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\star\title\star{Distillation\star of\star local\star purity\star from\star quantum\star states\star}\star \star\date\star{\star\today\star}\star \star \star\author\star{I\star.\star Devetak\star}\star \star\email\star{devetak\star@csi\star.usc\star.edu\star}\star \star\affiliation\star{IBM\star T\star.\star~J\star.\star~Watson\star Research\star Center\star,\star PO\star Box\star 218\star,\star Yorktown\star Heights\star,\star NY\star 10598\star,\star USA\star}\star
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\star\begin\star{abstract\star}\star Recently\star Horodecki\star et\star al\star.\star \star[Phys\star.\star Rev\star.\star Lett\star.\star 90\star,\star 100402\star \star(2003\star)\star]\star introduced\star an\star important\star quantum\star information\star processing\star paradigm\star,\star in\star which\star two\star parties\star sharing\star many\star copies\star of\star the\star same\star bipartite\star quantum\star state\star \star distill\star local\star pure\star states\star,\star by\star means\star of\star \star local\star unitary\star operations\star \star assisted\star by\star a\star one\star-way\star \star(two\star-way\star)\star completely\star dephasing\star channel\star.\star \star Local\star pure\star states\star are\star a\star valuable\star resource\star from\star a\star thermodynamical\star \star point\star of\star view\star,\star since\star they\star allow\star thermal\star energy\star to\star be\star converted\star into\star work\star by\star local\star quantum\star heat\star engines\star.\star We\star give\star a\star simple\star information\star-theoretical\star characterization\star of\star the\star one\star-way\star distillable\star local\star purity\star,\star which\star turns\star out\star to\star be\star closely\star related\star \star to\star a\star previously\star known\star operational\star \star measure\star of\star classical\star correlations\star,\star the\star one\star-way\star distillable\star common\star randomness\star.\star \star\end\star{abstract\star}\star \star\title{Distillation of local purity from quantum states}\star
\star
\star\vspace\star{2mm\star}\star
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\star\section\star{Introduction\star}\star
\star
One\star of\star the\star primary\star tasks\star of\star quantum\star information\star theory\star is\star to\star explore\star the\star operational\star reductions\star between\star information\star processing\star resources\star such\star as\star shared\star entanglement\star or\star quantum\star channels\star,\star \star including\star both\star the\star noisy\star and\star noiseless\star varieties\star.\star For\star instance\star,\star \star entanglement\star distillation\star \star\cite\star{bdsw\star}\star involves\star transforming\star a\star large\star number\star of\star noisy\star bipartite\star quantum\star states\star \star$\star\rho\star^\star{AB\star}\star$\star,\star \star shared\star between\star two\star distant\star parties\star Alice\star and\star Bob\star,\star into\star pure\star ebits\star \star$\star\ket\star{\star\Phi\star^\star+\star}\star \star=\star 1\star/\star\sqrt\star{2\star}\star \star(\star\ket\star{0\star}\star \star\ket\star{0\star}\star \star+\star \star \star\ket\star{1\star}\star \star\ket\star{1\star}\star)\star$\star at\star the\star best\star possible\star conversion\star rate\star.\star This\star conversion\star task\star is\star naturally\star defined\star within\star the\star LOCC\star \star(local\star operations\star and\star classical\star communication\star)\star paradigm\star:\star Alice\star and\star Bob\star are\star allowed\star \star\emph\star{at\star no\star cost\star}\star to\star i\star)\star locally\star add\star pure\star state\star ancillas\star to\star their\star quantum\star systems\star,\star \star ii\star)\star perform\star local\star unitary\star operations\star and\star iii\star)\star communicate\star classically\star.\star \star In\star a\star slight\star refinement\star of\star this\star paradigm\star,\star one\star could\star assign\star a\star cost\star for\star one\star-way\star classical\star communication\star,\star leading\star to\star trade\star-offs\star between\star the\star amount\star of\star entanglement\star distilled\star and\star the\star classical\star communication\star invested\star \star\cite\star{family\star}\star.\star \star The\star communication\star theorist\star still\star feels\star at\star home\star with\star this\star modification\star:\star after\star all\star,\star classical\star communication\star is\star a\star valuable\star \star bipartite\star resource\star,\star and\star should\star not\star be\star taken\star for\star granted\star.\star It\star is\star only\star recently\star that\star attention\star has\star been\star given\star to\star \star\emph\star{local\star}\star resources\star,\star in\star particular\star local\star pure\star states\star \star\cite\star{hhhhoss\star}\star.\star \star
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Local\star pure\star states\star can\star be\star seen\star as\star valuable\star from\star a\star \star \star\emph\star{thermodynamical\star}\star perspective\star.\star Although\star we\star use\star the\star language\star of\star quantum\star states\star,\star the\star phenomenon\star is\star essentially\star classical\star.\star Landauer\star \star\cite\star{landauer\star}\star \star was\star the\star first\star to\star observe\star that\star work\star was\star required\star to\star erase\star a\star bit\star of\star information\star,\star i\star.e\star.\star to\star reset\star a\star system\star from\star an\star unknown\star state\star to\star a\star known\star \star(pure\star)\star state\star.\star Conversely\star,\star a\star supply\star of\star pure\star states\star can\star be\star used\star as\star \star`\star`fuel\star'\star'\star to\star increase\star the\star amount\star of\star useful\star work\star extractable\star from\star a\star system\star \star at\star non\star-zero\star temperature\star \star\cite\star{szilard\star,\star bennett\star}\star.\star \star This\star is\star achieved\star by\star reversibly\star transferring\star entropy\star from\star the\star system\star to\star \star the\star pure\star states\star,\star thereby\star \star`\star`cooling\star'\star'\star the\star system\star \star\cite\star{lloyd\star}\star.\star
\star
Having\star an\star appreciation\star for\star the\star value\star of\star pure\star states\star,\star it\star is\star natural\star to\star ask\star about\star the\star different\star ways\star \star in\star which\star they\star can\star be\star \star produced\star.\star In\star \star\cite\star{ohhh\star,\star hhhhoss\star}\star the\star idea\star of\star manipulating\star and\star \star concentrating\star \star`\star`purity\star'\star'\star already\star existing\star in\star a\star diluted\star form\star,\star rather\star than\star performing\star work\star to\star create\star it\star,\star was\star introduced\star.\star This\star is\star very\star much\star analogous\star to\star entanglement\star distillation\star:\star given\star a\star noisy\star resource\star one\star wishes\star to\star remove\star impurities\star from\star it\star.\star \star \star There\star is\star a\star local\star and\star distributed\star version\star of\star this\star problem\star.\star In\star the\star local\star scenario\star,\star which\star we\star call\star \star \star\emph\star{purity\star concentration\star}\star,\star Alice\star is\star given\star a\star large\star supply\star of\star \star states\star \star$\star\rho\star^A\star$\star \star and\star her\star task\star is\star to\star extract\star pure\star qubit\star states\star \star using\star only\star unitary\star operations\star.\star The\star maximal\star asymptotic\star rate\star at\star which\star \star this\star can\star be\star done\star is\star given\star by\star the\star difference\star between\star the\star size\star of\star the\star \star system\star \star$A\star$\star \star(in\star qubits\star)\star and\star its\star von\star Neumann\star entropy\star \star\cite\star{hho\star}\star.\star In\star the\star distributed\star scenario\star \star-\star-\star \star \star\emph\star{local\star purity\star distillation\star}\star \star-\star-\star \star Alice\star and\star Bob\star share\star a\star supply\star of\star bipartite\star states\star \star$\star\rho\star^\star{AB\star}\star$\star and\star they\star wish\star to\star distill\star local\star pure\star states\star using\star CLOCC\star(\star\emph\star{closed\star}\star local\star operations\star and\star classical\star communication\star)\star\cite\star{ohhh\star}\star,\star a\star modification\star of\star the\star LOCC\star paradigm\star that\star disallows\star unrestricted\star consumption\star of\star local\star \star pure\star states\star.\star Horodecki\star et\star al\star.\star\cite\star{hhhhoss\star}\star had\star previously\star obtained\star some\star bounds\star on\star this\star problem\star,\star both\star for\star the\star one\star-way\star and\star two\star-way\star CLOCC\star case\star.\star \star
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In\star this\star paper\star we\star investigate\star the\star two\star scenarios\star \star in\star detail\star.\star Our\star main\star result\star pertains\star to\star the\star distributed\star case\star;\star we\star give\star an\star information\star theoretical\star expression\star for\star the\star optimal\star one\star-way\star \star distillable\star local\star purity\star.\star This\star quantity\star turns\star out\star to\star be\star related\star to\star a\star previously\star known\star operational\star measure\star of\star classical\star correlations\star,\star \star the\star one\star-way\star distillable\star common\star randomness\star \star\cite\star{cr\star}\star.\star
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Section\star II\star is\star devoted\star to\star establishing\star notation\star.\star Section\star III\star treats\star the\star local\star scenario\star,\star reproducing\star the\star results\star of\star \star\cite\star{hho\star}\star in\star a\star somewhat\star more\star rigorous\star coding\star-theoretical\star \star language\star.\star The\star two\star-party\star distributed\star scenario\star is\star considered\star in\star section\star IV\star and\star our\star main\star result\star is\star proved\star.\star Section\star V\star discusses\star how\star to\star embed\star purity\star distillation\star and\star the\star CLOCC\star \star paradigm\star in\star the\star existing\star formalism\star for\star quantum\star Shannon\star theory\star,\star and\star concludes\star with\star open\star questions\star.\star Appendix\star A\star collects\star \star a\star number\star of\star auxiliary\star inequalities\star used\star throughout\star the\star paper\star.\star
\star \star \star\section\star{Notation\star and\star definitions\star}\star
\star \star Recall\star the\star notion\star of\star an\star \star\emph\star{ensemble\star}\star of\star quantum\star states\star \star$\star\CE\star \star=\star \star(p\star(x\star)\star,\star \star\rho\star_x\star^\star{B\star}\star)\star_\star{x\star \star\in\star \star{\cal X}\star}\star$\star:\star the\star quantum\star system\star \star$B\star$\star is\star in\star the\star \star state\star \star$\star\rho\star_x\star^B\star$\star with\star probability\star \star$p\star(x\star)\star$\star.\star The\star ensemble\star \star$\star\CE\star$\star is\star equivalently\star represented\star by\star a\star \star\emph\star{classical\star-quantum\star}\star system\star \star\cite\star{cr\star}\star \star$X\star B\star$\star in\star the\star state\star \star\begin {equation}\star \star\rho\star^\star{XB\star}\star \star=\star \star\sum\star_\star{x\star \star\in\star \star{\cal X}\star}\star p\star(x\star)\star \star\ket\star{x\star}\star \star\bra\star{x\star}\star^X\star \star\otimes\star \star\rho\star^B\star_x\star,\star \star\label\star{koki\star}\star \star\end {equation}\star where\star \star$\star\CH\star_X\star$\star has\star a\star preferred\star orthonormal\star \star basis\star \star$\star\star{\star \star\ket\star{x\star}\star \star\star}\star_\star{x\star \star\in\star \star{\cal X}\star}\star$\star.\star \star$X\star$\star plays\star the\star dual\star role\star of\star \star an\star auxiliary\star quantum\star system\star in\star the\star state\star \star$\star\sum\star_x\star p\star(x\star)\star \star\ket\star{x\star}\star \star\bra\star{x\star}\star$\star and\star of\star \star a\star random\star variable\star with\star distribution\star \star$p\star$\star and\star cardinality\star \star
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\star \star It\star is\star not\star hard\star to\star see\star that\star the\star above\star protocol\star may\star be\star bootstrapped\star to\star make\star the\star catalyst\star rate\star arbitrarily\star small\star.\star Moreover\star,\star if\star \star$\star\kappa\star(\star\rho\star^A\star)\star \star>\star 0\star$\star a\star catalyst\star is\star not\star needed\star at\star all\star \star(see\star also\star \star\cite\star{hhosss\star}\star)\star.\star \star
\star
\star\section\star{Discussion\star}\star
\star
The\star question\star of\star counting\star local\star resources\star in\star \star standard\star quantum\star information\star theoretical\star tasks\star,\star \star such\star as\star entanglement\star distillation\star,\star was\star recently\star raised\star by\star Bennett\star \star\cite\star{charlie\star}\star.\star In\star particular\star,\star it\star is\star desirable\star to\star \star extend\star the\star theory\star of\star \star\emph\star{resource\star inequalities\star}\star \star\cite\star{family\star}\star to\star include\star the\star manipulation\star of\star local\star resources\star.\star \star Recall\star the\star notation\star from\star \star\cite\star{cr\star}\star in\star which\star \star \star$\star[c\star \star\rightarrow\star c\star]\star$\star,\star \star$\star[q\star \star\rightarrow\star q\star]\star$\star and\star \star$\star[q\star \star q\star]\star$\star stand\star for\star \star a\star bit\star of\star classical\star communication\star,\star a\star qubit\star of\star quantum\star communication\star and\star and\star ebit\star of\star entanglement\star,\star respectively\star.\star There\star it\star was\star implicit\star that\star local\star pure\star ancillas\star could\star be\star added\star for\star free\star,\star which\star makes\star a\star classical\star channel\star and\star a\star dephasing\star quantum\star channel\star operationally\star equivalent\star.\star \star To\star define\star \star a\star \star`\star`closed\star'\star'\star version\star of\star this\star formalism\star,\star one\star must\star identify\star \star$\star[c\star \star\rightarrow\star c\star]\star$\star with\star a\star dephasing\star qubit\star channel\star,\star and\star introduce\star a\star new\star \star resource\star:\star a\star \star\emph\star{pbit\star}\star of\star purity\star,\star defined\star as\star a\star local\star pure\star qubit\star state\star \star$\star\ket\star{0\star}\star$\star w\star.l\star.o\star.g\star.\star in\star Bob\star's\star posession\star.\star A\star pbit\star may\star be\star written\star as\star either\star \star$\star[q\star]\star$\star or\star \star$\star[c\star]\star$\star,\star as\star there\star is\star little\star distinction\star between\star classical\star and\star quantum\star for\star strictly\star local\star resources\star.\star \star The\star main\star result\star of\star our\star paper\star may\star be\star written\star succinctly\star as\star \star$\star$\star \star\star{\star qq\star \star\star}\star \star+\star H\star(A\star)\star_\star\rho\star \star\star,\star[c\star \star\rightarrow\star c\star]\star \star \star\geq\star \star \star \star\kappa\star_\star{\star\rightarrow\star}\star(\star\rho\star^\star{AB\star}\star)\star \star\star,\star \star[q\star]\star,\star \star$\star$\star where\star \star \star$\star\star{\star q\star q\star \star\star}\star$\star represents\star the\star noisy\star static\star resource\star \star$\star\rho\star^\star{AB\star}\star$\star,\star and\star \star$\star\kappa\star_\star{\star\rightarrow\star}\star(\star\rho\star^\star{AB\star}\star)\star$\star is\star given\star by\star theorem\star 2\star.\star Regarding\star entanglement\star distillation\star,\star \star closer\star inspection\star of\star the\star optimal\star one\star-way\star \star protocol\star from\star \star\cite\star{devetak\star:winter\star}\star reveals\star that\star \star \star\begin\star{itemize\star}\star \star\item\star only\star a\star negligible\star rate\star of\star pure\star state\star \star ancillas\star need\star be\star \star consumed\star \star\item\star \star moreover\star,\star the\star locally\star concentrable\star purity\star \star \star$\star\kappa\star(\star\rho\star^A\star)\star \star+\star \star\kappa\star(\star\rho\star^B\star)\star$\star is\star available\star without\star affecting\star the\star entanglement\star distillation\star rate\star.\star \star\end\star{itemize\star}\star Whether\star the\star above\star holds\star for\star general\star quantum\star Shannon\star theoretic\star problems\star remains\star to\star be\star investigated\star.\star \star
\star
We\star conclude\star with\star a\star list\star of\star open\star problems\star.\star \star \star\begin\star{enumerate\star}\star \star\item\star It\star would\star be\star interesting\star to\star find\star the\star optimal\star trade\star-off\star between\star the\star local\star purity\star distilled\star and\star the\star one\star-way\star classical\star communication\star \star(dephasing\star)\star invested\star.\star In\star particular\star,\star does\star the\star problem\star \star reduce\star to\star the\star 1\star-DCR\star trade\star-off\star curve\star from\star \star\cite\star{cr\star}\star?\star Also\star,\star one\star could\star consider\star \star purity\star distillation\star assited\star by\star quantum\star communication\star \star \star\cite\star{phase\star}\star.\star \star\item\star We\star have\star seen\star that\star purity\star distillation\star and\star common\star randomness\star distillation\star are\star intimately\star related\star.\star Is\star there\star a\star non\star-trivial\star trade\star-off\star between\star the\star two\star,\star or\star it\star is\star always\star optimal\star to\star \star(linearly\star)\star \star interpolate\star between\star the\star known\star purity\star distillation\star and\star common\star randomness\star distillation\star protocols\star?\star One\star could\star also\star consider\star the\star simultaneous\star distillation\star of\star purity\star and\star other\star resources\star,\star such\star as\star entanglement\star \star \star(see\star \star\cite\star{ohhhh\star}\star)\star.\star \star \star \star\item\star Clearly\star,\star one\star would\star like\star a\star formula\star for\star the\star two\star-way\star distillable\star local\star purity\star.\star Solving\star this\star problem\star in\star the\star sense\star of\star the\star present\star paper\star appears\star to\star be\star difficult\star;\star \star\cite\star{hhosss\star}\star gives\star a\star formula\star involving\star \star maximizations\star over\star a\star class\star of\star states\star which\star is\star,\star alas\star,\star \star rather\star hard\star to\star characterize\star.\star A\star more\star tractable\star question\star is\star whether\star the\star relationship\star established\star between\star distillable\star purity\star and\star distillable\star common\star randomness\star carries\star over\star to\star the\star two\star-way\star scenario\star.\star \star
\star \star\end\star{enumerate\star}\star \star\acknowledgments\star We\star are\star grateful\star to\star Charles\star Bennett\star,\star Guido\star Burkard\star,\star David\star DiVincenzo\star,\star \star Aram\star Harrow\star,\star Barbara\star Terhal\star and\star John\star Smolin\star for\star useful\star discussions\star.\star We\star also\star thank\star Micha\star{\star\l\star}\star and\star Pawe\star{\star\l\star}\star Horodecki\star,\star \star Jonathan\star Oppenheim\star and\star Barbara\star Synak\star for\star comments\star on\star the\star manuscript\star and\star sharing\star their\star \star unpublished\star results\star on\star purity\star distillation\star \star\cite\star{hhosss\star,synak\star}\star.\star This\star work\star was\star supported\star in\star part\star by\star the\star NSA\star under\star the\star US\star Army\star Research\star \star Office\star \star(ARO\star)\star,\star \star grant\star numbers\star DAAG55\star-98\star-C\star-0041\star and\star DAAD19\star-01\star-1\star-06\star.\star
\star
\star\appendix\star
\star
\star\section\star{Miscellaneous\star \star inequalities\star}\star For\star states\star \star$\star\rho\star$\star,\star \star$\star\omega\star$\star and\star \star$\star\sigma\star$\star,\star \star the\star triangle\star inequality\star holds\star:\star \star\begin {equation}\star
\star\star|\star\rho\star \star-\star \star\omega\star \star\star|\star_1\star \star+\star \star\star|\star\omega\star \star-\star \star\sigma\star \star\star|\star_1\star
\star\geq\star \star\star|\star\rho\star \star-\star \star\sigma\star \star\star|\star_1\star.\star \star \star\label\star{tri\star}\star \star\end {equation}\star The\star following\star bound\star \star\cite\star{fuchs\star}\star relates\star trace\star distance\star and\star fidelity\star:\star \star\begin {equation}\star
\star\star|\star \star\rho\star-\star \star\proj\star{\star\phi\star}\star \star\star|\star_1\star \star\leq\star 2\star \star\sqrt\star{1\star \star-\star \star\bra\star{\star\phi\star}\star \star\rho\star \star\ket\star{\star\phi\star}\star}\star.\star \star\label\star{rut\star}\star \star\end {equation}\star The\star gentle\star operator\star lemma\star \star\cite\star{strong\star}\star says\star that\star a\star POVM\star element\star that\star \star succeeds\star on\star a\star state\star with\star high\star probability\star does\star not\star disturb\star it\star much\star.\star \star\begin\star{lemma\star}\star
\star \star \star\label\star{tender\star}\star
\star \star For\star a\star state\star \star$\star\rho\star$\star and\star \star operator\star \star$0\star\leq\star \star\Lambda\star\leq\star\1\star$\star,\star
\star \star if\star \star$\star \star{\star\rm\star{Tr\star \star\star,\star}\star}\star(\star\rho\star \star\Lambda\star)\star\geq\star 1\star-\star\lambda\star$\star,\star then\star
\star \star \star$\star$\star\left\star\star|\star\rho\star-\star\sqrt\star{\star\Lambda\star}\star\rho\star\sqrt\star{\star\Lambda\star}\star\right\star\star|\star_1\star\leq\star \star\sqrt\star{8\star\lambda\star}\star.\star$\star$\star
\star \star The\star same\star holds\star if\star \star$\star\rho\star$\star is\star only\star a\star subnormalized\star
\star \star density\star operator\star.\star
\star \star \star\qed\star \star\end\star{lemma\star}\star For\star two\star states\star \star$\star\rho\star$\star and\star \star$\star\omega\star$\star defined\star on\star \star a\star \star$d\star$\star-dimensional\star Hilbert\star space\star,\star \star Fannes\star'\star inequality\star \star\cite\star{fannes\star}\star reads\star:\star \star\begin {equation}\star
\star|\star H\star(\star\rho\star)\star \star-\star H\star(\star\sigma\star)\star \star|\star \star\leq\star \star\frac\star{1\star}\star{e\star}\star
\star \star+\star \star\log\star d\star \star\star|\star\rho\star \star-\star \star\omega\star\star|\star_1\star.\star \star\label\star{fannes\star}\star \star\end {equation}\star An\star important\star property\star of\star von\star Neumann\star entropy\star is\star \star\emph\star{subadditivity\star}\star \star\begin {equation}\star H\star(B\star)\star \star\geq\star H\star(AB\star)\star \star-\star H\star(A\star)\star.\star \star\label\star{sub\star}\star \star\end {equation}\star
\star
\star\section\star{Proof\star of\star lemma\star \star\ref\star{aw\star}\star}\star By\star the\star proof\star of\star the\star measurement\star compression\star theorem\star \star\cite\star{winter\star}\star,\star for\star any\star \star$\star\epsilon\star,\star \star\delta\star \star>\star 0\star$\star and\star sufficiently\star large\star \star$n\star$\star \star there\star is\star an\star ensemble\star of\star rank\star-1\star sub\star-POVMs\star \star \star$\star(p\star_s\star,\star \star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star:\star \star\CH\star_\star{A\star^n\star}\star \star\rightarrow\star \star\CH\star_K\star)\star_s\star$\star \star and\star a\star classical\star map\star \star$g\star:\star \star\CH\star_S\star \star\otimes\star \star\CH\star_K\star \star\rightarrow\star \star \star\CH\star_\star{X\star^n\star}\star$\star such\star that\star \star\begin\star{itemize\star}\star \star\item\star \star \star$\star\sum\star_k\star \star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star_k\star \star\leq\star \star\Pi\star$\star,\star where\star the\star index\star \star$k\star$\star ranges\star over\star \star$\star[2\star^\star{n\star[H\star(A\star)\star \star+\star \star\delta\star]\star}\star]\star$\star,\star and\star \star$\star\Pi\star$\star is\star a\star \star projector\star commuting\star with\star \star$\star(\star\rho\star^\star{A\star}\star)\star^\star{\star\otimes\star n\star}\star$\star such\star that\star \star$\star \star{\star\rm\star{Tr\star \star\star,\star}\star}\star \star\Pi\star \star\leq\star 2\star^\star{n\star[H\star(A\star)\star \star+\star \star\delta\star]\star}\star$\star and\star \star \star$\star \star{\star\rm\star{Tr\star \star\star,\star}\star}\star \star(\star\rho\star^\star{A\star}\star)\star^\star{\star\otimes\star n\star}\star \star\Pi\star \star\geq\star 1\star \star-\star \star\epsilon\star$\star.\star \star\item\star \star \star\begin {equation}\star
\star\left\star\star|\star \star(\star\rho\star^\star{XB\star}\star)\star^\star{\star\otimes\star n\star}\star \star-\star \star\sigma\star^\star{X\star^n\star B\star^n\star}\star \star \star\right\star\star|\star_1\star \star\leq\star \star\epsilon\star,\star \star\label\star{bogy\star}\star \star\end {equation}\star where\star \star \star\begin\star{eqnarray\star*\star}\star \star\sigma\star^\star{X\star^nB\star^n\star}\star \star&\star \star=\star \star \star&\star \star(g\star \star\otimes\star \star\1\star^\star{B\star^n\star}\star)\star \star\Omega\star^\star{S\star K\star B\star^n\star}\star\star \star \star\Omega\star^\star{S\star K\star B\star^n\star}\star \star&\star \star=\star \star&\star \star\sum\star_s\star p\star(s\star)\star \star\proj\star{s\star}\star^S\star \star\otimes\star
\star \star[\star(\star\Lambda\star^\star{\star(s\star)\star}\star \star\otimes\star \star\1\star^\star{B\star^n\star}\star)\star(\star\rho\star^\star{AB\star}\star)\star^\star{\star\otimes\star n\star}\star]\star,\star \star\end\star{eqnarray\star*\star}\star for\star some\star probability\star distribution\star \star$p\star(s\star)\star$\star.\star \star\end\star{itemize\star}\star Each\star sub\star-POVM\star \star$\star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star$\star may\star be\star augmented\star by\star no\star more\star than\star \star \star$2\star^\star{n\star[H\star(A\star)\star \star+\star \star\delta\star]\star}\star$\star rank\star-1\star elements\star to\star satisfy\star equality\star \star \star$\star$\star\sum\star_k\star \star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star_k\star \star=\star \star\Pi\star.\star$\star$\star The\star proof\star of\star lemma\star 1\star and\star \star Fannes\star'\star inequality\star \star \star(\star\ref\star{fannes\star}\star)\star imply\star the\star existence\star of\star \star a\star decomposition\star \star$A\star^\star{n\star}\star \star=\star A\star_1\star A\star_2\star$\star such\star that\star \star \star$\star$\star H\star(A\star_1\star)\star \star\leq\star \star\frac\star{1\star}\star{e\star}\star \star+\star \star n\star \star\epsilon\star \star\log\star d\star_A\star,\star \star$\star$\star while\star \star$\star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star$\star is\star now\star viewed\star as\star a\star rank\star-1\star POVM\star \star on\star \star$A\star_2\star$\star such\star that\star \star(\star\ref\star{bogy\star}\star)\star still\star holds\star for\star the\star \star$\star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star$\star.\star \star
\star
\star
Define\star \star$\star\epsilon\star'\star \star=\star \star \star\frac\star{3\star}\star{ne\star}\star \star+\star 2\star \star\epsilon\star \star\log\star \star(d\star_X\star d\star_B\star)\star$\star.\star Then\star \star\begin\star{eqnarray\star*\star}\star n\star I\star(X\star;\star B\star)\star_\star\rho\star \star&\star \star\leq\star \star&\star I\star(X\star^n\star;\star B\star^n\star)\star_\star\sigma\star \star-\star n\star \star\epsilon\star'\star \star\star \star&\star \star\leq\star \star&\star I\star(KS\star;\star B\star^n\star)\star_\star\Omega\star \star-\star \star n\star \star\epsilon\star'\star \star\star
\star&\star \star=\star \star&\star I\star(S\star;\star B\star^n\star)\star_\star\Omega\star \star+\star I\star(K\star;\star B\star^n\star|\star S\star)\star_\star\Omega\star \star-\star \star n\star \star\epsilon\star'\star \star\star
\star&\star \star=\star \star&\star I\star(K\star;\star B\star^n\star|\star S\star)\star_\star\Omega\star \star-\star \star n\star \star\epsilon\star'\star.\star \star\end\star{eqnarray\star*\star}\star The\star first\star inequality\star is\star a\star triple\star application\star of\star \star Fannes\star'\star inequality\star,\star and\star the\star second\star is\star by\star the\star data\star processing\star inequality\star \star(see\star e\star.g\star.\star \star\cite\star{nie\star&chuang\star}\star)\star.\star The\star last\star line\star is\star by\star locality\star:\star the\star state\star of\star \star$B\star^n\star$\star is\star independent\star of\star which\star measurement\star \star$\star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star$\star gets\star applied\star to\star \star$A\star^n\star$\star \star.\star Thus\star there\star exists\star a\star particular\star \star$s\star$\star such\star that\star \star(\star\ref\star{due\star}\star)\star is\star satisfied\star for\star \star \star$\star\widetilde\star{\star\Lambda\star}\star \star=\star \star\widetilde\star{\star\Lambda\star}\star^\star{\star(s\star)\star}\star$\star.\star \star\qed\star
\star
\star\vspace\star{4mm\star}\star
\star
\star\begin\star{thebibliography\star}\star{99\star}\star \star\bibitem\star{bdsw\star}\star C\star.\star H\star.\star Bennett\star,\star D\star.\star P\star.\star DiVincenzo\star,\star J\star.\star A\star.\star Smolin\star and\star W\star.\star K\star.\star Wootters\star,\star \star Phys\star.\star Rev\star.\star A\star 54\star,\star 3824\star \star(1996\star)\star \star \star\bibitem\star{family\star}\star I\star.\star Devetak\star,\star A\star.\star W\star.\star Harrow\star,\star A\star.\star Winter\star,\star quant\star-ph\star/0308044\star \star(2003\star)\star \star \star\bibitem\star{hhhhoss\star}\star M\star.\star Horodecki\star,\star K\star.\star Horodecki\star,\star P\star.\star Horodecki\star,\star \star R\star.\star Horodecki\star,\star J\star.\star Oppenheim\star,\star A\star.\star Sen\star \star(De\star)\star,\star U\star.\star Sen\star,\star Phys\star.\star Rev\star.\star Lett\star.\star 90\star,\star 100402\star \star \star(2003\star)\star \star\bibitem\star{landauer\star}\star R\star.\star Landauer\star,\star IBM\star J\star.\star Res\star.\star Dev\star.\star 5\star,\star 183\star \star(1961\star)\star \star\bibitem\star{szilard\star}\star L\star.\star Szilard\star,\star Zeitschrift\star fur\star Physik\star 53\star,\star 840\star \star(1929\star)\star \star \star\bibitem\star{bennett\star}\star C\star.\star H\star.\star Bennett\star,\star Int\star.\star J\star.\star Phys\star 21\star,\star 905\star \star(1982\star)\star \star\bibitem\star{lloyd\star}\star S\star.\star Lloyd\star,\star Phys\star.\star Rev\star.\star A\star 56\star,\star 3374\star \star(1997\star)\star \star\bibitem\star{ohhh\star}\star J\star.\star Oppenheim\star,\star M\star.\star Horodecki\star,\star P\star.\star Horodecki\star,\star \star R\star.\star Horodecki\star,\star Phys\star.\star Rev\star.\star Lett\star.\star 89\star,\star 180402\star \star(2002\star)\star \star\bibitem\star{hho\star}\star M\star.\star Horodecki\star,\star P\star.\star Horodecki\star,\star J\star.\star Oppenheim\star,\star Phys\star.\star Rev\star.\star A\star 67\star 062104\star \star(2003\star)\star \star
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Home Journals JNMES Performance Evaluation of a Solar PV/T Water Heater Integrated with Inorganic Salt Based Energy Storage Medium
Performance Evaluation of a Solar PV/T Water Heater Integrated with Inorganic Salt Based Energy Storage Medium
Prakash R* | Meenakshipriya B | Vijayan S | Kumaravelan R
Department of Mechanical Engineering, Velalar College of Engineering and Technology, Thindal, Erode 638012, Tamilnadu, India
Department of Mechatronics Engineering, Kongu Engineering College, Perundurai, Erode 638052, Tamilnadu, India
Department of Mechanical Engineering, National Institute of Technology, Tiruchirappalli, 620015, Tamilnadu, India
[email protected]
Thermal and Electrical performance of solar PV/T hybrid water heating system using salt mixture phase change materials in storage tank is analyzed in this study. Compare to all conventional type heaters, the solar PV/T hybrid module collector has ability to produces both electrical energy from PV module and utilizes incident solar energy to heat the water. The sheet and tube type absorber is used to heat up the tube which is attached at the back side of PV module and transfer the heat to flowing water and the electrical energy is tested by connecting the DC load on the PV terminals under glazed and unglazed modes respectively. To enhance the thermal performance, energy storage medium is used as phase change materials at good proportion in the tank. The thermo physical properties of PCM are analyzed by Differential Scanning Calorimetry. This experimental testing is conducted from 8.00 to 17.00 IST in various sunny days and results are compared for glazed and unglazed conditions. The results shows that the average water temperature easily reaches 38-45°C and the final temperature of water never dropped below 34°, the temperature of PCM is 45.6oC, which is 5oC higher than outlet. The amount of heat stored using PCM in tank is 16.86% greater than no-PCM in the tank for constant 0.01 kg/s mass flow rate. The daily average electrical efficiency is 6.4% under glazed mode and 8.8% under unglazed conditions.
PV/T hybrid module, phase change materials, salt mixture, differential scanning calorimetry
Energy offers a remarkable contribution in improving the life standard of human beings. Major part of the energy requirements for the domestic and industrial applications are met with fossil fuels. Unfortunately, the sole dependency of the fossil fuels has led to unimaginable rise in price and threatening environmental pollution. Nowadays, renewable energy sources are being emphasized to address the aforementioned problems. Solar energy seems to be the promising choice as it is clean and freely available everywhere. Solar thermal energy from sunlight is used to heat water or other fluids, and can also power solar cooling system applications. Numerous thermal management technologies in PV/T are explored, and various specifications with performance enhancement techniques were incorporated in the PV/T collectors. Generally, they are subdivided further into air-heating PV/T and water-heating PV/T based on the variance (medium of heat transfer) involved in the process. In the solar PV/T hybrid module, both electrical and thermal energy are extracted. Approximately, 75% of solar radiation falling on a surface is reflected as loss or absorbed as heat sources. Thermal management in solar photovoltaic panel is considered for improving the electrical performance of the module and the PV module cooling with a fluid stream like air or water, the electricity yield can be improved. At the same time, the heat pick-up from the fluid can be used to support domestic or residential hot-water systems.
