %if \vi doesn't work \letThereBe{\dvii}{1}{\vii{\dot{#1}}} %if \dvi doesn't work \letThereBe{\rnd}{1}{\mathup{#1}} %random variable \letThereBe{\vr}{1}{\mathbf{#1}} %random vector or matrix \letThereBe{\vrr}{1}{\boldsymbol{#1}} %random vector if \vr doesn't work \letThereBe{\dvr}{1}{\vr{\dot{#1}}} %differentiated vector or matrix \letThereBe{\vb}{1}{\pmb{#1}} %#TODO \letThereBe{\dvb}{1}{\vb{\dot{#1}}} %#TODO \letThereBe{\oper}{1}{\mathsf{#1}} \letThereBe{\quotient}{2}{{^{\displaystyle #1}}/{_{\displaystyle #2}}} % Basic notation - general \letThereBe{\set}{1}{\left\{#1\right\}} \letThereBe{\seqnc}{4}{\set{#1_{#2}}_{#2 = #3}^{#4}} \letThereBe{\Seqnc}{3}{\set{#1}_{#2}^{#3}} \letThereBe{\brackets}{1}{\left( #1 \right)} \letThereBe{\parentheses}{1}{\left[ #1 \right]} \letThereBe{\dom}{1}{\mcal{D}\, \brackets{#1}} \letThereBe{\complexConj}{1}{\overline{#1}} \LetThereBe{\divider}{\; \vert \;} \LetThereBe{\gets}{\leftarrow} \letThereBe{\rcases}{1}{\left.\begin{aligned}#1\end{aligned}\right\}} \letThereBe{\rcasesAt}{2}{\left.\begin{alignedat}{#1}#2\end{alignedat}\right\}} \letThereBe{\lcases}{1}{\begin{cases}#1\end{cases}} \letThereBe{\lcasesAt}{2}{\left\{\begin{alignedat}{#1}#2\end{alignedat}\right.} \letThereBe{\evaluateAt}{2}{\left.#1\right|_{#2}} \LetThereBe{\Mod}{\;\mathrm{mod}\;} \LetThereBe{\bigO}{O} \letThereBe{\BigO}{1}{\bigO\brackets{#1}} % Special symbols \LetThereBe{\const}{\mathrm{const}} \LetThereBe{\konst}{\mathrm{konst.}} \LetThereBe{\vf}{\varphi} \LetThereBe{\ve}{\varepsilon} \LetThereBe{\tht}{\theta} \LetThereBe{\Tht}{\Theta} \LetThereBe{\after}{\circ} \LetThereBe{\lmbd}{\lambda} \LetThereBe{\Lmbd}{\Lambda} % Shorthands \LetThereBe{\xx}{\vi x} \LetThereBe{\yy}{\vi y} \LetThereBe{\XX}{\vi X} \LetThereBe{\AA}{\vi A} \LetThereBe{\bb}{\vi b} \LetThereBe{\vvf}{\vi \vf} \LetThereBe{\ff}{\vi f} \LetThereBe{\gg}{\vi g} % Basic functions \letThereBe{\absval}{1}{\left| #1 \right|} \LetThereBe{\id}{\mathrm{id}} \letThereBe{\floor}{1}{\left\lfloor #1 \right\rfloor} \letThereBe{\ceil}{1}{\left\lceil #1 \right\rceil} \declareMathematics{\image}{im} %image \declareMathematics{\domain}{dom} %image \declareMathematics{\tg}{tg} \declareMathematics{\sign}{sign} \declareMathematics{\card}{card} %cardinality \letThereBe{\setSize}{1}{\left| #1 \right|} \LetThereBe{\countElements}{\#} \declareMathematics{\exp}{exp} \letThereBe{\Exp}{1}{\exp\brackets{#1}} \LetThereBe{\ee}{\mathrm{e}} \letThereBe{\indicator}{1}{\mathbb{I}_{#1}} \declareMathematics{\arccot}{arccot} \declareMathematics{\gcd}{gcd} % Greatest Common Divisor \declareMathematics{\lcm}{lcm} % Least Common Multiple \letThereBe{\limInfty}{1}{\lim_{#1 \to \infty}} \letThereBe{\limInftyM}{1}{\lim_{#1 \to -\infty}} % Useful commands \letThereBe{\onTop}{2}{\mathrel{\overset{#2}{#1}}} \letThereBe{\onBottom}{2}{\mathrel{\underset{#2}{#1}}} \letThereBe{\tOnTop}{2}{\mathrel{\overset{\text{#2}}{#1}}} \letThereBe{\tOnBottom}{2}{\mathrel{\underset{\text{#2}}{#1}}} \LetThereBe{\EQ}{\onTop{=}{!}} \LetThereBe{\letDef}{:=} %#TODO: change the symbol \LetThereBe{\isPDef}{\onTop{\succ}{?}} \LetThereBe{\inductionStep}{\tOnTop{=}{induct. step}} \LetThereBe{\fromDef}{\triangleq} % Optimization \declareMathematicsStar{\argmin}{argmin} \declareMathematicsStar{\argmax}{argmax} \letThereBe{\maxOf}{1}{\max\set{#1}} \letThereBe{\minOf}{1}{\min\set{#1}} \declareMathematics{\prox}{prox} \declareMathematics{\loss}{loss} \declareMathematics{\supp}{supp} \letThereBe{\Supp}{1}{\supp\brackets{#1}} \LetThereBe{\constraint}{\text{s.t.