\(\newcommand{\footnotename}{footnote}\)\(\def \LWRfootnote {1}\)\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)\(\newcommand \ensuremath [1]{#1}\)\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)\(\newcommand {\setlength }[2]{}\)\(\newcommand {\addtolength }[2]{}\)\(\newcommand {\setcounter }[2]{}\)\(\newcommand {\addtocounter }[2]{}\)\(\newcommand {\cline }[1]{}\)\(\newcommand {\directlua }[1]{\text {(directlua)}}\)\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)\(\newcommand {\protect }{}\)\(\def \LWRabsorbnumber #1 {}\)\(\def \LWRabsorbquotenumber "#1 {}\)\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)\(\def \mathcode #1={\mathchar }\)\(\let \delcode \mathcode \)\(\let \delimiter \mathchar \)\(\let \LWRref \ref \)\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)\(\newcommand {\mathllap }[2][]{{#1#2}}\)\(\newcommand {\mathrlap }[2][]{{#1#2}}\)\(\newcommand {\mathclap }[2][]{{#1#2}}\)\(\newcommand {\mathmbox }[1]{#1}\)\(\newcommand {\clap }[1]{#1}\)\(\newcommand {\LWRmathmakebox }[2][]{#2}\)\(\newcommand {\mathmakebox }[1][]{\LWRmathmakebox }\)\(\newcommand {\cramped }[2][]{{#1#2}}\)\(\newcommand {\crampedllap }[2][]{{#1#2}}\)\(\newcommand {\crampedrlap }[2][]{{#1#2}}\)\(\newcommand {\crampedclap }[2][]{{#1#2}}\)\(\newenvironment {crampedsubarray}[1]{}{}\)\(\newcommand {\crampedsubstack }{}\)\(\newcommand {\smashoperator }[2][]{#2\limits }\)\(\newcommand {\adjustlimits }{}\)\(\newcommand {\SwapAboveDisplaySkip }{}\)\(\require {extpfeil}\)\(\Newextarrow \xleftrightarrow {10,10}{0x2194}\)\(\Newextarrow \xLeftarrow {10,10}{0x21d0}\)\(\Newextarrow \xhookleftarrow {10,10}{0x21a9}\)\(\Newextarrow \xmapsto {10,10}{0x21a6}\)\(\Newextarrow \xRightarrow {10,10}{0x21d2}\)\(\Newextarrow \xLeftrightarrow {10,10}{0x21d4}\)\(\Newextarrow \xhookrightarrow {10,10}{0x21aa}\)\(\Newextarrow \xrightharpoondown {10,10}{0x21c1}\)\(\Newextarrow \xleftharpoondown {10,10}{0x21bd}\)\(\Newextarrow \xrightleftharpoons {10,10}{0x21cc}\)\(\Newextarrow \xrightharpoonup {10,10}{0x21c0}\)\(\Newextarrow \xleftharpoonup {10,10}{0x21bc}\)\(\Newextarrow \xleftrightharpoons {10,10}{0x21cb}\)\(\newcommand {\LWRdounderbracket }[3]{\underset {#3}{\underline {#1}}}\)\(\newcommand {\LWRunderbracket }[2][]{\LWRdounderbracket {#2}}\)\(\newcommand {\underbracket }[1][]{\LWRunderbracket }\)\(\newcommand {\LWRdooverbracket }[3]{\overset {#3}{\overline {#1}}}\)\(\newcommand {\LWRoverbracket }[2][]{\LWRdooverbracket {#2}}\)\(\newcommand {\overbracket }[1][]{\LWRoverbracket }\)\(\newcommand {\LATEXunderbrace }[1]{\underbrace {#1}}\)\(\newcommand {\LATEXoverbrace }[1]{\overbrace {#1}}\)\(\newenvironment {matrix*}[1][]{\begin {matrix}}{\end {matrix}}\)\(\newenvironment {pmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)\(\newenvironment {bmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)\(\newenvironment {Bmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)\(\newenvironment {vmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)\(\newenvironment {Vmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)\(\newenvironment {smallmatrix*}[1][]{\begin {matrix}}{\end {matrix}}\)\(\newenvironment {psmallmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)\(\newenvironment {bsmallmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)\(\newenvironment {Bsmallmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)\(\newenvironment {vsmallmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)\(\newenvironment {Vsmallmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)\(\newenvironment {psmallmatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)\(\newenvironment {bsmallmatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)\(\newenvironment {Bsmallmatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)\(\newenvironment {vsmallmatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)\(\newenvironment {Vsmallmatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)\(\let \LWRorigshoveleft \shoveleft \)\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)\(\let \LWRorigshoveright \shoveright \)\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)\(\newenvironment {dcases}{\begin {cases}}{\end {cases}}\)\(\newenvironment {dcases*}{\begin {cases}}{\end {cases}}\)\(\newenvironment {rcases}{\begin {cases}}{\end {cases}}\)\(\newenvironment {rcases*}{\begin {cases}}{\end {cases}}\)\(\newenvironment {drcases}{\begin {cases}}{\end {cases}}\)\(\newenvironment {drcases*}{\begin {cases}}{\end {cases}}\)\(\newenvironment {cases*}{\begin {cases}}{\end {cases}}\)\(\newcommand {\MoveEqLeft }[1][]{}\)\(\def \LWRAboxed #1!|!{\fbox {\(#1\)}&\fbox {\(#2\)}} \newcommand {\Aboxed }[1]{\LWRAboxed #1&&!|!} \)\( \newcommand {\LWRABLines }[1][\Updownarrow ]{#1 \notag \\}\newcommand {\ArrowBetweenLines }{\ifstar \LWRABLines \LWRABLines } \)\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)\(\newcommand {\vdotswithin }[1]{\hspace {.5em}\vdots }\)\(\newcommand {\LWRshortvdotswithinstar }[1]{\vdots \hspace {.5em} & \\}\)\(\newcommand {\LWRshortvdotswithinnostar }[1]{& \hspace {.5em}\vdots \\}\)\(\newcommand {\shortvdotswithin }{\ifstar \LWRshortvdotswithinstar \LWRshortvdotswithinnostar }\)\(\newcommand {\MTFlushSpaceAbove }{}\)\(\newcommand {\MTFlushSpaceBelow }{\\}\)\(\newcommand \lparen {(}\)\(\newcommand \rparen {)}\)\(\newcommand {\ordinarycolon }{:}\)\(\newcommand {\vcentcolon }{\mathrel {\mathop \ordinarycolon }}\)\(\newcommand \dblcolon {\vcentcolon \vcentcolon }\)\(\newcommand \coloneqq {\vcentcolon =}\)\(\newcommand \Coloneqq {\dblcolon =}\)\(\newcommand \coloneq {\vcentcolon {-}}\)\(\newcommand \Coloneq {\dblcolon {-}}\)\(\newcommand \eqqcolon {=\vcentcolon }\)\(\newcommand \Eqqcolon {=\dblcolon }\)\(\newcommand \eqcolon {\mathrel {-}\vcentcolon }\)\(\newcommand \Eqcolon {\mathrel {-}\dblcolon }\)\(\newcommand \colonapprox {\vcentcolon \approx }\)\(\newcommand \Colonapprox {\dblcolon \approx }\)\(\newcommand \colonsim {\vcentcolon \sim }\)\(\newcommand \Colonsim {\dblcolon \sim }\)\(\newcommand {\nuparrow }{\mathrel {\cancel {\uparrow }}}\)\(\newcommand {\ndownarrow }{\mathrel {\cancel {\downarrow }}}\)\(\newcommand {\bigtimes }{\mathop {\Large \times }\limits }\)\(\newcommand {\prescript }[3]{{}^{#1}_{#2}#3}\)\(\newenvironment {lgathered}{\begin {gathered}}{\end {gathered}}\)\(\newenvironment {rgathered}{\begin {gathered}}{\end {gathered}}\)\(\newcommand {\splitfrac }[2]{{}^{#1}_{#2}}\)\(\let \splitdfrac \splitfrac \)\(\newcommand {\LWRoverlaysymbols }[2]{\mathord {\smash {\mathop {#2\strut }\limits ^{\smash {\lower 3ex{#1}}}}\strut }}\)\(\newcommand{\alphaup}{\unicode{x03B1}}\)\(\newcommand{\betaup}{\unicode{x03B2}}\)\(\newcommand{\gammaup}{\unicode{x03B3}}\)\(\newcommand{\digammaup}{\unicode{x03DD}}\)\(\newcommand{\deltaup}{\unicode{x03B4}}\)\(\newcommand{\epsilonup}{\unicode{x03F5}}\)\(\newcommand{\varepsilonup}{\unicode{x03B5}}\)\(\newcommand{\zetaup}{\unicode{x03B6}}\)\(\newcommand{\etaup}{\unicode{x03B7}}\)\(\newcommand{\thetaup}{\unicode{x03B8}}\)\(\newcommand{\varthetaup}{\unicode{x03D1}}\)\(\newcommand{\iotaup}{\unicode{x03B9}}\)\(\newcommand{\kappaup}{\unicode{x03BA}}\)\(\newcommand{\varkappaup}{\unicode{x03F0}}\)\(\newcommand{\lambdaup}{\unicode{x03BB}}\)\(\newcommand{\muup}{\unicode{x03BC}}\)\(\newcommand{\nuup}{\unicode{x03BD}}\)\(\newcommand{\xiup}{\unicode{x03BE}}\)\(\newcommand{\omicronup}{\unicode{x03BF}}\)\(\newcommand{\piup}{\unicode{x03C0}}\)\(\newcommand{\varpiup}{\unicode{x03D6}}\)\(\newcommand{\rhoup}{\unicode{x03C1}}\)\(\newcommand{\varrhoup}{\unicode{x03F1}}\)\(\newcommand{\sigmaup}{\unicode{x03C3}}\)\(\newcommand{\varsigmaup}{\unicode{x03C2}}\)\(\newcommand{\tauup}{\unicode{x03C4}}\)\(\newcommand{\upsilonup}{\unicode{x03C5}}\)\(\newcommand{\phiup}{\unicode{x03D5}}\)\(\newcommand{\varphiup}{\unicode{x03C6}}\)\(\newcommand{\chiup}{\unicode{x03C7}}\)\(\newcommand{\psiup}{\unicode{x03C8}}\)\(\newcommand{\omegaup}{\unicode{x03C9}}\)\(\newcommand{\Alphaup}{\unicode{x0391}}\)\(\newcommand{\Betaup}{\unicode{x0392}}\)\(\newcommand{\Gammaup}{\unicode{x0393}}\)\(\newcommand{\Digammaup}{\unicode{x03DC}}\)\(\newcommand{\Deltaup}{\unicode{x0394}}\)\(\newcommand{\Epsilonup}{\unicode{x0395}}\)\(\newcommand{\Zetaup}{\unicode{x0396}}\)\(\newcommand{\Etaup}{\unicode{x0397}}\)\(\newcommand{\Thetaup}{\unicode{x0398}}\)\(\newcommand{\Varthetaup}{\unicode{x03F4}}\)\(\newcommand{\Iotaup}{\unicode{x0399}}\)\(\newcommand{\Kappaup}{\unicode{x039A}}\)\(\newcommand{\Lambdaup}{\unicode{x039B}}\)\(\newcommand{\Muup}{\unicode{x039C}}\)\(\newcommand{\Nuup}{\unicode{x039D}}\)\(\newcommand{\Xiup}{\unicode{x039E}}\)\(\newcommand{\Omicronup}{\unicode{x039F}}\)\(\newcommand{\Piup}{\unicode{x03A0}}\)\(\newcommand{\Varpiup}{\unicode{x03D6}}\)\(\newcommand{\Rhoup}{\unicode{x03A1}}\)\(\newcommand{\Sigmaup}{\unicode{x03A3}}\)\(\newcommand{\Tauup}{\unicode{x03A4}}\)\(\newcommand{\Upsilonup}{\unicode{x03A5}}\)\(\newcommand{\Phiup}{\unicode{x03A6}}\)\(\newcommand{\Chiup}{\unicode{x03A7}}\)\(\newcommand{\Psiup}{\unicode{x03A8}}\)\(\newcommand{\Omegaup}{\unicode{x03A9}}\)\(\newcommand{\alphait}{\unicode{x1D6FC}}\)\(\newcommand{\betait}{\unicode{x1D6FD}}\)\(\newcommand{\gammait}{\unicode{x1D6FE}}\)\(\newcommand{\digammait}{\mathit{\unicode{x03DD}}}\)\(\newcommand{\deltait}{\unicode{x1D6FF}}\)\(\newcommand{\epsilonit}{\unicode{x1D716}}\)\(\newcommand{\varepsilonit}{\unicode{x1D700}}\)\(\newcommand{\zetait}{\unicode{x1D701}}\)\(\newcommand{\etait}{\unicode{x1D702}}\)\(\newcommand{\thetait}{\unicode{x1D703}}\)\(\newcommand{\varthetait}{\unicode{x1D717}}\)\(\newcommand{\iotait}{\unicode{x1D704}}\)\(\newcommand{\kappait}{\unicode{x1D705}}\)\(\newcommand{\varkappait}{\unicode{x1D718}}\)\(\newcommand{\lambdait}{\unicode{x1D706}}\)\(\newcommand{\muit}{\unicode{x1D707}}\)\(\newcommand{\nuit}{\unicode{x1D708}}\)\(\newcommand{\xiit}{\unicode{x1D709}}\)\(\newcommand{\omicronit}{\unicode{x1D70A}}\)\(\newcommand{\piit}{\unicode{x1D70B}}\)\(\newcommand{\varpiit}{\unicode{x1D71B}}\)\(\newcommand{\rhoit}{\unicode{x1D70C}}\)\(\newcommand{\varrhoit}{\unicode{x1D71A}}\)\(\newcommand{\sigmait}{\unicode{x1D70E}}\)\(\newcommand{\varsigmait}{\unicode{x1D70D}}\)\(\newcommand{\tauit}{\unicode{x1D70F}}\)\(\newcommand{\upsilonit}{\unicode{x1D710}}\)\(\newcommand{\phiit}{\unicode{x1D719}}\)\(\newcommand{\varphiit}{\unicode{x1D711}}\)\(\newcommand{\chiit}{\unicode{x1D712}}\)\(\newcommand{\psiit}{\unicode{x1D713}}\)\(\newcommand{\omegait}{\unicode{x1D714}}\)\(\newcommand{\Alphait}{\unicode{x1D6E2}}\)\(\newcommand{\Betait}{\unicode{x1D6E3}}\)\(\newcommand{\Gammait}{\unicode{x1D6E4}}\)\(\newcommand{\Digammait}{\mathit{\unicode{x03DC}}}\)\(\newcommand{\Deltait}{\unicode{x1D6E5}}\)\(\newcommand{\Epsilonit}{\unicode{x1D6E6}}\)\(\newcommand{\Zetait}{\unicode{x1D6E7}}\)\(\newcommand{\Etait}{\unicode{x1D6E8}}\)\(\newcommand{\Thetait}{\unicode{x1D6E9}}\)\(\newcommand{\Varthetait}{\unicode{x1D6F3}}\)\(\newcommand{\Iotait}{\unicode{x1D6EA}}\)\(\newcommand{\Kappait}{\unicode{x1D6EB}}\)\(\newcommand{\Lambdait}{\unicode{x1D6EC}}\)\(\newcommand{\Muit}{\unicode{x1D6ED}}\)\(\newcommand{\Nuit}{\unicode{x1D6EE}}\)\(\newcommand{\Xiit}{\unicode{x1D6EF}}\)\(\newcommand{\Omicronit}{\unicode{x1D6F0}}\)\(\newcommand{\Piit}{\unicode{x1D6F1}}\)\(\newcommand{\Rhoit}{\unicode{x1D6F2}}\)\(\newcommand{\Sigmait}{\unicode{x1D6F4}}\)\(\newcommand{\Tauit}{\unicode{x1D6F5}}\)\(\newcommand{\Upsilonit}{\unicode{x1D6F6}}\)\(\newcommand{\Phiit}{\unicode{x1D6F7}}\)\(\newcommand{\Chiit}{\unicode{x1D6F8}}\)\(\newcommand{\Psiit}{\unicode{x1D6F9}}\)\(\newcommand{\Omegait}{\unicode{x1D6FA}}\)\(\let \digammaup \Digammaup \)\(\renewcommand {\digammait }{\mathit {\digammaup }}\)\(\newcommand {\smallin }{\unicode {x220A}}\)\(\newcommand {\smallowns }{\unicode {x220D}}\)\(\newcommand {\notsmallin }{\LWRoverlaysymbols {/}{\unicode {x220A}}}\)\(\newcommand {\notsmallowns }{\LWRoverlaysymbols {/}{\unicode {x220D}}}\)\(\newcommand {\rightangle }{\unicode {x221F}}\)\(\newcommand {\intclockwise }{\unicode {x2231}}\)\(\newcommand {\ointclockwise }{\unicode {x2232}}\)\(\newcommand {\ointctrclockwise }{\unicode {x2233}}\)\(\newcommand {\oiint }{\unicode {x222F}}\)\(\newcommand {\oiiint }{\unicode {x2230}}\)\(\newcommand {\ddag }{\unicode {x2021}}\)\(\newcommand {\P }{\unicode {x00B6}}\)\(\newcommand {\copyright }{\unicode {x00A9}}\)\(\newcommand {\dag }{\unicode {x2020}}\)\(\newcommand {\pounds }{\unicode {x00A3}}\)\(\newcommand {\iddots }{\unicode {x22F0}}\)\(\newcommand {\utimes }{\overline {\times }}\)\(\newcommand {\dtimes }{\underline {\times }}\)\(\newcommand {\udtimes }{\overline {\underline {\times }}}\)\(\newcommand {\leftwave }{\left \{}\)\(\newcommand {\rightwave }{\right \}}\)\(\newcommand {\toprule }[1][]{\hline }\)\(\let \midrule \toprule \)\(\let \bottomrule \toprule \)\(\newcommand {\cmidrule }[2][]{}\)\(\newcommand {\morecmidrules }{}\)\(\newcommand {\specialrule }[3]{\hline }\)\(\newcommand {\addlinespace }[1][]{}\)\(\newcommand {\LWRsubmultirow }[2][]{#2}\)\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)\(\newcommand {\multirow }[2][]{\LWRmultirow }\)\(\newcommand {\mrowcell }{}\)\(\newcommand {\mcolrowcell }{}\)\(\newcommand {\STneed }[1]{}\)\( \newcommand {\multicolumn }[3]{#3}\)\(\newcommand {\tothe }[1]{^{#1}}\)\(\newcommand {\raiseto }[2]{{#2}^{#1}}\)\(\newcommand {\ang }[2][]{(\mathrm {#2})\degree }\)\(\newcommand {\num }[2][]{\mathrm {#2}}\)\(\newcommand {\si }[2][]{\mathrm {#2}}\)\(\newcommand {\LWRSI }[2][]{\mathrm {#1\LWRSInumber \,#2}}\)\(\newcommand {\SI }[2][]{\def \LWRSInumber {#2}\LWRSI }\)\(\newcommand {\numlist }[2][]{\mathrm {#2}}\)\(\newcommand {\numrange }[3][]{\mathrm {#2\,\unicode {x2013}\,#3}}\)\(\newcommand {\SIlist }[3][]{\mathrm {#2\,#3}}\)\(\newcommand {\SIrange }[4][]{\mathrm {#2\,#4\,\unicode {x2013}\,#3\,#4}}\)\(\newcommand {\tablenum }[2][]{\mathrm {#2}}\)\(\newcommand {\ampere }{\mathrm {A}}\)\(\newcommand {\candela }{\mathrm {cd}}\)\(\newcommand {\kelvin }{\mathrm {K}}\)\(\newcommand {\kilogram }{\mathrm {kg}}\)\(\newcommand {\metre }{\mathrm {m}}\)\(\newcommand {\mole }{\mathrm {mol}}\)\(\newcommand {\second }{\mathrm {s}}\)\(\newcommand {\becquerel }{\mathrm {Bq}}\)\(\newcommand {\degreeCelsius }{\unicode {x2103}}\)\(\newcommand {\coulomb }{\mathrm {C}}\)\(\newcommand {\farad }{\mathrm {F}}\)\(\newcommand {\gray }{\mathrm {Gy}}\)\(\newcommand {\hertz }{\mathrm {Hz}}\)\(\newcommand {\henry }{\mathrm {H}}\)\(\newcommand {\joule }{\mathrm {J}}\)\(\newcommand {\katal }{\mathrm {kat}}\)\(\newcommand {\lumen }{\mathrm {lm}}\)\(\newcommand {\lux }{\mathrm {lx}}\)\(\newcommand {\newton }{\mathrm {N}}\)\(\newcommand {\ohm }{\mathrm {\Omega }}\)\(\newcommand {\pascal }{\mathrm {Pa}}\)\(\newcommand {\radian }{\mathrm {rad}}\)\(\newcommand {\siemens }{\mathrm {S}}\)\(\newcommand {\sievert }{\mathrm {Sv}}\)\(\newcommand {\steradian }{\mathrm {sr}}\)\(\newcommand {\tesla }{\mathrm {T}}\)\(\newcommand {\volt }{\mathrm {V}}\)\(\newcommand {\watt }{\mathrm {W}}\)\(\newcommand {\weber }{\mathrm {Wb}}\)\(\newcommand {\day }{\mathrm {d}}\)\(\newcommand {\degree }{\mathrm {^\circ }}\)\(\newcommand {\hectare }{\mathrm {ha}}\)\(\newcommand {\hour }{\mathrm {h}}\)\(\newcommand {\litre }{\mathrm {l}}\)\(\newcommand {\liter }{\mathrm {L}}\)\(\newcommand {\arcminute }{^\prime }\)\(\newcommand {\minute }{\mathrm {min}}\)\(\newcommand {\arcsecond }{^{\prime \prime }}\)\(\newcommand {\tonne }{\mathrm {t}}\)\(\newcommand {\astronomicalunit }{au}\)\(\newcommand {\atomicmassunit }{u}\)\(\newcommand {\bohr }{\mathit {a}_0}\)\(\newcommand {\clight }{\mathit {c}_0}\)\(\newcommand {\dalton }{\mathrm {D}_\mathrm {a}}\)\(\newcommand {\electronmass }{\mathit {m}_{\mathrm {e}}}\)\(\newcommand {\electronvolt }{\mathrm {eV}}\)\(\newcommand {\elementarycharge }{\mathit {e}}\)\(\newcommand {\hartree }{\mathit {E}_{\mathrm {h}}}\)\(\newcommand {\planckbar }{\mathit {\unicode {x210F}}}\)\(\newcommand {\angstrom }{\mathrm {\unicode {x212B}}}\)\(\let \LWRorigbar \bar \)\(\newcommand {\bar }{\mathrm {bar}}\)\(\newcommand {\barn }{\mathrm {b}}\)\(\newcommand {\bel }{\mathrm {B}}\)\(\newcommand {\decibel }{\mathrm {dB}}\)\(\newcommand {\knot }{\mathrm {kn}}\)\(\newcommand {\mmHg }{\mathrm {mmHg}}\)\(\newcommand {\nauticalmile }{\mathrm {M}}\)\(\newcommand {\neper }{\mathrm {Np}}\)\(\newcommand {\yocto }{\mathrm {y}}\)\(\newcommand {\zepto }{\mathrm {z}}\)\(\newcommand {\atto }{\mathrm {a}}\)\(\newcommand {\femto }{\mathrm {f}}\)\(\newcommand {\pico }{\mathrm {p}}\)\(\newcommand {\nano }{\mathrm {n}}\)\(\newcommand {\micro }{\mathrm {\unicode {x00B5}}}\)\(\newcommand {\milli }{\mathrm {m}}\)\(\newcommand {\centi }{\mathrm {c}}\)\(\newcommand {\deci }{\mathrm {d}}\)\(\newcommand {\deca }{\mathrm {da}}\)\(\newcommand {\hecto }{\mathrm {h}}\)\(\newcommand {\kilo }{\mathrm {k}}\)\(\newcommand {\mega }{\mathrm {M}}\)\(\newcommand {\giga }{\mathrm {G}}\)\(\newcommand {\tera }{\mathrm {T}}\)\(\newcommand {\peta }{\mathrm {P}}\)\(\newcommand {\exa }{\mathrm {E}}\)\(\newcommand {\zetta }{\mathrm {Z}}\)\(\newcommand {\yotta }{\mathrm {Y}}\)\(\newcommand {\percent }{\mathrm {\%}}\)\(\newcommand {\meter }{\mathrm {m}}\)\(\newcommand {\metre }{\mathrm {m}}\)\(\newcommand {\gram }{\mathrm {g}}\)\(\newcommand {\kg }{\kilo \gram }\)\(\newcommand {\of }[1]{_{\mathrm {#1}}}\)\(\newcommand {\squared }{^2}\)\(\newcommand {\square }[1]{\mathrm {#1}^2}\)\(\newcommand {\cubed }{^3}\)\(\newcommand {\cubic }[1]{\mathrm {#1}^3}\)\(\newcommand {\per }{/}\)\(\newcommand {\celsius }{\unicode {x2103}}\)\(\newcommand {\fg }{\femto \gram }\)\(\newcommand {\pg }{\pico \gram }\)\(\newcommand {\ng }{\nano \gram }\)\(\newcommand {\ug }{\micro \gram }\)\(\newcommand {\mg }{\milli \gram }\)\(\newcommand {\g }{\gram }\)\(\newcommand {\kg }{\kilo \gram }\)\(\newcommand {\amu }{\mathrm {u}}\)\(\newcommand {\nm }{\nano \metre }\)\(\newcommand {\um }{\micro \metre }\)\(\newcommand {\mm }{\milli \metre }\)\(\newcommand {\cm }{\centi \metre }\)\(\newcommand {\dm }{\deci \metre }\)\(\newcommand {\m }{\metre }\)\(\newcommand {\km }{\kilo \metre }\)\(\newcommand {\as }{\atto \second }\)\(\newcommand {\fs }{\femto \second }\)\(\newcommand {\ps }{\pico \second }\)\(\newcommand {\ns }{\nano \second }\)\(\newcommand {\us }{\micro \second }\)\(\newcommand {\ms }{\milli \second }\)\(\newcommand {\s }{\second }\)\(\newcommand {\fmol }{\femto \mol }\)\(\newcommand {\pmol }{\pico \mol }\)\(\newcommand {\nmol }{\nano \mol }\)\(\newcommand {\umol }{\micro \mol }\)\(\newcommand {\mmol }{\milli \mol }\)\(\newcommand {\mol }{\mol }\)\(\newcommand {\kmol }{\kilo \mol }\)\(\newcommand {\pA }{\pico \ampere }\)\(\newcommand {\nA }{\nano \ampere }\)\(\newcommand {\uA }{\micro \ampere }\)\(\newcommand {\mA }{\milli \ampere }\)\(\newcommand {\A }{\ampere }\)\(\newcommand {\kA }{\kilo \ampere }\)\(\newcommand {\ul }{\micro \litre }\)\(\newcommand {\ml }{\milli \litre }\)\(\newcommand {\l }{\litre }\)\(\newcommand {\hl }{\hecto \litre }\)\(\newcommand {\uL }{\micro \liter }\)\(\newcommand {\mL }{\milli \liter }\)\(\newcommand {\L }{\liter }\)\(\newcommand {\hL }{\hecto \liter }\)\(\newcommand {\mHz }{\milli \hertz }\)\(\newcommand {\Hz }{\hertz }\)\(\newcommand {\kHz }{\kilo \hertz }\)\(\newcommand {\MHz }{\mega \hertz }\)\(\newcommand {\GHz }{\giga \hertz }\)\(\newcommand {\THz }{\tera \hertz }\)\(\newcommand {\mN }{\milli \newton }\)\(\newcommand {\N }{\newton }\)\(\newcommand {\kN }{\kilo \newton }\)\(\newcommand {\MN }{\mega \newton }\)\(\newcommand {\Pa }{\pascal }\)\(\newcommand {\kPa }{\kilo \pascal }\)\(\newcommand {\MPa }{\mega \pascal }\)\(\newcommand {\GPa }{\giga \pascal }\)\(\newcommand {\mohm }{\milli \ohm }\)\(\newcommand {\kohm }{\kilo \ohm }\)\(\newcommand {\Mohm }{\mega \ohm }\)\(\newcommand {\pV }{\pico \volt }\)\(\newcommand {\nV }{\nano \volt }\)\(\newcommand {\uV }{\micro \volt }\)\(\newcommand {\mV }{\milli \volt }\)\(\newcommand {\V }{\volt }\)\(\newcommand {\kV }{\kilo \volt }\)\(\newcommand {\W }{\watt }\)\(\newcommand {\uW }{\micro \watt }\)\(\newcommand {\mW }{\milli \watt }\)\(\newcommand {\kW }{\kilo \watt }\)\(\newcommand {\MW }{\mega \watt }\)\(\newcommand {\GW }{\giga \watt }\)\(\newcommand {\J }{\joule }\)\(\newcommand {\uJ }{\micro \joule }\)\(\newcommand {\mJ }{\milli \joule }\)\(\newcommand {\kJ }{\kilo \joule }\)\(\newcommand {\eV }{\electronvolt }\)\(\newcommand {\meV }{\milli \electronvolt }\)\(\newcommand {\keV }{\kilo \electronvolt }\)\(\newcommand {\MeV }{\mega \electronvolt }\)\(\newcommand {\GeV }{\giga \electronvolt }\)\(\newcommand {\TeV }{\tera \electronvolt }\)\(\newcommand {\kWh }{\kilo \watt \hour }\)\(\newcommand {\F }{\farad }\)\(\newcommand {\fF }{\femto \farad }\)\(\newcommand {\pF }{\pico \farad }\)\(\newcommand {\K }{\mathrm {K}}\)\(\newcommand {\dB }{\mathrm {dB}}\)\(\newcommand {\kibi }{\mathrm {Ki}}\)\(\newcommand {\mebi }{\mathrm {Mi}}\)\(\newcommand {\gibi }{\mathrm {Gi}}\)\(\newcommand {\tebi }{\mathrm {Ti}}\)\(\newcommand {\pebi }{\mathrm {Pi}}\)\(\newcommand {\exbi }{\mathrm {Ei}}\)\(\newcommand {\zebi }{\mathrm {Zi}}\)\(\newcommand {\yobi }{\mathrm {Yi}}\)\(\require {mhchem}\)\(\require {cancel}\)\(\newcommand {\fint }{âĺŊ}\)\(\newcommand {\hdots }{\cdots }\)\(\newcommand {\mathnormal }[1]{#1}\)\(\newcommand {\vecs }[2]{\vec {#1}_{#2}}\)\(\newcommand {\B }{\mathcal {B}}\)\(\newcommand {\BB }{\mathbb {B}}\)\(\newcommand {\DD }{\mathbb {D}}\)\(\newcommand {\e }{\mathrm {e}}\)\(\renewcommand {\L }{\mathcal {L}}\)\(\renewcommand {\O }{\mathcal {O}}\)\(\newcommand {\Q }{\mathcal {Q}}\)\(\newcommand {\R }{\mathcal {R}}\)\(\newcommand {\grad }{\operatorname {grad}}\)\(\renewcommand {\div }{\operatorname {div}}\)\(\newcommand {\const }{\operatorname {const}}\)\(\newcommand {\supp }{\operatorname {supp}}\)\(\newcommand {\dist }{\operatorname {dist}}\)\(\newcommand {\trace }{\operatorname {trace}}\)\(\renewcommand {\theta }{\vartheta }\)\(\newcommand {\name }[1]{\textsc {#1}}\)\(\newcommand {\smallpmatrix }[1]{\left (\begin {smallmatrix}#1\end {smallmatrix}\right )}\)\(\newcommand {\matlab }{{\fontfamily {bch}\scshape \selectfont {}Matlab}}\)\(\newcommand {\innerproduct }[1]{\left \langle {#1}\right \rangle }\)\(\newcommand {\norm }[1]{\left \Vert {#1}\right \Vert }\)\(\renewcommand {\natural }{\mathbb {N}}\)\(\newcommand {\integer }{\mathbb {Z}}\)\(\newcommand {\rational }{\mathbb {Q}}\)\(\newcommand {\real }{\mathbb {R}}\)\(\newcommand {\complex }{\mathbb {C}}\)\(\renewcommand {\d }{\mathop {}\!\mathrm {d}}\)\(\newcommand {\dr }{\d {}r}\)\(\newcommand {\ds }{\d {}s}\)\(\newcommand {\dt }{\d {}t}\)\(\newcommand {\du }{\d {}u}\)\(\newcommand {\dv }{\d {}v}\)\(\newcommand {\dw }{\d {}w}\)\(\newcommand {\dx }{\d {}x}\)\(\newcommand {\dy }{\d {}y}\)\(\newcommand {\dz }{\d {}z}\)\(\newcommand {\dsigma }{\d {}\sigma }\)\(\newcommand {\dphi }{\d {}\phi }\)\(\newcommand {\dvarphi }{\d {}\varphi }\)\(\newcommand {\dtau }{\d {}\tau }\)\(\newcommand {\dxi }{\d {}\xi }\)\(\newcommand {\dtheta }{\d {}\theta }\)\(\newcommand {\tp }{\mathrm {T}}\)
Das Spline-Konzept
Polynome stellen zwar gute lokale Approximationen für glatte Funktionen dar, allerdings kann die Genauigkeit auf großen Intervallen sehr klein sein. Außerdem haben lokale Änderungen einen globalen Einfluss.Daher ist der Übergang zu stückweise Polynomen sozusagen „natürlich“.
Spline: Ein Spline vom Grad \(\le n\) mit Gitterweite \(h\) ist \((n - 1)\)-fach stetig differenzierbar und stimmt auf jedem Gitterintervall \([i, i+1]h\) desParameterintervalls \(D\) mit einem Polynom vom Grad \(\le n\) überein.
Diese Definition eignet sich natürlich nicht für numerische Berechnungen, daher muss eine lokale Basis konstruiert werden. Die Basis, die hier verwendet wird, kann durch den linearen Fall (Hut-Funktionen)motiviert werden.
Definition und grundlegende Eigenschaften
B-Spline: Der uniforme B-Spline vom Grad \(n\) ist definiert durch die Rekursion
\(\seteqnumber{0}{}{0}\)
\begin{align*}b^n(x) := \int _{x-1}^x b^{n-1},\end{align*}beginnend mit der charakteristischen Funktion \(b^0\) des Einheitsintervalls \([0, 1)\). Äquivalent ist die Rekursion
\(\seteqnumber{0}{}{0}\)
\begin{align*}\frac {d}{dx} b^n(x) := b^{n-1}(x) - b^{n-1}(x - 1)\end{align*}mit \(b^n(0) = 0\).
Eigenschaften von B-Splines: B-Splines erfüllen die folgenden Eigenschaften:
Positivität und lokaler Träger: \(b^n\) ist positiv auf \((0, n + 1)\) und verschwindet außerhalb dieses Intervalls (außer für \(n = 0\), hier gilt \(b^0(0) = 1\)).
Glattheit: \(b^n\) ist \((n - 1)\)-fach stetig differenzierbar, wobei die \(n\)-te Ableitung in den Knotenpunkten \(0, \dotsc , n + 1\) unstetig ist.
Struktur als stückweises Polynom: \(b^n\) ist auf jedem Intervall \([k, k + 1)\), \(k = 0, \dotsc , n\), ein Polynom vom Grad \(n\).
Symmetrie und Monotonie: Der B-Spline vom Grad \(n\) ist symmetrisch, d. h.
\(\seteqnumber{0}{}{0}\)
\begin{align*}b^n(x) = b^n(n + 1 - x),\end{align*}und auf \([0, (n + 1)/2]\) und \([(n + 1)/2, n + 1]\) strikt monoton.