Recent researchers have suggested that the incorporation of phase change materials (PCM) is also a favourable option to improve the performance PV/T collectors. PCMs are incorporated in the solar thermal water tank to withhold the thermal energy in the form of sensible and latent heat. He et al. [1] studied the thermal performance of solar flat-box type hybrid solar collector using aluminum-alloy and achieved 40% on daily basis. Khalifa et al. [2] designed the solar collector using paraffin wax as phase change materials in the copper pipes of diameter 80 mm. By the use of PCMs in the storage media will enhance the heat transfer performance even at no sunlight and acts as the heat source in the late evenings. Mosaffa et al. [3] tested the multiple PCMs rather than single PCM in the latent heat storage system. It was found that the heat transfer performance shows greater improvement using metal fins kept inside metal screens and beads and impregnation on the porous materials. Allan et al. [4] experimented serpentine type solar collector and found that the overall efficiency is 61% higher and electrical efficiency is 8% lower than other system. Preet et al. [5] experimented the water based solar PV/T collector using paraffin wax as phase change material and investigated the effect of mass flow rate. They found that the thermal efficiency of the system increases with increase in flow rate of water. Al-Waeli et al. [6] reviewed the various techniques and methods used in solar PV/T system for enhancing the thermal and electrical efficiency using phase change materials. It has shown more interest in research to increase the productivity and in by reducing the cost. Yang et al. [7] conducted daily tests for comparing the performance of PV/T system and PV/T system with PCM. The result found that thermal and electrical efficiency by using PCM will be improved compared to conventional system and the simulation analysis are also checked using TRNSYS. Smyth et al. [8] designed a new type solar PV/T integrated facade collector system and tested at indoor solar simulator conditions. The efficiency of the new system is 5-10% better than a traditional unglazed hybrid system and calculated the heat retention efficiency for single and double-glazed condition. Browne et al. [9] tested the hybrid PV/T solar heater at three different condition PV/T system with PCM called Capric:Palmitic acid, PV/T system without PCM and PV module alone. The results show that by using PCM in the hybrid system, it can store more heat compared to traditional one. Browne et al. [10] discussed the various types of solar collector by adopting phase change materials at varying environment. In all types, the performance will be improved by using PCM. Venkatesh and Vijayan [11] designed and analysed the performance of the multi-purpose solar water heater. Iserval et al. [12] and kumar et al. [13], solar collectors for heating water. Stritih and Stritih [14] modified the solar PV system using phase change materials RT28HC and found that the maximum temperature of the panel is 35.6℃ higher than without using PCM and the same is also simulated using TRNSYS software and validated the results. Browne et al. [15] experimented the PCM called fatty acid eutectic, capric-palmitic in the solar hybrid system and showed that the performance is improved than without using PCM. Browne et al. [16] investigated the application of phase change materials for the thermal performance improvement. The temperature is about 50℃ regulated using PCM. Zhou and Eames [17] examined the thermal properties of inorganic salts like lithium nitrate and sodium chloride as PCMs. The DSC results proved that the thermal conductivity was 0.56-0.6 W/mk. The above salts were tested in three various materials like copper and stainless steel 304 and 316 grade for indirect thermal storage applications. In stainless steel 316 grade, the overall thermal performance is better compared to others.
The literature review revealed that the PV/T solar water heating system has many advantageous features and the incorporation of energy storage materials enhance the thermal as well as electrical performance. In an open literature survey, it is noticed that the effect of inorganic salt hydrates as PCM on solar PV/T water heating system performance has not been explored much. Therefore, in this work an attempt is made to study the thermal and electrical performance under glazed and unglazed conditions.
2.1 Experimental Setup Description
The block diagram of solar PV/T hybrid water heating system is shown in Figure 1. The hybrid PVT module is set by connecting a sheet and tube absorber at the backside of PV module. 20Wp poly crystalline type module is used in this experiment. The copper sheet of 36SWG is used as fin and it is painted with black to enhance the solar insolation absorptivity. Copper tube having diameter of 12.7 mm is used as header and 6.35 mm copper tubes are used as risers. The centre to centre spacing between the risers is kept as 50 mm. The useful surface area in hybrid module for heat absorption is 0.18 m2 and total PV module cell area is 0.135 m2. To prevent convective heat loss to the ambient, insulation material - glass wool of 25 mm thickness is used in rear portions of PV/T module and total setup is enclosed in a metallic container. The additional glazing is given at the top of the metal box using glass (solar grade sheet) of thickness 4 mm. This module is also made vapor tight using rubber beading. The photograph of the experimental setup is shown in below Figure 2. The PV/T hybrid module is inclined towards south facing and twisted at an angle of 30° to the ground.
The storage tank is made up of a double shell arrangement in cylindrical shape using SS304 grade sheet. It is enclosed by glass wool material at inner and outer of the tank to reduce heat loss to the environment. The storage collector of 50 litres capacity is used. The tank was fixed at 3 feet elevation from the floor on a metallic stand to ensure the proper circulation of the water through the system. Two numbers of thermometers are used to measure the temperature deviation at various sections such as the centre and the outlet of the storage tank. The heat energy loss to atmosphere is minimized by using the rubber beading at the outer surface of the tubes and the arrangement of PCM aluminium canisters in rack are shown in Figure 3.
Figure 1. Block diagram of solar PV/T hybrid water heater
Figure 2. Experimental setup
Figure 3. PCM containers
2.2 Phase Change Materials (PCMs)
The two inorganic salts hydrates like sodium thiosulphate pentahydrate (Na2SO3.5H2O) and magnesium nitrate hexahydrate (Mg(NO3)2.6H2O) is selected based on the literature study. The purchased salts from the manufacture are used directly without any purification. The salt mixtures are taken in the proportion 80% of Mg(NO3)2.6H2O+20% of Na2SO3.5H2O in the study. For the experiment, 2.5 kg of salt mixture PCM is used. During preparation, the two salts were physically mixed and then treated under hot air oven above 100℃. During the heat treatment, the salt mixture should undergo the heating process above melting range, and converts into a liquid solution. Then the solution was stirred at a speed of 450 rpm for 30 min with the help of stirrer. Then the hot liquid was allowed to solidify in room temperature in a closed container to prevent unwanted chemical reactions with the ambient air. Due to this, the two different salts are completely mixed with each other and it becomes homogeneous. Finally, by using the agate mortar, the salt mixture solid is altered into powder form. The complete preparation process of 20 gm salt mixture PCM is given in Figure 4.
Differential Scanning Calorimetry (DSC) is used for the evaluating the thermophysical properties like phase transition temperature, latent heat energy and heat stored capacity of PCM after the complete test runs. This testing facility is available at CSIR-CLRI Chennai, Tamilnadu, India. During testing, a small amount of prepared salt 10 gm is used for analysis. After the completion of test runs, DSC provides thermal properties of PCM. The photography of the DSC apparatus during testing is shown in Figure 5. The DSC results of the salt mixture are given in Table 1. PCMs salt was packed into aluminium canisters to avoid heat loss during higher temperature and arranged inside the storage tank.
Figure 4. Melting Process of 80:20 salt mixture PCMs
Figure 5. DSC apparatus during sample testing
Table 1. Thermo physical properties of salt PCM at 80:20 mixtures during DSC test
Phase transition temperature
34.2°C -
Latent heat energy
169.91 kJ/kg
Heat storage capacity
15.95 kJ/kgK
2.3 Instrumentation and Experimentation Procedure
The fabricated experimental setup is placed at Erode District (11°34'N, 77°71'E), Tamilnadu, India during testing. The experimental studies are conducted during the month of November 2019 from 08:00 hrs to 17:00 hrs every day. During the experimental testing, the data recordings are done manually by using the instruments such as solar power meter, bulb type thermometer, K-type thermocouple, digital thermometer, multimeter and flowmeter after accurate calibration. A solar power meter is utilized to measure the incident solar radiation at experiment site. The daily measured solar radiations are compared with the data's available in the Tamilnadu Meteorological Weather Report. The bulb thermometers are used to measure the tank temperature at various positions. To measure the ambient air temperature, Ktype thermocouple is used. In order to record the temperature of PV/T module, module temperature is measured by using digital thermometer. Two digital multimeters are connected across the PV panel output terminals to know the varying voltage and current during the glazed and unglazed conditions. The flowmeter is used to measure the mass flow rates of water. The inlet water to the storage tank was collected from an overhead water tank which is at 6 feet from the ground. The water from the tank flows into the hybrid module gets heated and rises to the tank again. Based on thermosyphon principle, the hot water increases due to lower density. The parameters like water temperature (at inlet, at outlet, on PCM container), ambient temperature, module temperature on top & bottom, tank temperature, voltage and current across the PV module are noted. By the measured quantities, the thermal, electrical and primary energy saving efficiencies are analyzed.
3. Data Reduction
The experimental setup is tested on daily basis from morning 8 am to evening 5 pm and the variation of the water temperature is recorded at a periodic interval. The electrical performance of solar photovoltaic like power, voltage and current production is noted at every 30 minutes time duration and the average results are used for analysis.
The thermal efficiency of the water heater fully depends on solar isolation, ambient temperature, wind speed etc. The formula used to calculate ηth is:
$\eta_{t h}=\frac{m \times C_{p} \times\left(T_{f}-T_{i n}\right)}{H \times A_{c}}$ (1)
where, m is the mass of water used in kg, Cp is the specific heat of water in J/kgK, H be the overall solar isolation on the collector surface in W/m2, Tin and Tf are the initial and final water temperatures in K and Ac is the collector surface area in m2. The formula used to calculate the thermal performance of the system; when the solar absorptance (α), overall system loss coefficient (U) and the average ambient temperature (Tamb) are known:
$\eta_{t h}=\alpha+U \times\left(\frac{T_{i}-T_{a m b}}{H}\right)$ (2)
The daily electrical efficiency ηel is calculated using the below equation,
$\eta_{\text {el }}=\int_{\text {day }}\left(\frac{V_{L} \times I_{L}}{A_{p v} \times H}\right)$ (3)
where VL, IL, Apv are load voltage, load current and radiation collecting area of the PV module respectively. In this expression, H value should be taken in the unit kWh/m2.
4. Results and Discussion
Experimental results of hybrid PV/T module under various conditions such as glazed and unglazed, with and without using PCM in storage tank have been reported and discussed. The thermal and electrical performances of the system and temperature variation on the PV/T module have been analyzed in the present section.
4.1 Variation of temperature of PV/T module with solar radiation
Figure 6. Temperature profile during load test
Figure 7. Temperature profile during no-load test
The outlet water temperature during load and no-load test on time variation are shown in Figures 6 and 7. The load test is carried out by connecting the electrical terminals of the hybrid module. In this case, both thermal and electrical energy is produced simultaneously. The temperature on PV and PV/T module are measured separately on the same site location and at variation positions like PV/T top & bottom, PV top & bottom and ambient are analyzed. From Figure 6, it is observed that the higher temperature is at PV/T top because of combined power generation during PV process and reelected radiation within the module and followed by PV/T bottom due to blocking of incoming solar radiation by copper. Next higher temperature is at PV top and bottom, because the transparency in copper sheet is always higher than tedlar sheet. The maximum temperature differences of 7.34℃ and 6.28℃ between top and bottom layers on PV and PV/T module are noted respectively. The temperature difference between the top and bottom layer is minimum during peak solar energy compared to the PV module. This is because of lower thermal resistance of the PV/T module and higher thermal conductivity. During the no-load test, an electrical terminal of PV module is disconnected. Hence, it acts in open-circuit condition; no current will be produced, only as heat energy is generated. Due to isolated conditions of electrical terminals, PV/T bottom has higher temperature compared to PV/T top. Since there will be some emission of radiation in copper sheet, it stands second. The peak temperature variation between the top and bottom layers in PV and PV/T module is 5.20℃ and 4.72℃. It is found that, PV and PV/T module temperature rise is constant and slightly higher in PV module because of greater thermal resistance.
4.2 Thermal performance of storage tank with PCMs
The temperatures of the water storage tank at various location of the tank such as top, middle and bottom portions are measured. Figure 8 shows the measurement of temperature for every 10 minutes time period without any hot water removal. The temperature reached the maximum of 69℃, 61℃ and 55℃ for the tank top, tank middle and tank bottom locations respectively, in one day. The top layer and middle layer of the thermal storage tank showed a steep increase in the temperature build-up, but the tank bottom layer temperature change was more gradual. This provides cooler water to the inlet of the solar collector at bottom. This results in the improved conversion efficiency of the PV/T module.
From Figure 9, it is seen that the average temperature difference of 4.28℃ was obtained during the period from 10:30 am to 01:00 pm, with a maximum temperature difference of 8.46℃ between the outlet fluid and inlet fluid of the solar collector. The maximum temperature in PCM container is 54.5℃, which is 10.77% higher than the outlet water temperature without PCM. Among all the positions of the tank, PCM container holds higher temperature. This type of PCM has an ability to store the latent heat.
Figure 8. Storage tank temperature performance
Figure 9. Temperature of tank at various positions
Figure 10. Heat stored performance
Figure 11. Effect of mass flow rate on temperature
Figure 10 shows the heat stored performance of tank with PCM. A significant improvement of 16.86% of amount of heat energy stored at same climatic condition. This clearly proved that, by the use of PCM in the tank the heat stored and storage time of heat energy has been increased. So, this type of salt PCM will be highly recommended for the use of heat energy during half sunshine hours. The performance of the hybrid water heater is analyzed by varying the flow rates from 0.01 kg/s to 0.0667 kg/s as shown in Figure 11. The maximum temperature of the water in the tank was attained at lower mass flow rate of 0.016 kg/s and the time taken to attain the stratified temperature of 50°C was 9 hours, whereas for 0.016 kg/s the time taken to attain the stratified temperature was 5 hours. It clearly depicts that the amount of heat transferred increases with increase in mass flow rate of water. Further, it leads to reduction in reduced stratified temperature in the tank. It revealed that by the addition of PCM in the tank shows better improvement in results. The thermal efficiency of the collector increases with the increase in the rate of water flow, whereas the average temperature rise falls with the rise in the mass flow rate. From the above results, it has been noted that the incorporation of inorganic salt hydrates inside the storage tank has significant effect on the performance by lowering the water temperature to the collector. Hence further experimental studies to investigate the performance under glazed and unglazed conditions. Throughout the studies mass flow rate has been maintained as 0.01 kg/s.
4.3 Thermal performance of PV/T module with PCMs
The semi-empirical system efficiency model that correlates the daily efficiency test results with reduced temperature (Ti–Tamb)/H are calculated from the equation (2). In this equation, α represents the system efficiency, when Ti=Tamb. The values of α and U can be determined by linear regression analysis, which has been used in this experimentation. The plots of ηth against reduced temperature (Ti-Tamb)/H, for both glazed and unglazed modes, are given in Figure 12. The corresponding linear regression line equations were found to be,
$\text { For glazed mode } \eta_{\mathrm{th}}=30.01-9.142\left(\mathrm{T}_{\mathrm{i}}-\mathrm{T}_{\mathrm{amb}}\right) / \mathrm{H}$ (4)
$\text { For unglazed mode } \eta_{\mathrm{th}}=16.36-15.09\left(\mathrm{T}_{\mathrm{i}}-\mathrm{T}_{\mathrm{amb}}\right) / \mathrm{H}$ (5)
The 'α' value was found to be lower than a conventional solar thermal collector for both the modes. This is because, the PV module is fixed over the thermal absorber sheet and it resists the falling solar radiation on the absorber. The emissivity of a PV module is much higher compared to the absorber of the traditional solar collector which has a selective absorptive layer coated on it. Hence, the heat loss between PV/T collector and environment is much higher than that of the traditional solar collector and the absorbed solar energy by the PV/T collector is much less than the traditional solar collector. The additional thermal resistance exists between the PV/T surface exposed to solar radiation and water path in tube due to aluminium absorber plate and an adhesive layer.
Figure 12. Thermal efficiency vs reduced temperature during glazed and unglazed conditions
Figure 13. Hourly variations in water temperature in glazed and unglazed conditions
The hourly variations in the water temperature for glazed and unglazed modes are shown in Figure 13. The peak water temperature during glazed and unglazed modes is 60℃ and 50℃ respectively. There will be 10℃ temperature rise in glazed compared to unglazed operation. This is because; the extra glazing will reduce the heat loss and in turn increases the PV/T module temperature. It is also found that the water temperature increased till 14:30 hrs after which there was only a marginal rise in it. It showed that the active heating took place between 10:30 hrs to 14:30 hrs during which much of the heating had occurred.
4.4 Electrical performance of PV/T module with PCMs
The 12V DC bulb is used as electrical load. Two digital multimeters are used for measuring the voltage, current and power of the system. The electrical power is calculated by using the formula P = V x I, whereas the voltage and current are measured from hybrid module. This PV/T hybrid module is tested on both glazed and unglazed conditions separately. The hourly electrical performances of a solar PV/T module have been noted on vary with the solar radiation and shown in Figures 14 and 15. The instant maximum electrical efficiency is achieved at 9.14% for glazed and 11.94% for unglazed modes. The average electrical efficiency is 6.4% and 8.8% during glazed and unglazed operations. It also found that the daily average electrical efficiency is always smaller than the instantaneous range. In the unglazed modes, more amount of output power is observed compared to glazed modes. In case of glazing modes, due to glazing effect, there will be more reflection losses and hence produces lesser power output and efficiency.
Figure 14. Electrical Power output with solar radiation
Figure 15. Electrical Efficiency of PV module
Figure 16. Load voltage vs Time
Figure 17. Load current vs Time
Figure 18. Module temperature vs Time
The electrical efficiency of the PV module is compared with PV/T hybrid collector and the results are analyzed. The PV module temperature load voltage and load current at different operating modes such as glazed, unglazed and PV alone are measured and shown in Figure 16, 17 and 18.
The results found that the electrical efficiency of the PV module is 8.1%. It is observed that the solar PV module temperature will easily attain 53°C under free convection. The electrical efficiency of the solar hybrid PV/T module should be higher than the PV module; because of the flowing water through the pipes removes the heat directly from the PV module and consequently the temperature of PV module decreases and therefore efficiency will be increased. When there is a good thermal insulation between the PV module and the absorber plate, the water flows through the pipes will increase the heat dissipation from the PV module, and hence heat convection between the PV module and the ambient air will be decreased roughly. In such case, the electrical efficiency is found to be lesser. When the temperature of the flowing water is colder, then the heat transfer between the module and water is increased. In such case, the temperature of the water is increased slowly. Hence, the temperature of PV module in the PV/T hybrid module will be superior to the temperature of a PV module alone. Thus, the electrical performance of hybrid PV/T collector is cheaper. For this reason, the extra glazing cover is required in need of higher power output and efficiency.
The performance of the solar PV/T water heating system is continuously improved by using various technologies. One such improvement is by using inorganic salt hydrate as PCM. The thermal performance of the hybrid heater is improved by using PCM in storage tank and also the electrical efficiency of the PV module is improved by using refractive mirror. From experimental results, the following conclusions are made:
Mg(NO3)2.6H2O+Na2SO3.5H2O salts with 80:20 percentage volume fraction has increased the storage tank useful temperature by 16.86% in thermosyphon mode.
The maximum practical water temperature in storage tank reached is 75.2℃.
The highest outlet temperature from the collector is 75oC using salt mixture as PCM in thermosyphon, while the lowest temperature is at 41.6℃ in 0.01 kg/s.
During load test, the maximum temperature difference is 7.34℃ and 6.28℃ and in no-load condition, it is 5.1℃ and 3.8℃ between top and bottom layers on PV and PV/T module
The average thermal efficiency using salt mixture PCM is 33.5% in 0.01 kg/s
The average electrical power output is 11.5W in PV/T module and 11.7W in PV module respectively. Hence, the electrical output of the PV/T is alike as PV module.
The maximum electrical efficiency of the PV/T module is 7.8% almost same as 8% in PV module in normal conditions.
The average daily electrical efficiency of the glazed system is observed at 6.4% and 8.8% for unglazed modes with PCM.
A significant thermal efficiency of 33.25% improvement is observed with glazing for 0.01 kg/s, which is 4% greater than without glazing using PCM.
The electrical efficiency is reduced as 15.78% in the presence of glazing with PCM.
In order to increase the yield of electrical performance, the use of automatic tracking controller at the proper tuning circuit is preferable. This is the future scope of the present work. By using salt mixture PCMs, the result shows a greater improvement in the overall thermal performance of the hybrid module and so it is highly suitable for domestic hot water applications.
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\begin{document}
\title{Self-Assembly of Infinite Structures}
\begin{abstract} We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent various notions of computation self-assemble. Several open questions are also presented and motivated. \end{abstract}
\section{Introduction}
The simplest mathematical model of nanoscale self-assembly is the Tile Assembly Model (TAM), an effectivization of Wang tiling \cite{Wang61,Wang63} that was introduced by Winfree \cite{Winf98} and refined by Rothemund and Winfree \cite{RotWin00,Roth01}. (See also \cite{Adle99,Reif02,SolWin07}.) As a basic model for the self-assembly of matter, the TAM has allowed researchers to explore an assortment of avenues into both laboratory-based and theoretical approaches to designing systems that self-assemble into desired shapes or autonomously coalesce into patterns that, in doing so, perform computations.
Actual physical experimentation has driven lines of research involving kinetic variations of the TAM to deal with molecular concentrations, reaction rates, etc. as in \cite{Winfree98simulationsof}, as well as work focused on error prevention and error correction \cite{ChenGoel04,WinBek03,SolWin05}. For examples of the impressive progress in the physical realization of self-assembling systems, see \cite{RoPaWi04,MajSahLaBRei06}.
Divergent from, but supplementary to, the laboratory work, much theoretical research involving the TAM has also been carried out. Interesting questions concerning the minimum number of tile types required to self-assemble shapes have been addressed in \cite{SolWin07,RotWin00,ACGHKMR02,AGKS04}. Different notions of running time and bounds thereof were explored in \cite{AdChGoHu01,BeckerRR06,CGM04}. Variations of the model where temperature values are intentionally fluctuated and the ensuing benefits and tradeoffs can be found in \cite{KS07,AGKS04}. Systems for generating randomized shapes or approximations of target shapes were investigated in \cite{KaoSchS08,BeckerRR06}. This is just a small sampling of the theoretical work in the field of algorithmic self-assembly.
However, as different as they may be, the above mentioned lines of research share a common thread. They all tend to focus on the self-assembly of \emph{finite} structures. Clearly, for experimental research, this is a necessary limitation. Further, if the eventual goal of most of the theoretical research is to enable the development of fully functional, real world self-assembly systems, a valid question is: ``Why care about anything other than finite structures?'' This is the question that we address in this paper.
This paper surveys a collection of recent findings related to the self-assembly of \emph{infinite} structures in the TAM. As a theoretical exploration of the TAM, this collection of results seeks to define absolute limitations on the classes of shapes that self-assemble. These results also help to explore how fundamental aspects of the TAM, such as the inability of spatial locations to be reused and their immutability, affect and limit the constructions and computations that are achievable.
In addition to providing concise statements and intuitive descriptions of results, we also define and motivate a set of open questions in the hope of furthering this line of research. First, we begin with some preliminary definitions and constructions that will be referenced throughout this paper.
\section{Preliminaries}
\subsection{The Tile Assembly Model}
This section provides a very brief overview of the TAM. See \cite{Winf98,RotWin00,Roth01,jSSADST} for other developments of the model. Our notation is that of \cite{jSSADST}. We work in the $2$-dimensional discrete space $\mathbb{Z}^2$. We write $U_2$ for the set of all {\it unit vectors}, i.e., vectors of length 1 in $\mathbb{Z}^2$. We write $[X]^2$ for the set of all $2$-element subsets of a set $X$. All {\it graphs} here are undirected graphs, i.e., ordered pairs $G = (V, E)$, where $V$ is the set of {\it vertices} and $E \subseteq [V]^2$ is the set of {\it edges}.
A {\it grid graph} is a graph $G = (V, E)$ in which $V \subseteq \mathbb{Z}^2$ and every edge $\{\vec{a}, \vec{b} \} \in E$ has the property that $\vec{a} - \vec{b} \in U_2$. The {\it full grid graph} on a set $V \subseteq \mathbb{Z}^2$ is the graph $\fgg{V} = (V, E)$ in which $E$ contains {\it every} $\{\vec{a}, \vec{b} \} \in [V]^2$ such that $\vec{a} - \vec{b} \in U_2$.
Intuitively, a tile type $t$ is a unit square that can be translated, but not rotated, having a well-defined ``side $\vec{u}$'' for each $\vec{u} \in U_2$. Each side $\vec{u}$ of $t$ has a ``glue'' of ``color'' $\textmd{col}_t(\vec{u})$ - a string over some fixed alphabet $\Sigma$ - and ``strength'' $\textmd{str}_t(\vec{u})$ - a natural number - specified by its type $t$. Two tiles $t$ and $t'$ that are placed at the points $\vec{a}$ and $\vec{a}+\vec{u}$ respectively, {\it bind} with {\it strength} $\textmd{str}_t\left(\vec{u}\right)$ if and only if $\left(\textmd{col}_t\left(\vec{u}\right),\textmd{str}_t\left(\vec{u}\right)\right) = \left(\textmd{col}_{t'}\left(-\vec{u}\right),\textmd{str}_{t'}\left(-\vec{u}\right)\right)$.
Given a set $T$ of tile types, an {\it assembly} is a partial function $\pfunc{\alpha}{\mathbb{Z}^2}{T}$.
An assembly is $\tau$-{\it stable}, where $\tau \in \mathbb{N}$, if it cannot be broken up into smaller assemblies without breaking bonds whose strengths sum to at least $\tau$.
Self-assembly begins with a {\it seed assembly} $\sigma$ and proceeds asynchronously and nondeterministically, with tiles adsorbing one at a time to the existing assembly in any manner that preserves stability at all times. A {\it tile assembly system} ({\it TAS}) is an ordered triple $\mathcal{T} = (T, \sigma, \tau)$, where $T$ is a finite set of tile types, $\sigma$ is a seed assembly with finite domain, and $\tau$ is the temperature. An {\it assembly sequence} in a TAS $\mathcal{T} = (T, \sigma, 1)$ is a (possibly infinite) sequence $\vec{\alpha} = ( \alpha_i \mid 0 \leq i < k )$ of assemblies in which $\alpha_0 = \sigma$ and each $\alpha_{i+1}$ is obtained from $\alpha_i$ by the ``$\tau$-stable'' addition of a single tile.
We write $\prodasm{T}$ for the {\it set of all producible assemblies of} $\mathcal{T}$. An assembly $\alpha$ is {\it terminal}, and we write $\alpha \in \termasm{\mathcal{T}}$, if no tile can be stably added to it. We write $\termasm{T}$ for the {\it set of all terminal assemblies of } $\mathcal{T}$. A TAS ${\mathcal T}$ is {\it directed}, or {\it produces a unique assembly}, if it has exactly one terminal assembly i.e., $|\termasm{T}| = 1$. The reader is cautioned that the term ``directed" has also been used for a different, more specialized notion in self-assembly \cite{AKKR02}.
A set $X \subseteq \mathbb{Z}^2$ {\it weakly self-assembles} if there exists a TAS ${\mathcal T} = (T, \sigma, 1)$ and a set $B \subseteq T$ such that $\alpha^{-1}(B) = X$ holds for every assembly $\alpha \in \termasm{T}$. A set $X$ {\it strictly self-assembles} if there is a TAS $\mathcal{T}$ for which every assembly $\alpha\in\termasm{T}$ satisfies ${\rm dom} \; \alpha = X$. The reader is encouraged to consult \cite{SolWin07} for a detailed discussion of {\it local determinism} - a general and powerful method for proving the correctness of tile assembly systems.
\subsection{Discrete Self-Similar Fractals} In this subsection we introduce discrete self-similar fractals, and zeta-dimension.
\begin{definition} \label{def-c-discrete-self-similar-fractal} Let $1 < c \in \mathbb{N}$, and $X\subsetneq \mathbb{N}^2$. We say that $X$ is a $c$-{\it discrete self-similar fractal}, if there is a (non-trivial) set $V \subseteq \{0,\ldots,c-1\}\times\{0,\ldots,c-1\}$ such that $\displaystyle X = \bigcup_{i=0}^{\infty}{X_i}$, where $X_i$ is the $i^{\textmd{th}}$ {\it stage} satisfying $X_0 = \{(0,0)\}$, and $X_{i+1} = X_i \cup \left(X_i + c^i V \right)$. In this case, we say that $V$ {\it generates} $X$. \end{definition}
\begin{figure}
\caption{\small Example of a $c$-discrete self-similar fractal ($c = 3$), the Sierpinski carpet}
\label{fig:fractal_stage1}
\label{fig:fractal_stage2}
\label{fig:fractal_stage3}
\label{fig_fractal_stage4}
\label{fig:fractals}
\end{figure}
The most commonly used dimension for discrete fractals is zeta-dimension, which we refer to in this paper.
\begin{definition}\cite{ZD} For each set $A \subseteq \mathbb{Z}^2$, the {\it zeta-dimension} of $A$ is \begin{gather*}
\textmd{Dim}_\zeta(A) = \limsup_{n \rightarrow \infty}\frac{ \log|A_{\le n}|}{\log n}, \end{gather*}
where $A_{\le n} = \{(k,l) \in A \mid |k|+|l| \le n\}$. \end{definition} It is clear that $0 \le \textmd{Dim}_\zeta(A) \le 2$ for all $A \subseteq \mathbb{Z}^2$.
\subsection{The Wedge Construction}
\begin{wrapfigure}{r}{.5\textwidth}
\includegraphics[width=2.5in]{wedge_example} \caption{Example of the first four rows of a sample wedge construction which is simulating a Turing machine $M$ on the input string `01'} \label{fig:wedge_construction}
\end{wrapfigure}
In order to perform universal computation in the TAM, we make use of a particular TAS called the ``wedge construction'' \cite{SADS}. The wedge construction, based on Winfree's proof of the universality of the TAM \cite{Winf98}, is used to simulate an arbitrary Turing machine $M = (Q, \Sigma, \Gamma, \delta, q_0, q_A, q_R)$ on a given input string $w \in \Sigma^*$ in a temperature 2 TAS.
The wedge construction works as follows. Every row of the assembly specifies the complete configuration of $M$ at some time step.
$M$ starts in its initial state with the tape head on the leftmost tape cell and we assume that the tape head never moves left off the left end of the tape. The seed row (bottommost) encodes the initial configuration of $M$. There is a special tile representing a blank tape symbol as the rightmost tile in the seed row. Every subsequent row grows by one additional cell to the right. This gives the assembly the wedge shape responsible for its name. Figure~\ref{fig:wedge_construction} shows the first four rows of a wedge construction for a particular TM, with arrows depicting a possible assembly sequence.
\section{Strict Self-Assembly}
The self-assembly of shapes (i.e., subsets of $\mathbb{Z}^2$) in the TAM is most naturally characterized by strict self-assembly.
In searching for absolute limitations of strict self-assembly in the TAM, it is necessary to consider infinite shapes because any finite, connected shape strictly self-assembles via a spanning tree construction in which there is a unique tile type created for each point. In this section we discuss (both positive and negative) results pertaining to the strict self-assembly of infinite shapes in the TAM.
\subsection{Pinch-point Discrete Self-Similar Fractals Do Not Strictly Self-Assemble}
In \cite{SADSSF}, Patitz and Summers defined a class $\mathcal{C}$ of (non-tree) ``pinch-point'' discrete self-similar fractals, and proved that if $X \in \mathcal{C}$, then $X$ does not strictly self-assemble.
\begin{definition} A {\it pinch-point discrete self-similar fractal} is a discrete self-similar fractal satisfying (1) $\{(0,0),(0,c-1),(c-1,0)\} \subseteq V$, (2) $V \cap (\{1,\ldots c-1\}\times \{c-1\}) = \emptyset$, (3), $V \cap (\{c-1\} \times \{1,\ldots, c-1\}) = \emptyset$, and $\fgg{V}$ is connected \end{definition}
A famous example of a pinch-point fractal is the standard discrete Sierpinski triangle $\mathbf{S}$. The impossibility of the strict self-assembly of $\mathbf{S}$ was first shown in \cite{jSSADST}. Figure~\ref{fig:pinch_points_highlighted} shows another example of a pinch-point discrete self-similar fractal. Note that any fractal $X$ such that $\fgg{X}$ is a tree is necessarily a pinch-point discrete self-similar fractal.
The following (slight) generalization to \cite{jSSADST} was shown in \cite{SADSSF}.
\begin{theorem} \label{firstmaintheorem} If $X \subsetneq \mathbb{N}^2$ is a pinch-point discrete self-similar fractal, then $X$ does not strictly self-assemble in the TAM. \end{theorem}
The idea behind the proof of Theorem~\ref{firstmaintheorem} can be seen in Figure~\ref{fig:pinch_points_highlighted}. Note that the black points are pinch-points in the sense that arbitrarily large aperidic sub-structures appear on the far-side of the black tile from the origin.
\begin{wrapfigure}{r}{.45\textwidth}
\begin{center}
\includegraphics[width=2.0in]{pinch_points_highlighted}
\caption{\label{fig:pinch_points_highlighted}\small An example of the first four stages of pinch-point fractal with the first three
pinch-points highlighted in black.}
\end{center} \end{wrapfigure}
Theorem~\ref{firstmaintheorem} motivates the following question.
\begin{openproblem}\label{no_fractals_assemble} Does any non-trivial discrete self-similar fractal strictly self-assemble in the TAM? We conjecture that the answer is `no', for any temperature $\tau \in \mathbb{N}$.
However, proving that there exists a (non-trivial) discrete self-similar fractal that does strictly self-assemble would likely involve a novel and useful algorithm for directing the behavior self-assembly. \end{openproblem}
\subsection{Strict Self-Assembly of Nice Discrete Self-Similar Fractals}
As shown above, there is a class of discrete self-similar fractals that do not strictly self-assemble (at any temperature) in the TAM. However, in \cite{SADSSF}, Patitz and Summers introduced a particular set of ``nice'' discrete self-similar fractals that contains some but not all pinch-point discrete self-similar fractals. Further, they proved that any element of the former class has a ``fibered'' version that strictly self-assembles.
\subsubsection{Nice Discrete Self-Similar Fractals}
\begin{definition} A {\it nice discrete self-similar fractal} is a discrete self-similar fractal such that $(\{0,\ldots,c-1\} \times \{0\}) \cup (\{0\}\times\{0,\ldots,c-1\}) \subseteq V$, and $\fgg{V}$ is connected. \end{definition}
See Figure~\ref{fig:nice_fractals} for examples of both nice, and non-nice discrete self-similar fractals.
\begin{figure}
\caption{\small Stage 2 of some discrete self-similar fractals.}
\label{fig:nice}
\label{fig:bad}
\label{fig:nice_fractals}
\end{figure}
\subsubsection{Nice Fractals Have Fibered Versions}
The inability of pinch-point fractals (and the conjectured inability of any discrete self-similar fractal) to strictly self-assemble in the TAM is based on the intuition that the necessary amount of information cannot be transmitted through available connecting tiles during self-assembly.
Thus, for any nice discrete self-similar fractal $X$, Patitz and Summers \cite{SADSSF} defined a fibered operator $\mathcal{F}(X)$ (a routine extension of \cite{jSSADST}) which adds, in a zeta-dimension-preserving manner, additional bandwidth to $X$. Strict self-assembly of $\mathcal{F}(X)$ is achieved via a ``modified binary counter'' algorithm that is embedded into the additional bandwidth of $\mathcal{F}(X)$.
For any nice discrete self-similar fractal $X$, $\mathcal{F}(X)$ is defined recursively. Figure~\ref{fig:fibered_example} shows an example of the construction of $\mathcal{F}(X)$, where $X$ is the discrete Sierpinski carpet.
\begin{figure}
\caption{\small Construction of the fibered Sierpinski carpet}
\label{fig:fibered_example}
\end{figure} Note that $\mathcal{F}(X)$ is only defined if $X$ is a nice discrete self-similar fractal. Moreover, it appears non-trivial to extend $\mathcal{F}$ to other discrete self-similar fractals such as the `H' fractal (the second-to-the-right most image in Figure~\ref{fig:nice_fractals}).
\begin{openproblem} Does there exist a zeta-dimension-preserving fibered operator $\mathcal{F}$ for a class of discrete self-similar fractals which is a superset of the nice discrete self-similar fractals (e.g. it also includes the `H' fractal)? The above open question is intentionally vague. Not only should $\mathcal{F}$ preserve zeta-dimension, but $\mathcal{F}(X)$ should also ``look'' like $X$ in some reasonable visual sense. \end{openproblem}
\section{Weak Self-Assembly}
Weak self-assembly is a natural way to define what it means for a tile assembly system to compute. There are examples of (decidable) sets that weakly self-assemble but do not strictly self-assemble (i.e., the discrete Sierpinski triangle \cite{jSSADST}). However, if a set $X$ weakly self-assembles, then $X$ is necessarily computably enumerable. In this section, we discuss results that pertain to the weak self-assembly of (1) discrete self-similar fractals \cite{SADSSF}, (2) decidable sets \cite{SADS}, and (3) computably enumerable sets \cite{CCSA}.