}\;} $$ $$ % Operators - Analysis \LetThereBe{\hess}{\nabla^2} \LetThereBe{\lagr}{\mcal L} \LetThereBe{\lapl}{\Delta} \declareMathematics{\grad}{grad} \declareMathematics{\divergence}{div} \declareMathematics{\Dgrad}{D} \LetThereBe{\gradient}{\nabla} \LetThereBe{\jacobi}{\nabla} \LetThereBe{\Jacobi}{\vi{\mathrm J}} \letThereBe{\jacobian}{2}{\Dgrad_{#1}\brackets{#2}} \LetThereBe{\d}{\mathrm{d}} \LetThereBe{\dd}{\,\mathrm{d}} \letThereBe{\partialDeriv}{2}{\frac {\partial #1} {\partial #2}} \letThereBe{\npartialDeriv}{3}{\partialDeriv{^{#1} #2} {#3^{#1}}} \letThereBe{\partialOp}{1}{\frac {\partial} {\partial #1}} \letThereBe{\npartialOp}{2}{\frac {\partial^{#1}} {\partial #2^{#1}}} \letThereBe{\pDeriv}{2}{\partialDeriv{#1}{#2}} \letThereBe{\npDeriv}{3}{\npartialDeriv{#1}{#2}{#3}} \letThereBe{\deriv}{2}{\frac {\d #1} {\d #2}} \letThereBe{\nderiv}{3}{\frac {\d^{#1} #2} {\d #3^{#1}}} \letThereBe{\derivOp}{1}{\frac {\d} {\d #1}\,} \letThereBe{\nderivOp}{2}{\frac {\d^{#1}} {\d #2^{#1}}\,} % Convergence \LetThereBe{\pointwiseTo}{\to} \LetThereBe{\uniformlyTo}{\rightrightarrows} \LetThereBe{\normallyTo}{\tOnTop{\longrightarrow}{norm}} \LetThereBe{\compactlyTo}{\tOnTop{\longrightarrow}{comp.}} \LetThereBe{\locallyUnifTo}{\tOnTop{\longrightarrow}{l.u.}} % Curves \letThereBe{\graphOf}{1}{\parentheses{#1}} \declareMathematics{\interior}{Int} % complex \LetThereBe{\Cinfty}{\tilde{\C}} \declareMathematics{\residual}{res} \letThereBe{\resAt}{1}{\residual_{#1}} \declareMathematics{\complexarg}{arg} \declareMathematics{\complexArg}{Arg} \LetThereBe{\carg}{\complexarg} \LetThereBe{\cArg}{\complexArg} \LetThereBe{\IM}{\mathfrak{Im}} \LetThereBe{\RE}{\mathfrak{Re}} \letThereBe{\imOf}{1}{\IM\,#1} \letThereBe{\reOf}{1}{\RE\,#1} \letThereBe{\ImOf}{1}{\IM \brackets{#1}} \letThereBe{\ReOf}{1}{\RE \brackets{#1}} $$ $$ % Linear algebra \letThereBe{\norm}{1}{\left\lVert #1 \right\rVert} \letThereBe{\seminorm}{1}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}} \letThereBe{\scal}{2}{\left\langle #1, #2 \right\rangle} \letThereBe{\avg}{1}{\overline{#1}} \letThereBe{\Avg}{1}{\bar{#1}} \letThereBe{\linspace}{1}{\mathrm{lin}\set{#1}} \letThereBe{\algMult}{1}{\mu_{\mathrm A} \brackets{#1}} \letThereBe{\geomMult}{1}{\mu_{\mathrm G} \brackets{#1}} \LetThereBe{\Nullity}{\mathrm{nullity}} \letThereBe{\nullity}{1}{\Nullity \brackets{#1}} \LetThereBe{\nulty}{\nu} \declareMathematics{\SpanOf}{span} \letThereBe{\Span}{1}{\SpanOf\set{#1}} \LetThereBe{\projection}{\Pi} % Linear algebra - Matrices \LetThereBe{\tr}{\top} \LetThereBe{\Tr}{^\tr} \LetThereBe{\pinv}{\dagger} \LetThereBe{\Pinv}{^\dagger} \LetThereBe{\Inv}{^{-1}} \LetThereBe{\ident}{\vi{I}} \letThereBe{\mtr}{1}{\begin{pmatrix}#1\end{pmatrix}} \letThereBe{\bmtr}{1}{\begin{bmatrix}#1\end{bmatrix}} \declareMathematics{\trace}{tr} \declareMathematics{\diagonal}{diag} \declareMathematics{\rank}{rank} % Multilinear algebra \LetThereBe{\tensorProd}{\otimes} \LetThereBe{\tprod}{\tensorProd} \LetThereBe{\extProd}{\wedge} \LetThereBe{\wdg}{\extProd} \LetThereBe{\wedges}{\wedge \dots \wedge} \declareMathematics{\altMap}{Alt} $$ $$ % Statistics \LetThereBe{\iid}{\overset{\text{i.i.d.}}{\sim}} \LetThereBe{\ind}{\overset{\text{ind}}{\sim}} \LetThereBe{\condp}{\,\vert\,} \letThereBe{\complementOf}{1}{{{#1}^c}} \LetThereBe{\acov}{\gamma} \LetThereBe{\acf}{\rho} \LetThereBe{\stdev}{\sigma} \LetThereBe{\procMean}{\mu} \LetThereBe{\procVar}{\stdev^2} \declareMathematics{\variance}{var} \letThereBe{\Variance}{1}{\variance \brackets{#1}} \declareMathematics{\cov}{cov} \declareMathematics{\corr}{cor} \letThereBe{\sampleVar}{1}{\rnd S^2_{#1}} \letThereBe{\populationVar}{1}{V_{#1}} \declareMathematics{\expectedValue}{\mathbb{E}} \declareMathematics{\rndMode}{Mode} \letThereBe{\RndMode}{1}{\rndMode\brackets{#1}} \letThereBe{\expect}{1}{\expectedValue #1} \letThereBe{\Expect}{1}{\expectedValue \brackets{#1}} \letThereBe{\expectIn}{2}{\expectedValue_{#1} #2} \letThereBe{\ExpectIn}{2}{\expectedValue_{#1} \brackets{#2}} \LetThereBe{\betaF}{\mathrm