Used Notation
$$
\newcommand{\LetThereBe}[2]{\newcommand{#1}{#2}}
\newcommand{\letThereBe}[3]{\newcommand{#1}[#2]{#3}}
% Declare mathematics (so they can be overwritten for PDF)
\newcommand{\declareMathematics}[2]{\DeclareMathOperator{#1}{#2}}
\newcommand{\declareMathematicsStar}[2]{\DeclareMathOperator*{#1}{#2}}
% striked integral
\newcommand{\avint}{\mathop{\mathchoice{\,\rlap{-}\!\!\int}
{\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int}
{\rlap{\raise.09em{\scriptscriptstyle -}}\!\int}
{\rlap{-}\!\int}}\nolimits}
% \d does not work well for PDFs
\LetThereBe{\d}{\differential}
\LetThereBe{\Im}{\mathrm{Im}}
\LetThereBe{\Re}{\mathrm{Re}}
\letThereBe{\imOf}{1}{\Im\,#1}
\letThereBe{\reOf}{1}{\Re\,#1}
\letThereBe{\ImOf}{1}{\Im \brackets{#1}}
\letThereBe{\ReOf}{1}{\Re \brackets{#1}}
\letThereBe{\linefrac}{2}{#1/#2}
$$
$$
% 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{\im}{\mathrm{i}}
% Sets from PDEs
\LetThereBe{\boundary}{\partial}
\letThereBe{\closure}{1}{\overline{#1}}
\letThereBe{\Contf}{1}{C^{#1}}
\letThereBe{\contf}{2}{\Contf{#2}(#1)}
\letThereBe{\compactContf}{2}{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}}
% 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}\;}
% 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}
% 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|}
\declareMathematics{\exp}{exp}
\letThereBe{\Exp}{1}{\exp\brackets{#1}}
\letThereBe{\indicator}{1}{\mathbb{I}_{#1}}
\declareMathematics{\arccot}{arccot}
\declareMathematics{\complexArg}{arg}
\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}}
% 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.}\;}
$$
$$
% Operators - Analysis
\LetThereBe{\hess}{\nabla^2}
\LetThereBe{\lagr}{\mcal L}
\LetThereBe{\lapl}{\Delta}
\declareMathematics{\grad}{grad}
\declareMathematics{\Dgrad}{D}
\LetThereBe{\gradient}{\nabla}
\LetThereBe{\jacobi}{\nabla}
\LetThereBe{\Jacobi}{\vi{\mathrm J}}
\letThereBe{\jacobian}{2}{\Dgrad_{#1}\brackets{#2}}
\LetThereBe{\d}{\mathrm{d}}
\LetThereBe{\dd}{\,\mathrm{d}}
\letThereBe{\partialDeriv}{2}{\frac {\partial #1} {\partial #2}}
\letThereBe{\npartialDeriv}{3}{\partialDeriv{^{#1} #2} {#3^{#1}}}
\letThereBe{\partialOp}{1}{\frac {\partial} {\partial #1}}
\letThereBe{\npartialOp}{2}{\frac {\partial^{#1}} {\partial #2^{#1}}}
\letThereBe{\pDeriv}{2}{\partialDeriv{#1}{#2}}
\letThereBe{\npDeriv}{3}{\npartialDeriv{#1}{#2}{#3}}
\letThereBe{\deriv}{2}{\frac {\d #1} {\d #2}}
\letThereBe{\nderiv}{3}{\frac {\d^{#1} #2} {\d #3^{#1}}}
\letThereBe{\derivOp}{1}{\frac {\d} {\d #1}\,}
\letThereBe{\nderivOp}{2}{\frac {\d^{#1}} {\d #2^{#1}}\,}
$$
$$
% Linear algebra
\letThereBe{\norm}{1}{\left\lVert #1 \right\rVert}
\letThereBe{\scal}{2}{\left\langle #1, #2 \right\rangle}
\letThereBe{\avg}{1}{\overline{#1}}
\letThereBe{\Avg}{1}{\bar{#1}}
\letThereBe{\linspace}{1}{\mathrm{lin}\set{#1}}
\letThereBe{\algMult}{1}{\mu_{\mathrm A} \brackets{#1}}
\letThereBe{\geomMult}{1}{\mu_{\mathrm G} \brackets{#1}}
\LetThereBe{\Nullity}{\mathrm{nullity}}
\letThereBe{\nullity}{1}{\Nullity \brackets{#1}}
\LetThereBe{\nulty}{\nu}