\subsection{Discrete Self-Similar Fractals}
Recall that pinch-point discrete self-similar fractals do not strictly self-assemble (at any temperature). Furthermore, Patitz and Summers \cite{SADSSF} proved that \emph{no} (non-trivial) discrete self-similar fractal weakly self-assembles in a locally deterministic \cite{SolWin07} temperature 1 tile assembly system.
\begin{theorem} \label{weaktheorem} If $X \subsetneq \mathbb{N}^2$ is a discrete self-similar fractal, and $X$ weakly self-assembles in the locally deterministic TAS $\mathcal{T}_X = (T,\sigma,\tau)$, where $\sigma$ consists of a single tile placed at the origin, then $\tau > 1$. \end{theorem}
Intuitively, the proof relies on the aperiodic nature of discrete self-similar fractals and the fact that the binding (a.k.a. adjacency) graph of the terminal assembly of $\mathcal{T}_X$ is an infinite tree, and every infinite branch is composed of an infinite, periodically repeating sequence of tile types.
\begin{openproblem} Does Theorem \ref{weaktheorem} hold for any directed (not necessarily locally deterministic) TAS? We conjecture that it does, and that such a proof would provide useful new tools for impossibility proofs in the TAM. \end{openproblem}
\subsection{Decidable Sets}
We now shift gears and discuss the weak self-assembly of sets at temperature~2.
\subsubsection{A Characterization of Decidable Sets of Natural Numbers} \label{decidable_in_N}
In \cite{SADS}, Patitz and Summers exhibited a novel characterization of decidable sets of positive integers in terms of weak self-assembly in the TAM, where they proved the following theorem.
\begin{theorem} \label{sads_theorem} Let $A \subseteq \mathbb{N}$. Then $A \subseteq \mathbb{N}$ is decidable if and only if $A \times \{0\}$ and $A^c \times \{0\}$ weakly self-assemble. \end{theorem}
Theorem~\ref{sads_theorem} is the ``self-assembly version'' of the classical theorem, which says that a set $A \subseteq \mathbb{N}$ is decidable if and only if $A$ and $A^c$ are computably enumerable. The following lemma makes the proof of the reverse direction of Theorem~\ref{sads_theorem} straight-forward.
\begin{lemma} \label{primitive_simulator} Let $X \subseteq \mathbb{Z}^2$. If $X$ weakly self-assembles, then $X$ is computably enumerable. \end{lemma}
The proof of Lemma~\ref{primitive_simulator} constructs a self-assembly simulator to enumerate $X$.
To prove the forward direction of Theorem~\ref{sads_theorem}, it suffices to construct an infinite stack of wedge constructions and simply propagate the halting signals down to the negative $y$-axis. This is illustrated in Figure~\ref{fig:decider_overview}.
\begin{figure}
\caption{\small The left-most (dark grey) vertical bars represent a binary counter that is embedded into the tile types of the TM; the darkest (black) rows represent the initial configuration of $M$ on inputs 0, 1, and 2; and the (light grey) horizontal rows that contain a white/black tile represent halting configurations of $M$. Although this image seems to imply that the embedded binary counter increases its width (to the left) each time it increments, this is not true in the construction. This image merely depicts the general shape of the counter as it increments.}
\label{fig:decider_overview}
\end{figure}
\subsubsection{Quadrant Optimality}
In addition to their positive result, Patitz and Summers \cite{SADS} established that any tile assembly system $\mathcal{T}$ that ``row-computes'' a decidable language $A \subseteq \mathbb{N}$ having sufficient space complexity must place at least one tile in each of two adjacent quadrants. A TAS $\mathcal{T}$ is said to \emph{row-compute} a language $A \subseteq \mathbb{N}$ if $\mathcal{T}$ simulates a TM $M$ with $L(M) = A$ on every input $n \in \mathbb{N}$, one row at a time, and uses single-tile-wide paths of tiles to propagate the answer to the question, ``does $M$ accept input $n$?" to the $x$-axis. Figure~\ref{fig:decider_overview} depicts the essence of what it means for a TAS to row-compute some language. This result, stated precisely, is as follows.
\begin{theorem}\label{quadrant_optimality} Let $A \subseteq \mathbb{N}$. If $A \not \in \textmd{DSPACE}\left(2^n\right)$, and $\mathcal{T}$ is any TAS that ``row-computes'' $A$, then every terminal assembly of $\mathcal{T}$ places at least one tile in each of two adjacent quadrants. \end{theorem}
\begin{openproblem} Let $A \subseteq \mathbb{N}$ with $A \not \in \textmd{DSPACE}\left(2^n\right)$. Is it possible to construct a directed TAS $\mathcal{T}$ in which the sets $A \times \{0\}$ and $A^c \times \{0\}$ weakly self-assemble, and every terminal assembly $\alpha \in \termasm{T}$ is contained in the first quadrant? We conjecture that the answer is `no', and any proof would account for all, possibly exotic methods of computation in the TAM, not only by row-computing. \end{openproblem}
\subsubsection{There Exists a Decidable Set That Does Not Weakly Self-Assemble}
In contrast to Theorem~\ref{sads_theorem}, Lathrop, Lutz, Patitz, and Summers \cite{CCSA} proved that there are decidable sets $D \subseteq \mathbb{Z}^2$ that do not weakly self-assemble. To see this, for each $r \in \mathbb{N}$, define $$
D_r = \{\left.(m,n) \in \mathbb{Z}^2 \; \right| \; |m|+|n|=r\}. $$ This set is a ``diamond'' in $\mathbb{Z}^2$ with radius $r$ and center at the origin. For each $A \subseteq \mathbb{N}$, let $$ D_A = \bigcup_{r\in A}{D_r}. $$ This set is the ``system of concentric diamonds'' centered at the origin with radii in $A$.
Using Lemma~\ref{primitive_simulator}, one can establish the following result.
\begin{lemma}\label{dtime} Let $A \in \mathbb{N}$. If $D_A$ weakly self-assembles, then there exists an algorithm that, given $r \in \mathbb{N}$, halts and accepts in time $O(2^{4n})$, where $n = \lfloor\lg r \rfloor + 1$, if and only if $r \in A$. \end{lemma}
The proof of Lemma~\ref{dtime} is based on the simple observation that each diamond is finite, and once a tile is placed at some point, it cannot be removed.
The time hierarchy theorem \cite{HarSte65} can be employed to show that there exists a set $A \in \mathbb{N}$ such that $A \in \textmd{DTIME}\left(2^{5n}\right)-\textmd{DTIME}\left(2^{4n}\right)$. Lemma~\ref{dtime} with $D=D_A$ is sufficient to prove the following theorem.
\begin{theorem}\label{CCSA_impossibility_proof} There is a decidable set $D \subseteq \mathbb{Z}^2$ that does not weakly self-assemble. \end{theorem}
It is easy to see that if $A \subseteq \mathbb{N}$, then $D_A \in \textmd{DTIME}\left(2^{\textmd{linear}}\right)$ because you can simulate self-assembly with a Turing machine. Is it possible to do better?
\begin{openproblem} \cite{CCSA} Is there a polynomial-time decidable set $D \in \mathbb{Z}^2$ such that $D$ does not weakly self-assemble? \end{openproblem}
\subsection{Computably Enumerable Sets}
The characterization of decidable sets in terms of weak self-assembly \cite{SADS} is closely related to the characterization of computably enumerable sets in terms of weak self-assembly due to Lathrop, Lutz, Patitz and Summers \cite{CCSA}.
Let $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ be a function such that for all $n \in \mathbb{N}$, $f(n) \geq n$ and $f(n) = O\left(n^2\right)$.
For each set $A\subseteq \mathbb{Z}^+$, the set $$ X_A = \left\{ (f(n),0) \mid n \in A \right\} $$ is thus a straightforward representation of $A$ as a set of points on the positive $x$-axis. The first main result of \cite{CCSA} is stated as follows.
\begin{theorem} \label{CCSA_firstmaintheorem} If $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ is a function as defined above, then, for all $A \subseteq \mathbb{Z}^+$, $A$ is computably enumerable if and only if the set $X_A = \{ (f(n), 0) \mid n \in A \}$ self-assembles. \end{theorem}
The reverse direction of the proof follows easily from Lemma~\ref{primitive_simulator}. To prove the forward direction, it is sufficient to exhibit, for any TM $M$, a directed TAS $\mathcal{T}_M$ that correctly simulates $M$ on all inputs $x \in \mathbb{Z}^+$ in $\mathbb{Z}^2$. A snapshot of the main construction of \cite{CCSA} is shown in Figure~\ref{fig:CCSA_construction}.
\begin{figure}
\caption{Simulation of $M$ on every input $x \in \mathbb{N}$. Notice that $M(2)$ halts - indicated by the black tile along the $x$-axis.}
\label{fig:CCSA_construction}
\end{figure}
Intuitively, $\mathcal{T}_M$ self-assembles a ``gradually thickening bar'' immediately below the positive $x$-axis with upward growths emanating from well-defined intervals of points. For each $x \in \mathbb{Z}^+$, there is an upward growth, in which a modifed wedge construction carries out a simulation of $M$ on $x$. If $M$ halts on $x$, then (a portion of) the upward growth associated with the simulation of $M(x)$ eventually stops, and sends a signal down along the right side of the upward growth via a one-tile-wide-path of tiles to the point $(f(x),0)$, where a black tile is placed.
Note that Theorem~\ref{sads_theorem} is exactly Theorem~\ref{CCSA_firstmaintheorem} with ``computably enumerable'' replaced with ``decidable,'' and $f(n) = n$.
\begin{openproblem} \cite{CCSA} Does Theorem~\ref{CCSA_firstmaintheorem} hold for any $f$ such that $f(n) = O(n)$? We conjecture that the answer is ``no'', and that the construction of \cite{CCSA} is effectively optimal. If the answer to this question is ``yes,'' then the proof would require a novel construction which manages to provide an infinite amount of space for each of an infinite number of perhaps non-halting computations in a more compact way than \cite{CCSA}.
\end{openproblem}
\section{Conclusion}
This paper surveyed a subset of recent theoretical results in algorithmic self-assembly relating to the self-assembly of infinite structures in the TAM. Specifically, in this paper we reviewed impossibility results with respect to the strict/weak self-assembly of various classes of discrete self-similar fractals \cite{SADSSF}, impossibility results for the weak self-assembly of exponential-time decidable sets \cite{CCSA}, characterizations of particular classes of languages in terms of weak self-assembly \cite{CCSA,SADS}, and the strict self-assembly of fractal-like structures. Finally, we believe that the benefit of continued research along these lines has the potential to shed light on the elusive relationship between geometry and computation.
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\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}
\end{document} | arXiv |
Observed information
In statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the "log-likelihood" (the logarithm of the likelihood function). It is a sample-based version of the Fisher information.
Definition
Suppose we observe random variables $X_{1},\ldots ,X_{n}$, independent and identically distributed with density f(X; θ), where θ is a (possibly unknown) vector. Then the log-likelihood of the parameters $\theta $ given the data $X_{1},\ldots ,X_{n}$ is
$\ell (\theta |X_{1},\ldots ,X_{n})=\sum _{i=1}^{n}\log f(X_{i}|\theta )$.
We define the observed information matrix at $\theta ^{*}$ as
${\mathcal {J}}(\theta ^{*})=-\left.\nabla \nabla ^{\top }\ell (\theta )\right|_{\theta =\theta ^{*}}$
$=-\left.\left({\begin{array}{cccc}{\tfrac {\partial ^{2}}{\partial \theta _{1}^{2}}}&{\tfrac {\partial ^{2}}{\partial \theta _{1}\partial \theta _{2}}}&\cdots &{\tfrac {\partial ^{2}}{\partial \theta _{1}\partial \theta _{p}}}\\{\tfrac {\partial ^{2}}{\partial \theta _{2}\partial \theta _{1}}}&{\tfrac {\partial ^{2}}{\partial \theta _{2}^{2}}}&\cdots &{\tfrac {\partial ^{2}}{\partial \theta _{2}\partial \theta _{p}}}\\\vdots &\vdots &\ddots &\vdots \\{\tfrac {\partial ^{2}}{\partial \theta _{p}\partial \theta _{1}}}&{\tfrac {\partial ^{2}}{\partial \theta _{p}\partial \theta _{2}}}&\cdots &{\tfrac {\partial ^{2}}{\partial \theta _{p}^{2}}}\\\end{array}}\right)\ell (\theta )\right|_{\theta =\theta ^{*}}$
Since the inverse of the information matrix is the asymptotic covariance matrix of the corresponding maximum-likelihood estimator, the observed information is often evaluated at the maximum-likelihood estimate for the purpose of significance testing or confidence-interval construction.[1] The invariance property of maximum-likelihood estimators allows the observed information matrix to be evaluated before being inverted.
Alternative definition
Andrew Gelman, David Dunson and Donald Rubin[2] define observed information instead in terms of the parameters' posterior probability, $p(\theta |y)$:
$I(\theta )=-{\frac {d^{2}}{d\theta ^{2}}}\log p(\theta |y)$
Fisher information
The Fisher information ${\mathcal {I}}(\theta )$ is the expected value of the observed information given a single observation $X$ distributed according to the hypothetical model with parameter $\theta $:
${\mathcal {I}}(\theta )=\mathrm {E} ({\mathcal {J}}(\theta ))$.
Comparison with the expected information
The comparison between the observed information and the expected information remains an active and ongoing area of research and debate. Efron and Hinkley[3] provided a frequentist justification for preferring the observed information to the expected information when employing normal approximations to the distribution of the maximum-likelihood estimator in one-parameter families in the presence of an ancillary statistic that affects the precision of the MLE. Lindsay and Li showed that the observed information matrix gives the minimum mean squared error as an approximation of the true information if an error term of $O(n^{-3/2})$ is ignored.[4] In Lindsay and Li's case, the expected information matrix still requires evaluation at the obtained ML estimates, introducing randomness.
However, when the construction of confidence intervals is of primary focus, there are reported findings that the expected information outperforms the observed counterpart. Yuan and Spall showed that the expected information outperforms the observed counterpart for confidence-interval constructions of scalar parameters in the mean squared error sense.[5] This finding was later generalized to multiparameter cases, although the claim had been weakened to the expected information matrix performing at least as well as the observed information matrix.[6]
See also
• Fisher information matrix
• Fisher information metric
References
1. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
2. Gelman, Andrew; Carlin, John; Stern, Hal; Dunson, David; Vehtari, Aki; Rubin, Donald (2014). Bayesian Data Analysis (3rd ed.). p. 84.
3. Efron, B.; Hinkley, D.V. (1978). "Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher Information". Biometrika. 65 (3): 457–487. doi:10.1093/biomet/65.3.457. JSTOR 2335893. MR 0521817.
4. Lindsay, Bruce G.; Li, Bing (1 October 1997). "On second-order optimality of the observed Fisher information". The Annals of Statistics. 25 (5). doi:10.1214/aos/1069362393.
5. Yuan, Xiangyu; Spall, James C. (July 2020). "Confidence Intervals with Expected and Observed Fisher Information in the Scalar Case": 2599–2604. doi:10.23919/ACC45564.2020.9147324. {{cite journal}}: Cite journal requires |journal= (help)
6. Jiang, Sihang; Spall, James C. (24 March 2021). "Comparison between Expected and Observed Fisher Information in Interval Estimation": 1–6. doi:10.1109/CISS50987.2021.9400253. {{cite journal}}: Cite journal requires |journal= (help)
| Wikipedia |
\begin{document}
\title{Equilibria of an aggregation model with linear diffusion in domains with boundaries}
\author{Daniel A. Messenger} \address{Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada.} \email{[email protected]\footnote{D.M.'s current affiliation: Department of Applied Mathematics, University of Colorado Boulder, 11 Engineering Dr., Boulder, CO 80309, USA.}}
\author{Razvan C. Fetecau} \address{Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada.} \email{[email protected]}
\maketitle
\begin{abstract} We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends explicitly on a certain volume filling property of the domain, and (ii) metastable mass translation occurs in domains without sufficient symmetry. From the first property, we present a sharp condition for existence (resp. non-existence) of global minimizers in a large class of domains, analogous to results in free space, and from the second property, we identify that external forces are necessary to confine the swarm and grant existence of global minimizers in general domains. We also introduce a numerical method for computing critical points of the energy and give examples to motivate further research. \end{abstract}
{\small {\bf Keywords: }Nonlocal modeling, swarm equilibria, domains with boundaries, energy minimizers, metastability, Wasserstein metric}
\section{Introduction} \label{sect:intro}
We consider minimizers of the following nonlocal and non-convex energy functional:
\begin{align} \label{eqn:energy} {\mathcal{E}}^\nu[\mu] &= \frac{1}{2}\int_D\int_D K(x-y)\,d\mu(x)\,d\mu(y)+\nu\int_D\log(\rho(x))\,d\mu(x)+\int_DV(x)\,d\mu(x),
\end{align} for measures $\mu$ that are absolutely continuous with respect to the Lebesgue measure ($\rho$ denotes the density of $\mu$) and for general domains $D\subset {\mathbb R}^d$ with smooth boundary. Here, $K$ and $V$ represent interaction and external potentials, respectively, and $\nu>0$ is the diffusion parameter.
Minimizers of the energy ${\mathcal{E}}^\nu$ relate to equilibria of the aggregation model with linear diffusion,
\begin{equation}\label{aggdD}
\begin{dcases}
\ppt \mu^{\nu}_t + \nabla \cdot \left(\mu^{\nu}_t \,v^\nu \right) = \nu\Delta \mu^{\nu}_t, & x\in D\\[8pt] \Big\langle n(x),\, \rho^\nu_t(x) \, v^\nu(x)+\nu\nabla \rho^\nu_t(x)\Big\rangle = 0, & x\in \partial D\\[8pt]
\mu^{\nu}_t\big\vert_{t=0} = \mu^\nu_0 \in {\mathcal{P}}_2(D),
\end{dcases}\\[2pt] \end{equation}
where $\rho^\nu_t$ is the density of $\mu^{\nu}_t$, $v^\nu$ denotes the swarm velocity
\begin{equation}\label{aggdvelocity} v^\nu(x) := -\int_D \nabla K(x-y)\,d\mu^{\nu}_t(y)-\nabla V(x), \end{equation}
and $n(x)$ denotes the unit outward normal to $\partial D$ at $x$. Specifically, weak-measure solutions of \eqref{aggdD} are 2-Wasserstein gradient flows of ${\mathcal{E}}^\nu$ on the space ${\mathcal{P}}_2(D)$ of probability measures with finite second moment \cite{ambrosio2008gradient,CaMcVi2006}, such that steady states of \eqref{aggdD} correspond to critical points of ${\mathcal{E}}^\nu$.
Aggregation-diffusion models of type \eqref{aggdD} and their associated energies of the form \eqref{eqn:energy} (with or without diffusion) appear in many phenomena, including biological swarms \cite{MoKe1999,LeToBe2009}, granular media \cite{carrillo2003kinetic}, self-assembly of nanoparticles \cite{HoPu2006} and opinion dynamics \cite{MotschTadmor2014}. In such applications the linear diffusion can model anti-crowding, locally repulsive interactions, or it can be the result of Brownian noise included in the model. On the other hand, the potential $K$ models nonlocal social interactions such as attraction and repulsion between the members of a group (e.g., individuals of a biological swarm). There has been extensive research on various aspects of this model, with the vast majority of these works being concerned with the model set up in free space ($D=\R^d$). We will briefly review below some of this literature.
In the absence of diffusion, the study of the evolution model in free space has been a very active area of research recently \cite{BertozziCarilloLaurent,BertozziLaurent,LeToBe2009,FeRa10,carrillo2011global}. The behaviour of its solutions relies fundamentally on the nature and properties of the interaction potential $K$. Consequently, attractive potentials lead to (finite or infinite-time) blow-up \cite{BertozziCarilloLaurent,HuBe2010}, while balancing attraction and repulsion can generate finite-size, confined aggregations \cite{FeHuKo11,LeToBe2009}. The model with diffusion has an extensive literature of its own; we refer here to \cite{carrillo2003kinetic,CaMcVi2006} for comprehensive studies on well-posedness of solutions to this model using the theory of gradient flows in probability spaces \cite{ambrosio2008gradient}, and to \cite{HoPu2006,HuFe2013} for studies on equilibria of the diffusive model with applications to aggregation/collective behaviour.
Critical points of the interaction energy \eqref{eqn:energy} (or equivalently, equilibria of the dynamic model) have been studied in various papers recently. For the case $\nu =0$, existence of global minimizers has been established in \cite{CaChHu2014,CaCaPa2015,ChFeTo2015,SiSlTo2015}, while qualitative properties such as dimensionality, size of the support, symmetry and stability have been investigated in \cite{balague2013dimensionality,BaCaLaRa2013,LeToBe2009,FeHuKo11,FeHu13}. A provoking gallery of such minimizers is presented for instance in \cite{KoSuUmBe2011}; it contains aggregations on disks, annuli, rings, soccer balls, and others. In the presence of diffusion ($\nu>0$), the focus is the competition between local repulsion effects (diffusion) and nonlocal attractive interactions that provides existence (or lack of) energy minimizers. This delicate balance of such forces was recently investigated by Carillo et al. in \cite{carrillo2018existence} for the case of free space; this work is central to our paper and we will return to it frequently throughout.
Our main interest in this paper lies in domains with boundaries, which are very relevant to many realistic physical settings (e.g., the boundary may be an obstacle in the environment, such as a river or the ground; the latter arises for instance in the locust model from \cite{ToDoKeBe2012}). Equilibria for the aggregation model without diffusion in domains with boundaries have been studied in \cite{bernoff2011primer}, while the well-posedness of its solutions (in the probability measure space) has been established in \cite{carrillo2014nonlocal,wu2015nonlocal}. Also, in a recent study \cite{fetecau2017swarm}, the authors identified a flaw of the aggregation equation with zero diffusion in domains with boundaries: its solutions can evolve into unstable equilibria. This is a surprising degeneracy of model \eqref{aggdD} without diffusion, given that it has a gradient flow formulation. From this perspective, adding diffusion can be seen as a regularizing mechanism \cite{evers2016metastable,zhang2017continuity,fetecau2018zero}.
We also mention that there has been extensive work on aggregation models with repulsive effects modelled by nonlinear diffusion (see for instance \cite{BeRoBe2011,CaHiVoYa2019} and references therein). The two modes of diffusion (linear vs. nonlinear) result in different features of equilibria/minimizers of the associated energy. In particular, nonlinear diffusion models admit compactly supported equilibria\cite{BurgerDiFrancescoFranek,BuFeHu14,CaHiVoYa2019}, in contrast with equilibria for linear diffusion which can only have full support within the domain (see Section \ref{sect:critical-points} of the present paper). Another important distinction is that the linear diffusion model has an underlying particle system associated to it (modelled by Brownian motion), while nonlinear diffusion does not, making the former more flexible in terms of numerical simulations \cite{fetecau2018zero}.
We discuss now the organization and the contributions of the present paper. Notation and assumptions are introduced in Section \ref{sect:prelim}. In Section \ref{sect:critical-points} we establish some useful properties of critical points of the energy ${\mathcal{E}}^\nu$, including a new energy argument showing that minimizers $\overline{\rho}$ satisfy $\supp{\overline{\rho}}=D$. Section \ref{sect:free-space} contains a brief review of various results in free space established in \cite{carrillo2018existence}, while Sections \ref{sect:non-exist} and \ref{sect:existence} present our results on energy minimization in domains with boundaries.
In Section \ref{sect:non-exist}, we identify two primary phenomena which lead to non-existence of global minimizers of ${\mathcal{E}}^\nu$ and depend on properties of the spatial domain. The first non-existence phenomenon (discussed in Section \ref{subsect:imbalance}) involves an imbalance of diffusive and aggregative forces and is analogous to non-existence results for free space found in \cite{carrillo2018existence}. An interesting novelty of our study, however, is that we find that non-existence as a result of diffusion-dominated spreading depends on the \textit{effective volume dimension} of the domain (defined in Section \ref{sect:prelim}) which informally describes the number of orthogonal directions in the domain which independently extend to infinity.
The second non-existence phenomenon (presented in Section \ref{subsect:escape}) is a result of asymmetries within $D$ and involves metastable translation of the swarm under diffusion-mediated repulsion from the boundary $\partial D$. This phenomenon rules out the existence of minimizers in the absence of an external potential ($V=0$) for large classes of unbounded domains. Specifically, the necessary condition in Theorem \ref{domassymthm} for critical points implies that if the domain is not suitably symmetric, arbitrarily large attraction at infinity (e.g., an attractive potential such as $K(x) = |x|^p/p$ with $p$ large) cannot contain the swarm regardless of how small the diffusion. Non-existence of this type necessitates the use of an external potential $V$ to confine the swarm, in contrast to results in free space, where existence of minimizers for $V=0$ is guaranteed for sufficiently strong attraction at infinity \cite{carrillo2018existence}.
Section \ref{sect:existence} is devoted to the existence of global minimizers in light of the phenomena discussed in Section \ref{sect:non-exist}. Theorem \ref{tightbound} establishes a sharp condition for existence in domains of type $D = F\times \mathbb{R}^{d-m}$ with an effective volume dimension of $d-m$, where $F\subset \mathbb{R}^m$ is a compact $m$-dimensional set with $0<m<d$. This result serves as a generalization of the sharp existence condition in free space established in \cite{carrillo2018existence}. Theorems \ref{existence1} and \ref{existence2} then provide sufficient conditions for existence in general domains with boundaries, which involve establishing a minimal set of requirements on $V$ for confinement, in light of the metastable translations discussed in Section \ref{subsect:escape} that occur when $V=0$.
Finally, in Section \ref{sect:numerics} we discuss some of the findings in the previous sections through numerical examples of critical points computed using a fixed-point iteration scheme with relaxation. We illustrate both purely attractive and attractive-repulsive interaction potentials, with the latter featuring non-uniqueness of critical points.
\section{Preliminaries and Assumptions} \label{sect:prelim}
In this section we provide some preliminaries and background, as well as list the assumptions we make on the potentials and the domain.
\subsection*{Notations}
Let ${\mathcal{B}}^d$ denote the Borel $\sigma$-algebra on $\R^d$. For $A\in {\mathcal{B}}^d$, $|A|$ denotes the volume of $A$ (with respect to the Lebesgue measure) and $\ind{A}$ represents the indicator function of $A$. Let $B_R(x)$ denote the $d$-dimensional Euclidean ball of radius $R$ centred at $x\in\R^d$. For $D\in {\mathcal{B}}^d$, let ${\mathcal{P}}(D)$ denote the set of Borel probability measures on $D$ and ${\mathcal{P}}^{ac}(D)\subset{\mathcal{P}}(D)$ the set of absolutely continuous measures with respect to the Lebesgue measure.
For $\mu\in {\mathcal{P}}(D)$, $x\in \R^d$ and $\gamma\in \mathbb{R}$, define the $\gamma^{th}$ moment of $\mu$ centred at $x$ as
\begin{equation}\label{moment}
M^x_\gamma(\mu) := \int_D|y-x|^\gamma \,d\mu(y), \end{equation}
with \begin{equation} \label{moment0} M_\gamma(\mu):=M^0_\gamma(\mu), \end{equation}
and the centre of mass of $\mu$ by
\begin{equation}\label{centreofmass} \CalC\left(\mu\right):=\int_D x\,d\mu(x). \end{equation}
\begin{rmrk} \normalfont
Throughout, we will often refer to an absolutely continuous measure directly by its density $\rho$, and by abuse of notation sometimes write $\rho\in {\mathcal{P}}^{ac}(D)$ to mean $d\rho(x) = \rho(x)\,dx$. \end{rmrk}
\subsection*{Weak-* Relative Compactness and Tightness}
\hspace{5mm} We say that a sequence $\left\{\mu_n\right\}_{n\geq 0} \subset {\mathcal{P}}(D)$ converges weakly-* to $\mu\in {\mathcal{P}}(D)$ and write $\mu_n\overset{*}{\rightharpoonup} \mu$ if for every bounded continuous function $f:D\to \mathbb{R}$ we have
\[\lim_{n\to \infty}\int_Df(x)\,d\mu_n(x) = \int_Df(x)\,d\mu(x).\]
A collection of measures ${\mathcal{F}}\subset {\mathcal{P}}(D)$ is said to be weakly-* relatively compact if for every sequence $\left\{\mu_n\right\}_{n\geq 0} \subset {\mathcal{F}}$ there exists a subsequence $\left\{\mu_{n_k}\right\}_{k\geq 0}$ which converges weakly-* to some $\mu\in {\mathcal{P}}(D)$. A collection of measures ${\mathcal{F}}\subset {\mathcal{P}}(D)$ is said to be \textit{tight} if for every $\epsilon>0$ there exists a compact set $K_\epsilon\subset D$ such that
\[\mu(K_\epsilon^c)<\epsilon\]
for every $\mu\in {\mathcal{F}}$, where $K_\epsilon^c$ denotes the complement of $K_\epsilon$ within $D$ (i.e. $K_\epsilon^c = D\setminus K_\epsilon$). Recall that weak-* relative compactness and tightness are related by Prokhorov's Theorem:
\begin{lemm}{(Prokhorov's theorem \cite[Chapter 1, Section 5]{billingsley2013convergence})} A collection of measures ${\mathcal{F}}\subset {\mathcal{P}}(D)$ is weakly-* relatively compact if and only if it is tight. \end{lemm}
\subsection*{The $p$-Wasserstein Space}
\noindent For $p\in [1,\infty)$, define the space
\begin{equation}\label{p1d} \mathcal{P}_p(D):=\left\{\mu\in\mathcal{P}(D):~M_p(\mu)<+\infty\right\}, \end{equation}
where $M_p(\mu)$ is defined \eqref{moment0}. The $p$-Wasserstein distance on $\mathcal{P}_p(D)$ is then
\begin{equation}\label{wass}
{\mathcal{W}}_p(\mu,\eta)=\left(\inf_{\pi\in\Lambda(\mu,~\eta)}\Big\{\int_{D\times D}|x-y|^p\,d\pi(x,y)\Big\}\right)^{\frac{1}{p}}=\left(\inf_{X\sim \mu,Y\sim \eta}\Big\{\mathbb{E}[|X-Y|^p]\Big\}\right)^{\frac{1}{p}}, \end{equation}
where $\Lambda(\mu,~\eta)$ is the set of joint probability measures on ${D}\times{D}$ with marginals $\mu$ and $\eta$, also known as \textit{transport plans}, and $(X,Y)$ ranges over all possible couplings of random variables $X$ and $Y$ with laws $\mu$ and $\eta$, respectively.
Recall that for each $p\in[1,\infty)$ the metric space $({\mathcal{P}}_p(D),{\mathcal{W}}_p)$ is complete and convergence in $({\mathcal{P}}_p(D),{\mathcal{W}}_p)$ is equivalent to weak-* convergence of measures. We also have the following useful upper bound on ${\mathcal{W}}_p(\mu,\eta)$ in terms of the total variation measure $d|\mu-\eta|$:
\begin{lemm}\label{ELlemm2} \cite[Ch. 6]{villani2008optimal} For all $p\in [1,\infty)$, any $x_0 \in D$ and $\mu,\eta \in {\mathcal{P}}_p(D)$,
\[{\mathcal{W}}_p^p(\mu,\eta)\leq 2^{(p-1)}\int_D|x-x_0|^pd|\mu-\eta|(x).\] \end{lemm}
\noindent We refer readers to the books \cite{ambrosio2008gradient,villani2008optimal} for further background on $p$-Wasserstein spaces.
\subsection*{Associated Energy}
The energy functional \eqref{eqn:energy} can be written as:
\begin{equation} \label{eqn:energy-3parts} {\mathcal{E}}^\nu[\mu] = {\mathcal{K}}[\mu] + \nu {\mathcal{S}}[\mu] + {\mathcal{V}}[\mu], \end{equation}
where ${\mathcal{K}}$, ${\mathcal{S}}$ and ${\mathcal{V}}$ are referred to as the interaction energy, entropy and potential energy, respectively, and are defined for all $\mu\in {\mathcal{P}}(D)$ by:
\begin{equation}\label{energyK} {\mathcal{K}}[\mu] := \frac{1}{2}\int_D\int_D K(x-y)\,d\mu(y)\,d\mu(x), \end{equation}
\begin{equation}\label{energyS} {\mathcal{S}}[\mu] := \begin{dcases} \int_D\rho(x)\log(\rho(x))\,d x, &\quad \text {if }\mu\in{\mathcal{P}}^{ac}(D) \text{ with } d\mu(x) = \rho(x)\,d x,\\ +\infty, & \quad \text{otherwise,} \end{dcases} \end{equation} and \begin{equation} {\mathcal{V}}[\mu] := \int_DV(x)\,d\mu(x). \end{equation}
In this way, the energy ${\mathcal{E}}^\nu$ is defined on the entire space ${\mathcal{P}}(D)$, but takes the value $+\infty$ on measures which are not absolutely continuous.
Extrema of ${\mathcal{E}}^\nu$ are defined as in \cite{carrillo2018existence}: for $r>0$ we define a ${\mathcal{W}}_p$-$r$ local minimizer of ${\mathcal{E}}^\nu$ to be a measure $\overline{\rho}$ such that
\begin{equation}\label{wprmin} {\mathcal{E}}^\nu[\overline{\rho}]\leq {\mathcal{E}}^\nu[\eta] \txt{0.5}{for all} \eta \in \ballp{p}{\overline{\rho}}{r}, \end{equation}
where $\ballp{p}{\overline{\rho}}{r}$ is the ball of radius $r$ in ${\mathcal{P}}_p(D)$ centred at $\overline{\rho}$. A ${\mathcal{W}}_p$-$r$ local maximizer is defined analogously by reversing the inequality. In what follows, we will use the terms \textit{minimizer}, \textit{maximizer} or \textit{extremizer} in reference to condition \eqref{wprmin} whenever the ${\mathcal{W}}_p$ metric has been established.
\begin{rmrk}\label{ac-gmin}
\normalfont In light of the definition of the entropy \eqref{energyS}, whenever a global minimizer $\overline{\rho}$ of ${\mathcal{E}}^\nu$ exists, it follows that $\overline{\rho}\in {\mathcal{P}}^{ac}(D)$, as one can always find a measure $\mu\in {\mathcal{P}}^{ac}(D)$ for which ${\mathcal{E}}^\nu[\mu]<+\infty$ (e.g. for a compact, $d$-dimensional set $F\subset D$, the energy ${\mathcal{E}}^\nu\left[\frac{1}{|F|}\ind{F}\right]$ is finite). \end{rmrk}
\subsection*{Assumptions}
\hspace{5mm} Throughout we will make the following minimal assumptions about potentials $K$ and $V$, and the spatial domain $D$.
\noindent \begin{assum}[Potentials]\label{assumKV} \normalfont \quad \begin{enumerate}[label=(\roman*)] \item (Local integrability) $K,V\in L^1_{loc}(\R^d)$. \item (Lower semicontinuity) $K$ and $V$ are lower semicontinuous. \item (Symmetry of $K$) $K(x) = K(-x)$ for all $x\in \mathbb{R}^d$.
\end{enumerate} \end{assum}
\begin{assum}[Domain]\label{assumD} \normalfont \quad \begin{enumerate}[label=(\roman*)]
\item (Domain topology) $D\in {\mathcal{B}}^d$ is closed, connected, and satisfies $|D|>0$. \item (Boundary regularity) There exists a unique outward normal vector $n(x)$ associated to almost every $x\in \partial D$.
\end{enumerate} \end{assum}
\subsection*{Effective Volume Dimension}
As we will show in Theorems \ref{NE} and \ref{tightbound}, the following property is a crucial component of the asymptotic upper bound on $K$ below which diffusion dominates and infinite spreading occurs, leading to non-existence of global minimizers of ${\mathcal{E}}^\nu$. Moreover, in domains of the form \eqref{Fdomain} below, this asymptotic bound is sharp.
For $D \in {\mathcal{B}}^d$ define the function
\begin{equation}\label{fDr} V_{_D}(r) = \sup_{x\in D}\left\vert D\cap B_r(x)\right\vert.\\[-3pt] \end{equation}
We then define the \textit{effective volume dimension} $\eff$ of $D$ by
\begin{equation}\label{effvoldim} \eff\ :=\ \sup\Bigl\{s \in \mathbb{R} \txt{0.2}{:} V_{_D}(r) \ \gtrsim\ r^s \txt{0.2}{as} r \to \infty\, \Bigr\}. \end{equation}
Consequently, for some $C>0$ and $r'>0$,
\begin{equation}\label{effvolbound} V_{_D}(r)\ \geq\ C\,r^{\eff} \txt{0.4}{for all} r>r'\,; \end{equation}
moreover, $\eff$ is the largest value such that \eqref{effvolbound} holds. In words, the largest volume of $D$ intersect a ball of radius $r$ grows proportionally to $r^{\eff}$. In this way, $\eff\in[0,d]$ and $\eff = 0$ if $D$ is bounded. For intuition in three dimensions, example domains with $\eff = 1$ and $\eff=2$ are an open right cylinder and the space between two infinite parallel planes, respectively.