B} \LetThereBe{\fisherMat}{J} \LetThereBe{\mutInfo}{I} \LetThereBe{\expectedGain}{I_e} \letThereBe{\KLDiv}{2}{D\brackets{#1 \parallel #2}} \LetThereBe{\entropy}{H} \LetThereBe{\diffEntropy}{h} \LetThereBe{\probF}{\pi} \LetThereBe{\densF}{\vf} \LetThereBe{\att}{_t} %at time \letThereBe{\estim}{1}{\hat{#1}} \letThereBe{\estimML}{1}{\hat{#1}_{\mathrm{ML}}} \letThereBe{\estimOLS}{1}{\hat{#1}_{\mathrm{OLS}}} \letThereBe{\estimMAP}{1}{\hat{#1}_{\mathrm{MAP}}} \letThereBe{\predict}{3}{\estim {\rnd #1}_{#2 | #3}} \letThereBe{\periodPart}{3}{#1+#2-\ceil{#2/#3}#3} \letThereBe{\infEstim}{1}{\tilde{#1}} \letThereBe{\predictDist}{1}{{#1}^*} \LetThereBe{\backs}{\oper B} \LetThereBe{\diff}{\oper \Delta} \LetThereBe{\BLP}{\oper P} \LetThereBe{\arPoly}{\Phi} \letThereBe{\ArPoly}{1}{\arPoly\brackets{#1}} \LetThereBe{\maPoly}{\Theta} \letThereBe{\MaPoly}{1}{\maPoly\brackets{#1}} \letThereBe{\ARmod}{1}{\mathrm{AR}\brackets{#1}} \letThereBe{\MAmod}{1}{\mathrm{MA}\brackets{#1}} \letThereBe{\ARMA}{2}{\mathrm{ARMA}\brackets{#1, #2}} \letThereBe{\sARMA}{3}{\mathrm{ARMA}\brackets{#1}\brackets{#2}_{#3}} \letThereBe{\SARIMA}{3}{\mathrm{ARIMA}\brackets{#1}\brackets{#2}_{#3}} \letThereBe{\ARIMA}{3}{\mathrm{ARIMA}\brackets{#1, #2, #3}} \LetThereBe{\pacf}{\alpha} \letThereBe{\parcorr}{3}{\rho_{#1 #2 | #3}} \LetThereBe{\noise}{\mathscr{N}} \LetThereBe{\jeffreys}{\mathcal J} \LetThereBe{\likely}{\mcal L} \letThereBe{\Likely}{1}{\likely\brackets{#1}} \LetThereBe{\loglikely}{\mcal l} \letThereBe{\Loglikely}{1}{\loglikely \brackets{#1}} \LetThereBe{\CovMat}{\Gamma} \LetThereBe{\covMat}{\vi \CovMat} \LetThereBe{\rcovMat}{\vrr \CovMat} \LetThereBe{\AIC}{\mathrm{AIC}} \LetThereBe{\BIC}{\mathrm{BIC}} \LetThereBe{\AICc}{\mathrm{AIC}_c} \LetThereBe{\nullHypo}{H_0} \LetThereBe{\altHypo}{H_1} \LetThereBe{\rve}{\rnd \ve} \LetThereBe{\rtht}{\rnd \theta} \LetThereBe{\rX}{\rnd X} \LetThereBe{\rY}{\rnd Y} \LetThereBe{\rZ}{\rnd Z} \LetThereBe{\rA}{\rnd A} \LetThereBe{\rB}{\rnd B} \LetThereBe{\vrZ}{\vr Z} \LetThereBe{\vrY}{\vr Y} \LetThereBe{\vrX}{\vr X} \LetThereBe{\rW}{\rnd W} \LetThereBe{\rS}{\rnd S} \LetThereBe{\rM}{\rnd M} \LetThereBe{\rtau}{\rnd \tau} % Bayesian inference \LetThereBe{\paramSet}{\mcal T} \LetThereBe{\sampleSet}{\mcal Y} \LetThereBe{\bayesSigmaAlg}{\mcal B} % Different types of convergence \LetThereBe{\inDist}{\onTop{\to}{d}} \letThereBe{\inDistWhen}{1}{\onBottom{\onTop{\longrightarrow}{d}}{#1}} \LetThereBe{\inProb}{\onTop{\to}{P}} \letThereBe{\inProbWhen}{1}{\onBottom{\onTop{\longrightarrow}{P}}{#1}} \LetThereBe{\inMeanSq}{\onTop{\to}{\ltwo}} \LetThereBe{\inltwo}{\onTop{\to}{\ltwo}} \letThereBe{\inMeanSqWhen}{1}{\onBottom{\onTop{\longrightarrow}{\ltwo}}{#1}} \LetThereBe{\convergeAS}{\tOnTop{\to}{a.s.}} \letThereBe{\convergeASWhen}{1}{\onBottom{\tOnTop{\longrightarrow}{a.s.}}{#1}} % Asymptotic qualities \LetThereBe{\simAsymp}{\tOnTop{\sim}{as.}} % Stochastic analysis \letThereBe{\diffOn}{2}{\diff #1_{[#2]}} % \LetThereBe{\timeSet}{\Theta} \LetThereBe{\eventSet}{\Omega} \LetThereBe{\filtration}{\mcal F} % TODO: Rename allFiltrations and the like \letThereBe{\allFiltrations}{1}{\set{\filtration_t}_{#1}} \letThereBe{\natFilter}{1}{\filtration_t^{#1}} \letThereBe{\NatFilter}{2}{\filtration_{#2}^{#1}} \letThereBe{\filterAll}{1}{\set{#1}_{t \geq 0}} \letThereBe{\FilterAll}{2}{\set{#1}_{#2}} \LetThereBe{\borelAlgebra}{\mcal B} \LetThereBe{\sAlgebra}{\mcal A} \LetThereBe{\quadVar}{Q} \LetThereBe{\totalVar}{V} \LetThereBe{\adaptIntProcs}{\mcal M} \letThereBe{\reflectProc}{2}{#1^{#2}} $$ $$ % Distributions \letThereBe{\WN}{2}{\mathrm{WN}\brackets{#1,#2}} \declareMathematics{\uniform}{Unif} \declareMathematics{\binomDist}{Bi} \declareMathematics{\negbinomDist}{NBi} \declareMathematics{\betaDist}{Beta} \declareMathematics{\betabinomDist}{BetaBin} \declareMathematics{\gammaDist}{Gamma} \declareMathematics{\igammaDist}{IGamma} \declareMathematics{\invgammaDist}{IGamma} \declareMathematics{\expDist}{Ex} \declareMathematics{\poisDist}{Po} \declareMathematics{\erlangDist}{Er} \declareMathematics{\altDist}{A} \declareMathematics{\geomDist}{Ge} \LetThereBe{\normalDist}{\mathcal N} %\declareMathematics{\normalDist}{N} \letThereBe{\normalD}{1}{\normalDist \brackets{#1}} \letThereBe{\mvnormalD}{2}{\normalDist_{#1} \brackets{#2}} \letThereBe{\NormalD}{2}{\normalDist \brackets{#1, #2}} \LetThereBe{\lognormalDist}{\log\normalDist} $$ $$ % Game Theory \LetThereBe{\doms}{\succ} \LetThereBe{\isdom}{\prec} \letThereBe{\OfOthers}{1}{_{-#1}} \LetThereBe{\ofOthers}{\OfOthers{i}} \LetThereBe{\pdist}{\sigma} \letThereBe{\domGame}{1}{G_{DS}^{#1}} \letThereBe{\ratGame}{1}{G_{Rat}^{#1}} \letThereBe{\bestRep}{2}{\mathrm{BR}_{#1}\brackets{#2}} \letThereBe{\perf}{1}{{#1}_{\mathrm{perf}}} \LetThereBe{\perfG}{\perf{G}} \letThereBe{\imperf}{1}{{#1}_{\mathrm{imp}}} \LetThereBe{\imperfG}{\imperf{G}} \letThereBe{\proper}{1}{{#1}_{\mathrm{proper}}} \letThereBe{\finrep}{2}{{#2}_{#1{\text -}\mathrm{rep}}} %T-stage game \letThereBe{\infrep}{1}{#1_{\mathrm{irep}}} \LetThereBe{\repstr}{\tau} %strategy in a repeated game \LetThereBe{\emptyhist}{\epsilon} \letThereBe{\extrep}{1}{{#1^{\mathrm{rep}}}} \letThereBe{\avgpay}{1}{#1^{\mathrm{avg}}} \LetThereBe{\succf}{\pi} %successor function \LetThereBe{\playf}{\rho} %player function \LetThereBe{\actf}{\chi} %action function $$ $$ \LetThereBe{\fourierOp}{\mcal{F}} \letThereBe{\fourier}{1}{\widehat{#1}} \letThereBe{\ifourier}{1}{\check{#1}} % Shortcuts \letThereBe{\FT}{1}{\fourier{#1}} \letThereBe{\iFT}{1}{\ifourier{#1}} \LetThereBe{\FTOp}{\fourierOp} \LetThereBe{\lspace}{\mcal L} \LetThereBe{\lone}{\lspace^{1}} \letThereBe{\Lone}{1}{\lone\brackets{#1}} \LetThereBe{\ltwo}{\lspace^2} \letThereBe{\Ltwo}{1}{\ltwo\brackets{#1}} \letThereBe{\lp}{1}{\lspace^{#1}} \letThereBe{\Lp}{2}{\lp{#1}\brackets{#2}} \LetThereBe{\linfty}{\lspace^{\infty}} \letThereBe{\Linfty}{1}{\linfty\brackets{#1}} \LetThereBe{\ltwoEq}{\onTop{=}{\ltwo}} \letThereBe{\decayContf}{1}{\mcal C_0\brackets{#1}} \letThereBe{\cinftyContf}{1}{\mcal C^{\infty}_c\brackets{#1}} \LetThereBe{\unitBall}{\mathbb{S}} \LetThereBe{\unitBallSurface}{S} \LetThereBe{\gammaF}{\Gamma} \letThereBe{\GammaF}{1}{\gammaF\brackets{#1}} \LetThereBe{\betaF}{B} \letThereBe{\BetaF}{1}{\betaF\brackets{#1}} \LetThereBe{\ofOrder}{\asymp} \declareMathematics{\vol}{vol} \LetThereBe{\holomorphic}{\mcal H} \LetThereBe{\hlmr}{\holomorphic} \LetThereBe{\schwartz}{\mcal S} \LetThereBe{\testSpace}{\mcal D} \letThereBe{\TestSpace}{1}{\testSpace\brackets{#1}} \LetThereBe{\bumpf}{\psi} \letThereBe{\Bumpf}{1}{\bumpf\brackets{#1}} \letThereBe{\reverse}{1}{\check{#1}} \LetThereBe{\translationOp}{\mcal{T}} \letThereBe{\translateBy}{1}{\translationOp_{#1}} \LetThereBe{\flcOp}{\mathscr{F}} \letThereBe{\flc}{1}{\flcOp\brackets{#1}} \LetThereBe{\hodge}{*} \LetThereBe{\tangent}{\mathrm{T}} \letThereBe{\tangentAt}{1}{\tangent_{#1}} \letThereBe{\lieBracket}{2}{\left[#1, #2\right]} \declareMathematics{\degree}{deg} \LetThereBe{\forms}{\mathscr{F}} \letThereBe{\kforms}{1}{\forms^{#1}} \letThereBe{\formsOver}{1}{\forms\brackets{#1}} \letThereBe{\kformsOver}{2}{\kforms{#1}\brackets{#2}} \letThereBe{\pullback}{1}{{#1}^*} \LetThereBe{\atlas}{\mcal A} \LetThereBe{\adjd}{\oper \delta} \LetThereBe{\connect}{\nabla} \LetThereBe{\christoffel}{\Gamma} \letThereBe{\lengthOf}{1}{\mathcal{l}\brackets{#1}} \LetThereBe{\HOT}{\mathrm{h.o.t.