\declareMathematics{\SpanOf}{span}
\letThereBe{\Span}{1}{\SpanOf\set{#1}}
% Linear algebra - Matrices
\LetThereBe{\tr}{\top}
\LetThereBe{\Tr}{^\tr}
\LetThereBe{\pinv}{\dagger}
\LetThereBe{\Pinv}{^\dagger}
\LetThereBe{\Inv}{^{-1}}
\LetThereBe{\ident}{\vi{I}}
\letThereBe{\mtr}{1}{\begin{pmatrix}#1\end{pmatrix}}
\letThereBe{\bmtr}{1}{\begin{bmatrix}#1\end{bmatrix}}
\declareMathematics{\trace}{tr}
\declareMathematics{\diagonal}{diag}
\declareMathematics{\rank}{rank}
$$
$$
% Statistics
\LetThereBe{\iid}{\overset{\text{i.i.d.}}{\sim}}
\LetThereBe{\ind}{\overset{\text{ind}}{\sim}}
\LetThereBe{\condp}{\,\vert\,}
\letThereBe{\complement}{1}{\overline{#1}}
\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}
\LetThereBe{\ltwo}{L^2}
\LetThereBe{\ltwoEq}{\onTop{=}{\ltwo}}
% 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}}
$$
$$
% Distributions
\letThereBe{\WN}{2}{\mathrm{WN}\brackets{#1,#2}}
\declareMathematics{\uniform}{Unif}
\declareMathematics{\binomDist}{Bi}
\declareMathematics{\negbinomDist}{NBi}
\declareMathematics{\betaDist}{Beta}
\declareMathematics{\betabinomDist}{BetaBin}
\declareMathematics{\gammaDist}{Gamma}
\declareMathematics{\igammaDist}{IGamma}
\declareMathematics{\invgammaDist}{IGamma}
\declareMathematics{\expDist}{Ex}
\declareMathematics{\poisDist}{Po}
\declareMathematics{\erlangDist}{Er}
\declareMathematics{\altDist}{A}
\declareMathematics{\geomDist}{Ge}
\LetThereBe{\normalDist}{\mathcal N}
%\declareMathematics{\normalDist}{N}
\letThereBe{\normalD}{1}{\normalDist \brackets{#1}}
\letThereBe{\mvnormalD}{2}{\normalDist_{#1} \brackets{#2}}
\letThereBe{\NormalD}{2}{\normalDist \brackets{#1, #2}}
\LetThereBe{\lognormalDist}{\log\normalDist}
$$
$$
% Game Theory
\LetThereBe{\doms}{\succ}
\LetThereBe{\isdom}{\prec}
\letThereBe{\OfOthers}{1}{_{-#1}}
\LetThereBe{\ofOthers}{\OfOthers{i}}
\LetThereBe{\pdist}{\sigma}
\letThereBe{\domGame}{1}{G_{DS}^{#1}}
\letThereBe{\ratGame}{1}{G_{Rat}^{#1}}
\letThereBe{\bestRep}{2}{\mathrm{BR}_{#1}\brackets{#2}}
\letThereBe{\perf}{1}{{#1}_{\mathrm{perf}}}
\LetThereBe{\perfG}{\perf{G}}
\letThereBe{\imperf}{1}{{#1}_{\mathrm{imp}}}
\LetThereBe{\imperfG}{\imperf{G}}
\letThereBe{\proper}{1}{{#1}_{\mathrm{proper}}}
\letThereBe{\finrep}{2}{{#2}_{#1{\text -}\mathrm{rep}}} %T-stage game
\letThereBe{\infrep}{1}{#1_{\mathrm{irep}}}
\LetThereBe{\repstr}{\tau} %strategy in a repeated game
\LetThereBe{\emptyhist}{\epsilon}
\letThereBe{\extrep}{1}{{#1^{\mathrm{rep}}}}
\letThereBe{\avgpay}{1}{#1^{\mathrm{avg}}}
\LetThereBe{\succf}{\pi} %successor function
\LetThereBe{\playf}{\rho} %player function
\LetThereBe{\actf}{\chi} %action function
$$
$$
% 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}
% 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_{\boundary \ball{#1}{#2}}}
\LetThereBe{\SurfAvg}{\surfAvg{x}{\ve}}
\LetThereBe{\corrF}{\varphi^{\times}}
\LetThereBe{\greenF}{G}
\letThereBe{\reflect}{1}{\tilde{#1}}
\letThereBe{\unitBall}{1}{\alpha(#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}{\mathcal C}
% Numerical methods
\declareMathematics{\globErr}{err}
\declareMathematics{\locErr}{le}