In Theorem \ref{tightbound} we consider domains of the form
\begin{equation}\label{Fdomain} D = F\times \mathbb{R}^{d-m}, \end{equation}
where $F$ is a compact subset of $\mathbb{R}^m$ for some $m\in\left\{1,\dots,d-1\right\}$ and satisfies $|F|>0$. In this case, $D$ has effective volume dimension $\eff = d-m$. To see this, note that
\[V_D(r)\leq |F|(2r)^{d-m},\]
and letting $H^{d-m}_a$ denote a $(d-m)$-dimensional hypercube of side-length $a$, for $r>\frac{\sqrt{d}}{2}\text{diam}(F)$ we have
\[V_D(r)\geq |F\times H^{d-m}_{\frac{2}{\sqrt{d}}r}| = |F|\left(\frac{2}{\sqrt{d}}r\right)^{d-m}.\]
Together this implies $V_{_D}(r) \sim Cr^{d-m}$ as $r\to\infty$. In this way, $\eff$ is exactly the number of orthogonal directions in $D$ which independently extend to infinity.
\section{Critical Points of the Energy} \label{sect:critical-points}
Interaction energies of type \eqref{eqn:energy} (with or without linear/nonlinear diffusion) have been studied extensively in free space ($D = \R^d$). It is well known that critical points of the energy ${\mathcal{E}}^\nu$ in free space satisfy the Euler-Lagrange equation \eqref{ELcond} given below. This equation had been derived in various papers, we refer for instance to \cite{balague2013dimensionality} for a derivation using the ${\mathcal{W}}_2$ metric for ${\mathcal{E}}^\nu$ without diffusion, and to \cite{carrillo2018existence} for interaction energies with general diffusion under the ${\mathcal{W}}_\infty$ metric. We include a derivation of \eqref{ELcond} here under general ${\mathcal{W}}_p$ metrics over general domains $D$, for lack of a direct reference, using the techniques in \cite{balague2013dimensionality} and \cite{carrillo2018existence}.
First, we highlight in Theorem \ref{suppmin} the property that minimizers of ${\mathcal{E}}^\nu$ for every $\nu>0$ are supported on the whole domain $D$. This is briefly mentioned in \cite{carrillo2018existence} for free space and is justified by the authors using the Euler-Lagrange equation. As there is no reason a priori why in general domains $D$ a minimizer $\overline{\rho}$ should simultaneously satisfy the Euler-Lagrange equation and have $\supp{\overline{\rho}}=D$, we present a proof of Theorem \ref{suppmin} without the Euler-Lagrange equation.
\begin{thm}\label{suppmin}[Minimizers have full support] Let Assumptions \ref{assumKV} and \ref{assumD} hold. In addition, assume that there exist constants $p_K \geq 0$, $R_K>0$, and $C_K>0$ such that
\begin{equation}\label{powerlaw}
0 \leq K(x) \leq C_K|x|^{p_K} \txt{0.5}{ for all } |x| > R_K. \end{equation}
If $\overline{\rho}\in {\mathcal{P}}^{ac}_p(D)$ is a ${\mathcal{W}}_p$-$r$ local minimizer of ${\mathcal{E}}^\nu$ where $p\in \big[\max\left\{1,p_{_K}\right\}, \infty\big]$, then $\supp{\overline{\rho}} = D$. \end{thm}
\begin{proof}
We proceed by contradiction. Let $\overline{\rho}$ be a ${\mathcal{W}}_p$-$r$ local minimizer and assume to the contrary that $\supp{\overline{\rho}} \subsetneq D$. By definition, $\supp{\overline{\rho}}$ is a closed subset of $D$, hence $D\,\setminus\,\supp{\overline{\rho}}$ must have positive Lebesgue measure. This implies that there exists a point $x_0 \in\partial\, \supp{\overline{\rho}}$ and $\delta>0$ such that the set $A = B_\delta(x_0)\cap \Big(D\setminus\supp{\overline{\rho}}\Big)$ has positive Lebesgue measure $|A|$. We now construct a measure $\eta$ with ${\mathcal{W}}_p(\overline{\rho},\eta)<r$ such that ${\mathcal{E}}^\nu[\eta]<{\mathcal{E}}^\nu[\overline{\rho}]$, contradicting the assumption that $\overline{\rho}$ is a ${\mathcal{W}}_p$-$r$ local minimizer of ${\mathcal{E}}^\nu$. Define
\[\eta = (1-\alpha)\overline{\rho} +\alpha\frac{1}{|A|}\ind{A},\]
where $\alpha\in (0,1)$ will be picked in two stages.
First, using Lemma \ref{ELlemm2} we have
\begin{align*}
{\mathcal{W}}^p_p(\overline{\rho},\eta)&\leq 2^{(p-1)}\int_D|x-x_0|^p\,d|\overline{\rho}-\eta|(x)\\
&=\alpha\,2^{(p-1)}\left(\int_D|x-x_0|^p\,d\overline{\rho}(x)+\frac{1}{|A|}\int_A|x-x_0|^p\,dx\right)\\ &\leq \alpha\,2^{(p-1)}\left(M^{x_0}_p(\overline{\rho})+\delta^p\right), \end{align*}
and so we choose
\begin{equation}\label{alphbound1} \alpha<\min\left\{1,\ \dfrac{r^p}{2^{(p-1)}\left(M^{x_0}_p(\overline{\rho})+\delta^p\right)}\right\} \end{equation}
to ensure that ${\mathcal{W}}_p(\overline{\rho},\eta) < r$.
Next, we find an additional constraint on $\alpha$ to ensure that ${\mathcal{E}}^\nu[\eta]<{\mathcal{E}}^\nu[\overline{\rho}]$ by bounding terms in the energy. For any $x\in \mathbb{R}$ we have $(1-\alpha)^2x < x+2\alpha|x|$, and so a direct calculation of the interaction energy yields the bound
\begin{align*}
{\mathcal{K}}[\eta] &= (1-\alpha)^2{\mathcal{K}}[\overline{\rho}] + \frac{\alpha(1-\alpha)}{|A|}\int_D\left(\int_A K(x-y)dy\right)\overline{\rho}(x)\,dx+\frac{\alpha^2}{2|A|^2}\int_A\int_A K(x-y)\,dxdy\\
&< {\mathcal{K}}[\overline{\rho}]+\alpha\left(2\left\vert{\mathcal{K}}[\overline{\rho}]\right\vert+\frac{1}{|A|}\underbrace{\int_D\left(\int_A \left\vert K(x-y)\right\vert\,dy\right)\overline{\rho}(x)\,dx}_{:= I}+\frac{1}{2|A|}\nrm{K}_{L^1(B_{2\delta}(0))}\right). \end{align*}
The integral $I$ is finite independently of $\alpha$, and hence so is the entire expression in parentheses, due to the power-law growth \eqref{powerlaw} and local integrability of $K$ together with the fact that $\overline{\rho}$ has finite $p_{_K}$-th moment. Indeed, fixing $R>{R}_{_{K}}+2\delta$, we partition the integral to get
\begin{align*}
I &=\int_{B_R(x_0)\cap D}\left(\int_A |K(x-y)|\, dy \right)\overline{\rho}(x)\,dx+\int_{B_R^c(x_0)\cap D}\left(\int_A |K(x-y)|\,dy \right) \overline{\rho}(x)\,dx\\[5pt]
&\leq \nrm{K}_{L^1(B_{R+\delta}(0))} \int_{B_R(x_0)\cap D}\overline{\rho}(x)\,dx+C_K|A|\int_{B_R^c(x_0)\cap D}(|x-x_0|+\delta)^{p_{_K}} \overline{\rho}(x)\,dx\\[5pt]
&\leq \nrm{K}_{L^1(B_{R+\delta}(0))}+C_K|A|\left(1+\frac{\delta}{R}\right)^{p_{_K}} M^{x_0}_{p_{_K}}(\overline{\rho}), \end{align*}
which is finite. From this bound we have that for some $C>0$ independent of $\alpha$,
\[{\mathcal{K}}[\eta] < {\mathcal{K}}[\overline{\rho}]+\alpha C.\]
For the entropy, since $A \cap \text{supp}(\overline{\rho}) = \emptyset$ we have
\begin{align*}
\mathcal{S}[\eta] &= (1-\alpha)\int_D\overline{\rho}(x)\log((1-\alpha)\overline{\rho}(x))\,dx+\frac{\alpha}{|A|}\int_A\log\left(\frac{\alpha}{|A|}\right)\,dx\\
&=(1-\alpha)\mathcal{S}[\overline{\rho}]+(1-\alpha)\log(1-\alpha)+\alpha\log\left(\frac{\alpha}{|A|}\right)\\
&< (1-\alpha)\mathcal{S}[\overline{\rho}]+\alpha\log\left(\frac{\alpha}{|A|}\right)\\
&= \mathcal{S}[\overline{\rho}] +\alpha\left(-\mathcal{S}[\overline{\rho}]+\log\left(\frac{\alpha}{|A|}\right)\right). \end{align*}
Together this allows us to bound the difference in energy as follows:
\begin{align*}
\mathcal{E}^\nu[\eta]-\mathcal{E}^\nu[\overline{\rho}] &< \alpha\left(C -\nu\mathcal{S}[\overline{\rho}]+ \nu\log\left(\frac{\alpha}{|A|}\right)-{\mathcal{V}}[\overline{\rho}]+\frac{1}{|A|}\int_AV(x)\,dx\right). \end{align*}
Now, choosing $\alpha$ such that
\[\alpha<|A|\,\exp\left(-\frac{C}{\nu}+\mathcal{S}[\overline{\rho}]+\frac{1}{\nu}{\mathcal{V}}[\overline{\rho}]-\frac{1}{\nu|A|}\int_AV(x)\,dx\right),\]
along with the constraint \eqref{alphbound1}, we see by the monotonicity of the logarithm that
\[\mathcal{E}^\nu[\eta]<\mathcal{E}^\nu[\overline{\rho}].\]
Since $\eta$ has lower energy than $\overline{\rho}$ and lives in the ball $\ballp{p}{\overline{\rho}}{r}$, $\overline{\rho}$ cannot be a ${\mathcal{W}}_p$-$r$ local minimizer, giving us the desired contradiction. Thus, the support of $\overline{\rho}$ must be the entire domain $D$.
\end{proof}
We now derive the Euler-Lagrange equation.
\begin{thm}\label{EL} [Euler-Lagrange equation] Let Assumptions \ref{assumKV} and \ref{assumD} hold. Suppose that $\overline{\rho}\in {\mathcal{P}}^{ac}_p(D)$ is a ${\mathcal{W}}_p$-$r$ local extremizer of ${\mathcal{E}}^\nu$ for some $p\in \big[1, \infty\big]$. Then there exists a constant $\lambda\in \mathbb{R}$ such that
\begin{equation}\label{ELcond} K*\overline{\rho}(x) +\nu\log(\overline{\rho}(x)) +V(x)= \lambda \quad \text{ for } \quad \overline{\rho}\text{-a.e. } x\in D. \end{equation}
\end{thm}
\begin{proof} Without loss of generality, assume $\overline{\rho}$ is a ${\mathcal{W}}_p$-$r$ local \textit{minimizer} (the case where $\overline{\rho}$ is a maximizer follows similarly by reversing the following inequality). As in \cite{carrillo2018existence}, it follows that
\[\frac{d}{dt}\mathcal{E}_\nu[\overline{\rho}+t(\eta-\overline{\rho})]\Bigg\vert_{t=0}\geq 0\]
for all $\eta\in \ballp{p}{\overline{\rho}}{r}$. From this a direct calculation then yields
\begin{equation} \label{ELbound}
\int_D \left(K*\overline{\rho}+\nu\log(\overline{\rho})+V\right)d\eta\geq \int_D \left(K*\overline{\rho}+\nu\log(\overline{\rho})+V\right)d\overline{\rho}. \end{equation}
We now construct a suitably general $\eta$ to deduce \eqref{ELcond}. Choose $\phi$ in $L^\infty(D;\,\overline{\rho})\cap L^1(D;\,\overline{\rho})$
and define
\[ \eta = \overline{\rho}+\epsilon\left(\phi-\int_D\phi \,d\overline{\rho}\right)\overline{\rho}, \]
where $\epsilon$ will be chosen such that $\eta \in \ballp{p}{\overline{\rho}}{r}$. It is clear that $\eta(D)=1$. To ensure that $\eta\geq 0$ and hence $\eta\in\mathcal{P}_p(D)$, it suffices to pick $\epsilon \leq \frac{1}{2\nrm{\phi}_\infty}$. Another application of Lemma \ref{ELlemm2} gives
\begin{align*}
{\mathcal{W}}_p^p(\overline{\rho},\eta)&\leq 2^{p-1}\int_D|x|^p\,d|\overline{\rho}-\eta|\\
&= \epsilon \,2^{p-1}\int_D|x|^p\left\vert\phi - \int_D\phi \,d\overline{\rho}\right\vert \,d\overline{\rho}\\ &\leq \epsilon\, 2^p\nrm{\phi}_{\infty}M_p(\overline{\rho}). \end{align*}
Hence, ${\mathcal{W}}_p(\overline{\rho},\eta) <r$ provided
\[\epsilon <\min\left\{\dfrac{r^p}{2^p \nrm{\phi}_{\infty} M_p(\overline{\rho})},\,\frac{1}{2\nrm{\phi}_\infty}\right\},\]
which guarantees that $\eta \in \ballp{p}{\overline{\rho}}{r}$. Substituting $\eta$ into \eqref{ELbound} then gives us
\[\int_D \left(\phi-\int_D\phi \,d\overline{\rho}\right)\left(K*\overline{\rho}+\nu\log(\overline{\rho})+V\right)\,d\overline{\rho}\geq 0.\]
The above calculations work for both $\phi$ and $-\phi$, hence upon multiplying by $-1$ we find that
\[ \int_D \left(\phi-\int_D\phi\, d\overline{\rho}\right)\left(K*\overline{\rho}+\nu\log(\overline{\rho})+V\right)\,d\overline{\rho} = 0.\]
Now, by setting $\phi = \ind{B}$ for any Borel set $B\subset \supp{\overline{\rho}}$ with $\overline{\rho}(B) >0$, we further have
\begin{equation}\label{ELcondvanish} \frac{1}{\overline{\rho}(B)}\int_B \left(\nu\log(\overline{\rho})+K*\overline{\rho}+V\right)d\overline{\rho}
= \int_D \left(\nu\log(\overline{\rho})+K*\overline{\rho}+V\right)d\overline{\rho}. \end{equation}
From this we deduce \eqref{ELcond} by contradiction. Define
\begin{equation}\label{Lambda} \Lambda(x) := K*\overline{\rho}(x)+\nu\log(\overline{\rho}(x))+V(x) \end{equation}
and assume that $\Lambda$ is not constant $\overline{\rho}$-a.e. Then there exists $\lambda^*\in \mathbb{R}$ such that the sets $B_1 = \left\{\Lambda< \lambda^*\right\}$ and $B_2 = \left\{\Lambda> \lambda^*\right\}$ satisfy $\overline{\rho}(B_1)>0$ and $\overline{\rho}(B_2)>0$.
Using $B=B_1$ and $B=B_2$ in \eqref{ELcondvanish} then gives us
\[\lambda^* \,>\,\int_D\Lambda(x)\,d\overline{\rho}\txt{0.5}{and}\lambda^* \,<\,\int_D\Lambda(x)\,d\overline{\rho},\]
respectively, which is a contradiction, thus $\Lambda$ must be constant $\overline{\rho}$--a.e. This completes the proof. \end{proof}
\paragraph{Fixed-Point Characterization.} The Euler-Lagrange equation \eqref{ELcond} can be recast in the following way if the critical point $\overline{\rho}$ satisfies $\supp{\overline{\rho}} =D$. Solving for $\overline{\rho}$ using the logarithm we have
\[\overline{\rho}(x) = \frac{1}{Z(\overline{\rho})}\exp\left(-\frac{K*\overline{\rho}(x)+V(x)}{\nu}\right),\] where
\begin{equation}\label{gibbstatZ} \hspace{4pt}Z(\overline{\rho}) := \int_D \exp\left(-\frac{K*\overline{\rho}(x)+V(x)}{\nu}\right)\, dx. \end{equation}
This motivates the following corollary which will be used below.
\begin{corr}\label{fixedpointmap} Let $\overline{\rho} \in {\mathcal{P}}^{ac}(D)$ have $\supp{\overline{\rho}} = D$. Then $\overline{\rho}$ satisfies \eqref{ELcond} if and only if $\overline{\rho}$ is a fixed point of the map $T: {\mathcal{P}}(D)\to{\mathcal{P}}^{ac}(D)$ defined by
\begin{equation}\label{gibbstat} T(\mu) = \frac{1}{Z(\mu)}\exp\left(-\frac{K*\mu(x)+V(x)}{\nu}\right) \end{equation}
for $Z$ defined in \eqref{gibbstatZ}. \end{corr}
By integrating \eqref{ELcond} against $d\overline{\rho}(x)$, we can also identify the constant $\lambda$ as
\[\lambda = {\mathcal{E}}^\nu[\overline{\rho}]+{\mathcal{K}}[\overline{\rho}] = -\nu\log\left(Z(\overline{\rho})\right).\]
In Section \ref{sect:numerics} we discretize \eqref{gibbstat} for numerical computation of critical points. We note that \eqref{gibbstat} has been used in the literature, for instance by Benachour et al. in \cite{benachour1998nonlinear} to show existence of stationary states for associated McKean-Vlasov processes on $D=\mathbb{R}$.
\section{Review: Existence of Global Minimizers in Free Space} \label{sect:free-space} To exhibit the role played by domain geometry in determining existence of global minimizers, we briefly review existence results in free space. In \cite{carrillo2018existence}, the authors show that when $D=\R^d$ and $V=0$, existence of a global minimizer is guaranteed as soon as the energy is bounded below. As we will show, this is not the case in domains with boundaries.
Unboundedness from below of the energy is shown in \cite{carrillo2018existence} to correspond to an imbalance of diffusive and aggregative forces. If local attractive forces are too strong with respect to local diffusive repulsion, then the energy is lowered to $-\infty$ as the swarm aggregates onto a discrete set of points. It is shown in Theorem 4.1 of \cite{carrillo2018existence} that if
\begin{equation}\label{NEaggreg}
\liminf_{|x|\to 0} \nabla K(x)\cdot x > 2d\nu, \end{equation}
then such aggregation-dominated contraction occurs and $\inf {\mathcal{E}}^\nu =-\infty$. Condition \eqref{NEaggreg} is shown to imply that
\[K \lesssim 2d\nu \log|x| \txt{0.4}{as} |x|\to 0,\]
hence aggregation-dominated contraction may occur unless $K$ is well-behaved at the origin; in particular $K$ cannot have a singularity at the origin worse than logarithmic.
If diffusion is too strong with respect to long-range attractive forces, then minimizing sequences of ${\mathcal{E}}^\nu$ vanish as diffusion causes infinite spreading of the swarm throughout the domain. For linear diffusion, Carrillo et al. \cite{carrillo2018existence} show that existence of global minimizers of ${\mathcal{E}}^\nu$ in free space for $V=0$ corresponds to the following conditions on $K$, $\nu$ and the dimension $d$, which prevents both diffusion-dominated spreading and aggregation-dominated contraction. Let $K$ satisfy Assumption \ref{assumKV} and be positive and differentiable away from the origin. If $K$ satisfies
\begin{equation}\label{nonexistcond1}
\limsup_{|x|\to \infty}\,\nabla K(x)\cdot x<2d\nu, \end{equation}
then ${\mathcal{E}}^\nu$ is not bounded below and a global minimizer does not exist. Alternatively, if
\begin{equation}\label{existcond1}
\liminf_{|x|\to \infty}\,\nabla K(x)\cdot x>2d\nu, \end{equation}
then ${\mathcal{E}}^\nu$ is bounded below and there exists $\overline{\rho} \in {\mathcal{P}}(\R^d)$ such that
\[{\mathcal{E}}^\nu(\overline{\rho})=\inf {\mathcal{E}}^\nu>-\infty.\]
By requiring $K$ to be positive, the condition \eqref{NEaggreg} is (sufficiently) prevented and aggregation-dominated contraction cannot occur. In addition, the constraint \eqref{existcond1}, shown in \cite{carrillo2018existence} to imply
\begin{equation}\label{existcond2}
K(x) \gtrsim 2d \nu \log|x| \txt{0.4}{as}|x|\to \infty, \end{equation}
prevents diffusion-dominated spreading by requiring that $K$ grows at least logarithmically as $|x|\to \infty$. Moreover, by inspection of condition \eqref{nonexistcond1}, we see that condition \eqref{existcond1} is sharp.
In summary, in free space, if the two force-imbalance pathologies of diffusion-dominated spreading or aggregation-dominated contraction are prevented, then the energy is bounded below, which immediately implies existence of a global minimizer. Moreover, condition \eqref{existcond1} for preventing diffusion-dominated spreading is sharp.
In domains with boundaries, unboundedness from below of the energy due to aggregation-dominated contraction occurs under the same condition as in free space, \eqref{NEaggreg}, while diffusion-dominated spreading occurs under a condition analogous to \eqref{nonexistcond1}, only with dependence on the effective volume dimension $\eff$ instead of the dimension $d$. Moreover, we will see that such conditions for spreading are sharp in the class of domains defined by \eqref{Fdomain}. In addition, we find that boundedness from below of the energy is not enough to grant existence of a global minimizer when the domain is not suitably symmetric.
\begin{rmrk} \normalfont Requirements on the interaction potential $K$ for existence or non-existence of global minimizers of ${\mathcal{E}}^\nu$ are presented in Theorems \ref{NE}, \ref{tightbound}, \ref{existence1} and \ref{existence2} in the form of asymptotic relations similar to \eqref{existcond2} which provide a more explicit characterization of $K$ than \eqref{existcond1}; however, we could have equivalently worked with conditions such as \eqref{existcond1} involving $\nabla K$. \end{rmrk}
\noindent The following lemmas from \cite{carrillo2018existence} will be used in the existence proofs in our paper.
\begin{lemm}\label{lowerSC} \cite{carrillo2018existence} Assume that $K$ and $V$ are both lower semicontinuous. Then ${\mathcal{E}}^\nu$ is weakly-* lower semicontinuous, in that for any sequence $\left\{\mu_n\right\}_{n\geq 0} \subset {\mathcal{P}}(D)$ such that $\mu_n\overset{*}{\rightharpoonup} \mu\in {\mathcal{P}}(D)$, it holds that \[\liminf_{n\to \infty} {\mathcal{E}}^\nu[\mu_n] \geq {\mathcal{E}}^\nu[\mu].\] \end{lemm}
\begin{lemm}{(Logarithmic Hardy-Littlewood-Sobolev (HLS) inequality \cite[Lemma 2.6]{carrillo2018existence})} \label{lemma:HLS}
Let $\rho\in {\mathcal{P}}^{ac}(\R^d)$ satisfy $\log(1+|\cdot|^2)\rho\in L^1(\R^d)$. Then there exists $C_0\in \mathbb{R}$ depending only on $d$ such that \begin{equation}\label{HLS}
-\int_{\R^d}\int_{\R^d}\log(|x-y|)\rho(x)\rho(y)\,dx\,dy\leq \frac{1}{d}\int_{\R^d}\rho(x)\log(\rho(x))\,dx+C_0. \end{equation} \end{lemm}
\begin{lemm}{\cite[Lemma 2.9]{carrillo2018existence}}\label{Ktight} Let $K(x) \in L^1_{loc}(\R^d)$ be positive, symmetric and satisfying
\[\lim_{|x|\to \infty} K(x) = +\infty.\] Given a sequence $\left\{\mu_n\right\}_{n\geq 0} \subset {\mathcal{P}}(D)$, if \[\liminf_{n\to\infty}\int_{\R^d}\int_{\R^d}K(x-y)\,d\mu_n(x)\,d\mu_n(y)<\infty,\] then $\left\{\mu_n\right\}_{n\geq 0} \subset {\mathcal{P}}(D)$ is weakly-* relatively compact up to translations. \end{lemm}
\section{Non-existence of Global Minimizers: Domains with Boundaries} \label{sect:non-exist} In this section we investigate various possible scenarios when global minimizers of the energy cannot exist. First we treat the force-imbalance pathologies from free space which could make the energy unbounded from below. We then introduce a new non-existence phenomenon which results entirely from asymmetries in the domain and only occurs in domains with boundaries.
\subsection{Imbalance of Forces} \label{subsect:imbalance} In comparison with the spreading case in free space \eqref{nonexistcond1}, the following result gives a relation between the diffusion parameter $\nu$, the interaction potential $K$ and the effective volume dimension $\eff$ (as opposed to the dimension of the space $d$) which guarantees non-existence of ground states of ${\mathcal{E}}^\nu$ in the form of diffusion-dominated spreading throughout the domain.
\begin{thm}\label{NE} [Non-existence: diffusion-dominated regime] Let Assumptions 1 and 2 hold with $V=0$ and $\eff$ defined by \eqref{effvoldim}. Then the energy ${\mathcal{E}}^\nu$ is not bounded below on ${\mathcal{P}}(D)$ provided there exists $\delta_0$ with
$0<\delta_0<1$, $C_0\in \mathbb{R}$ and $R_0>0$ such that
\begin{equation}\label{lemm1i}
K(x) \leq 2(1-\delta_0)\eff\nu\log|x| +C_0, \qquad \text{for all} \quad |x|>R_0. \end{equation} \end{thm}
\begin{proof} We will explicitly construct a sequence of measures which sends the energy to $-\infty$ by exploiting properties of the effective volume dimension. The supremum in the function $V_{_D}(r)$ defined in \eqref{fDr}
implies that for every $n\in \mathbb{N} $ there exists $x_n\in D$ such that
\[V_{_D}(n)>\left\vert D\cap B_n(x_n)\right\vert> \frac{1}{2} V_{_D}(n).\]
Define the sets $D_n := D\cap B_n(x_n)$ and the sequence of probability measures
\[\mu_n = \frac{1}{|D_n|}\ind{D_n}.\]
We first bound above the interaction energy of $\mu_n$ for $n\geq R_0$:
\begin{align*}
{\mathcal{K}}[\mu_n] &= \frac{1}{2|D_n|^2}\int_{D_n}\int_{D_n} K(x-y)\,dy\,dx \\
&=\frac{1}{2|D_n|^2}\int_{D_n}\left[\int_{D_n\cap B_{R_0}(x)}K(x-y)\,dy+ \int_{D_n\cap B^c_{R_0}(x)}K(x-y)\,dy\right]\,dx\\
&\leq \frac{1}{2|D_n|} \bigg[\nrm{K}_{L^1(B_{R_0}(0))}+\left\vert D_n\cap B^c_{R_0}\right\vert\Big(2(1-\delta_0)\eff\nu\log(2n) +C_0\Big)\bigg]\\[5pt] &\leq (1-\delta_0)\eff\nu\log(n) +\tilde{C}_1 \end{align*}
where
\[\tilde{C}_1 = \frac{1}{2|D_1|} \nrm{K}_{L^1(B_{R_0}(0))}+(1-\delta_0)\eff\nu\log(2) +\frac{1}{2}C_0.\]
Using the characterization \eqref{effvolbound} of the effective volume dimension, for $n>r'$ we have
\[|D_n| >\frac{1}{2} V_{_D}(n)\geq \frac{C}{2}n^{\eff},\]
and so the entropy of $\mu_n$ for $n>r'$ is bounded above as follows:
\[{\mathcal{S}}[\mu_n] = -\log|D_n| \leq -\log\left(\frac{C}{2}n^{\eff}\right)=-\eff\log\left(n\right)-\log\left(\frac{C}{2}\right).\]
Hence, for $n>\max\left\{R_0,r'\right\}$, the total energy of $\mu_n$ satisfies
\[{\mathcal{E}}^\nu[\mu_n]\ \ \leq\ \ (1-\delta_0)\eff\nu\log(n) - \eff\nu\log\left(n\right) +\tilde{C}_1-\nu\log\left(\frac{C}{2}\right)\ \ =\ \ -\delta_0 \eff\nu\log(n) + \tilde{C},\]\\[-10pt]
for $\tilde{C}\in \mathbb{R}$. Hence, $\lim_{n\to \infty} {\mathcal{E}}^\nu[\mu_n] = -\infty$, which concludes the proof. \end{proof}
The interpretation of Theorem \ref{NE} is that if attraction forces are too weak (at large distances), then diffusion dominates and spreading occurs. In free space ($D=\R^d$), the result is consistent with that derived by Carrillo et al. in \cite{carrillo2018existence} since $\eff=d$. In more general unbounded domains the relevant factor in the logarithmic bound \eqref{lemm1i} includes the effective volume dimension, more specifically the product $\eff \nu$. With the interpretation that the effective dimension specifies the number of orthogonal directions that independently extend to infinity in $D$, the factor $\eff \nu$ suggests a minimal balance between the diffusion $\nu$ and the number of directions $\eff$ in which mass can escape to infinity, so that a global minimizer can exist.
\begin{rmrk}
\normalfont As mentioned in Section \ref{sect:free-space}, non-existence due to aggregation-dominated contraction occurs in domains with boundaries under the same conditions as in free space (i.e. condition \eqref{NEaggreg}), since contraction to a point can occur in any domain $D\in {\mathcal{B}}^d$ satisfying $|D|>0$. \end{rmrk}
\subsection{Escaping Mass Phenomena}
\label{subsect:escape} As mentioned above, boundedness from below of the energy is not sufficient to guarantee existence of a minimizer in domains with boundaries. To begin this discussion, we present the following theorem, where \eqref{existcond1} is clearly satisfied, hence $\inf{\mathcal{E}}^\nu>-\infty$, yet no energy minimizer exists.
\begin{thm}\label{COM1} Let $D = [0, +\infty)$, $K(x) = \frac{1}{2}x^2$ and $V = 0$. Then the energy ${\mathcal{E}}^\nu$ has no minimizers. \end{thm}
\begin{proof}
\noindent The energy is given by
\begin{equation}\label{en} {\mathcal{E}}^\nu[\rho] = \frac{1}{4}\int_0^\infty\int_0^\infty (x-y)^2\,\rho(x)\rho(y)\,dx\,dy+\nu\int_0^\infty\rho(x)\log(\rho(x))\,dx. \end{equation}
We proceed by contradiction. Assume that a minimizer $\overline{\rho}$ of \eqref{en} exists. Then $\overline{\rho}\in {\mathcal{P}}_2^{ac}(D)$ due to the growth of $K$. Since $\overline{\rho}$ has $\supp{\overline{\rho}} = D$ by Theorem \ref{suppmin} and $\overline{\rho}$ satisfies the Euler-Lagrange equation \eqref{ELcond}, by Corollary \ref{fixedpointmap} one has
\[\overline{\rho}(x) = Z^{-1}\exp\left(-\frac{K*\overline{\rho}(x)}{\nu}\right),\]
where
\[Z = \int_0^\infty \exp\left(-\frac{K*\overline{\rho}(x)}{\nu}\right)\,dx.\]
From an elementary calculation,
\begin{align*} K*\overline{\rho}(x) &= \frac{1}{2}\int_0^\infty(x-y)^2\overline{\rho}(y)\,dy\\
&= \frac{1}{2}(x-M_1(\overline{\rho}))^2-\frac{1}{2}\left(M_1(\overline{\rho})^2-M_2(\overline{\rho})\right), \end{align*}
hence $\overline{\rho} = \rho_c$ for some $c\in \mathbb{R}$, where
\[\rho_c(x) = A(c)\,\exp\left(-\frac{1}{2\nu}\left(x-c\right)^2\right)\]
is a shifted and truncated Gaussian. Here $c = M_1(\overline{\rho})$ and $A(c)$ is the normalization constant
\[A(c) = \frac{2/\sqrt{2\pi\nu}}{1+\text{erf}(c/\sqrt{2\nu})},\]
where $\text{erf}(x)$ denotes the error function.
Let $\Gamma_c = \left\{\rho_c\right\}_{c\geq 0}$ be the family of shifted and truncated Gaussians on $[0,+\infty)$. Then since $\overline{\rho}\in\Gamma_c$ and $\overline{\rho}$ is a critical point of ${\mathcal{E}}^\nu$ over ${\mathcal{P}}(D)$, $\overline{\rho}$ is a critical point of ${\mathcal{E}}^\nu$ over $\Gamma_c$ as well, and so the function $c\to{\mathcal{E}}^\nu[\rho_c]$ has a critical point at some $c \in \mathbb{R}$. By direct calculation of ${\mathcal{E}}^\nu[\rho_c]$, we now show that no such critical point exists.
For the entropy, we have
\begin{align*} {\mathcal{S}}[\rho_c] &= A(c) \int_0^\infty \exp\left(-\frac{1}{2\nu}\left(x-c\right)^2\right)\left(-\frac{1}{2\nu}\left(x-c\right)^2+\log(A(c))\right)\,dx\\ &=\log(A(c))-\frac{1}{2\nu}\underbrace{A(c)\int_0^\infty (x-c)^2\exp\left(-\frac{1}{2\nu}(x-c)^2\right)\,dx}_{I}.\\
\intertext{For the interaction energy, we get}
{\mathcal{K}}[\rho_c] &= \frac{1}{4}A(c)^2\int_0^\infty\int_0^\infty(x-y)^2 \exp\left\{-\frac{1}{2\nu}(x-c)^2-\frac{1}{2\nu}(y-c)^2\right\} \,dx\,dy\\ &= \frac{1}{2}\underbrace{A(c)\int_0^\infty (x-c)^2\exp\left(-\frac{1}{2\nu}(x-c)^2\right)\,dx}_{I}\\ &\mathbin{\color{white}===}-\frac{1}{2}\left[A(c)\int_0^\infty (x-c)\exp\left(-\frac{1}{2\nu}(x-c)^2\right)\,dx\right]^2\\ &= \frac{1}{2}I - \frac{\nu^2}{2}\left(A(c) \exp\left(-\frac{c^2}{2\nu}\right)\right)^2. \end{align*}
The total energy ${\mathcal{E}}^\nu[\rho_c]$ then reduces to
\begin{align*} {\mathcal{E}}^\nu[\rho_c] &= {\mathcal{K}}[\rho_c]+\nu{\mathcal{S}}[\rho_c]\\ &= \nu\log(A(c)) -\frac{\nu^2}{2}\left(A(c) \exp\left(-\dfrac{c^2}{2\nu}\right)\right)^2\\ &= \nu\log\left(\frac{2}{\sqrt{2\pi\nu}}\right)-\nu\log\left(1+\text{erf}\left(\frac{c}{\sqrt{2\nu}}\right)\right) -\frac{\nu}{4}\left(\dfrac{\dfrac{2}{\sqrt{\pi}}\exp\left(-\dfrac{c^2}{2\nu}\right)}{1+\text{erf}\left(\dfrac{c}{\sqrt{2\nu}}\right)}\right)^2. \end{align*}
By letting $\tilde{c} = c/{\sqrt{2\nu}}$, one can also write:
\begin{equation}\label{COMenergy1} {\mathcal{E}}^\nu[\rho_c]=\nu\log\left(\frac{2}{\sqrt{2\pi\nu}}\right) - \nu\left(f(\tilde{c})+\frac{1}{4}f'(\tilde{c})^2\right), \end{equation}
where
\[f(\tilde{c}) = \log\left(1+\text{erf}(\tilde{c})\right).\]
Since $c\to {\mathcal{E}}^\nu[\rho_c]$ has a critical point (by hypothesis) and is a smooth function, we have
\begin{equation}\label{COMcont} \frac{d}{dc}{\mathcal{E}}^\nu[\rho_c] = -\sqrt{\frac{\nu}{2}} f'(\tilde{c})\left(1+\frac{1}{2}f''(\tilde{c})\right) = 0, \\[5pt] \end{equation} for some $\tilde{c}\in\mathbb{R}$. However, $f' > 0$ and $\min f'' = -\frac{4}{\pi}>-2$ together imply that \eqref{COMcont} has no solutions. This contradicts the assumption that ${\mathcal{E}}^\nu$ has a critical point.