}} $$ $$ % ODEs \LetThereBe{\timeInt}{\mcal I} \LetThereBe{\stimeInt}{\mcal J} \LetThereBe{\Wronsk}{\mcal W} \letThereBe{\wronsk}{1}{\Wronsk \parentheses{#1}} \LetThereBe{\prufRadius}{\rho} \LetThereBe{\prufAngle}{\vf} \LetThereBe{\weyr}{\sigma} \LetThereBe{\linDifOp}{\mathsf{L}} \LetThereBe{\Hurwitz}{\vi H} \letThereBe{\hurwitz}{1}{\Hurwitz \brackets{#1}} % Cont. Models \LetThereBe{\dirac}{\delta} \LetThereBe{\torus}{\mathbb{T}} % PDEs % \avint -- defined in format-respective tex files \LetThereBe{\fundamental}{\Phi} \LetThereBe{\fund}{\fundamental} \letThereBe{\normaDeriv}{1}{\partialDeriv{#1}{\vec{n}}} \letThereBe{\volAvg}{2}{\avint_{\ball{#1}{#2}}} \LetThereBe{\VolAvg}{\volAvg{x}{\ve}} \letThereBe{\surfAvg}{2}{\avint_{\boundaryOf \ball{#1}{#2}}} \LetThereBe{\SurfAvg}{\surfAvg{x}{\ve}} \LetThereBe{\corrF}{\varphi^{\times}} \LetThereBe{\greenF}{G} \letThereBe{\reflect}{1}{\tilde{#1}} \LetThereBe{\conv}{*} \letThereBe{\dotP}{2}{#1 \cdot #2} \letThereBe{\translation}{1}{\tau_{#1}} \declareMathematics{\dist}{dist} \letThereBe{\regularizef}{1}{\eta_{#1}} \letThereBe{\fourier}{1}{\widehat{#1}} \letThereBe{\ifourier}{1}{\check{#1}} \LetThereBe{\fourierOp}{\mcal F} \LetThereBe{\ifourierOp}{\mcal F^{-1}} \letThereBe{\FourierOp}{1}{\fourierOp\set{#1}} \letThereBe{\iFourierOp}{1}{\ifourierOp\set{#1}} \LetThereBe{\laplaceOp}{\mcal L} \letThereBe{\LaplaceOp}{1}{\laplaceOp\set{#1}} \letThereBe{\Norm}{1}{\absval{#1}} % SINDy \LetThereBe{\Koop}{\mcal K} \letThereBe{\oneToN}{1}{\left[#1\right]} \LetThereBe{\meas}{\mathrm{m}} \LetThereBe{\stateLoss}{\mcal J} \LetThereBe{\lagrm}{p} % Stochastic analysis \LetThereBe{\RiemannInt}{(\mcal R)} \LetThereBe{\RiemannStieltjesInt}{(\mcal {R_S})} \LetThereBe{\LebesgueInt}{(\mcal L)} \LetThereBe{\ItoInt}{(\mcal I)} \LetThereBe{\Stratonovich}{\circ} \LetThereBe{\infMean}{\alpha} \LetThereBe{\infVar}{\beta} % Dynamical systems \LetThereBe{\nUnit}{\mathrm N} \LetThereBe{\timeUnit}{\mathrm T} % Masters thesis \LetThereBe{\evolOp}{\oper{\vf}} \letThereBe{\obj}{1}{\mathbb{#1}} \LetThereBe{\timeSet}{\obj T} \LetThereBe{\stateSpace}{\obj X} \LetThereBe{\contStateSpace}{\stateSpace_{C}} \LetThereBe{\orbit}{Or} \letThereBe{\Orbit}{1}{\orbit\brackets{#1}} \LetThereBe{\limitSet}{\obj \Lambda} \LetThereBe{\crossSection}{\obj \Sigma} \declareMathematics{\codim}{codim} % Left and right closed-or-open intervals \LetThereBe{\lco}{\langle} \LetThereBe{\rco}{\rangle} \letThereBe{\testInt}{1}{\mathrm{Int}_{#1}} \letThereBe{\evalOp}{1}{\oper{\eta}_{#1}} \LetThereBe{\nonzeroEl}{\bullet} \LetThereBe{\zeroEl}{\circ} \LetThereBe{\solOp}{\oper{S}} \LetThereBe{\infGen}{\oper{A}} \LetThereBe{\indexSet}{\mcal I} \letThereBe{\indicesOf}{1}{\indexSet\parentheses{#1}} \letThereBe{\IndicesOf}{2}{\indexSet_{#2}\parentheses{#1}} \LetThereBe{\meshGrid}{\obj M} \declareMathematics{\starter}{starter} \declareMathematics{\indexer}{indx} \declareMathematics{\enumerator}{enum} \LetThereBe{\inSS}{_{\infty}} \LetThereBe{\manifold}{\mcal M} \LetThereBe{\curve}{\mcal C} % Numerical methods \declareMathematics{\globErr}{err} \declareMathematics{\locErr}{le} \declareMathematics{\locTrErr}{lte} \declareMathematics{\estimErr}{est} \declareMathematics{\incrementFunc}{Inc} \letThereBe{\incrementF}{1}{\incrementFunc \brackets{#1}} \LetThereBe{\discreteNodes}{\mcal T} \LetThereBe{\stableFunc}{R} \letThereBe{\stableF}{1}{\stableFunc\brackets{#1}} \LetThereBe{\stableRegion}{\Omega} %Stochastic analysis \LetThereBe{\RiemannInt}{(\mcal R)} \LetThereBe{\RiemannStieltjesInt}{(\mcal {R_S})} \LetThereBe{\LebesgueInt}{(\mcal L)} \LetThereBe{\ItoInt}{(\mcal I)} \LetThereBe{\Stratonovich}{\circ} \LetThereBe{\infMean}{\alpha} \LetThereBe{\infVar}{\beta} %Optimization \LetThereBe{\goldRatio}{\tau} %Interpolation \LetThereBe{\lagrPoly}{l} $$