\declareMathematics{\locTrErr}{lte}
\declareMathematics{\incrementFunc}{Inc}
\letThereBe{\incrementF}{1}{\incrementFunc \brackets{#1}}
%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}
$$
For easier orientation in the text, we include a basic list of used notation throughout the text:
| Symbol | Meaning |
|---|---|
| \(\R, \R^+\) | set of all real numbers, \(\set{x \in \R \divider x > 0}\) |
| \(\N, \N_0\) | set of all natural numbers, \(\N \cup \set{0}\) |
| \(\norm{\cdot}_k\) | appropriate (vector, function, etc.) \(L^k\)-norm |
| \(x,\vi x,\rnd X, \vr X\) | scalar, vector, random scalar, random vector variable, respectively |
| \(\scal {\vi u} {\vi v}\) | scalar (dot) product between vectors \(\vi u\) and \(\vi v\) |
| \(f, \vi f\) | scalar-valued, vector-valued function |
| \(f(x)'\) | first derivative of a scalar-valued function \(f\) with respect to scalar variable \(x\) |
| \(\dot{x}(t), \dvi x(t)\) | derivative of a scalar, vector variable (dependent on time) with respect to time \(t\) |
| \(f^{(m)}(x), \vi f^{(m)}(x)\) | \(m\)-th derivatives of a scalar-valued and vector-valued function with respect to a scalar variable \(x\), respectively |
| \(\jacobi f, \jacobi \vi f, \jacobi_{\vi y} \vi f(\vi x, \vi y)\) | gradient of \(f\), Jacobian matrix of \(\vi f\), Jacobian matrix of \(\vi f\) with respect to the 2nd vector argument \(\vi y\) |
| \(\Jacobi\) | Jacobian matrix of a corresponding function \(\vi f\) |
| \(\reOf{x}, \imOf{x}\) | real and imaginary parts of complex variable \(x\), respectively |
| \(\oneToN{n}\) | \(\set{i \in \N | i \leq n} = \set{1,2,\dots,n-1, n}\) |
| \(\stateSpace, \contStateSpace\) | state space of a dynamical system, often \(\R^n\), state space of a semidynamical system, often \(\contf{[-\tau, 0], \stateSpace}{}\) |
| \(\timeSet\) | algebraic structure capturing time for (semi)dynamical system, often \((\R, +)\) |
| \(\vi M\) | monodromy matrix of a cycle |
| \(\vi x_t\) | DDE history segment, \(\vi x_t(s) \letDef \vi x(t + s)\), \(-\tau \leq s \leq 0\) |
| \(\jacobi_{\tau_i} \vi f\) | Jacobian matrix of \(\vi f\) with respect to the state variable corresponding to delay \(\tau_i\) |
| \(\incrementFunc\) | increment function associated with a given numerical ODE solver |
| \(\globErr, \locErr, \locTrErr\) | global, local and local truncation error of a numerical integration method, respectively |
| \(\meshGrid, \indicesOf{\meshGrid}\) | discretization of a given section of 2-parameter plane to a grid, its 2D indices |
| \(\manifold, \curve\) | manifold, typically of a cycle, curve |
| \((a,b\rco\) | interval from \(a\) to \(b\) open on the left and open or closed on the right |
| \(\Phi\) | phase condition for a collocation problem |
| \(\solOp, \infGen\) | solution operator, infinitesimal generator of a linearized DDE |
| \(\evolOp\) | evolution operator of a (semi)dynamical system |
| \(\normalD{\mu, \sigma^2}\) | normal distribution with expected value \(\mu\) and variance \(\sigma^2\) |
| \(\rW(t)\) | Wiener process |
| \(\borelAlgebra\) | Borel \(\sigma\)-algebra |