Figure \ref{energy_quad_attract} shows the monotonically decreasing profile of $c\to {\mathcal{E}}^\nu[\rho_c]$ together with energy plots with an added external potential $V(x)=gx$, which will be addressed in Theorem \ref{COM2}. In particular, it shows the case $g=0$ corresponding to \eqref{COMenergy1}. \end{proof}
\begin{figure}\label{energy_quad_attract}
\end{figure}
The non-existence result in Theorem \ref{COM1} is an example of a more general phenomenon which we refer to as the \textit{escaping mass phenomenon}. Sequences such as $\Gamma_c$ in Theorem \ref{COM1} are \textit{escaping} in the sense that the centre of mass $\CalC\left(\rho_c\right)$ reaches infinity without the measures vanishing in the traditional sense. This phenomenon manifests in dynamics as the persistent, metastable translation of the centre of mass of the swarm (see Remark \ref{COMdyn}).
Geometrically, asymmetries in $D$ cause the energy ${\mathcal{E}}^\nu$ to lose translation invariance, and so the same tight-up-to-translations arguments from free space do not apply. To enforce the existence of a global minimizer, we can add a confining potential $V$ and exploit any symmetries within $D$. This process is described in Section \ref{sect:existence}. To complete the discussion on escaping mass, Theorem \ref{domassymthm} below provides a necessary condition for the existence of a minimizer which comes as a direct corollary of the Euler-Lagrange equation.
\begin{thm}\label{domassymthm} Let Assumptions 1 and 2 be satisfied and let $K,V\in W_{loc}^{1,1}(D)$. Then if $\overline{\rho}$ is a critical point of the energy $\,{\mathcal{E}}^\nu$ with $\supp{\overline{\rho}} = D$, then $\overline{\rho}$ satisfies
\begin{equation}\label{boundaryV} \nu\int_{\partial D} n(x)\,\overline{\rho}(x)\,dS(x) = -\int_D \overline{\rho}(x)\nabla V(x)\,dx, \end{equation}
\end{thm}
\begin{proof} Assume that $\overline{\rho}$ is such a critical point.
Since $K,V\in W_{loc}^{1,1}(D)$, $\overline{\rho}$ is differentiable almost everywhere. Taking the gradient of both sides of the Euler-Lagrange equation \eqref{ELcond} and integrating against $\overline{\rho}(x)\,dx$ then gives us
\begin{equation}\label{domassymthm_grad} \nu\int_D \nabla \overline{\rho}(x) \,dx = -\underbrace{\int_D \overline{\rho}(x) \nabla K*\overline{\rho}(x)\, dx}_{=:I}- \int_D \overline{\rho}(x)\nabla V(x)\,dx. \end{equation}
The anti-symmetry of $\nabla K$ implies that
\[I = \int_D\int_D \nabla K(x-y)\,\overline{\rho}(y)\,\overline{\rho}(x)\,dy\,dx = -\int_D\int_D \nabla K(y-x)\,\overline{\rho}(y)\,\overline{\rho}(x)\,dy\,dx =-I,\]
and hence $I=0$. To integrate the left-hand side of \eqref{domassymthm_grad}, we use the divergence theorem. Consider a sequence of bounded sets $A_n \subset D$ with smooth boundary such that $\lim_{n\to\infty} A_n = D$ and $\lim_{n\to\infty} \partial A_n = \partial D$. Then for any fixed $\vec{a}\in \R^d$,
\begin{align*} \vec{a}\cdot \int_{A_n}\nabla \overline{\rho}(x)\,dx &= \int_{A_n}\nabla\cdot\left(\overline{\rho}(x)\vec{a}\right)\,dx\\ &= \int_{\partial A_n}n(x)\cdot \left(\overline{\rho}(x)\vec{a}\right)\,dS(x)\\ &= \vec{a}\cdot\int_{\partial A_n}n(x)\,\overline{\rho}(x)\,dS(x). \end{align*}
Since this holds for any $\vec{a}\in \R^d$, for each $n$ we have
\[\int_{A_n} \nabla \overline{\rho}(x)\,dx = \int_{\partial A_n} n(x)\overline{\rho}(x)\,dS(x),\]
and so
\[\int_D \nabla \overline{\rho}(x)\,dx = \lim_{n\to \infty} \int_{A_n} \nabla \overline{\rho}(x)\,dx = \lim_{n\to \infty} \int_{\partial A_n} n(x)\overline{\rho}(x)\,dS(x) = \int_{\partial D} n(x)\, \overline{\rho}(x)\,dS(x).\]
Since $n(x)$ is defined for almost every $x\in \partial D$ (Assumption \ref{assumD}) and $\overline{\rho}\in L^1(D)$, we can apply classical trace theorems \cite[Ch. 5]{evans10} to conclude that the right-most integral in \eqref{domassymthm_grad} is finite. This yields the result. \end{proof}
\begin{rmrk}\label{nonexist} \normalfont Theorem \ref{domassymthm} indicates that minimizers of ${\mathcal{E}}^\nu$ cannot exist under zero external potential in a large class of domains (see the Examples below). Indeed, $V=0$ implies that the right-hand side of \eqref{boundaryV} is zero, yet the left-hand side of \eqref{boundaryV} is nonzero: the formula for critical points $\overline{\rho}$ with $\supp{\overline{\rho}}=D$,
\[\overline{\rho}(x) = Z^{-1}\exp\left(-\frac{K*\overline{\rho}(x)}{\nu}\right),\]
implies that $\overline{\rho}(x)>0$ for all $x\in \partial D$. For this reason, \eqref{boundaryV} cannot hold in many infinite domains. \end{rmrk}
\begin{rmrk}\label{COMdyn} \normalfont Condition \eqref{boundaryV} relates to the dynamics of the aggregation-diffusion model \eqref{aggdD} in the following way. Consider the evolution in time of the centre of mass:
\begin{align*} \frac{d}{dt}\mathcal{C}(\mu^{\nu}_t) &=\int_Dx\left(\ppt\rho^\nu_t(x)\right)\, dx\\ &= \int_Dx \nabla \cdot\bigg(\rho^\nu_t(x)\Big(\nabla K*\rho^\nu_t(x)+\nabla V(x)\Big)+\nu\nabla \rho^\nu_t(x)\bigg)\,dx\\ &= -\int_D\nabla K*\rho^\nu_t(x) \rho^\nu_t(x)\,dx-\int_D\nabla V(x)\rho^\nu_t(x)\,dx-\nu\int_{D} \nabla \rho^{\nu}_t(x)\,dx\\
\intertext{(integrating by parts and utilizing the boundary conditions)}
&=-\int_D \nabla V(x)\rho^\nu_t(x)\,dx-\nu\int_{\partial D} n(x) \rho^{\nu}_t(x) \,dS(x). \end{align*} For $V=0$, this is exactly
\begin{equation}\label{comtrans} \frac{d}{dt}\CalC\left(\mu^{\nu}_t\right) = -\nu\int_{\partial D} n(x) \rho^{\nu}_t(x) \,dS(x), \end{equation}
hence the swarm translates in the direction opposite the average outward normal vector with speed proportional to the mass along the boundary, weighted by $\nu$. Unless the domain is bounded or symmetric enough that mass may be distributed along the boundary in such a way that the right-hand side of \eqref{comtrans} is zero, translation will occur indefinitely, further justifying the terminology ``escaping-mass phenomenon''. Clearly this takes effect as soon as $\nu>0$. \end{rmrk} \subsubsection{Examples}
\hspace{12pt} The following are a few example domains where a minimizer $\overline{\rho}$ cannot exist by the argument in Remark \ref{nonexist}.
\begin{enumerate} \item Half-space: Here $D = \R^d_+:= \mathbb{R}^{d-1}\times [0,\infty)$ where $n(x) = -\hat{e}_d$ is constant for all $x\in \partial D$. This gives \[\int_{\partial D}n(x)\,\overline{\rho}(x)\,dS(x) \,=\, -\hat{e}_d\int_{\mathbb{R}^{d-1}}\overline{\rho}(x)\,dx_1\dots \,dx_{d-1} \,<\, 0.\] Note that Theorem \ref{COM1} demonstrates this case for $d=1$. \item Wedge domain: $D = \left\{x\in \mathbb{R}^2 \txt{0.2}{:} 0\leq x_2\leq \tan(\phi) x_1\right\}$ for $\phi\in (0,\pi/2)$. Then
\[\int_{\partial D}n(x)\,\overline{\rho}(x)\,dS(x) \,=\, (N_1,N_2)\] where \[N_1 = -\sin(\phi)\int_0^\infty \overline{\rho}(z,\tan(\phi)z)\,dz<0.\]
\item Paraboloid: Let $x = (x_1,\dots,x_{d-1},x_d) = (x',x_d)\in \R^d$ and define
\[D = \left\{x\in \R^d \txt{0.2}{:} x_d \geq |x'|^2\right\}.\]
Then $n(x) = \dfrac{1}{\sqrt{|x'|^2+\frac{1}{4}}}\left(x', -\frac{1}{2}\right)$ and so
\[\int_{\partial D} n(x)\,\overline{\rho}(x) \, dS(x) = (N', N_d)\]
where again $N_d$ cannot be zero. \end{enumerate}
In the next section we establish a relation between the domain geometry and the external potential $V$, motivated by Theorem \ref{domassymthm}, that ensures existence of a minimizer.
\section{Existence of Global Minimizers} \label{sect:existence}
\subsection{A sharp existence condition for certain domains} \label{subsect:tight-exist}
Recall that in free space we have a sharp condition for existence of global minimizers. More specifically, in free space the sharp condition \eqref{existcond1} (resp. \eqref{nonexistcond1}) determines whether the energy ${\mathcal{E}}^\nu$ is bounded (resp. unbounded) below, and boundedness from below is all that is needed to guarantee existence of minimizers in free space. Here we show that for a wide class of domains with boundaries (i.e. those of the form \eqref{Fdomain}), the analogous condition \eqref{kbound} (resp. \eqref{lemm1i}) for granting boundedness (resp. unboundedness) from below of the energy is also sharp, and depends explicitly on the effective volume dimension $\eff$ of the domain.
\begin{thm}\label{tightbound} Let $D\subset \R^d$ have the form \eqref{Fdomain} and $\nu>0$. Suppose that $V=0$ and $K$ is positive and satisfies Assumption \ref{assumKV}. In addition, suppose that for some $\delta_1>0$ and $C_1\in \mathbb{R}$,
\begin{equation}\label{kbound}
K(x) \geq 2(1+\delta_1)\eff\nu\log|x|+C_1\txt{0.4}{for all} x\in D-D, \end{equation}
where $D-D := \left\{x-y \in \R^d\txt{0.2}{:} x,y\in D\right\}$. Then the energy ${\mathcal{E}}^\nu$ is bounded below on ${\mathcal{P}}(D)$. Moreover, there exists a global minimizer $\overline{\rho} \in {\mathcal{P}}^{ac}(D)$ of ${\mathcal{E}}^\nu$, that is
\[{\mathcal{E}}^\nu[\overline{\rho}] = \inf_{\rho\in {\mathcal{P}}(D)} {\mathcal{E}}^\nu[\rho] >-\infty.\] \end{thm}
\begin{proof} First we establish some notation. Recall that $D=F\times \mathbb{R}^{d-m}$ where $F\subset \mathbb{R}^m$ is compact and $m$-dimensional for $m\in \left\{1,\dots,d-1\right\}$. Denote $x = (x_1,x_2,\dots,x_d) \in D$ by $x=(\overline{x},\tilde{x})$ for $\overline{x}\in F$ and $\tilde{x}\in \mathbb{R}^{d-m}$. For $\rho\in {\mathcal{P}}^{ac}(D)$, define the $\overline{x}$-marginal $\rho_{_{F}} \in {\mathcal{P}}^{ac}(\mathbb{R}^{d-m})$ of $\rho$ by
\begin{equation} \rho_{_{F}}(\tilde{x}) = \int_F \rho(x)\,d\overline{x}.
\end{equation}
\textit{Step 1:} For any $\rho\in {\mathcal{P}}^{ac}(D)$, we have
\begin{equation}\label{step1}
{\mathcal{S}}[\rho]\geq {\mathcal{S}}[\rho_{_{F}}]-\log|F|, \end{equation}
where
\[{\mathcal{S}}[\rho_{_{F}}] = \int_{\mathbb{R}^{d-m}}\rho_{_{F}}(\tilde{x})\log(\rho_{_{F}}(\tilde{x}))\,d\tilde{x}.\]
To show \eqref{step1}, by Fubini's theorem we have
\[{\mathcal{S}}[\rho] =\int_D\rho(x)\log(\rho(x))\,dx= \int_{{\mathbb R}^{d-m}}\left(\int_F\rho(\overline{x},\tilde{x})\log\left(\rho(\overline{x},\tilde{x})\right)\,d\overline{x}\right)\,d\tilde{x}.\]
We now claim that for almost every $\tilde{x}\in \mathbb{R}^{d-m}$,
\begin{equation}\label{claim1} \int_F\rho(\overline{x},\tilde{x})\log\left(\rho(\overline{x},\tilde{x})\right)\,d\overline{x} \ \geq\
\rho_{_{F}}(\tilde{x})\log\left(\frac{\rho_{_{F}}(\tilde{x})}{|F|}\right). \end{equation}
Assuming the claim, we then have
\[{\mathcal{S}}[\rho] \geq \int_{\mathbb{R}^{d-m}} \rho_{_{F}}(\tilde{x})\log\left(\frac{\rho_{_{F}}(\tilde{x})}{|F|}\right) \,d\tilde{x} = {\mathcal{S}}[\rho_{_{F}}] - \log|F|,\] showing \eqref{step1}.
We now prove claim \eqref{claim1} using convexity. For almost every $\tilde{x}\in \mathbb{R}^{d-m}$, the function $f({\overline{x}}):=\rho({\overline{x}},\tilde{x})$ is defined for $\overline{x}\in F$ up to a set of measure zero and satisfies $\nrm{f}_{L^1(F)} = \rho_{_{F}}(\tilde{x})$. For $\nrm{f}_{L^1(F)} = 0$ or $\nrm{f}_{L^1(F)} = +\infty$, the claim \eqref{claim1} trivially holds with equality. For $0<\nrm{f}_{L^1(F)}<+\infty$, by the convexity of $U(x) = x\log(x)$ we have for almost every $x \in [0,\,\infty)$ and every $y\in [0,\infty)$:
\[U(y) \geq U(x)+U'(x)(y-x).\]
Letting $y = f({\overline{x}})$ and $x = \frac{\nrm{f}_{L^1(F)}}{|F|}$, we have for almost every $\overline{x}\in F$,
\[U\left(f({\overline{x}})\right)\geq U\left(\frac{\nrm{f}_{L^1(F)}}{|F|}\right)+U'\left(\frac{\nrm{f}_{L^1(F)}}{|F|}\right)\,\left(f(\overline{x})-\frac{\nrm{f}_{L^1(F)}}{|F|}\right).\]
Integrating over $F$ we then get
\begin{align*}
\int_FU\left(f({\overline{x}})\right)\,d\overline{x}&\geq \int_FU\left(\frac{\nrm{f}_{L^1(F)}}{|F|}\right)\,d\overline{x}\\
&= \int_F \left(\frac{\nrm{f}_{L^1(F)}}{|F|}\right)\log\left(\frac{\nrm{f}_{L^1(F)}}{|F|}\right)\,d\overline{x}\\
&= \rho_{_{F}}(\tilde{x})\log\left(\frac{\rho_{_{F}}(\tilde{x})}{|F|}\right), \end{align*} which proves the claim.\\[5pt]
We briefly note that intuition for \eqref{claim1} comes from the case $\rho_{_{F}}(\tilde{x}) = 1$, which implies that $\rho(\overline{x},\tilde{x})$ is a probability density on $F$. The inequality \eqref{claim1} is then equivalent to the uniform distribution on $F$ being the global minimizer of ${\mathcal{S}}$ over ${\mathcal{P}}(F)$, which is intuitively clear from an information perspective: the uniform distribution corresponds to the state with \textit{least information}, or maximum entropy $-{\mathcal{S}}$.
\noindent \textit{Step 2.} There exists $\tilde{C}\in \mathbb{R}$ such that for any $\rho\in {\mathcal{P}}^{ac}(D)$, we have
\begin{equation}\label{step2} {\mathcal{K}}[\rho] \geq -(1+\delta_1)\nu\,{\mathcal{S}}[\rho_{_{F}}] + \tilde{C}. \end{equation}
To show this, first note that for any $x = (\overline{x},\tilde{x})$ and $y = (\overline{y},\tilde{y})$ in $D$:
\[\log\left(|x-y|\right)\geq \log\left(|\tilde{x}-\tilde{y}|\right).\]
Using this, together with the lower bound \eqref{kbound} on $K$, Fubini's theorem, and the logarithmic-HLS inequality on the space ${\mathcal{P}}^{ac}(\mathbb{R}^{d-m})$ (see Lemma \ref{lemma:HLS}), we get:
\begin{align*} {\mathcal{K}}[\rho] &= \frac{1}{2}\int_D\int_D K(x-y)\rho(x)\rho(y)\,dx\,dy\\
&\geq (1+\delta_1)(d-m)\nu\int_D\int_D\log\left(|x-y|\right)\rho(x)\rho(y)\,dx\,dy+ \frac{C_1}{2}\\
&\geq (1+\delta_1)(d-m)\nu\int_{\mathbb{R}^{d-m}}\int_{\mathbb{R}^{d-m}}\log\left(|\tilde{x}-\tilde{y}|\right)\left(\int_F\rho(x)\,d\overline{x}\right)\left(\int_F\rho(y)\,d\overline{y}\right)\,d\tilde{x}\,d\tilde{y}+ \frac{C_1}{2}\\
&= (1+\delta_1)(d-m)\nu\int_{\mathbb{R}^{d-m}}\int_{\mathbb{R}^{d-m}}\log\left(|\tilde{x}-\tilde{y}|\right)\rho_{_{F}}(\tilde{x})\rho_{_{F}}(\tilde{y})\,d\tilde{x}\,d\tilde{y}+ \frac{C_1}{2}\\ &\geq -(1+\delta_1)\nu{\mathcal{S}}[\rho_{_{F}}] -(1+\delta_1)(d-m)\nu\,C_0+ \frac{C_1}{2}. \\[-10pt] \end{align*}
\textit{Step 3.} We now show that $\inf{\mathcal{E}}^\nu>-\infty$. \\
Consider a minimizing sequence $\left\{\rho^n\right\}_{n\geq 0}$ of ${\mathcal{E}}^\nu$ and without loss of generality assume that $\sup_n\left\{{\mathcal{E}}^\nu[\rho^n]\right\}<+\infty$. By Step 1 and the positivity of $K$, for every $n$ we have
\[\nu {\mathcal{S}}[\rho_{_{F}}^n]\leq \nu{\mathcal{S}}[\rho^n]+\nu\log|F|\leq {\mathcal{E}}^\nu[\rho^n]+\nu\log|F|<+\infty\]
and so $\sup_n\left\{{\mathcal{S}}[\rho_{_{F}}^n]\right\}<+\infty$. Putting Steps 1 and 2 together (for $\tilde{C}$ different from above), we get
\[{\mathcal{E}}^\nu[\rho^n] \geq -(1+\delta_1)\nu{\mathcal{S}}[\rho_{_{F}}^n] +\nu {\mathcal{S}}[\rho_{_{F}}^n]+\tilde{C} = - \left(\delta_1\nu\right) {\mathcal{S}}[\rho_{_{F}}^n] +\tilde{C},\]
which implies
\[\inf_{\rho\in{\mathcal{P}}(D)}{\mathcal{E}}^\nu[\rho] =\lim_{n\to \infty} {\mathcal{E}}^\nu[\rho^n] \geq - \left(\delta_1\nu\right) \sup_n\left\{{\mathcal{S}}[\rho_{_{F}}^n]\right\} >-\infty.\]
\textit{Step 4.} Minimizing sequences are tight-up-to-translations in $\mathbb{R}^{d-m}$.\\[5pt]
The inequality \eqref{step2} in Step 2 can be rewritten as
\[\nu{\mathcal{S}}[\rho_{_{F}}]\geq -\frac{1}{1+\delta_1}\left({\mathcal{K}}[\rho]-\tilde{C}\right).\]
Combined with Step 1 and the positivity of $K$, this gives us
\[{\mathcal{E}}^\nu[\rho] \geq \frac{\delta_1}{1+\delta_1}{\mathcal{K}}[\rho]+\tilde{C}\geq \tilde{C},\]
for $\tilde{C}\in \mathbb{R}$ different from above. The boundedness of $\left\{{\mathcal{E}}^\nu[\rho^n]\right\}_{n\geq 0}$ implies that $\left\{{\mathcal{K}}[\rho^n]\right\}_{n\geq 0}$ is bounded: it was shown above that energy is bounded below, while the interaction energy has to be bounded above, and so by Lemma \ref{Ktight}, there exists a sequence $\left\{\rho^n\right\}_{n\geq 0}$ which is tight up to translations in $\R^d$. Let $\left\{\tilde{\rho}^n\right\}_{n\geq 0}$ be a translated version of the sequence that is tight in ${\mathcal{P}}(\R^d)$ and given by
\[\tilde{\rho}^n(x) = \rho^n(x-x^n).\]
Without loss of generality, we may assume by the compactness of $F$ that the translations $x^n$ satisfy $x^n_i = 0$ for $i=1,\dots,m$, which implies that (i) $\left\{\tilde{\rho}^n\right\}_{n\geq 0}$ is tight in ${\mathcal{P}}(D)$ and (ii) for each $n$, ${\mathcal{E}}^\nu[\rho^n] = {\mathcal{E}}^\nu[\tilde{\rho}^n]$, since the energy is invariant to translations in the last $d-m$ coordinates. By (i) and Prokhorov's theorem, we are guaranteed existence of a subsequence $\left\{\tilde{\rho}^{n_k}\right\}_{k\geq 0}$ which converges weakly-* to some $\overline{\rho}\in {\mathcal{P}}(D)$, and by (ii) and the lower semicontinuity of the energy (Lemma \ref{lowerSC}), we have that
\[{\mathcal{E}}^\nu[\overline{\rho}]\leq \liminf_{n\to \infty}{\mathcal{E}}^\nu[\tilde{\rho}^n] = \lim_{n\to \infty} {\mathcal{E}}^\nu[\rho^n] = \inf_{\rho\in {\mathcal{P}}(D)}{\mathcal{E}}^\nu[\rho],\]
and so $\overline{\rho}$ realizes the infimum of ${\mathcal{E}}^\nu$. Since $\inf_{\rho\in {\mathcal{P}}(D)}{\mathcal{E}}^\nu<+\infty$, we have that $\overline{\rho}$ is absolutely continuous with respect to the Lebesgue measure (see Remark \ref{ac-gmin}). \end{proof}
\begin{rmrk}
\normalfont Boundedness from below of the energy in Theorem \ref{tightbound} can be extended to domains $D = F\times H$ where $F\subset \mathbb{R}^{m}$ is compact and $m$-dimensional and $H= H_1\times \dots\times H_{d-m}$ where each $H_i\subset \mathbb{R}$ satisfies $|H_i| = +\infty$ and is given by the closure of a disjoint union of intervals:
\[H_i = \overline{\cup_{k=0}^\infty I_k^i}.\]
In particular, we could have $H_i = [0,\infty)$. Introducing domains with such asymmetries, however, leads us again into the dilemma of the escaping mass phenomenon, and so boundedness from below of the energy may not be enough to grant existence. \end{rmrk}
\subsection{Existence of Minimizers via Confining Potentials} \label{subsect:V-existence} In this section we establish sufficient conditions for existence of global minimizers of the energy that take into account the escaping mass phenomenon. Given the considerations above, in many canonical domains we have no global minimizer despite ${\mathcal{E}}^\nu$ being bounded below. With insight from Theorem \ref{domassymthm}, we present here an approach for guaranteeing existence of a global minimizer through the addition of a suitable external potential $V$. For a simple example, we first return to the case $K(x) = \frac{1}{2}x^2$ and $D = [0,\infty)$ from Section \ref{subsect:escape}, adding a potential $V(x) = gx$. In Theorem \ref{existence1} we then establish a condition on $V$ (see \eqref{Vbound1}) that is in some sense minimal and guarantees existence of a global minimizer of ${\mathcal{E}}^\nu$ for general domains $D$ satisfying Assumption 2. Finally, Theorem \ref{existence2} provides a weaker set of requirements on $V$ which takes advantage of symmetries within the domain.
\begin{thm}\label{COM2}
Let $D = [0, +\infty)$, $K(x) = \frac{1}{2}x^2$ and $V(x) = gx$ for $g>0$. Then for any $g>0$, there exists a unique critical point $\overline{\rho}$ for ${\mathcal{E}}^\nu$ in the space of measures in ${\mathcal{P}}_2^{ac}(D)$ having support equal to $D$.
\end{thm}
\begin{proof}
\noindent The energy is given by
\begin{equation}\label{en-g} {\mathcal{E}}^\nu[\rho] = \frac{1}{4}\int_0^\infty\int_0^\infty (x-y)^2\,\rho(x)\rho(y)\,dx\,dy+\nu\int_0^\infty\rho(x)\log(\rho(x))\,dx+g\int_0^\infty x\rho(x)\,dx. \end{equation}
We proceed as in Theorem \ref{COM1} and look for energy minimizers using the fixed-point characterization of critical points \eqref{gibbstat}, only now we show that the map $T(\rho)$ has a unique fixed point. Borrowing from the calculations in Theorem \ref{COM1}, for any $\rho\in{\mathcal{P}}^{ac}_2(D)$ we have:
\[T(\rho) = Z^{-1}\exp\left(-\frac{K*\rho(x)+V(x)}{\nu}\right) = A(c)\,\exp\left(-\frac{1}{2\nu}\left(x-c\right)^2\right),\]
where now $c = M_1(\rho)-g$. Since $T$ maps ${\mathcal{P}}^{ac}_2(D)$ into $\Gamma_c$, by Corollary \ref{fixedpointmap} it suffices to look for critical points in $\Gamma_c$.
We then proceed as above and first attempt to satisfy the necessary condition $\frac{d}{dc}{\mathcal{E}}^\nu[\rho_c] = 0$. With $g>0$ the energy \eqref{COMenergy1} becomes
\begin{equation}\label{energy_quad_attract_formula} {\mathcal{E}}^\nu[\rho_c] = \nu\log\left(\frac{2}{\sqrt{2\pi\nu}}\right) - \nu\left(f({\tilde{c}})+\frac{1}{4}f'({\tilde{c}})^2\right) + g\left(\sqrt{\frac{\nu}{2}} f'({\tilde{c}}) +\sqrt{2\nu}\,{\tilde{c}}\right), \end{equation}
whereby solving $\frac{d}{dc}{\mathcal{E}}^\nu[\rho_c] = 0$ reduces to finding a root $\tilde{c}$ to
\[\left(\sqrt{\frac{2}{\nu}}g - f'(\tilde{c})\right)\left(1+\frac{1}{2}f''(\tilde{c})\right) = 0.\]
From \eqref{COMcont}, we know that the second term is strictly positive, so we may divide by it and further reduce the problem to solving
\begin{equation}\label{COMroot} f'(\tilde{c}) = \sqrt{\frac{2}{\nu}}\,g. \end{equation}
For any $g,\nu>0$, \eqref{COMroot} has a unique solution since $f':{\mathbb R}\to[0,\infty)$ is smooth and monotonically decreasing, so we have that there exists a unique \textit{candidate} critical point $\rho_{c^*}\in \Gamma_c$ where $\frac{c^*}{\sqrt{2\nu}}$ solves \eqref{COMroot}; $\rho_{c^*}$ is then a critical point of ${\mathcal{E}}^\nu$ over the space $\Gamma_c$.
All that remains is to show that $T(\rho_{c^*})=\rho_{c^*}$ to conclude that $\rho_{c^*}$ is in fact a critical point of ${\mathcal{E}}^\nu$ over all of ${\mathcal{P}}^{ac}_2(D)$. Indeed, since $T$ maps ${\mathcal{P}}^{ac}_2(D)$ into $\Gamma_c$, we have $T(\rho_{c^*})=\rho_{c'} \in \Gamma_c$ for some $c'\in\mathbb{R}$, and by direct calculation,
\[c' = M_1(\rho_{c^*})-g = \sqrt{\frac{\nu}{2}}f'\left(\frac{c^*}{\sqrt{2\nu}}\right)+c^*-g = c^*,\]
since $\frac{c^*}{\sqrt{2\nu}}$ solves \eqref{COMroot}. This shows that $\rho_{c'} = \rho_{c^*}$ since every member of $\Gamma_c$ is uniquely determined by its shift $c$. This completes the proof. We refer the reader back to Figure \ref{energy_quad_attract} for a comparison of $c\to {\mathcal{E}}^\nu[\rho_c]$ for several values of $g>0$, with each plot showing one global minimum. \end{proof}
\begin{rmrk} \label{rmk:gc} \normalfont For $g=g_c := \sqrt{\frac{2\nu}{\pi}}$, the solution $c$ to \eqref{COMroot} is exactly $c=0$, which implies that the critical point $\overline{\rho}$ is exactly a half-Gaussian, and for $g\geq g_c$, the maximum of $\overline{\rho}$ lies at $x=0$. This is used to benchmark the numerical method in Section \ref{sect:numerics}, in Figure \ref{kp2}. \end{rmrk}
\begin{thm}\label{existence1} Suppose that Assumptions \ref{assumKV} and \ref{assumD} are satisfied and that $K$ and $V$ are positive. In addition, suppose that for some $\delta > 0$ and $C_K\in \mathbb{R}$,
\begin{equation}\label{existFree}
K(x) \geq 2(1+\delta)d\nu \log|x| + C_K, \end{equation}
and that for some $x_0\in D$, $V$ satisfies
\begin{equation}\label{Vbound1} \lim_{R\to \infty}\left( \inf_{x\in B_R^c(x_0)} V(x)\right) = +\infty. \end{equation}
Then there exists a global minimizer $\overline{\rho} \in {\mathcal{P}}^{ac}(D)$ of ${\mathcal{E}}^\nu$. \end{thm}
\begin{proof} We will first show that the energy ${\mathcal{E}}^\nu$ is bounded below and then prove that minimizing sequences are tight. Indeed, the boundedness from below of ${\mathcal{E}}^\nu$ follows from results in free space. By Theorem \ref{tightbound} above (along with \cite{carrillo2018existence}), relation \eqref{existFree} between $K$ and $\nu$ is sufficient to guarantee that ${\mathcal{E}}^\nu$ is bounded below over ${\mathcal{P}}(\R^d)$ by a constant $C\in \mathbb{R}$ when $V=0$.
Since $|D|>0$ and we are not requiring any regularity of measures other than absolute continuity with respect to Lebesgue measure, for any $\mu \in {\mathcal{P}}^{ac}(D)$ with density $\rho$ we can define a measure $\mu_0\in {\mathcal{P}}^{ac}(\R^d)$ with density $\rho_0(x)$ by extending $\rho$ by zero:
\begin{equation}\label{rhozero} \rho_0(x) = \begin{cases} \rho(x), & x\in D \\ 0, & x \in D^{\,c}.\end{cases} \end{equation}
For each $\mu \in {\mathcal{P}}^{ac}(D)$ we then have the lower bound
\begin{align*} {\mathcal{E}}^\nu[\mu] &=\frac{1}{2}\int_D\int_DK(x-y)\,d\mu(x)d\mu(y)+\nu\int_D\rho(x)\log(\rho(x))\,dx+\int_DV(x)\,d\mu(x) \\ &= \frac{1}{2}\int_{\R^d}\int_{\R^d} K(x-y)\,d\mu_0(x)d\mu_0(y)+\nu\int_{\R^d}\rho_0(x)\log(\rho_0(x))\,dx+\int_{D} V(x)\,d\mu(x)\\ &>C + \int_{D} V(x)\,d\mu(x), \end{align*}
which implies
\begin{equation}\label{Vbound2} \int_DV(x)\,d\mu(x) < {\mathcal{E}}^\nu[\mu]-C. \end{equation}
Now consider a minimizing sequence $\left\{\mu_n\right\}_{n\geq 0}\subset {\mathcal{P}}^{ac}(D)$ of ${\mathcal{E}}^\nu$. The following argument shows that $\left\{\mu_n\right\}_{n\geq 0}$ is tight. Since $\left\{\mu_n\right\}_{n\geq 0}$ is minimizing, we can assume $\left\{{\mathcal{E}}^\nu[\mu_n]\right\}_{n\geq 0}$ is bounded above, hence \eqref{Vbound2} implies
\begin{equation}\label{Vbound3} \sup_n\int_D V(x)\,d\mu_n(x) < M \end{equation}
for some $M\in \mathbb{R}$. Fix $\epsilon>0$ and let $L>0$ be large enough that $M/L<\epsilon$. From \eqref{Vbound1}, to $L$ there corresponds an $R$ such that
\[\inf_{x\in B^c_R(x_0)}V(x)>L.\]
For each $\mu_n$ we then have
\[L\int_{B^c_R(x_0)\cap D}\,d\mu_n(x)\leq \int_{B^c_R(x_0)\cap D}V(x)\,d\mu_n(x)\leq \int_DV(x)\,d\mu_n(x)<M,\]
hence for the compact set $K_\epsilon = B_R(x_0)\cap D$,
\[\mu_n(K_\epsilon)> 1-\epsilon.\]
This shows that the minimizing sequence $\left\{\mu_n\right\}_{n\geq 0}$ is tight. By Prokhorov's theorem we may then extract a subsequence $\left\{\mu_{n_k}\right\}_{k\geq 0}$ which converges in the weak-* topology of measures to some $\overline{\rho}\in {\mathcal{P}}(D)$. It follows from the weak-* lower semicontinuity of ${\mathcal{E}}^\nu$ (Lemma \ref{lowerSC}) that
\[{\mathcal{E}}^\nu[\overline{\rho}] \leq \liminf_{k\to \infty}{\mathcal{E}}^\nu[\rho_{n_k}] = \lim_{n\to \infty}{\mathcal{E}}^\nu[\rho_n] = \inf_{\rho \in {\mathcal{P}}(D)} {\mathcal{E}}^\nu[\rho],\]
and so $\overline{\rho}$ realizes the infimum. Moreover, by Remark \ref{ac-gmin}, $\overline{\rho}\in {\mathcal{P}}^{ac}(D)$. \end{proof}
The previous theorem provides a way to guarantee existence of a minimizer in all domains $D$ satisfying Assumption 2, simply by adding an external potential to contain the mass and enforce tightness. As the following theorem shows, in many domains a less restrictive external potential is needed to ensure a minimizer.