Theorem 4.1 (Monotone convergence [1]) Let $(S, \Sigma, \mu)$ be a measure space and $X \in \Sigma$. If $(f_n)$ is a sequence of non-negative measurable functions on $X$ such that \[ 0 \leq f_1(x) \leq f_2(x) \leq \dots \] for all $x \in X$, i.e., $(f_n)$ is monotone increasing, then the point-wise supremum $f \letDef \sup_{n} f_n$ is measurable and \[ \int_X f \dd \mu = \lim_{n \to \infty} \int_X f_n \dd \mu = \sup_{n} \int_X f_n \dd \mu. \]

Theorem 4.2 (Dominated convergence [2]) Let $(f_n)$ be a sequence of complex-valued measurable functions on a given measure space $(S, \Sigma, \mu)$. Suppose that $f_n \to f$, i.e., $f_n$ converges point-wise to $f$, \[ \lim_{n \to \infty} f_n(x) = f(x) \] for every $x \in S$. Assume that $f_n$ is dominated by some integrable function $g$ (called majorant) in the sense \[ \absval{f_n(x)} \leq g(x) \] for all points $x \in S$ and indices $n$. Then $f_n$, $f$ are integrable and \[ \lim_{n \to \infty} \int_S f_n \dd \mu = \int_S \lim_{n \to \infty} f_n \dd \mu = \int_S f \dd \mu. \]