Rekursionsformel
Rekursionsformel: Der B-Spline \(b^n\) ist eine gewichtete Summe von B-Splines vom Grad \(n - 1\):
\(\seteqnumber{0}{}{0}\)
\begin{align*}b^n(x) = \frac {x}{n} b^{n-1}(x) + \frac {n + 1 - x}{n} b^{n-1}(x - 1).\end{align*}
Taylor-Koeffizienten: Die \(n + 1\) polynomialen Segmente
\(\seteqnumber{0}{}{0}\)
\begin{align*}a_{k,0}^n + a_{k,1}^n t + \dotsb + a_{k,n}^n t^n,\quad t = x - k \in [0, 1),\end{align*}des B-Splines \(b^n\) können mit der Rekursion
\(\seteqnumber{0}{}{0}\)
\begin{align*}a_{k,\ell }^n = \frac {k}{n} a_{k,\ell }^{n-1} + \frac {1}{n} a_{k,\ell -1}^{n-1} + \frac {n + 1 - k}{n} a_{k-1,\ell }^{n-1} - \frac {1}{n} a_{k-1,\ell -1}^{n-1}\end{align*}berechnet werden, wobei \(a_{0,0}^1 := 1\) und \(a_{k,\ell }^n := 0\) für \(k \notin \{0, \dotsc , n\}\) oder \(\ell \notin \{0, \dotsc , n\}\).
Darstellung von Polynomen
kardinale Splines: Für \(h > 0\) und \(k \in \integer \) sind
\(\seteqnumber{0}{}{0}\)
\begin{align*}b_{k,h}^n(x) := b^n(x/h - k)\end{align*}B-Splines auf dem Gitter \(h\integer \). Ihre Linearkombinationen \(\sum _{k \in \integer } c_k b_{k,h}^n\) heißen kardinaleSplines vom Grad \(\le n\) mit Gitterweite \(h\).
Marsden-Identität: Für \(x, t \in \real \) gilt
\(\seteqnumber{0}{}{0}\)
\begin{align*}(x - t)^n = \sum _{k \in \integer } \psi _{k,h}^n(t) b_{k,h}^n(x),\end{align*}wobei \(\psi _{k,h}^n(t) := h^n (k + 1 - t/h) \dotsm (k + n - t/h)\).
lineare Unabhängigkeit: Für jedes Gitterintervall \([\ell , \ell + 1)h\) sind die B-Splines \(b_{k,h}\),
\(k = \ell - n, \dotsc , \ell \), die auf diesem Intervall nicht verschwinden, linear unabhängig.
Subdivision
Gitterverfeinerung: Der B-Spline \(b_{k,h}^n\) kann als Linearkombination von B-Splines mit Gitterweite \(h/2\) geschrieben werden:
\(\seteqnumber{0}{}{0}\)
\begin{align*}b_{k,h}^n = 2^{-n} \sum _{\ell =0}^{n+1} \binom {n+1}{\ell } b_{2k+\ell ,h/2}^n.\end{align*}
Subdivisionsalgorithmus: Die Koeffizienten \(c_\ell ’\) eines kardinalen Splines \(\sum _k c_k b_{k,h}^n\) bzgl. der halben Gitterweite \(h/2\) können wie folgt berechnet werden:
Zunächst setzt man \(c_{2k}’ := c_{2k+1}’ := c_k\).
Anschließend bildet man simultan Mittelwerte, d. h. \(c_\ell ’ \leftarrow \frac {1}{2} (c_\ell ’ + c_{\ell -1}’)\), \(\ell \in \integer \).
Dieser Schritt wird \(n\)-mal insgesamt wiederholt.
Skalarprodukte
Faltung: Die Faltung zweier B-Splines ist ein B-Spline höheren Grades: \(\seteqnumber{0}{}{0}\)
\begin{align*}b^{m+n+1}(x) = \int _\real b^m(x - y) b^n(y) \dy .\end{align*}
Skalarprodukte: Die Skalarprodukte der B-Splines \(b_{k,h}^n\) und \(b_{\ell ,h}^n\) und ihrer Ableitungen sind
\(\seteqnumber{0}{}{0}\)
\begin{align*}s_{k-\ell }^n &:= h b^{2n+1} (n + 1 + k - \ell ),\\ d_{k-\ell }^n &:= h^{-2} (2s_{k-\ell }^{n-1} - s_{k-\ell -1}^{n-1} - s_{k-\ell +1}^{n-1}).\end{align*}
Tabelle: Skalarprodukte der B-Splines und ihrer Ableitungen für \(h = 1\):
\(\seteqnumber{0}{}{0}\)
\begin{align*}\begin{array}{c||cccc|cccc} n & s_0^n & s_1^n & s_2^n & s_3^n & d_0^n & d_1^n & d_2^n & d_3^n\\\hline 1 & \frac {2}{3} & \frac {1}{6} & & &2 & -1 & &\\ 2 & \frac {11}{20} & \frac {13}{60} & \frac {1}{120} & & 1 & -\frac {1}{3} & -\frac {1}{6} &\\ 3 & \frac {151}{315} & \frac{397}{1680} & \frac {1}{42} & \frac {1}{5040} & \frac {2}{3} & -\frac {1}{8} & -\frac {1}{5} & -\frac {1}{120} \end {array}\end{align*}Wegen Symmetrie gilt \(s_i^n = s_{-i}^n\) und \(d_i^n = d_{-i}^n\).
Skalarprodukte von Ableitungen höherer Ordnung: Skalarprodukte mit höheren Ableitungen können durch die Differentiationsformel
\(\seteqnumber{0}{}{0}\)
\begin{align*}\frac {d^\alpha }{dx^\alpha } b_{k,h}^n(x) = h^{-\alpha } \sum _{\nu =0}^\alpha (-1)^\nu \binom {\alpha }{\nu } b_{k+\nu ,h}^{n-\alpha }\end{align*}errechnet werden.