We will need some terminology for the next theorem. Define a \textit{band} $S^i_a$ in $\R^d$ by
\[S^i_a = \left\{x\in \R^d \,:\, |x_i|<a\right\}.\]
Also, we define a function $f:\R^d\to {\mathbb R}$ to be \textit{discrete-translation invariant in} $u \in \R^d$, if for any $m\in \mathbb{Z}$,
\[f(x+m u) = f(x), \qquad \text{for all } x\in \R^d.\] A subset $D\subset \R^d$ is called discrete-translation invariant in $u \in \R^d$ if its indicator function $\ind{D}$ is discrete-translation invariant in $u$ by definition above.
\begin{thm}\label{existence2} Let $(x_1,\dots,x_d)$ be a fixed orthogonal coordinate system for $\R^d$. Suppose the hypotheses of Theorem \ref{existence1} are satisfied, except that \eqref{Vbound1} is replaced with the following: for each coordinate $x_i$, at least one of the following holds:
\begin{enumerate}[label=(\roman*)] \item $D$ is bounded in $x_i$. \item $V$ is unbounded in $x_i$ of the form \begin{equation}\label{Vcoordbound} \lim_{a\to \infty}\left( \inf_{x\in (S^i_a)^c} V(x)\right) = +\infty. \end{equation} \item $D$ and $V$ are discrete-translation invariant in $s_i\hat{e}_i$ for some $s_i>0$. \end{enumerate}
Then there exists a global minimizer $\overline{\rho}\in {\mathcal{P}}^{ac}(D)$ of ${\mathcal{E}}^\nu$. \end{thm}
\begin{proof} As before, we consider a minimizing sequence $\left\{\mu_n\right\}_{n\geq 0} \subset {\mathcal{P}}^{ac}(D)$ for ${\mathcal{E}}^\nu$ over ${\mathcal{P}}(D)$, where we assume that $\left\{{\mathcal{E}}^\nu[\mu_n]\right\}_{n\geq 0}$ is bounded above by some $\tilde{M}>0$. Again, \eqref{existFree} implies that $\left\{{\mathcal{E}}^\nu[\mu_n]\right\}_{n\geq 0}$ is bounded below, and so the upper bound \eqref{Vbound3} on $\left\{{\mathcal{V}}[\mu_n]\right\}_{n\geq 0}$ still holds. As in \eqref{rhozero}, an absolutely continuous measure $\mu\in {\mathcal{P}}^{ac}(D)$ may be extended by zero to a probability measure on $\R^d$, so with some abuse of notation we will refer to $\mu\in {\mathcal{P}}^{ac}(\R^d)$ as a probability measure on $D$ whenever $\mu(D^c) = 0$.
Since we can no longer extract tightness just from $V$, we will instead exploit the fact that the interaction energy ${\mathcal{K}}$ is bounded and use Lemma \ref{Ktight} to conclude that $\left\{\mu_n\right\}_{n\geq 0}$ is tight-up-to-translations in ${\mathcal{P}}(\R^d)$. Then, with each coordinate $x_i$ satisfying at least (i), (ii) or (iii), we will show that a translated sequence $\left\{\tilde{\mu}_n\right\}_{n\geq 0}$ exists that lies in ${\mathcal{P}}(D)$ and remains minimizing.
To see that the interaction portion of the energy is bounded, we reuse some arguments from \cite{carrillo2018existence}. Namely, the logarithmic HLS inequality (Lemma \ref{lemma:HLS}) together with \eqref{existFree} imply that for each $\mu\in {\mathcal{P}}^{ac}(D)$ with $d\mu(x) = \rho(x)\,dx$,
\begin{align*}
\nu{\mathcal{S}}[\mu] &\geq -\nu d\int_{\R^d}\int_{\R^d}\log(|x-y|)\rho(x)\rho(y)\,dx\,dy-\nu dC_0\\ &\geq -\frac{1}{2(1+\delta)} \int_{\R^d}\int_{\R^d}K(x-y)\,d\mu(x)\,d\mu(y)-\nu dC_0-\frac{1}{2(1+\delta)}C_K\\ &= - \frac{1}{1+\delta}{\mathcal{K}}[\mu]-\tilde{C} \end{align*}
for $\tilde{C} = \nu dC_0+\frac{1}{2(1+\delta)}C_K$. By the positivity of $V$, for each $\mu_n$ we have
\[\frac{\delta}{1+\delta}{\mathcal{K}}[\mu_n]\ \ \leq\ \ {\mathcal{K}}[\mu_n]+\nu{\mathcal{S}}[\mu_n]+{\mathcal{V}}[\mu_n] + \tilde{C}\ \ =\ \ {\mathcal{E}}^\nu[\mu_n]+\tilde{C}\ <\ \tilde{M}+\tilde{C},\]
hence $\left\{{\mathcal{K}}[\mu_n]\right\}_{n\geq 0}$ is bounded above. By Lemma \ref{Ktight} we now have that $\left\{\mu_n\right\}_{n\geq 0}$ is tight up to translations in free space.
We now construct a tight, translated version of $\left\{\mu_n\right\}_{n\geq 0}$ that retains the property $\mu_n(D)=1$ and remains energy minimizing. To do so we address each coordinate $x_i$ and consider the three cases above. Let $\epsilon>0$ be given. \\
\noindent (i) For each $x_i$ in which $D$ is bounded, let $L_i = \displaystyle\sup_{x\in D}|x_i|$ and note that for each $n$
\[\mu_n(S^i_{L_i}) = 1 > 1-\epsilon.\]
\noindent (ii) Similarly, for each $x_i$ in which $V$ satisfies \eqref{Vcoordbound}, there exists $L_i>0$ such that
\[\mu_n(S^i_{L_i})>1-\epsilon\]
uniformly in $n$ by a similar argument as in Theorem \ref{existence1}. Indeed, since $\left\{{\mathcal{V}}[\mu_n]\right\}_{n\geq 0}$ is bounded above by some $M>0$, let $L$ be large enough that $M/L<\epsilon$. Then there exists $L_i>0$ such that
\[\inf_{x\in \bigl(S^i_{L_i}\bigr)^c}V(x)>L,\] hence
\[\int_{\bigl(S^i_{L_i}\bigr)^c\cap D}\,d\mu_n(x)\leq \frac{1}{L}\int_{\bigl(S^i_{L_i}\bigr)^c\cap D}V(x)\,d\mu_n(x)\leq \frac{1}{L}\int_DV(x)\,d\mu_n(x)< \epsilon.\]
\noindent (iii) Now consider the index set $I$ of coordinates $x_i$ for which $D$ and $V$ are discrete translation invariant in $s_i\hat{e}_i$ for some $s_i>0$. First we note that if $D$ is discrete translation invariant, then so are ${\mathcal{K}}$ and ${\mathcal{S}}$ by a change of variables. If $V$ is also discrete translation invariant, then so is ${\mathcal{V}}$, hence for each $i\in I$, the energy ${\mathcal{E}}^\nu$ is discrete translation invariant in $s_i\hat{e}_i$.
Let $\left\{\mu^1_n\right\}_{n\geq 0} = \left\{\mu_n(x-x^{n,1})\right\}_{n\geq 0} \subset {\mathcal{P}}^{ac}(\R^d)$ be a translated sequence which is tight but may no longer satisfy $\mu^1_n(D)=1$. Without loss of generality we have $x^{n,1}_i = 0$ for $i\notin I$ using the arguments above for (i) and (ii), so translations have only occurred in coordinates $x_i$ for $i\in I$.
Now define another translated sequence $\left\{\mu^2_n\right\}_{n\geq 0}$ by
\[\mu^2_n := \mu^1_n(x+\tilde{x}^{n,1}) = \mu_n(x-x^{n,2})\]
where the translations are defined by
\[x^{n,2} := x^{n,1} - \sum_{\substack{i=1\\ i\in I}}^d\text{mod}\left(x^{n,1}_i,s_i\right)\hat{e}_i := x^{n,1}-\tilde{x}^{n,1}\]
where
\[\text{mod}\left(x^{n,1}_i,s_i\right) := x^{n,1}_i - \Big\lfloor\frac{x^{n,1}_i}{s_i}\Big\rfloor s_i.\]
From this we get for each $i\in I$ that
\[x^{n,2}_i = \Big\lfloor\frac{x^{n,1}_i}{s_i}\Big\rfloor s_i = m^n_i s_i, \qquad \text{for some } m^n_i\in \mathbb{Z}.\]
Hence by discrete translation invariance,
\[\mu^2_n(D) = \mu_n\left(D - x^{n,2}\right) = \mu_n\left(D - \sum_{\substack{i=1\\ i\in I}}^dm^n_i s_i\hat{e}_i\right) = \mu_n(D)=1,\]
and so $\left\{\mu^2_n\right\}_{n\geq 0}$ lies in ${\mathcal{P}}(D)$. Similarly,
\[{\mathcal{E}}^\nu[\mu^2_n] = {\mathcal{E}}^\nu\left[\mu_n\left(x - \sum_{\substack{i=1\\ i\in I}}^dm_i s_i\hat{e}_i\right)\right] = {\mathcal{E}}^\nu[\mu_n],\]
thus $\left\{\mu^2_n\right\}_{n\geq 0}$ retains the minimizing property of the original sequence $\left\{\mu_n\right\}_{n\geq 0}$. To see that $\left\{\mu^2_n\right\}_{n\geq 0}$ is tight, we can use the fact that $\left\{\mu^1_n\right\}_{n\geq 0}$ is tight to find a compact set $K^1_\epsilon\subset \R^d$ for which $\mu^1_n(K^1_\epsilon)>1-\epsilon$ for each $n$. Since
\[\left\vert x^{n,2} - x^{n,1}\right\vert \leq \sqrt{d}\max_{i\in I}s_i,\]
the compact set
\[K^2_\epsilon = \left\{x\in D \,:\, \dist{x}{K^1_\epsilon}\leq \sqrt{d}\max_{i\in I}s_i\right\}\]
satisfies $\mu_n^2(K^2_\epsilon)>1-\epsilon$ for each $n$. We may now apply Prokhorov's theorem and lower semicontinuity of the energy to extract a convergent subsequence $\left\{\mu^2_{n_k}\right\}_{k\geq 0}$ such that $\mu^2_{n_k}\overset{*}{\rightharpoonup} \overline{\rho} \in {\mathcal{P}}(D)$ and ${\mathcal{E}}^\nu[\overline{\rho}] = \inf{\mathcal{E}}^\nu>-\infty$. Finally, as above, Remark \ref{ac-gmin} implies that $\overline{\rho}\in {\mathcal{P}}^{ac}(D)$, which completes the proof. \end{proof}
\begin{rmrk} \normalfont Theorem \ref{existence2}, although more technical, is designed to capture many practical cases. As it reads, one such case is that of the half-space domain $D = \mathbb{R}^{d-1}\times [0, \infty)$ together with a potential $V$ of the form
\[V(x) = V(x_n) \leq Cx_n^p,\]
for $p>0$, which only depends on the final coordinate $x_n$. This is the case commonly considered when modeling a swarm in a gravitational field. Another case is that of an infinite channel $D = B_R^{d-1}\times\mathbb{R}$ where $B_R^{d-1}$ is a $(d-1)$ dimensional ball of radius $R$. Since the infinite channel is either bounded or translation invariant in each coordinate, a global minimizer exists for $V=0$, consistent with Theorem \ref{tightbound}. \end{rmrk}
\section{Numerical Computation of Critical Points} \label{sect:numerics}
\hspace{5mm} We compute critical points of ${\mathcal{E}}^\nu$ under purely attractive power-law potentials and attractive-repulsive potentials on an interval $D= [0,L]$, using an iterative method to find fixed points of \eqref{gibbstat}. In all cases, we check that the Euler-Lagrange equation \eqref{ELcond} is satisfied to within the error tolerance of the iterative method. Due to the exponential decay of critical points it can be assumed that for sufficiently large $L$, critical points computed on the interval $D = [0,L]$ are good approximations of critical points on $[0,\infty)$ (when the latter exist). In light of Theorem \ref{domassymthm}, which implies that for $V=0$ no critical points exist on the half-line, computations with $V=0$ should be interpreted as approximations to critical points in free space ($D=\mathbb{R}$), while computations made with $V\neq 0$ should be interpreted as approximations to critical points on the half line $D = [0,\infty)$.
\subsection{Numerical Method} \paragraph{Fixed-Point Iterator.} The following scheme computes critical points of ${\mathcal{E}}^\nu$ by discretizing the map ${T}:{\mathcal{P}}(D)\to{\mathcal{P}}^{ac}(D)$ given in \eqref{gibbstat}. Recall that
fixed points of ${T}$ are critical points of ${\mathcal{E}}^\nu$ (in particular, the set of fixed points of ${T}$ are exactly the critical points of ${\mathcal{E}}^\nu$ which are absolutely continuous and supported on the whole domain). We use the iterative scheme
\begin{equation}\label{fxptscheme} \rho^{n+1} = (1-{\tau}_n)\rho^n+{\tau}_n{T}(\rho^n), \end{equation}
where
\begin{equation} {\tau}_n =\begin{cases} 1, &\text{if }{\mathcal{E}}^\nu\left[{T}(\rho^n)\right]<{\mathcal{E}}^\nu[\rho^n],\\[2pt] {\tau}_c,&\text{otherwise},\end{cases} \end{equation}
with inputs ${\tau}_c \in (0,1)$ and $\rho^0\in {\mathcal{P}}^{ac}(D)$.
In words, each iteration produces an absolutely continuous probability measure $\rho^{n+1}$ that is a convex combination of the previous iterate $\rho^n$ and its image under ${T}$, unless the energy of ${T}(\rho^n)$ is lower than that of $\rho^n$, in which case $\rho^{n+1} = {T}(\rho^n)$. Each step requires computation of the integral terms in ${T}(\rho^n)$ and ${\mathcal{E}}^\nu[\rho^n]$, which for $D=[0,L]$ is done by discretizing the interval into $N$ quadrature nodes and numerically integrating. For uniform grids, we use MATLAB's \texttt{conv} function to compute $K*\rho^n$, while for non-uniform grids we use trapezoidal integration. The scheme is terminated when
\begin{equation}\label{fxptconv} \nrm{\rho^n-{T}(\rho^n)}_{L^1(D)} < \txt{0.05}{tol} \qquad \text{or} \qquad n>N_{\max}, \end{equation}
where tol and $N_{\max}$ are specified by the user. In what follows, we denote by $\gmin_{_{FP}}$ the numerical solution produced by the fixed-point iterator upon convergence.
\paragraph{Stability Constraints.} The scheme \eqref{fxptscheme} has many benefits. It is explicit, so only numerical integration is required at each step, which allows for flexibility of the spatial grid. It is also positivity preserving. Due to the explicit nature, however, there are a few stability constraints.
{\em Oscillations.} The first stability constraint prevents spurious oscillations and can be explained by casting the scheme as a discretization of the following integro-differential equation: assuming ${\tau}_n \ll 1$, \eqref{fxptscheme} can be viewed as a forward-Euler discretization of\\[-10pt]
\begin{equation}\label{diffmap} \begin{dcases} \ppt \rho(x,t) = {T}(\rho(x,t))-\rho(x,t), & (x,t)\in D\times (0,\infty),\\ \hspace{3.75mm}\rho(x,0) = \rho_0(x) \in {\mathcal{P}}^{ac}(D), & x\in D,\\[1pt] \end{dcases} \end{equation}
whose steady states are exactly the fixed points of ${T}$.
For any point $x^*\in D$, the time evolution of $\rho(x^*,t)$ under \eqref{diffmap} is such that $\rho(x^*,t)$ increases when $\rho(x^*,t)<{T}(\rho(x^*,t))$ and decreases when $\rho(x^*,t)>{T}(\rho(x^*,t))$. Analytically, if $\rho_0$ lies in the basin of attraction of some fixed point $\overline{\rho}$ of $T$, we expect pointwise convergence $\lim_{t\to \infty} \rho(x^*,t)=\overline{\rho}(x^*)$. If $\rho(x^*,t)$ oscillates around $\overline{\rho}(x^*)$ as it approaches $\overline{\rho}(x^*)$, numerically one can expect spurious growth of such oscillates. Indeed, oscillations do appear in the fixed-point method \eqref{fxptscheme} for ``timesteps'' ${\tau}_c$ that are too large, in which case the iterates $\rho^n$ cycle indefinitely through a finite set of measures.
To arrive at a stable value of ${\tau}_c$ which prevents oscillations, we examine a bound on the $L^1$-Lipschitz constant $L_{{T}}$ of ${T}$ (derived in \cite{messenger2019aggregation} assuming $D$ is bounded, $K$ is bounded below, and $V$ is positive):
\begin{equation}\label{lipboundfp} L_{{T}} \leq\frac{2}{\nu} \nrm{\tilde{K}}_{L^\infty(D-D)} \exp\left(\frac{1}{\nu}\nrm{\tilde{K}}_{L^\infty(D-D)}\right), \end{equation}
where $\tilde{K}:=K - \min_{x\in D-D}\left\{K(x)\right\}$. Due to the exponential dependence on $\nrm{K}_{\infty}$, \eqref{lipboundfp} may not be a very encouraging bound, but it does suggest that ${\tau}_c$ should be proportional to $\nu$. Indeed, we see convergence of the scheme for ${\tau}_c = \mathcal{O}(\nu)$ and in all computations below set ${\tau}_c = 5\nu$. Direct dependence of ${\tau}_c$ on $\nrm{K}_{\infty}$ was not observed.
{\em Normalization and Underflow.} Another numerical issue is round-off error. Assuming for the moment that $K$ and $V$ are both positive, when $\nu$ is small the argument of the exponent in ${T}$ is negative and large in magnitude. This results in underflow of digits when calculating $Z(\rho)$ and subsequent division by a small quantity. To avoid this, we exploit the fact that the set of critical points of ${\mathcal{E}}^\nu$ is unchanged by adding a constant to $K$ and at each step normalize the argument of the exponent by adding to $K$ the factor $c_n := - \nu \log Z(\rho^{n-1})$. The potential used in simulations then changes at each iteration and is given by $K_n(x) = K_{n-1}(x)+c_n$ with $K_0 = K$. For $Z(\rho^n)$ we then have
\[Z(\rho^n) = Z(\rho^{n-1}) \int_D \exp\left(\displaystyle -\frac{K_{n-1}*\rho^n(x)+V}{\nu}\right) dx,\]
and so as $\rho^n\to \overline{\rho}$ we see that $Z(\rho^n)\to 1$. This normalization turns out to stabilize the problem, and results in the constant on the right-hand side of the Euler-Lagrange equation \eqref{ELcond} conveniently converging to zero, since the true value $\lambda$ is equal to $-\nu\log Z(\overline{\rho})$.
\paragraph{Continuation.} The scaling ${\tau}_c = \mathcal{O}(\nu)$ adopted in light of the bound \eqref{lipboundfp} implies that the number of iterations required for convergence is large for small $\nu$. To prevent this, we use continuation on $\nu$: we let $\left\{\nu_j\right\}_{j=0}^J$ be a decreasing sequence of diffusion parameters and for each $\nu_j$ we set the initial guess $\rho^0_j$ for the fixed-point iterator to the output $\gmin_{_{FP}}^{j-1}$ of the fixed-point iterator with $\nu=\nu_{j-1}$. This is very effective for reducing the total iteration count and is also seen in Figure \ref{multistate} to be crucial for revealing non-uniqueness of critical points: different sequences of diffusion parameters (sharing the same final value of $\nu$) can produce different critical points.
\paragraph{Convergence Criteria.} We are primarily concerned with satisfying the Euler-Lagrange equation in its original form,
\[\Lambda(x) := K*\overline{\rho}(x) + \nu \log(\overline{\rho}(x)) + V(x) = \lambda, \qquad \text{ for all } x\in [0, L],\]
where $\lambda = {\mathcal{E}}^\nu[\overline{\rho}]+{\mathcal{K}}[\overline{\rho}]$, and so we check that the quantity
\begin{equation}\label{laminf} \Lambda_\infty = \nrm{\Lambda - {\mathcal{E}}^\nu[\gmin_{_{FP}}] - {\mathcal{K}}[\gmin_{_{FP}}]}_\infty \end{equation}
is below the chosen error tolerance for each numerical solution $\gmin_{_{FP}}$. We also check that the boundary condition \eqref{boundaryV} derived in Theorem \ref{domassymthm} is satisfied, which reads
\[\overline{\rho}(0)-\overline{\rho}(L) = \frac{1}{\nu}\int_0^L V'(x)\overline{\rho}(x)\,dx.\]
However, in all numerical experiments we use $V(x) = gx$ and choose $L$ large enough that $\overline{\rho}(L)$ is negligible, so this reduces to
\begin{equation}\label{reducedbc} \overline{\rho}(0) = g/\nu\ , \end{equation}
which is exact for $D=[0,\infty)$. Thus, we also assess the relative error
\begin{equation}\label{bccheck} {E_0} := \frac{\left\vert \gmin_{_{FP}}(0) - g/\nu\right\vert}{g/\nu}. \end{equation}
\subsection{Purely Attractive Interaction Potential} \label{subsect:attraction} The first class of potentials we examine are purely attractive, power-law potentials
\[K_p(x) := \frac{1}{p}|x|^p,\]
for $p > 0$, where repulsive forces are present in the swarm only in the form of diffusion. Without diffusion, for all $p >0$ the global minimizer is a single $\delta$-aggregation with location determined by $V$. The effect of switching on diffusion is to smooth out the $\delta$-aggregation. Indeed, Figures \ref{kp2}--\ref{kplarge} show critical points which are continuous and unimodal, but are supported on the whole domain with fast-decaying tails.
First we examine the case $p=2$ in detail given the results in Theorem \ref{COM2}, and compare with other small values of $p$. Then we look into the limit of large $p$, which is motivated by the fact that minimizers of ${\mathcal{E}}^\nu$ are supported on the entire domain regardless of the attraction strength (see Theorem \ref{suppmin}).
\begin{rmrk} \normalfont For uniformly convex interaction potentials (i.e $K_p$ with $p\geq 2$), it can be shown using displacement convexity as in \cite{mccann1997convexity} and \cite{carrillo2003kinetic} that for $V= gx$ the global minimizer is the unique critical point of ${\mathcal{E}}^\nu$ for $D=[0,\infty)$. Numerics suggest that uniqueness holds for general power-law, purely-attractive $K$ when (i) $D = \mathbb{R}$ with $V=0$ and (ii) $D = [0,+\infty)$ with $V$ convex and satisfying condition \eqref{Vbound1}. In this way, convexity of $K$ may be relaxed if $K$ remains purely attractive. However, Figure \ref{multistate} below shows that for attractive-repulsive (non-convex) interaction potentials, minimizers are not unique. \end{rmrk}
\paragraph{Moderate Attraction Strength and Connection to Theorem \ref{COM2}.} The case $p=2$ is used to benchmark the fixed-point iterator. Theorems \ref{COM2} and \ref{existence1} together imply that for every $g>0$ and $\nu>0$ the global minimizer of ${\mathcal{E}}^\nu$ under $K_2$ and $V=gx$ is the unique critical point; moreover, we have an explicit formula for the global minimizer, up to solving equation \eqref{COMroot} (e.g. with MATLAB's \texttt{fzero} command). Figure \ref{kp2} contains computed solutions under $K_2$ for several values of $g$ along with convergence data. Agreement with the exact solution $\overline{\rho}_{_{ex}}$ and the Euler-Lagrange equation as measured by $\Lambda_\infty$ and the boundary condition ${E_0}$ are all on the order of the chosen error tolerance of $\sci{1}{6}$. We see especially good agreement with the Euler-Lagrange equation, gaining two orders of accuracy relative to the error tolerance.
\begin{figure}\label{kp2}
\end{figure}
Figure \ref{kpsmall} shows computed critical points for $p\in(1,8]$ to compare with the case $p=2$, demonstrating that increasing $p$ decreases the maximum height of the solution. We examine the cases $g=0$ and $g=\nu$, the former resulting in critical points which are symmetric about the center of the domain while the latter causes clustering near the domain boundary at $x=0$. Notice that with $g=\nu$, \eqref{reducedbc} implies that $\overline{\rho}(0) = 1$, which is clearly represented on the rightmost plot.
\begin{figure}\label{kpsmall}
\end{figure}
\paragraph{Limit of Large Attraction.} We now examine numerically the limit of large $p$, which is motivated by the fact that minimizers $\overline{\rho}$ of ${\mathcal{E}}^\nu$ satisfy $\supp{\overline{\rho}} =D$ regardless of how strong the (power-law) attraction is (see Theorem \ref{suppmin}). This is a striking feature because intuitively one might expect that for very large attraction the swarm would be confined to a compact set. Only as $p\to+\infty$, however, do we reach a state with compact support. We derive this family of compactly supported states below in one dimension and compute critical points for powers up to $p=256$ to suggest convergence to the compactly supported states included in Figure \ref{kplarge}.
The limit as $p\to \infty$ is clearly singular, as the limiting interaction potential $K_\infty$ defined by
\[\lim_{p\to \infty} K_p(x) = K_\infty(x) := \begin{cases} 0, & x\in [-1,1]\\+\infty, & x\notin [-1,1]\end{cases}\]
is no longer locally integrable. As such, the space of probability measures on which the resulting energy is finite is very limited. Despite this, we can still determine minimizers for ${\mathcal{E}}^\nu$ under $K_\infty$. It is not hard to show that the corresponding interaction energy ${\mathcal{K}}_\infty$ satisfies
\[{\mathcal{K}}_\infty[\mu] = \begin{dcases} 0, & \txt{0.1}{if}\mu\txt{0.1}{is supported on a unit interval,} \\ +\infty, & \text{otherwise,}\end{dcases}\]
and so the space we should be looking for minimizers in is
\[\left\{\mu\in {\mathcal{P}}_\infty(D) \txt{0.2}{:} \supp{\mu}\subset [a,1+a]\txt{0.2}{for some} a\in \mathbb{R}\right\}.\]
To arrive at this, for the interaction energy we have
\[{\mathcal{K}}_\infty[\mu] = \frac{1}{2}\int_D\int_DK_\infty(x-y)\,d\mu(y)\,d\mu(x) = \frac{1}{2}\int_{\supp{\mu}}
K_\infty*\mu(x) \,d\mu(x),\]
which is finite if and only if $K_\infty*\mu$ is finite $\mu$-a.e. By computing
\begin{align*} K_\infty* \mu(x) &= \int_{\supp{\mu}}K_\infty(x-y)\,d\mu(y)\\ &= \int_{\supp{\mu}\cap [x-1,x+1]^c }K_\infty(x-y)\,d\mu(y)\\ &= \begin{cases} 0, &\mu([x-1,x+1]^c)=0\\ +\infty, & \text{otherwise},\end{cases} \end{align*}
we see that ${\mathcal{K}}_\infty[\mu] = +\infty$ unless $\mu([x-1,x+1]^c)=0$ for $\mu$-a.e. $x\in D$, which is equivalent to $\mu$ having support on a unit interval. From this we deduce that a minimizer $\overline{\rho}_\infty$ has support on a unit interval and satisfies $K_\infty* \overline{\rho}_\infty(x) = 0$ for $\overline{\rho}_\infty$-a.e.\ $x$. Hence the Euler-Lagrange equation reads
\[\nu\log(\overline{\rho}_\infty(x))+V(x) = \lambda, \qquad \overline{\rho}_\infty\text{-a.e. }x\in D,\]
or, taking $\supp{\overline{\rho}_\infty} = [0,1]$,
\begin{equation}\label{pinf} \overline{\rho}_\infty = \begin{dcases} \ind{[0,1]} & \txt{0.2}{for} V=0 \\[5pt] Z^{-1} e^{-V/\nu}\ind{[0,1]} & \txt{0.2}{for} V\neq 0.\end{dcases} \end{equation}
Figure \ref{kplarge} shows critical points for $K_p$ and $V=gx$ for larger values of $p$ together with the corresponding limiting measure $\overline{\rho}_\infty$ derived above. For $g=0$, as $p$ increases we see solutions increasing to $\overline{\rho}_\infty$ inside $[0.5,1.5]$ and decreasing to zero elsewhere. For $g=\nu$ the boundary condition \eqref{reducedbc} again reduces to $\overline{\rho}(0)=1$, which is satisfied through increasingly sharp transitions as $p$ increases, and is not satisfied in the limit by $\overline{\rho}_\infty$. We still see $\Lambda_\infty$ values near the error tolerance, except for $p=256$, where the method clearly breaks down, as the scheme converges in fewer than $N_{\max}$ iterations yet $\Lambda_\infty$ is $\mathcal{O}(1)$.
\begin{figure}\label{kplarge}
\end{figure}
\subsection{Non-Uniqueness under Attractive-Repulsive Potentials} \label{subsect:att-rep} The second class of interaction potentials we consider involve attraction at large distances and repulsion at short distances. So-called attractive-repulsive potentials have been the subject of a substantial amount of research in recent years (see \cite{balague2013dimensionality,bernoff2011primer,evers2016metastable,fellner2010stable,fetecau2017swarm,fetecau2017swarming,mogilner2003mutual}) for their use in modelling biological swarms, which predominantly seem to obey the following basic rules: if two individuals are too close, increase their distance, if too far away, decrease their distance. We show here through a numerical example that for such potentials, uniqueness of critical points does not hold in general.
We examine a regularization of the potential
\[K_{QANR}(x) = \frac{1}{2}|x|^2+2\phi(x),\]
which features quadratic attraction and Newtonian repulsion given by the free-space Green's function $\phi(x) = -\frac{1}{2}|x|$ for the negative Laplacian $-\Delta$ in one dimension.
Specifically, we consider the $C^1$ regularized versions of $K_{QANR}$ in the form of the one-parameter family
\begin{equation}
K_\epsilon(x) := \frac{1}{2}x^2 + 2\phi_\epsilon(x):= \frac{1}{2}x^2 + \begin{dcases} -|x|, & \qquad |x|>\epsilon, \\ -\frac{\epsilon}{2}- \frac{1}{2\epsilon}x^2, &\qquad |x|\leq \epsilon,\end{dcases} \end{equation}
for $\epsilon\in (0, 1]$.
One might expect that for each $\epsilon$, switching on diffusion selects a unique number of aggregates in all minimizing states. Similar results have been documented: Evers and Kolokolnikov establish in \cite{evers2016metastable} that adding any level of diffusion to an equilibrium consisting of two aggregates of unequal mass for the plain aggregation model under the double-well potential $K(x) = -\frac{1}{2}x^2+\frac{1}{4}x^4$, causes the state to become metastable, where mass is transferred between the two aggregates until their masses equilibrate, which only happens in infinite time. As evidenced by the numerical example in Figure \ref{multistate}, where a four-aggregate and a five-aggregate state both exist as critical points for the same $\epsilon$ and $\nu$ values, it seems that diffusion does not guarantee a unique number of aggregates. It is clear that the four-aggregate state is preferred, as it has lower energy and requires fewer iterations of the fixed-point iterator.
Using continuation on the diffusion parameter, as in Figure \ref{multistate}, suggests a method for computing the globally-minimizing configuration for each $\epsilon$. Both the four-aggregate and five-aggregate state are computed with final diffusion $\nu = 2^{-13}$, but the four-aggregate state is reached using continuation from initial diffusion $\nu_0 = 10\nu$, whereas the five-aggregate state uses $\nu_0= 2\nu$. The more energy-favourable state is reached from a larger starting $\nu_0$, which suggests that continuation from larger diffusion might be a mechanism for extracting the global minimizer. Ice crystallization provides a physical analogy: more imperfections form in ice crystals when water is frozen abruptly, indicating a non-energy-minimizing configuration, than when water is frozen slowly (see for instance \cite{kono2017effects}).
\begin{figure}\label{multistate}
\end{figure}
{\large \bf Acknowledgements } D.M. would like to thank his co-supervisor at Simon Fraser University, Ralf Wittenberg, for invaluable guidance during the writing of his Master's thesis. Ralf's thorough review of D.M.'s thesis directly led to a much better exposition of the material presented in this paper. The authors would also like to acknowledge Wittenberg for his insight during conversations related to the present article, and in particular, suggestions which led to the idea of the effective volume dimension for connecting diffusion-dominated spreading to domain geometry. R.F. was supported by NSERC Discovery Grant PIN-341834 during this research.
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Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach
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November 2013, 33(11&12): 5189-5202. doi: 10.3934/dcds.2013.33.5189
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
Irena Lasiecka 1, and Mathias Wilke 2,
Department of Mathematics, University of Virginia, Charlottesville, VA 22903
Institut für Mathematik, Martin-Luther Universität Halle-Wittenberg, 06099 Halle
Received November 2011 Revised October 2012 Published May 2013
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
Keywords: existence and uniqueness of strong solutions, maximal regularity, Quasilinear thermoelastic plates, exponential decay..
Mathematics Subject Classification: Primary: 74F05; Secondary: 35B30, 35B40, 74H4.
Citation: Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189
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Irena Lasiecka Mathias Wilke | CommonCrawl |
Let $\omega$ be a complex number such that $\omega^3 = 1.$ Find all possible values of
\[\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}.\]Enter all the possible values, separated by commas.
We can write
\begin{align*}
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} &= \frac{1 + \omega^2 + 1 + \omega}{(1 + \omega)(1 + \omega^2)} \\
&= \frac{2 + \omega + \omega^2}{1 + \omega + \omega^2 + \omega^3} \\
&= \frac{2 + \omega + \omega^2}{2 + \omega + \omega^2} \\
&= \boxed{1}.
\end{align*} | Math Dataset |
What is the meaning of Einstein's field equation in terms of source and its effects on curvature?
The Einstein's Field Equation is $$R_{\mu\nu}-(1/2)g_{\mu\nu}R=-8\pi T_{\mu\nu},$$ where the left hand side is the curvature term and the right hand side is the source term (see, Hartle). Now, in the case of Schwarzschild Exterior Solution (solution for empty space-time outside of a spherical body) we are taking the energy-momentum tensor $T_{\mu\nu}$ to be zero as there is no mass or energy outside the spherical body; and we are writing the field equation as $$R_{\mu\nu}=0.$$ My confusion is that what is the source of curvature here? Undoubtedly, here the source of curvature is the spherical body which have non-zero $T_{\mu\nu}$. So, does not the energy-momentum tensor refer to the source of curvature? What does it refer to then? Please clarify the meaning of Einstein's Field Equation in terms of source and its effects on space-time curvature.
general-relativity curvature vacuum stress-energy-momentum-tensor
VacuuMVacuuM
$\begingroup$ The source of the curvature would be the spherically symmetric central body. If we are outside of the body, then there is no matter source, but that doesn't mean that there is no matter anywhere. $\endgroup$
$\begingroup$ Side note, with $T_{\mu\nu}=0$, you get $R_{\mu\nu}=\frac{1}{2}Rg_{\mu\nu}$, which only equals zero iff $R=R^{\mu}_{\,\mu}=0$ $\endgroup$
$\begingroup$ Sorry, but you did not get me. I haven't said that there is no matter anywhere; rather I said there in no matter outside the spherical body. And if the spherical body is the source of space-time curvature then why we don't take the energy-momentum tensor of that body in the field equation? I may have misconception about what is meant by $T_{\mu\nu}$ in field equation? please clarify. @Jerry $\endgroup$
– VacuuM
$\begingroup$ Sorry, but the condition $T_{\mu\nu}=0$ automatically, implies that $R=R_{\mu}^{\mu}=0$, that's why we can write, $R_{\mu\nu}=0$ @Jimdalf $\endgroup$
$\begingroup$ Remember that curvature has to be continuous. Spacetime is supposed to be a manifold. So when you get outside the body, the source of curvature in a given region is the curvature on the boundaries of that region. Infinitely far away from the body, spacetime is flat an Minkowski. So each point in between has to continuously adapt the curvature from what it is inside the body to what it is at infinity $\endgroup$
The fact that the energy-momentum tensor is called the source of curvature doesn't mean that there can't be any curvature where there is no energy-momentum. In fact, even if $T_{\mu\nu}=0$ across all spacetime, there are still nontrivial solutions of Einstein's equations, in the form of gravitational waves.
You should remember that $T_{\mu\nu}$ is a function of $x$, and can be nonzero inside a body but zero outside. Suppose we have a spherically symmetric star. Inside of it, the energy momentum tensor will be a very complicated object, with all sorts of pressures and flows and whatever. But outside of it, $T_{\mu\nu}$ will be simply zero. So we have divided space into two regions:
$$ \begin{cases} R_{\mu\nu}-\frac12 g_{\mu\nu} R = T_{\mu\nu} & \text{Inside} \\ R_{\mu\nu} = 0 & \text{Outside} \end{cases} $$
When we solve the outside equation assuming spherical symmetry and a static solution, we get Schwarzchild's solution, and we find that we don't need to know the metric inside the star in detail. All we need is a single number, $M$, that tells us everything we need to konw about the curvature outside of the star.