Proposition 4.1 (Jensen’s inequality [3]) Let $(\Omega, \mcal A, \mu)$ be a probability space. Let $f : \Omega \to \R$ be a $\mu$-measurable function and $\vf : \R \to \R$ convex. Then \[ \vf\brackets{\int_{\Omega} f \dd \mu} \leq \int_{\Omega} \vf \after f \dd \mu, \] and, in particular, \[ \int_{\Omega} f(x) \dd \mu(x) \leq \brackets{\int_{\Omega} \absval{f(x)}^p \dd \mu(x)}^{\frac 1 p}. \]

Lemma 4.1 (Fatou’s [4]) Given a measure space $(\Omega, \mcal A, \mu)$ and a set $X \in \mcal A$, let $(f_n)$ be a sequence of $\mu$-measurable non-negative functions on $X$. Define $f(x) = \liminf_{n \to \infty} f_n(x)$ for every $x \in X$. Then $f$ is $\mu$-measurable, and \[ \int_X f \dd \mu \leq \liminf_{n \to \infty} \int_X f_n \dd \mu. \]

Theorem 4.3 (Riesz-Fischer [5]) For $1 \leq p < \infty$, $\lp{p}(E)$ is a Banach space¹. Furthermore, if $f_n \onBottom{\to}{\lp{p}} f$, then $(f_n)_n$ has a subsequence that converges to $f$ point-wise almost everywhere on $E$.