The Schwarzchild metric satisfies $R_{\mu\nu}=0$, because outside of the star (or black hole or whatever), there are no sources. But when we solve an differential equation, we need boundary conditions. And the boundary conditions at the star's surface (or at $r=0$ for a black hole) are the energy-momentum tensor's way of telling us that there is matter somewhere, even if the region where we solved the equation is empty.
JavierJavier
$\begingroup$ So, is it like that both $R_{\mu\nu}-(1/2)g_{\mu\nu}R$ and $T_{\mu\nu}$ are functions of $(x,t)$ and we have to consider what is the value of $R_{\mu\nu}-(1/2)g_{\mu\nu}R$ and $T_{\mu\nu}$ at the same event and then equate them. @Javier Badia $\endgroup$
$\begingroup$ @VikramadityaMondal: Yes, that's exactly it. Everything that we're considering here is a function of $x^\mu$, and Einstein's equations are partial differential equations beteen tensors that are functions of space and time. $\endgroup$
– Javier
We first note that the vanishing of the Ricci tensor does not imply the vanishing of the Riemann tensor. Thus the vacuum equations, $R_{\mu\nu}=0$, do not imply that spacetime is flat. The vacuum equations tell us that certain linear combination of components of the Riemann tensor vanishes.
When solving differential equations, one usually has to worry about the boundary conditions. The EFEs are just differential equations of the metric, to bo be solved for the components $g_{\mu\nu}$. Let $D$ be a region of spacetime containing matter, i.e. $T|_D\ne 0$. General relativity seeks to solve the problem
$$ \begin{cases} G_{\mu\nu}(x) = 8 \pi T_{\mu\nu}(x) & x\in D \\ R_{\mu\nu}(x) = 0 & x\notin D \end{cases} $$ We have two solutions: $$ \begin{cases} \tilde{g}_{\mu\nu}(x) & x\in D \\ \bar{g}_{\mu\nu}(x) & x\notin D \end{cases} $$ The relevant boundary conditions are $$\tilde{g}_{\mu\nu}(p)=\bar{g}_{\mu\nu}(p),\quad\forall p\in \partial D.$$ Furthermore, we require that this be a smooth transition. The metric of spacetime is thus $$g_{\mu\nu}=\chi_D \tilde{g}_{\mu\nu}+(1-\chi_D)\bar{g}_{\mu\nu},$$ where $\chi_D$ is the characteristic function of $D$.
Let us consider a concrete example. We shall look at a spherically symmetric static star$^{1}$. By the Birkhoff theorem$^2$, we already know that the exterior solution is the Schwarzschild solution. We take the star to have a perfect fluid energy-momentum tensor in the interior. In the interior, the metric will still have spherical symmetry because of the properties of perfect fluids. Thus we must find $A$ and $B$ in $$\mathrm{d}s^2=-B(r)\mathrm{d}t^2+A(r)\mathrm{d}r^2+r^2\mathrm{d}\Omega^2.$$
The solution for $A(r)$ is $$A(r)=\left[1-\frac{2G\mathcal{M}(r)}{r}\right]^{-1},\quad\mathcal{M}(r)=\int^r \rho \,\mathrm{d}V=\int_0^r 4\pi r'^2\rho(r')\,\mathrm{d}r.$$ The solution for $B(r)$ is $$B(r)=\exp\left\{-\int_r^\infty \frac{2G}{r'^2}[\mathcal{M}(r')+4\pi r'^3 P(r')]A(r')\,\mathrm{d}r'\right\}.$$ In these equations $\rho$ is the density of the star and $P$ its pressure.
Clearly, the support$^3$ of the density and pressure is contained within the star. If the star has radius $R$, then $\mathcal{M}(R)=M$ is the total mass. Let us analyze our solution at $r=R$. For $A(r)$ we get just the standard Schwarzschild metric component. A simple exercise in calculus is to check that $$B(R)=A^{-1}(R)$$ and that $B(r)=A^{-1}(r)$ for $r\ge R$. We thus see the above general situation: we have an interior solution and an exterior vacuum solution, which are equal at the boundary of the source's support.
$^1$ See [1] section 11.1 or [2] section 6.2 for the full calculation.
$^2$ Proved in [1] on page 337.
$^3$ A quantity $f$ is said to have support in $X$ if $f(x)=0$ for all $x\notin X$.
[1] S. Weinberg, Gravitation and Cosmology (1972).
[2] R. M. Wald, General Relativity (1984).
Ryan UngerRyan Unger
The equations $R_{\mu\nu} = 0$ alone don't define a well-posed problem since you need to add boundary conditions. Moreover the curvature of a manifold is measured by the Riemann tensor and $R_{\mu\nu}=0$ doesn't imply ${R^{\mu}}_{\nu\sigma\rho} = 0$.
Phoenix87Phoenix87
It's helpful to look at an analogous situation in Classical Electrodynamics where the same issues come up.
In General Relativity, the source term is the stress-energy tensor. In Classical Electrodynamics the source terms are the charge and the current. When there are no sources, one possible vacuum solution is a constant uniform nonzero electric field. Another possible vacuum solution is a constant uniform zero electric field. We can piece together vacuum solution and sew them together, and this will no longer be a vacuum solution. But for instance if you take a field uniform in the $\hat{x}$ direction $\vec{E}=E\hat{x}$ from $x=0$ to $x=1m$, and piece it together with $\vec{E}=\vec{0}$ for $x<0$ and $\vec{E}=\vec{0}$ for $x> 1m$, then this can still be a solution to Maxwell, with a source term. In this case you need an infinite sheet of surface charge in the $x=0$ plane, and an equal and opposite surface charge on an infinite sheet in the $x=1m$ plane.
So let's try this with General Relativity, you can take two Schwarzschild solutions for different mass parameters, $M$ and $m$, with $M$>$m$. As embeddings they might look like funnels. If you a surface of constant areal coordinate, you get a thin spherical shell, pick a surface with surface area $4\pi R^2$. For the solution with $M$ remove the inside interior to the shell. Repeat for the solution with $m$ find the thin shell with the exact same surface area $4\pi R^2$ (with the same numerical value of $R$) and this time remove the outside exterior to the shell. So the one looks like a funnel with the tip cut off, and the other looks like a tip to a funnel. Since they have the same surface area and are just spherical shells, we can sew them together. We now have a manifold that looks like a funnel with a kink.
Each vacuum solution was a source free solution, and we sewed them together in a surface where the geometry of the surface matched. But the result (just like for the electrodynamics case) is not a source free solution. In fact this is a solution where there is a thin shell of mass at the surface area $4\pi R^2$ spherical shell surface. It is OK for spacetime to be curved (just like it is OK to have a travelling wave in electromagnetism or a constant electric field). But only certain curvature is allowed, what a source term does is allow different kinds of curvature.
And this is not contrived in any way. If you repeated that trick with the two solutions you could cut out the inside of the $m$ solution and put a $\mu$ solution inside it and then have a solution with two places with source, one at surface $4\pi R_1^2$ and another at $4\pi R_2^2$. As you place shells at more and more radii you can approach the solution to a realistic nonrotating star or planet. All the source term does is allow you to have a continuous limit of this process, the process of sewing vacuum solutions together.
Think of $G_{\mu \nu}=0$ as a natural allowed way for spacetime to curve (just as vacuum waves or static fields are allowed electromagnetic fields) and that curvature is allowed to deviate from these natural allowed curvatures as long as there is a source term there to give the appropriate OK. Just like an electric field line isn't allowed to terminate unless there is a charge there to approve that message.
TimaeusTimaeus
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Checking in on an inflation forecast
I made a forecast of PCE inflation using the dynamic information equilibrium model described in this post at the beginning of the year, and so far the model is doing well — new monthly core PCE data came out this morning:
Posted by Jason Smith at 11:26 AM No comments:
A forecast validation bonanza
New NGDP numbers are out today for the US, so that means I have to check several forecasts for accuracy. I would like to lead with a model that I seem to have forgotten to update all year: dynamic equilibrium for the ratio of nominal output to total employed (i.e. nominal output per employed, as I write N/L):
This one is particularly good because the forecast was made near what the model saw as a turnaround point in N/L (similar to the case of Bitcoin below, also forecast near a turnaround point) saying we should expect a return towards the trend growth rate of N/L of 3.8% per annum. This return appears to be on track.
The forecast of NGDP using the information equilibrium (IE) monetary model (i.e. a single factor of production where money — in this case physical currency — is that factor of production) is also "on track":
The interesting part of this forecast is that the log-linear models are basically rejected.
In addition to NGDP, quarterly [core] PCE inflation was updated today. The NY Fed DSGE model forecast (as well as FOMC forecast) was for this data, and it's starting to do worse compared to the IE monetary model (now updated with monthly core PCE number as well):
I've also checked my forecast for the Bitcoin exchange rate using the dynamic equilibrium model (which needs to be checked often because of how fast it evolves — it's time scale is -2.6/y so it should fall by about 1/2 over a quarter). It is also going well:
Update + 2 hours
Also, the S&P 500:
Posted by Jason Smith at 12:00 PM No comments:
Different unemployment rates do not contain different information
I saw this tweet today, and it just kind of frustrated me as a researcher. Why do we need yet another measure of unemployment? These measures all capture exactly the same information.
I mentioned this before, however I thought I'd be much more thorough this time. I used the dynamic equilibrium model on the U3 (i.e. "headline"), U4, U5, and U6 unemployment rates and looked a the parameters.
Here are the model fits along with an indicator of where the different centers of the shocks appear as well as a set of lines showing the different dynamic equilibrium slopes (which are -0.085/y, -0.084/y, -0.082/y, and -0.078/y, respectively):
Within the error of estimating these parameters, they are all the same. How about shock magnitudes? Again, within the error of estimating the shock magnitude parameters, they are all the same:
Basically, you need one model of one rate plus 3 scale factors to describe the data. There appears to be no additional information in the U4, U5, or U6 rates. Actually in information theory this is explicit. Let's call the U3 rate random variable U3 (and likewise for the others). Now U6 = f(U3), therefore:
H(U6) ≤ H(U3)
Update 13 October 2017
The St. Louis Fed just put out a tweet that contains another alternative unemployment measure. It also does not contain any additional information:
Was the Phillips curve due to women entering the workforce?
Janet Yellen in her Fed briefing from last week said "Our understanding of the forces driving inflation is imperfect." At least one aspect that's proven particularly puzzling is the relationship between inflation and unemployment: the Phillips curve. In an IMF working paper from November of 2015, Blanchard, Cerutti, and Summers show the gradual fall in the slope of the Phillips curve from the 1960s to the present. I discussed it in January of 2016 in a post here. A figure is reproduced below:
Since that time, I've been investigating the dynamic equilibrium model and one thing that I noticed is that there appears to be a Phillips curve-like anti-correlation signal if you look at PCE inflation data and unemployment data:
See here for more about that graph. It was also consistent with a "fading" Phillips curve. While I was thinking about the unemployment model today, I realized that the Phillips curve might be directly connected with women entering the workforce and the impact it had on inflation via the employment population ratio. I put the fading Phillips curve on the dynamic equilibrium view of the employment population ratio for women:
We see the stronger Phillips curve signal in the second graph above (now marked with asterisks in this graph) follows the "non-equilibrium" shock of women entering the workforce. After that non-equilibrium shock fades, the employment population ratio for women starts to become highly correlated with the ratio for men — showing almost identical recession shocks.
This suggests that the Phillips curve is not just due to inflation resulting from increasing employment, but rather inflation resulting from new people entering the labor force. The Phillips curve disappears when we reach an employment-population ratio equilibrium. This would explain falling inflation since the 1990s as the employment-population ratio has been stable or falling.
Now I don't necessarily want to say the mechanism is the old "wage-price spiral" — or at least the old version of it. What if the reason is sexism? Let me explain.
A recent study showed that men self-cite more often that women in academic journals, but the interesting aspect for me was that this appears to increase right around the time of women entering the workforce:
What if the wage-price spirals of the strong Phillips curve era were due to men trying (and succeeding) to negotiate even higher salaries than women (who were now more frequently in similar jobs)? As the labor market tightens during the recovery from a recession, managers who gave a woman in the office a raise might then turn around and give a man an even larger raise. The effect of women in the workforce would be to amplify what might be an otherwise undetectable Phillips curve effect into a strong signal in the 1960s, 70s and 80s. While sexism hasn't gone away, this effect may be attenuated today from its height in that period. This "business cycle" component of inflation happens on top of an overall surge in inflation due to an increasing employment population ratio (see also Steve Randy Waldman on the demographic explanation of 1970s inflation).
Whether sexism is really the explanation, the connection betweem women entering the workforce and the Phillips curve is intriguing. It would also mean that the fading of the Phillips curve might be a more permanent feature of the economy until some other demographic phenomenon occurs.
Mutual information and information equilibrium
Natalie Wolchover has a nice article on the information bottleneck and how it might relate to why deep learning works. Originally described in this 2000 paper by Tishby et al, the information bottleneck was a new variational principle that optimized a functional of mutual information as a Lagrange multiplier problem (I'll use their notation):
\mathcal{L}[p(\tilde{x} | x)] = I(\tilde{X} ; X) - \beta I(\tilde{X} ; Y)
The interpretation is that we pass the information the random variable $X$ has about $Y$ (i.e. the mutual information $I(X ; Y)$) through the "bottleneck" $\tilde{X}$.
Now I've been interested in the connection between information equilibrium, economics, and machine learning (e.g. GAN's real data, generated data, and discriminator have a formal similarity to information equilibrium's information source, information destination, and detector — the latter I use as a possible way of understanding of demand, supply, and price on this blog). I'm always on the lookout for connections to information equilibrium. This is a work in progress, but I first thought it might be valuable to understand information equilibrium in terms of mutual information.
The best way to illustrate this is with a Venn diagram:
If we have two random variables $X$ and $Y$, then information equilibrium is the condition that:
H(X) = H(Y)
Without loss of generality, we can identify $X$ as the information source (effectively a sign convention) and say in general:
H(X) \geq H(Y)
We can say mutual information is maximized when $Y = f(X)$. The diagram above represents a "noisy" case where either noise (or another random variable) contributes to $H(Y)$ (i.e. $Y = f(X) + n$). Mutual information cannot be greater than the information in $X$ or $Y$. And if we assert a simple case of information equilibrium (with information transfer index $k = 1$), e.g.:
p_{xy} = p_{x}\delta_{xy} = p_{y}\delta_{xy}
\begin{align}
I(X ; Y) & = \sum_{x} \sum_{y} p_{xy} \log \frac{p_{xy}}{p_{x}p_{y}}\\
& = \sum_{x} \sum_{y} p_{x} \delta_{xy} \log \frac{p_{x} \delta_{xy} }{p_{x}p_{y}}\\
& = \sum_{x} \sum_{y} p_{x} \delta_{xy} \log \frac{\delta_{xy} }{p_{y}}\\
& = \sum_{x} p_{x} \log \frac{1}{p_{x}}\\
& = -\sum_{x} p_{x} \log p_{x}\\
& = H(X)
Note that in the above, the information transfer index accounts for the "mismatch" in dimensionality in the Kronecker delta (i.e. a die roll that determines the outcome of a coin flip such that a roll of 1, 2, or 3 yields heads and 4, 5, or 6 yields tails).
Basically, information equilibrium is the case where $H(X)$ and $H(Y)$ overlap, $Y = f(X)$, and mutual information is maximized.
My introductory chapter on economics
I was reading the new CORE economics textbook. It starts off with some bold type stating "capitalism revolutionized the way we live", effectively defining economics as the study of capitalism as well as "other economic systems". The first date in the first bullet point is the 1700s (the first sentence mentions an Islamic scholar of the 1300s discussing India). This started me thinking: how would I start an economics textbook based on the information-theoretic approach I've been working on?
Well, this was the result:
An envelope-tablet (bulla) and tokens ca. 4000-3000 BCE from the Louvre.
© Marie-Lan Nguyen / Wikimedia Commons
Commerce and information
While she was studying the uses of clay before the development of pottery in Mesopotamian culture in the early 1970s, Denise Schmandt-Besserat kept encountering small dried clay objects in various shapes and sizes. They were labelled with names like "enigmatic objects" at the time because there was no consensus theory of what they were. Schmandt-Besserat first cataloged them as "geometric objects" because they resembled cones and disks, until ones resembling animals and tools began to emerge. Realizing they might have symbolic purpose, she started calling them tokens. That is what they are called today.
The tokens appear in the archaeological record as far back as 8000 BCE, and there is evidence they were fired which would make them some of the earliest fired ceramics known. They appear all over Iran, Iraq, Syria, Turkey, and Israel. Most of this was already evident, but unexplained, when Schmandt-Besserat began her work. Awareness of the existence of tokens, in fact, went back almost all the way to the beginning of archaeology in the nineteenth century.
Tokens were found inside one particular "envelope-tablet" (hollow cylinders or balls of clay — called bullae) found in the 1920s at a site near ancient Babylon. It had a cuneiform inscription on the outside that read: "Counters representing small cattle: 21 ewes that lamb, 6 female lambs... " and so on until 49 animals were described. The bulla turned out to contain 49 tokens.
In a 1966 paper Pierre Amiet suggested the tokens represented specific commodities, citing this discovery of 49 tokens and the speculation that the objects were part of an accounting system or other record-keeping. Similar systems are used to this day. For example, in parts of Iraq pebbles are used as counters to keep track of sheep.
But because this was the only such envelope-tablet known, it seemed a stretch to reconstruct a entire system of token-counting based on one single piece of evidence. But, as Schmandt-Besserat later noted, the existence of many tokens having the same shape but in different sizes is suggestive that they belonged to an accounting system of some sort. With no further evidence however, this theory remained just one possible explanation for the function of the older tokens that pre-dated writing.
In 2013, fresh evidence emerged from bullae dated to ca. 3300 BCE. Through use of CT scanning and 3D modelling to see inside unbroken clay balls, researchers discovered that the bullae contained a variety of geometric shapes consistent with Schmandt-Besserat's tokens.
CT scan of Choga Mish bulla and Denise Schmandt-Besserat
While these artifacts were of great interest in Schmandt-Besserat's hypothesis about the origin of writing, they fundamentally represent economic archaeological artifacts.
Could a system function with just a single type of token? Cowrie shells seemed to provide a similar accounting function around the Pacific and Indian oceans (because of this, the Latin name of the specific species is Monetaria moneta, "the money cowrie"). The distinctive cowrie shape was even cast in copper and bronze in China as early as 700 BCE, making it an early form of metal coinage. The earliest known metal coins along the Mediterranean come from Lydia from before 500 BCE.
Bronze cowrie shells from the Shang dynasty (1600-1100 BCE)
The 2013 study of the Mesopotamian bullae was touted in the press as the "very first data storage system", and (given the probability distributions of finding various tokens) a bulla containing a particular set of tokens represents a specific amount of information by Claude Shannon's definition in his 1948 paper establishing information theory.
In this light, commerce can be seen as an information processing system whose emergence is deeply entwined with the emergence of civilization. It is also deeply entwined with modern mathematics.
Fibonacci today is most associated with the Fibonacci sequence of integers. His Liber Abaci ("Book of Calculation", 1202) introduced Europe to Hindu-Arabic numerals in its first section, and in the second section illustrated the usefulness of these numerals (instead of the Roman numerals used at the time) to businessmen in Pisa with examples of calculations involving currency, profit, and interest. In fact, Fibonacci's work spread back into Arabic business as the Arabic numerals had mostly been used by Arabic astronomers and scientists.
The accouting system and other economic data is often described using these numerals today — at least where they need to be accessed by humans. In reality, the vast majority of commerce is conducted using abstract bullae that either contain a token or not: bits. These on/off states (bullae that contain a token or not) that are the fundamental units of information theory also represent the billions of transactions and other information flowing from person to person or firm to firm.
At its heart, economics is the study of this information processing system.
Update: See also Eric Lonergan's fun blog experiment (click on the last link) on money and language. Additionally, Kocherlakota's Money is Memory [pdf] is relevant.
I took liberally (maybe too liberally) from the first link here, and added to it.
https://www.usu.edu/markdamen/1320Hist&Civ/chapters/16TOKENS.htm
http://sites.utexas.edu/dsb/tokens/
https://oi.uchicago.edu/sites/oi.uchicago.edu/files/uploads/shared/docs/nn215.pdf
Dynamic equilibrium versus the Federal Reserve
The Fed is out with its forecast from its September meeting, and it effectively projects constant inflation and unemployment over the next several years. One thing that I did think was interesting is that if the Fed forecast is correct, then the dynamic equilibrium recession detection algorithm would predict a recession in 2019 (which is where it currently forecasts one unless the unemployment rate continues to fall):
This happens to be where the two forecasts diverge (the dynamic equilibrium model is a conditional forecast in the counterfactual world where there is no recession in the next several years [1]).
Here is an update versus the FRBSF forecast as well (post-forecast data in black):
[1] While it can't predict the timing of a recession, it does predict the form: the data will rise several percent above the path (depending on the size of the recession) continuing the characteristic "stair-step" appearance of the log-linear transform of the data:
Information, real, nominal, and Solow
John Handley has the issue that anyone with a critical eye has about the Solow model's standard production function and total factor productivity: it doesn't make sense once you start to compare countries or think about what the numbers actually mean.
My feeling is that the answer to this in economic academia is a combination of "it's a simple model, so it's not supposed to stand up to scrutiny" and "it works well enough for an economic model". Dietz Vollrath essentially makes the latter point (i.e. the Kaldor facts, which can be used to define the Solow production function, aren't rejected) in his pair of posts on the balanced growth path that I discuss in this post.
I think the Solow production function represents an excellent example of how biased thinking can lead you down the wrong path; I will attempt to illustrate the implicit assumptions about production that go into its formulation. This thinking leads to the invention of "total factor productivity" to account for the fact that the straitjacket we applied to the production function (for the purpose of explaining growth, by the way) makes it unable to explain growth.
Let's start with the last constraint applied to the Cobb-Douglas production function: constant returns to scale. Solow doesn't really explain it so much as assert it in his paper:
Output is to be understood as net output after making good the depreciation of capital. About production all we will say at the moment is that it shows constant returns to scale. Hence the production function is homogeneous of first degree. This amounts to assuming that there is no scarce nonaugmentable resource like land. Constant returns to scale seems the natural assumption to make in a theory of growth.
Solow (1956)
But constant returns to scale is frequently justified by "replication arguments": if you double the factory machines (capital) and the people working them (labor), you double output.
Already there's a bit of a 19th century mindset going in here: constant returns to scale might be true to a decent approximation for drilling holes in pieces of wood with drill presses. But it is not obviously true of computers and employees: after a certain point, you need an IT department to handle network traffic, bulk data storage, and software interactions.
But another reason we think constant returns to scale is a good assumption also involves a bit of competition: firms that have decreasing returns to scale must be doing something wrong (e.g. poor management), and therefore will lose out in competition. Increasing returns to scale is a kind of "free lunch" that also shouldn't happen.
Underlying this, however, is that Solow's Cobb-Douglas production function is thought of in real terms: actual people drilling actual holes in actual pieces of wood with actual drill presses. But capital is bought, labor is paid, and output is sold with nominal dollars. While some firms might adjust for inflation in their forecasts, I am certain that not every firm makes production decisions in real terms. In a sense, in order to have a production function in terms of real quantities, we must also have rational agents computing the real value of labor and capital, adjusting for inflation.
The striking thing is that if we instead think in nominal terms, the argument about constant returns to scale falls apart. If you double the nominal value of capital (going from 1 million dollars worth of drill presses to 2 million) and the labor force, there is no particular reason that nominal output has to double.
Going from 1 million dollars worth of drill presses to 2 million dollars at constant prices means doubling the number of drill presses. If done in a short enough period of time, inflation doesn't matter. In that case, there is no difference in the replication argument using real or nominal quantities. But here's where we see another implicit assumption required to keep the constant returns to scale — there is a difference over a long period of time. Buying 1000 drill presses in 1957 is very different from buying 1000 drill presses in 2017, so doubling them over 60 years is different. But that brings us back to constant returns to scale: constant returns to scale tells us that doubling in 1957 and doubling over 60 years both result in doubled output.
That's where part of the "productivity" is supposed to come in: 1957 vintage drill presses aren't as productive as 2017 drill presses given the same labor input. Therefore we account for these "technological microfoundations" (that seem to be more about engineering than economics) in terms of a growing factor productivity. What comes next requires a bit of math. Let's log-linearize and assume everything grows approximately exponentially (with growth rates $\eta$, $\lambda$, and $\kappa$). Constant returns to scale with constant productivity tells us:
\eta = (1-\alpha) \lambda + \alpha \kappa
Let's add an exponentially growing factor productivity to capital:
\eta = (1-\alpha) \lambda + \alpha (\kappa + \pi_{\kappa})
Now let's re-arrange:
\eta = & (1-\alpha) \lambda + \alpha \kappa (1 + \pi_{\kappa}/\kappa)\\
= & (1-\alpha) \lambda + \alpha' \kappa
Note that $1 - \alpha + \alpha' \neq 1$. We've now just escaped the straitjacket of constant returns to scale by adding a factor productivity we had to add in order to fit data because of our assumption of constant returns to scale. Since we've given up on constant returns to scale when including productivity, why not just add total factor productivity back in:
\eta = \pi_{TFP} + (1-\alpha) \lambda + \alpha \kappa
let's arbitrarily partition TFP to labor and capital (in half, but doesn't matter). Analogous to the capital productivity, we obtain:
\eta = \beta' \lambda + \alpha' \kappa
\beta' = & (1 - \alpha) + \pi_{TFP}/(2 \lambda)\\
\alpha' = & \alpha + \pi_{TFP}/(2 \kappa)
In fact, if we ignore constant returns to scale and allow nominal quantities (because you're no longer talking about the constant returns to scale of real quantities) you actually get a pretty good fit with constant total factor productivity [1]:
Now I didn't just come by this because I carefully thought out the argument against using constant returns to scale and real quantities. Before I changed my paradigm, I was as insistent on using real quantities as any economist.
What changed was that I tried looking at economics using information theory. In that framework, economic forces are about communicating signals and matching supply and demand in transactions. People do not buy things in real dollars, but in nominal dollars. Therefore those signals are going to be in terms of nominal quantities.
And this comes to our final assumption underlying the Solow model: that nominal price and the real value of a good are separable. Transforming nominal variables into real ones by dividing by the price level is taken for granted. It's true that you can always mathematically assert:
N = PY
But if $N$ represents a quantity of information about the goods and services consumed in a particular year, and $Y$ is supposed to represent effectively that same information (since nominal and real don't matter at a single point in time), can we really just divide $N$ by $P$? Can I separate the five dollars from the cup of coffee in a five dollar cup of coffee? The transaction events of purchasing coffee happen in units of five dollar cups of coffee. At another time, they happened in one dollar cups of coffee. But asserting that the "five dollar" and the "one dollar" can be removed such that we can just talk about cups of coffee (or rather "one 1980 dollar cups of coffee") is saying something about where the information resides: using real quantities tell us its in the cups of coffee, not the five dollars or in the holistic transaction of a five dollar cup of coffee.
Underlying this is an assumption about what inflation is: if the nominal price is separable, then inflation is just the growth of a generic scale factor. Coffee costs five dollars today rather than the one dollar it cost in 1980 because prices rose by about a factor of five. And those prices rise for some reason unrelated to cups of coffee (perhaps monetary policy). This might make some sense for an individual good like a cup of coffee, but it is nonsense for an entire economy. GDP is 18 trillion dollars today rather than 3 trillion in 1980 because while the economy grew by factor of 2, prices grew by a factor of 3 for reasons unrelated to the economy growing by a factor of 2 or changes in the goods actually produced?
To put this mathematically, when we assume the price is separable, we assume that we don't have a scenario where [2]
N = P(Y) Y(P)
because in that case, the separation doesn't make any sense.
One thing I'd like to emphasize is that I'm not saying these assumptions are wrong, but rather that they are assumptions — assumptions that didn't have to be made, or made in a particular way.
The end result of all these assumptions — assumptions about rational agents, about the nature of inflation, about where the information resides in a transaction, about constant returns to scale — led us down a path towards a production function where we have to invent a new mysterious fudge factor we call total factor productivity in order to match data. And it's a fudge factor that essentially undoes all the work being done by those assumptions. And it's for no reason because the Cobb-Douglas production function, which originally didn't have a varying TFP by the way, does a fine job with nominal quantities, increasing returns to scale, and a constant TFP as shown above.
It's one of the more maddening things I've encountered in my time as a dilettante in economic theory. Incidentally, this all started when Handley asked me on Twitter what the information equilibrium approach had to say about growth economics. Solow represents a starting point of growth economics, so I feel a bit like Danny in The Shining approaching the rest of the field:
Update #1: 20 September 2017
John Handley has a response:
Namely, [Smith] questions the attachment to constant returns to scale in the Solow model, which made me realize (or at least clarified my thinking about the fact that) growth theory is really all about increasing returns to scale.
That's partially why it is so maddening to me. Contra Solow's claim that constant returns to scale is "the natural assumption", it is in fact the most unnatural assumption to make in a theory of economic growth.
Sri brings up a great point on Twitter from the history of economics — that this post touches on the so-called "index number problem" and the "Cambridge capital controversy". I actually have posts on resolving both using information equilibrium (INP and CCC, with the latter being a more humorous take). However, this post intended to communicate that the INP is irrelevant to growth economics in terms of nominal quantities, and that empirically there doesn't seem to be anything wrong with adding up capital in terms of nominal value. The CCC was about adding real capital (i.e. espresso machines and drill presses) which is precisely Joan Robinson's "adding apples and oranges" problem. However, using nominal value of capital renders this debate moot as it becomes a modelling choice and shifts the "burden of proof" to comparison with empirical data.
Much like how the assumptions behind the production function lead you down the path of inventing a "total factor productivity" fudge factor because the model doesn't agree with data on its own, they lead you to additional theoretical problems such as the index number problem and Cambridge capital controversy.
[1] Model details and the Mathematica code can be found on GitHub in my informationequilibrium repository.
[2] Funny enough, the information equilibrium approach does mix these quantities in a particular way. We'd say:
N = & \frac{1}{k} P Y\\
= & \frac{1}{k} \frac{dN}{dY} Y
or in the notation I show $N = P(Y) Y$.
Ideal and non-ideal information transfer and demand curves
I created an animation to show how important the assumption of fully exploring the state space (ideal information transfer i.e. information equilibrium) is to "emergent" supply and demand. In the first case, we satisfy the ideal information transfer requirement that agent samples from the state space accurately reproduce that state space as the number of samples becomes large:
This is essentially what I described in this post, but now with an animation. However, if agents "bunch up" in one part of state space (non-ideal information transfer), then you don't get a demand curve:
Marking my S&P 500 forecast to market
Here's an update on how the S&P 500 forecast is doing (progressively zooming out in time):
Before people say that I'm just validating a log-linear forecast, it helps to understand that the dynamic equilibrium model says not just that in the absence of shocks the path of a "price" will be log-linear, but will also have the same log-linear slope before and after those shocks. A general log-linear stochastic projection will have two parameters (a slope and a level) [1], the dynamic equilibrium model has one. This is the same as saying the data will have a characteristic "stair-step" appearance [2] after a log-linear transformation (taking the log and subtracting a line of constant slope).
[1] An ARIMA model will also have a scale that defines the rate of approach to that log-linear projection. More complex versions will also have scales that define the fluctuations.
[2] For the S&P 500, it looks like this (steps go up or down, and in fact exhibit a bit of self-similarity at different scales):
The long trend in energy consumption
I came across this from Steve Keen on physics and economics, where he says:
... both [neoclassical economists and Post Keynesians] ignore the shared weakness that their models of production imply that output can be produced without using energy—or that energy can be treated as just a form of capital. Both statements are categorically false according to the Laws of Thermodynamics, which ... cannot be broken.
He then quotes Eddington, ending with "But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation." After this, Keen adds:
Arguably therefore, the production functions used in economic theory—whether spouted by mainstream Neoclassical or non-orthodox Post Keynesians—deserve to "collapse in deepest humiliation".
First, let me say that the second law of thermodynamics applies to closed systems, which the Earth definitely isn't (it receives energy from the sun). But beyond this, I have no idea what Keen is trying to prove here. Regardless of what argument you present, a Cobb-Douglas function cannot be "false" according to thermodynamics because it is just that: a function. It's like saying the Riemann zeta function violates the laws of physics.
The Cobb-Douglas production functions also do not imply output can be produced without energy. The basic one from the Solow model implies labor and capital are inputs. Energy consumption is an implicit variable in both labor and capital. For example, your effective labor depends on people being able to get to work, using energy. But that depends on the distribution of firms and people (which in the US became very dependent on industrial and land use policy). That is to say
Y = A(t) L(f_{1}(E, t), a, b, c ..., t)^{\alpha} K(f_{2}(E, t), x, y, z ..., t)^{\beta}
where $f_{1}$ and $f_{2}$ are complex functions of energy and time. Keen is effectively saying something equivalent to: "Production functions ignore the strong and weak nuclear forces, and therefore violate the laws of physics. Therefore we should incorporate nucleosynthesis in our equations to determine what kinds of metals are available on Earth in what quantities." While technically true, the resulting model is going to be intractable.
The real question you should be asking is whether that Cobb Douglas function fits the data. If it does, then energy must be included implicitly via $f_{1}$ and $f_{2}$. If it doesn't, then maybe you might want to consider questioning the Cobb-Douglas form itself? Maybe you can come up with a Cobb-Douglas function that includes energy as a factor of production. If that fits the data, then great! But in any of the cases, it wasn't because Cobb-Douglas production functions without explicit energy terms violate the laws of physics. They don't.
However, what I'm most interested in comes in at the end. Keen grabs a graph from this blog post by physicist Tom Murphy (who incidentally is a colleague of a friend of mine). Keen's point is that maybe the Solow residual $A(t)$ is actually energy, but doesn't actually make the case except by showing Murphy's graph and waving his hands. I'm actually going to end up making a better case.
Now the issues with Murphy's post in terms of economics were pretty much handled already by Noah Smith. I'd just like to discuss the graph, as well as note that sometimes even physicists can get things wrong. Muphy fits a line to a log plot of energy consumption with a growth rate of 2.9%. I've put his line (red) along with the data (blue) on this graph:
Ignore any caveats about measuring energy consumption in the 17th century and take the data as given. Now immediately we can ask a question of Murphy's model: energy consumption goes up by 2.9% per year regardless of technology from the 1600s to the 2000s? The industrial revolution didn't have any effect?
As you can already see, I tried the dynamic equilibrium model out on this data (green), and achieved a pretty decent fit with four major shocks. They're centered at 1707.35, 1836.93, 1902.07, and 1959.69. My guess is that these shocks are associated with the industrial revolution, railroads, electrification, and the mass production of cars. YMMV. Let's zoom in on the recent data (which is likely better) [1]:
Instead of Murphy's 2.9% growth rate that ignores technology (and gets the data past 1980 wrong by a factor of 2), we have a equilibrium growth rate of 0.5% (which incidentally is about half the US population growth rate during this period). Instead of Muphy's e-folding time of about 33 years, we have an e-folding time of 200 years. Muphy uses the doubling time of 23 years in his post, which becomes 139 years. This pushes boiling the Earth in 400 years (well, at a 2.3% growth rate per Murphy) to about 1850 years.
Now don't take this as some sort of defense of economic growth regardless of environmental impact (the real problem is global warming which is a big problem well before even that 400 year time scale), but rather as a case study where your conclusions depend on how you see the data. Murphy sees just exponential growth; I see technological changes that lead to shocks in energy consumption. The latter view is amenable to solutions (e.g. wind power) while Murphy thinks alternative energy isn't going to help [2].
Speaking of technology, what about Keen's claim that total factor productivity (Solow residual) might be energy? Well, I checked out some data from John Fernald at the SF Fed on productivity (unfortunately I couldn't find the source so I had to digitize the graph at the link) and ran it through the dynamic equilibrium model:
Interestingly the shocks to energy consumption and the shock to TFP line up almost exactly (TFP in 1958.04, energy consumption in 1959.69). The sizes are different (the TFP shock is roughly 1/3 the size of the energy shock in relative terms), and the dynamic equilibria are different (a 0.8% growth rate for TFP). These two pieces of information mean that it is unlikely we can use energy as TFP. The energy shock is too big, but we could fix that by decreasing the Cobb-Douglas exponent of energy. However, we need to increase the Cobb-Douglas exponent in order to match the growth rates making that already-too-big energy shock even bigger.