Theorem 4.4 (Cauchy-Schwarz inequality [6]) For all vectors $\vi u, \vi v$ of an inner product space, the following holds \[ \absval{\scal{\vi u}{\vi v}} \leq \norm{u} \norm{v}. \]

Theorem 4.5 (Riesz representation [7]) Let $H$ be a Hilbert space[^hilbert] whose inner product $\scal {x} {y}$ is linear in its first argument and anti-linear in its second argument. For every continuous linear functional $\vf \in H^*$, there exists a unique vector $f_{\vf} \in H$, called the Riesz representation of $\vf$, such that \[ \vf(x) = \scal x {f_{\vf}} \quad \forall x \in H. \] Importantly, for complex HIlbert spaces, $f_{\vf}$ is always located in the anti-linear coordinate of the inner product.

Definition 4.1 (Gamma function) We define a factorial extension to complex numbers $\gammaF$ as \[ \GammaF{z} = \int_0^{\infty} t^{z - 1} \ee^{-t} \dd t, \] for $\reOf{z} > 0$. Moreover, it holds that \[ \GammaF{z + 1} = z \GammaF{z} \]

Theorem 4.6 (Stone-Weierstrass) Suppose $X$ is a compact Hausdorff space² and $A$ is subalgebra of $\contf{X, \R}{}$ which contains a non-zero constant function. Then $A$ is dense in $\contf{X, \R}{}$ if and only if it separates points.

Theorem 4.7 (Liouville’s) Bounded entire function is constant.

Theorem 4.8 (Morera’s) Let $f$ be continuous on an open subset $D \subseteq \C$ such that $\oint_{\gamma} f(z) \dd z = 0$ for every closed piecewise-$\mcal C^1$ curve $\gamma$ in $D$. Then $f$ is holomorphic on $D$.

Theorem 4.9 (Cauchy’s Integral Theorem) Let $D \subseteq \C$ be simply connected open subset and $f$ is holomorphic in $D$. Then for every smoothed closed curve $\gamma$ in $D$ (i.e., $[\gamma] \subseteq D$) holds $\oint_{\gamma} f(z) \dd z = 0$.

Theorem 4.10 (Maximum principle) Let $f$ be holomorphic on bounded connected open subset $D \subset C$ and continuous on $\closureOf{D}$, i.e., $f \in \hlmr(D) \cap \contf{\closureOf{D}}{}$. If $\absval{f(z)} \leq M$ for all $z \in \boundaryOf D$ and some $M > 0$, then $\absval{f(z)} \leq M$ for all $z \in D$ such that $\absval{f(z)} = M$ occurs for $z \in D$ only if $f$ is constant on $D$.

Lemma 4.2 (Zorn’s) Let $P$ be a partially ordered set that satisfies the following two properties:

$P$ is nonempty;
Every chain in $P$ has an upper bound in $P$.

Then $P$ has at least one maximal element.

Banach space is a complete normed space.↩︎
Hausdorff space is a topological space where distinct points have disjoint neighborhoods↩︎