But the empirical match up between TFP and the energy shock in time is intriguing [3]. It also represents an infinitely better case for including energy in production functions than Keen's argument that they violate the laws of thermodynamics.
[1] Here is the derivative (Murphy's 2.9% in red, the dynamic equilibrium in gray and the model with the shock in green):
[3] Murphy actually says his conclusion is "independent of technology", but that's only true in the worst sense that his conclusion completely ignores technology. If you include technology (i.e. those shocks in the dynamic equilibrium), the estimate of the equilibrium growth rate falls considerably.
[2] It's not really that intriguing because I'm not sure TFP is really a thing. I've shown that if you look at nominal quantities, Solow's Cobb-Douglas production function is an excellent match to data with a constant TFP. There's no TFP function to explain.
The long trend in labor hours
Branko Milanovic posted a chart on Twitter showing the average annual hours worked showing, among other things, that people worked twice as many hours during the industrial revolution:
From 1816 to 1851, the number of hours fell by about 0.14% per year:
100 (Log[3185] − Log[3343])/(1851-1816) = − 0.138
This graph made me check out the data on FRED for average working hours (per week, indexed to 2009 = 100). In fact, I checked it out with the dynamic equilibrium model:
Any guess what the dynamic equilibrium rate of decrease is? It's − 0.132% — almost the same rate as in the 1800s! There was a brief period of increased decline (a shock to weekly labor hours centered at 1973.4) that roughly coincides with women entering the workforce and the inflationary period of the 70s (that might be all part of the same effect).
Bitcoin update
I wanted to compare the bitcoin forecast to the latest data even though I updated it only last week since according to the model, it should move fast (logarithmic decline of -2.6/y). Even including the "news shock" blip (basically noise) from Jamie Dimon's comments today, the path is on the forecast track:
This is a "conditional" forecast -- it depends on whether there is a major shock (positive or negative, but nearly all have been positive so far for bitcoin).
Update 18 September 2017
Over the past week (possibly due to Dimon's comments), bitcoin took a dive. It subsequently recovered to the dynamic equilibrium trend:
Continuing comparison of forecast to data:
Another few days of data:
There appears to have been yet another shock, so I considered this a failure of model usefulness. See more here.
JOLTS leading indicators update
New JOLTS data is out today. In a post from a couple of months ago, I noted that the hires data was a bit of a leading indicator for the 2008 recession and so decided to test that hypothesis by tracking it and looking for indications of a recession (i.e. a strong deviation from the model forecast requiring an additional shock — a recession — to explain).
Here is the update with the latest data (black):
This was a bit of mean reversion, but there remains a correlated negative deviation. In fact, most of the data since 2016 ([in gray] and all of the data since the forecast was made [in black]) has been below the model:
That histogram is of the data deviations from the model. However, we still don't see any clear sign of a recession yet — consistent with the recession detection algorithm based on unemployment data.
Search and matching II: theory
I think Claudia Sahm illustrates the issue with economics I describe in Part I with her comments on the approach to models of unemployment. Here are a few of Sahm's tweets:
I was "treated" to over a dozen paper pitches that tweaked a Mortensen-Pissarides [MP] labor search model in different ways, this isn't new ... but it is what science looks like, I appreciate broad summary papers and popular writing that boosts the work but this is a sloooow process ... [to be honest], I've never been blown away by the realism in search models, but our benchmark of voluntary/taste-based unemployment is just weird
Is the benchmark approach successful? If it is, then the lack of realism should make you question what is realistic and therefore the lack of realism of MP shouldn't matter. If it isn't, then why is it the benchmark approach? An unrealistic and unsuccessful approach should have been dropped long ago. Shouldn't you be fine with any other attempt at understanding regardless of realism?
This discussion was started by Noah Smith's blog post on search and matching models of unemployment, which are generally known as Mortensen-Pissarides models (MP). He thinks that they are part of a move towards realism:
Basically, Ljungqvist and Sargent are trying to solve the Shimer Puzzle - the fact that in classic labor search models of the business cycle, productivity shocks aren't big enough to generate the kind of employment fluctuations we see in actual business cycles. ... Labor search-and-matching models still have plenty of unrealistic elements, but they're fundamentally a step in the direction of realism. For one thing, they were made by economists imagining the actual process of workers looking for jobs and companies looking for employees. That's a kind of realism.
One of the unrealistic assumptions MP makes is the assumption of a steady state. Here's Morentsen-Pissarides (1994) [pdf]:
The analysis that follows derives the initial impact of parameter changes on each conditional on current unemployment, u. Obviously, unemployment eventually adjusts to equate the two in steady state. ... the decrease in unemployment induces a fall in job creation (to maintain v/u constant v has to fall when u falls) and an increase in job destruction, until there is convergence to a new steady state, or until there is a new cyclical shock. ... Job creation also rises in this case and eventually there is convergence to a new steady state, where although there is again ambiguity about the final direction of change in unemployment, it is almost certainly the case that unemployment falls towards its new steady-state value after its initial rise.
It's entirely possible that an empirically successful model has a steady state (the information equilibrium model described below asymptotically approaches one at u = 0%), but a quick look at the data (even data available in 1994) shows us this is an unrealistic assumption:
Is there a different steady state after every recession? Yet this particular assumption is never mentioned (in any of the commentary) and we have e.g. lack of heterogeneity as the go-to example from Stephen Williamson:
Typical Mortensen-Pissarides is hardly realistic. Where do I see the matching function in reality? Isn't there heterogeneity across workers, and across firms in reality? Where are the banks? Where are the cows? It's good that you like search models, but that can't be because they're "realistic."
Even Roger Farmer only goes so far as to say there are multiple steady states (multiple equilibria, using a similar competitive search [pdf] and matching framework). Again, it may well be that an empirically successful model will have one or more steady states, but if we're decrying the lack of realism of the assumptions shouldn't we decry this one?
I am going to construct the information equilibrium approach to unemployment trying to be explicit and realistic about my assumptions. But a key point I want to make here is that "realistic" doesn't necessarily mean "explicitly represent the actions of agents" (Noah Smith's "pool players arm"), but rather "realistic about our ignorance".
What follows is essentially a repeat of this post, but with more discussion of methodology along the way.
Let's start with an assumption that the information required to specify the economic state in terms of the number of hires ($H$) is equivalent to the information required to specify the economic state in terms of the number of job vacancies ($V$) when that economy is in equilibrium. Effectively we are saying there are, for example, two vacancies for every hire (more technically we are saying that the information associated with the probability of two vacancies is equal to the information associated with the probability of a single hire). Some math lets us then say:
\text{(1) } \; \frac{dH}{dV} = k_{v} \; \frac{H}{V}
This is completely general in the sense that we don't really need to understand the details of the underlying process, only that the agents fully explore every state in the available state space. We're assuming ignorance here and putting forward the most general definition of equilibrium consistent with conservation of information and scale invariance (i.e. doubling $H$ and $V$ gives us the same result).
If we have a population that grows a few percent a year, we can say our variables are proportional to an exponential function. If $V \sim \exp v t$ with growth rate $v$, then according to equation (1) above $H \sim \exp k_{v} v t$ and (in equilibrium):
\frac{d}{dt} \log \frac{H}{V} \simeq (k_{v} - 1) v
When the economy is in equilibrium (a scope condition), we should have lines of constant slope if we plot the time series $\log H/V$. And we do:
This is what I've called a "dynamic equilibrium". The steady state of an economy is not e.g. some constant rate of hires, but rather more general — a constant decrease (or increase) in $H/V$. However the available data does not appear to be equilibrium data alone. If the agents fail to fully explore the state space (for example, firms have correlated behavior where they all reduce the number of hires simultaneously), we will get violations of the constant slope we have in equilibrium.
In general it would be completely ad hoc to declare "2001 and 2008 are different" were it not for other knowledge about the years 2001 and 2008: they are the years of the two recessions in the data. Since we don't know much about the shocks to the dynamic equilibrium, let's just say they are roughly Gaussian in time (start of small, reach a peak, and then fall off). Another way to put this is that if we subtract the log-linear slope of the dynamic equilibrium, the resulting data should appear to be logistic step functions:
The blue line shows the fit of the data to this model with 90% confidence intervals for the single prediction errors. Transforming this back to the original data representation gives us a decent model of $H/V$:
Now employment isn't just vacancies that are turned into hires: there have to be unemployed people to match with vacancies. Therefore there should also be a dynamic equilibrium using hires and unemployed people ($U$). And there is:
Actually, we can rewrite our information equilibrium condition for two "factors of production" and obtain the Cobb-Douglas form (solving a pair of partial differential equations like Eq. (1) above):
H(U, V) = a U^{k_{u}} V^{k_{v}}
This is effectively a matching function (not the same as in the MP paper, but discussed in e.g. Petrongolo and Pissarides (2001) [pdf]). We should be able to fit this function to the data, but it doesn't quite:
Now this should work fine if the shocks to $H/U$ and shocks to $H/V$ are all the shocks. However it looks like we acquire a constant error coming through the 2008 recession:
Again, it makes sense to call the recession a disequilibrium process. This means there are additional shocks to the matching function itself in addition to the shocks to $H/U$ and $H/V$. We'd interpret this as a recession representing a combination of fewer job postings, fewer hires per available worker, as well as fewer matches given openings and available workers. We could probably work out some plausible psychological story (unemployed workers during a recession are seen as not as desirable by firms, firms are reticent to post jobs, and firms are more reticent to hire for a given posting). You could probably write up a model where firms suddenly become more risk averse during a recession. However you'd need many, many recessions in order to begin to understand if recessions have any commonalities.
Another way to put this is that given the limited data, it is impossible to specify the underlying details of the dynamics of the matching function during a recession. During "normal times", the matching function is boring — it's fully specified by a couple of constants. All of your heterogeneous agent dynamics "integrate out" (i.e. aggregate) into a couple of numbers. The only data that is available to work out agent details is during recessions. The JOLTS data used above has only been recorded since 2000, leaving only about 30 monthly measurements taken during two recessions to figure out a detailed agent model. At best, you could expect about 3 parameters (which is in fact how many parameters are used in the Gaussian shocks) before over-fitting sets in.
But what about heterogenous agents (as Stephen Williamson mentions in his comment on Noah Smith's blog post)? Well, the information equilibrium approach generalizes to ensembles of information equilibrium relationships such that we basically obtain
H_{i}(U_{i}, V_{i}) = a U_{i}^{\langle k_{u} \rangle} V_{i}^{\langle k_{v} \rangle}
where $\langle k \rangle$ represents an ensemble average over different firms. In fact, the $\langle k \rangle$ might change slowly over time (roughly the growth scale of the population, but it's logarithmic so it's a very slow process). The final matching function is just a product over different types of labor indexed by $i$:
H(U, V) = \prod_{i} H_{i}(U_{i}, V_{i})
Given that the information equilibrium model with a single type of labor for a single firm appears to explain the data about as well as it can be explained, adding heterogenous agents to this problem serves only to reduce information criterion metrics for explanation of empirical data. That is to say Williamson's suggestion is worse than useless because it makes us dumber about the economy.
And this is a general issue I have with economists not "leaning over backward" to reject their intuitions. Because we are humans, we have a strong intuition that tells us our decision-making should have a strong effect on economic outcomes. We feel like we make economic decisions. I've participated in both sides of the interview process, and I strongly feel like the decisions I made contributed to whether the candidate was hired. They probably did! But millions of complex hiring decisions at the scale of the aggregate economy seem to just average out to roughly a constant rate of hiring per vacancy. Noah Smith, Claudia Sahm, and Stephen Williamson (as well as the vast majority of mainstream and heterodox economists) all seem to see "more realistic" being equivalent to "adding parameters". But if realism isn't being measured by accuracy and information content compared to empirical data, "more realistic" starts to mean "adding parameters for no reason". Sometimes "more realistic" should mean a more realistic understanding of our own limitations as theorists.
It is possible those additional parameters might help to explain microeconomic data (as Noah mentions in his post, this is "most of the available data"). However, since the macro model appears to be well described by only a few parameters, this implies a constraint on your micro model: if you aggregate it, all of the extra parameters should "integrate out" to single parameter in equilibrium. If they do not, they represent observable degrees of freedom that are for some reason not observed. This need to agree with not just the empirical data but its information content (i.e. not have too many parameters than are observable at the macro scale) represents "macrofoundations" for your micro model. But given the available macro data, leaning over backwards requires us to give up on models of unemployment and matching that have more than a couple parameters — even if they aren't rejected.
Different unemployment rates do not contain differ...
Was the Phillips curve due to women entering the w...
Ideal and non-ideal information transfer and deman...
Search and matching I: methodology
Random agents: one tool in the toolbox
Markups, productivity, and maximum entropy
Demand curves under the microscope
Bitcoin (fork) and SP500 forecast model updates
Solow has science backward
Random agents and political identity
Date update: civilian labor force
Recession detection algorithm: update | CommonCrawl |
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Elham Rahmani8,
P. Read Montague1,4 &
Craig Ramey1
Early childhood educational investment produces positive effects on cognitive and non-cognitive skills, health, and socio-economic success. However, the effects of such interventions on social decision-making later in life are unknown. We recalled participants from one of the oldest randomized controlled studies of early childhood investment—the Abecedarian Project (ABC)—to participate in well-validated interactive economic games that probe social norm enforcement and planning. We show that in a repeated-play ultimatum game, ABC participants who received high-quality early interventions strongly reject unequal division of money across players (disadvantageous or advantageous) even at significant cost to themselves. Using a multi-round trust game and computational modeling of social exchange, we show that the same intervention participants also plan further into the future. These findings suggest that high quality early childhood investment can result in long-term changes in social decision-making and promote social norm enforcement in order to reap future benefits.
Early childhood investment improves the development of disadvantaged children through affecting a variety of cognitive and non-cognitive skills, often translating into better outcomes during adulthood1,2,3. The Abecedarian Project (ABC)—one of the world's oldest high-quality experiments of early childhood intervention—enrolled newborns from low-income, multi-risk families in Orange County, North Carolina, between 1972 and 1977, and provided intensive early childhood education intervention from the first few months of life until school entry. Follow-up studies have provided mounting evidence for positive cognitive4, educational5, economic6, and physical health7 outcomes into adulthood for participants who were exposed to this intervention. However, possible effects of early childhood interventions on social decision-making strategies have not yet been investigated in this population. This is an important issue as certain social decision-making strategies could benefit an individual, including later financial, educational, social, and health outcomes. One such strategy is to choose actions that enforce social norms such as equality. Social norm enforcement, which often entails a cost8, is thought to be motivated by the fact that it can result in long-term positive effects on cooperation9 and thus lead to future benefit—outweighing the immediate cost—for the individual. Since the development of social decision-making styles can be traced back to early childhood10,11, it is therefore essential to investigate if early childhood intervention can impact social decision-making later in adulthood. In the current study, we used two economic games to probe decision-making during social interactions in ABC participants at ages 39–45: the ultimatum game (UG) particularly effective at measuring enforcement of social norms of equality and fairness12 and the multi-round trust game (MRT) measuring the process of cooperation forming/rupture through iterated social exchanges13,14.
In the UG aimed at probing social decision-making related to norm enforcement, one player (Proposer) has to decide how to split a sum of money with another player (Responder). The Responder's acceptance results in each party receiving the allocated money whereas rejection results in both receiving no money15. The UG builds a context in which players have to make trade-offs between self-interest and social norms of equality—rejections being a way to punish behaviors that transgress such social norms16. A prominent hypothesis for explaining such behavior is the Fehr–Schmidt inequality aversion model, which proposes that people use a utility function that expresses preferences for equality and away from inequality, in both disadvantageous (i.e., Responder gets less than the Proposer) and advantageous (i.e., Responder gets more than the Proposer) situations17. The Fehr–Schmidt model is consistent with the fact that rejection of disadvantageous (low) offers has been consistently observed across studies15,18. However, behavior towards advantageous offers is more variable19,20,21. A recent study found participants rejected both disadvantageous and advantageous offers more than equal offers when playing the UG as a third party not involved in the distributive outcome. In contrast, when playing as Responders whose own benefit was affected by their choices, these same participants did not reject advantageous offers more than equal offers22. This finding suggests that even if individuals aspire to promote and enforce an "equal world", self-interest can often overcome inequality aversion.
The UG focuses on equality, but ignores strategic considerations that arise when repeatedly interacting with the same player. These involve planning over multiple rounds, along with the necessity of characterizing one's partners and modeling the iterative exchange. To examine this, each participant also played an MRT of 10 rounds with the same partner. In each round, one player (Investor) received $20 and had to choose to invest any portion of it. This amount of money was tripled and sent to the other player (Trustee) who decided how much of it to repay the Investor13. A recent model of preference and mental states has provided a quantitative way to model behavior during this task23. This model assumes that players compute the long-run utilities of the available options to guide decisions24 and it allows us to investigate both immediate reactions such as inequality aversion and their capacity to plan ahead—operationalized by the model's planning horizon. The planning horizon is important since failing to plan ahead during a social exchange can lead to a rupture in cooperation14.
In this study we showcase how combining economic games with sophisticated computational models of behavior can provide sensitive indicators to assess the long-term effects—more than 40 years later—of early educational interventions on social decision-making. With this ecological approach, as we show next, such intervention can result in long-term changes in social decision-making and promote social norm enforcement possibly for the consideration of future benefits.
ABC project overview
The ABC project was conducted with children from low-income families in the 1970s (see Supplementary Materials in ref. 7 for details). It was designed to examine the impact of intensive early childhood education on preventing developmental delays and academic failure. One hundred and twenty families were recruited in the ABC Project with Scores of High Risk Index25 as eligibility criteria (infants with a score higher than 11 were eligible4). Enrolled families were paired on High Risk Index scores and then one family from each pair was randomly assigned to either the intervention or the control group. The base sample included 111 children from 109 families that accepted their randomization assignment and accepted to participate (one family with a pair of twins and another with a pair of siblings). Among them, 57 were assigned to the intervention group; the other 54 to the control group. At the preschool stage (from 2 months to 5 years of life), for both intervention and control groups, a standard intervention including nutritional, health care, and family social support services was given, while the intervention group received an extra educational intervention. The educational intervention included cognitive and social stimulation, caregiving, and supervised play throughout a full 8-h day (5 days per week, 50 weeks per year) during the first 5 years, emphasizing language, emotional regulation, and cognitive skill development26,27. This intervention employed a curriculum including a series of "educational games" developed by Sparling and Lewis26. Each teacher was in charge of child care for three infants, which ensured intensive interactions between children and teachers. In addition, the intervention group was also provided with free primary pediatric care and nutrition, including routine screenings, immunizations, pediatric care staff visits, and laboratory tests. Another randomization was implemented when children entered schools when they were five years old, with about half of the children in the intervention or control group during the preschool stage assigned to another three-year intervention28,29. Analyses in the current paper focused on the outcomes for intervention in the preschool phase, irrespective of the assignments at the school-age stage5.
From the original 111 ABC participants, 78 took part in the current study, 36 (ABC Controls) received basic supports (i.e., nutritional, health care, and family social support services) from birth to age 5, while 42 (ABC Interventions) received these supports along with a 5-year, high-quality educational intervention focusing on cognition and social–emotional development (see Table 1 and Supplementary Table 1 showing attrition was comparable in both groups). We also tested an independent control group of 252 adults (Roanoke Controls) recruited from Roanoke, Virginia, who did not receive any controlled intervention in early childhood.
Table 1 Participant retention and attrition
Stronger norm enforcement in ABC Interventions in UG
Our participants played 60 rounds of UG as the Responder deciding whether to accept or reject offers about how to split $20 with different Proposers (Fig. 1a). Unknown to the participants, offers were generated by a computer algorithm which selected the offers from three different truncated Gaussian distributions: Low offers (mean = $4), Medium offers (mean = $8), or High offers (mean = $12), each with a standard deviation = $1.5. Participants were randomly assigned to one of two conditioning types: the medium-high–medium (MHM) type and the medium-low–medium (MLM) type. During the first 20 rounds, both types received offers taken from the Medium distribution. During the next 20 rounds, the MHM type received offers from the High distribution while the MLM type received offers taken from the Low distribution. For the last 20 rounds, both types received offers from the Medium distribution (Fig. 1a). All participants were asked to report their emotional reaction towards the current offer from unhappy to happy on a 1–9 scale in 60% of the rounds. To examine social behavior towards both disadvantageous and advantageous inequality, the MHM type was the main focus of our analysis on rejection rate and emotion rating, since advantageous offers (higher than $10) were rarely displayed in MLM type (see Supplementary Figs. 1 and 2, Supplementary Table 2 and Supplementary Note 1 for results for MLM type).
The ultimatum game (UG) and multi-round trust game (MRT) in the current study. a Procedure for UG comprising of 60 rounds, with Mean and SD of offer size in each round across Medium-Low-Medium (MLM) or Medium-High-Medium (MHM) conditioning type. Each participant was told he/she was the Responder in this game who decided to accept or reject the offer (s) from different Proposers in each round. Offers were sampled from one of the three Gaussian distributions: low offers (mean $4, SD $1.5); medium offers (mean $8, SD $1.5); and high offers (mean $12, SD $1.5). b Procedure for MRT comprising 10 consecutive rounds. Participants were told that they, as the Investor, were playing with the same Trustee across the whole game. In each round, the participant received $20 and decided how much of it to send to the Trustee. This amount of money (I) received by the Trustee was tripled (3*I) and any portion of it was then repaid to the investor (R*3*I). SD standard deviation
Using Group (ABC Intervention vs. ABC Control vs. Roanoke Control) × Equality (Disadvantageous Unequal vs. Equal vs. Advantageous Unequal) analyses of covariance (ANCOVA) with Gender as the covariate and Bonferroni-corrected post hoc t-tests, we found that ABC Controls who underwent the MHM conditioning type rejected disadvantageous offers (mean ± s.e.m., 42.7 ± 8.5% of all disadvantageous offers for one participant) more than equal offers (1.0 ± 1.0%), p < 0.001, but did not reject advantageous offers (4.5 ± 2.5%) more than equal offers (1.0 ± 1.0%), p = 1.000. This pattern of response is in line with previous work that reported behavior driven by self-interest in the face of advantageous offers (i.e., low rejection rates)22,30 and very similar to the pattern we observed in Roanoke Controls. In stark contrast, along with rejecting disadvantageous offers (48.3 ± 7.2%), ABC Interventions (in MHM type) rejected advantageous offers (43.4 ± 8.5%) more than equal offers (5.0 ± 2.8%), p's < 0.001. This difference in rejection pattern was confirmed by a significant Group × Equality interaction, F(3,258) = 13.464, p < 0.001, η2p = 0.140 (Fig. 2b), and the fact that ABC Interventions rejected advantageous offers more than ABC Controls (p < 0.001) and Roanoke Controls (p < 0.001). A significant Group × Offer Size interaction was also found, F(5,411) = 8.229, p < 0.001, η2p = 0.090 (see Supplementary Note 2 for details). In keeping with this, only the ABC Interventions showed a "V shape" rejection rate pattern, with rejection increasing as a function of inequality, regardless of whether the inequality was personally advantageous or disadvantageous (Fig. 2c and Table 2). Since rejecting offers in the UG is akin to punishing the Proposer31, this rejection pattern can be considered a strong social signal aimed at enforcing equality during exchanges.
Offer distribution, rejection rates, and model-based parameters in the ultimatum game. a Distribution of offer size for Medium-High-Medium (MHM) conditioning type. The frequency is the average occurrence for each offer size across participants in MHM. b Rejection rates for MHM grouped by level of equality. All groups had higher rejection rates for disadvantageous offers than equal offers. Advantageous offers were not rejected more than equal offers in ABC Control and Roanoke Control, while ABC Interventions rejected advantageous offers more than equal offers. c Rejection rates for MHM grouped by offer size. Only ABC Interventions increased rejection rates as a function of inequality, regardless of them being personally advantageous or disadvantageous, presenting a "V shape" pattern. d Parameter estimates from the behavioral modeling using a Fehr–Schmidt inequality aversion model. Both ABC Interventions and ABC Controls have a higher level of envy (unwillingness to accept unequal offers which are disadvantageous to the participant) than Roanoke Controls. The ABC Interventions had a higher guilt (unwillingness to accept unequal offers which are advantageous to the participant) than ABC Controls and Roanoke Controls. e The horizontal axis presents different levels of equality: DU disadvantageous unequal, E equal, AU advantageous unequal. The vertical axis presents the disutility defined by the inequality aversion (IA) model (i.e., sensitivity × inequality). The slope of each line presents the sensitivity for inequality aversion (IA; envy for DU and guilt for AU). A steeper slope corresponds to higher inequality aversion. Compared with the control groups, ABC Interventions presented a much more symmetric IA pattern (i.e., the same level of envy and guilt). Shaded areas are bounded by mean ± s.e.m. *p < 0.05, **p < 0.001 (post hoc t-test p-values). Error bar represents s.e.m.
Table 2 Rejection rates between offer sizes in Medium-High-Medium conditioning type
Taking advantage of behavioral modeling, we also estimated individual sensitivity to advantageous and disadvantageous inequality in the UG using the Fehr–Schmidt inequality aversion utility function17 where the utility of an offer is discounted by "envy" (unwillingness to accept disadvantageous offer) and "guilt" (unwillingness to accept advantageous offers). In line with our rejection rate results, an ANCOVA with Gender as the covariate and Bonferroni-corrected post hoc t-tests found that while ABC Interventions had similar envy coefficients as ABC Controls (p = 1.000), they had higher guilt coefficients than both ABC Controls (p < 0.001) and Roanoke Controls (p < 0.001) (Fig. 2d), suggesting heightened sensitivity to advantageous inequality (see Supplementary Note 3 for details). Furthermore, the similar envy and guilt coefficients in ABC Interventions suggest that, as a group, they displayed a unique symmetric disutility for advantageous and disadvantageous offers which is in accordance with their "V shape" rejection pattern (Fig. 2e). Finally, when asked to report their feelings towards the offers, only ABC Interventions rated equal offers as more pleasant than both advantageous and disadvantageous offers (p's < 0.001), consistent with symmetric sensitivity to inequality and the "V shape" rejection pattern (Fig. 3 and Table 3). Overall, these results suggest that an early childhood intervention program can profoundly alter sensitivity to norm deviation in adulthood and promote social norm enforcement.
Emotion rating across 60 rounds in Medium-High-Medium (MHM) type in the ultimatum game. a Emotion rating for MHM grouped by level of equality. ABC Interventions rated equal offers as more pleasant than both disadvantageous and advantageous offers. ABC Controls rated disadvantageous offers as less pleasant than equal offers but did not report different feelings about equal and advantageous offers. Roanoke Controls rated disadvantageous offers as less pleasant than equal offers but advantageous offers as more pleasant than equal offers. b Emotion rating across 60 rounds in MHM grouped by offer size. ABC Interventions rated equal offer as more pleasant than each disadvantageous offer as well as than each advantageous unequal offer, while ABC Controls decreased emotion rating for more disadvantageous offers but reported no difference between equal and advantageous offers. Compared with ABC Controls and Roanoke Controls, ABC Interventions rated each advantageous offer as significantly less pleasant. *p < 0.05, **p < 0.001 (post hoc t-test p-values). Error bar represents s.e.m.
Table 3 Emotion ratings between offer sizes in Medium-High-Medium conditioning type
ABC Interventions planned further into future in MRT
In the MRT, two players—an Investor and a Trustee—engage in 10 rounds of economic exchange game. In the current study, each participant played 10 consecutive rounds as the Investor (Fig. 1b). A model based on interactive partially observable Markov decision processes32 (iPOMDP) was used to characterize preference and mental states during social interaction within the MRT. The model assumes that players compute the long-run utilities (called Q-values) of the available options to guide decisions24. A self-consistency condition for the Q-values over successive rounds is prescribed by the Bellman equation33. Based on extensive validation23, this model has two structural characteristics. First, since the value of a player's action depends on the future decisions of the partner, players are assumed to develop a model of their partners—which assumes players have distinct levels of theory of mind and follow a cognitive hierarchy theory34. Specifically, a player of type k assumes that the partner has type k-1. Second, players are assumed to model Q-values only a certain number (the planning horizon) of rounds into the future, substituting default values thereafter (Fig. 4a).
The Planning horizon in the multi-round trust game (MRT). a In each round, the Investor (played by the participant; indicated in blue) received $20 and decided how much of it to send to the Trustee (played by a computer algorithm; indicated in black). The amount (I) the Investor sent was tripled and delivered to the Trustee, who decided what fraction (R) of this total (3*I) to send back to the Investor. The Investor ended up with 20-I+R*3*I; the Trustee with (1-R)*3*I. Planning horizon quantifies how many steps of the future interactions the participant took into account when assessing the results of his/her investment during the MRT. b Distribution of the future planning capacity parameter (planning horizon) in ABC Intervention, ABC Control, and Roanoke Control group, respectively
We first assessed possible group difference with one-way analyses of variance on overall behavior (fractional investments) and performance (total earnings) across the MRT. Average fractional investments were similar across groups, F(2,327) = 2.488, p = 0.085, η2p = 0.015 (ABC Interventions, 50.6 ± 3.2%; ABC Controls, 45.9 ± 3.2%; Roanoke Controls, 54.5 ± 1.5%). The total earnings were also not different among groups, F(2,327) = 2.443, p = 0.088, η2p = 0.015 (ABC Interventions, 215.67 ± 6.02; ABC Controls, 210.5 ± 7.60; Roanoke Controls, 225.07 ± 2.68).
However, using model-based analysis, we were able to highlight group differences in decision-making strategies, with ABC Interventions having a higher level of planning horizon compared to Roanoke Controls and a lower level of ToM compared to ABC Controls. To be more specific, with independent-samples Kruskal–Wallis H-test and the follow-up two-tailed Mann-Whitney U-tests, we found a significant main effect of group for planning horizon, H(2) = 6.849, p = 0.033, with ABC Interventions (2.45 ± 0.17) having a higher level of planning horizon than Roanoke Controls (2.01 ± 0.07, Bonferroni corrected p = 0.027) but not significantly different from ABC Controls (2.08 ± 0.20, Bonferroni-corrected p = 0.269) (Fig. 4b). This result suggests that social decision-making in ABC Interventions, compared to Roanoke Controls, seems to be particularly influenced by future and long-term outcomes. The main effect of group was also found to be significant for theory of mind (ToM), H(2) = 6.701, p = 0.035, with ABC Controls (2.83 ± 0.24) having a higher level of ToM than ABC Interventions (2.05 ± 0.24, Bonferroni-corrected p = 0.047) but not different from Roanoke Controls (2.62 ± 0.09, Bonferroni corrected p = 0.974), suggesting that ABC Controls might have adopted a different social decision-making strategy (i.e., modeling the other) compared to ABC Interventions' strategy (i.e., planning ahead). See Supplementary Note 4 for details about other parameters in this model.
The ABC project and its follow-up studies have shown that early childhood intervention can result in short- and long-term effects including important and substantial cognitive, health, and educational benefits5,7,29. In line with these results, now over 40 years after, ABC Interventions in the current study, compared to ABC Controls, reported higher levels of "very close" relationships with their parents (85.7% vs. 58.3%, p < 0.001), higher levels of educational attainment (97.6% vs. 75.0% completed high school, p < 0.001 and had four times the rate of college graduation), and more of them had saving accounts (92.9% vs. 66.7%, p < 0.001) (see Supplementary Table 3 and Supplementary Note 5 for midlife demographic information). These real-life outcomes might profit greatly from the ability to attend to social norms, establish and maintain positive social interactions, and plan into the future, which are dynamic and complex processes related to social decision-making.
The current study revealed a boost in social norm enforcement behaviors and sensitivity to social norm violation in middle-age adults from the ABC project who received a high-quality educational intervention during the first 5 years of their lives. Playing the role of the Responder in an UG, these individuals rejected unequal offers more than equal offers regardless of whether the split was disadvantageous or advantageous to them—displaying symmetric inequality aversion. Since rejecting the offers of the Proposers in the UG is akin to punishing the Proposer31, this rejection pattern can be considered as a strong social signal aimed at enforcing equality during social exchanges. Indeed, a study investigating the motivations of rejecting advantageous offers by analyzing verbal data during the discussion between pairs of players in the UG21 found that most participants claimed to enforce the norm of equality when they rejected offers that were advantageous to them.
Our results revealed that ABC Interventions rejected advantageous offers more than the two control groups while both ABC Interventions and ABC Controls rejected disadvantageous offers more than Roanoke Controls. This result for disadvantageous offers could be related to differences between the ABC and Roanoke samples: ABC groups had much more experience with psychological testing than the Roanoke group, ABC participants are African-American while the Roanoke Controls are more racially diverse, and the samples grew up in different states (North Carolina vs. Virginia). However, the amount of testing in the follow-up studies of the ABC project was equivalent between ABC Interventions and ABC Controls, and importantly, neither the intervention nor the follow-up tests involved economics games such as the UG (or MRT) for either ABC group. Hence the higher rejection rate on advantageous offers compared to the two control groups and the unique symmetric inequality aversion of ABC Intervention is more likely driven by the effects of the educational intervention during the first five years of their lives. We note that these differences are based on a relatively small sample which might raise the possibility of it being a false positive. However, the amount of convergent evidence, including the unique V shape rejection rate pattern of ABC Interventions, the group difference on model-based parameters and the corresponding emotion rating patterns, suggests that our result of a difference in equality-based decision-making between ABC Interventions and Controls is a robust finding.
When resources are divided unequally, people typically exhibit norm enforcement in which they punish the person responsible for the unequal split, even at a cost35. However, self-interest often overweighs the willingness to enforce this social norm22—unequal splits that are advantageous are less punished than disadvantageous ones30. Indeed, contrary to ABC Interventions who showed symmetric inequality aversion, our two control groups were unwilling to forgo the possibility of enjoying the personal benefits of advantageous offers which they rarely rejected. This important difference suggests that early educational interventions—in this case, for children from deeply impoverished backgrounds—can contribute to stronger norm enforcement during social exchanges in adulthood. Our results are in line with recent findings from another early childhood education program showing that preschool education made children more egalitarian at 7–8 years old36. Importantly, our results demonstrate that such changes can extend into adulthood—many decades after the intervention. It was recently shown that disadvantaged children whose mothers are more prosocial (measured by higher levels of altruism, trust, and other-regarding preference) had a higher increase in prosociality after receiving interventions that enriched their social environment37. Our results, showing that early investment in children can influence social decision-making (i.e., higher other-regarding and inequality-aversive in terms of monetary distribution) during adulthood thus suggest that facilitating prosociality behavior through early childhood investments could have a cascading effect—through intergenerational transmission—by potentializing the impacts of subsequent interventions in future generations. It is notable that the effect found in the current study may be relatively restricted in terms of its translation to everyday behaviors. It would be interesting to look in future work at commonplace measures that are closely related to inequality (such as charitable giving) in order to see if our laboratory findings are paralleled by social decision-making in their daily life.
In the MRT, additionally, although no differences were found on overall investment and performance between groups, using computational modeling we were able to highlight differences in social decision-making strategies. On the one hand, we observed that ABC Interventions planned further into the future in the MRT compared to Roanoke Controls. On the other hand, the higher ToM parameter for ABC Controls indicated that this group utilized more mentalization steps than ABC Interventions during this game. These findings suggest that the similar overall behavior of the two ABC groups might be motivated by different social decision-making strategies. It is possible that ABC Interventions focused more on future social interactions while ABC Controls took other's mental states into more consideration, potentially indicative of a preference to reap short-term benefits. These findings also reveal the advantage of our approach which provides information about the underlying mechanism (or strategy) of social decision-making, and thus go beyond the study of the outcome (or overall behavior) of this process. It would be important for future studies to examine the role of these underlying strategies (e.g., planning horizon and ToM) on social decision-making using experimental designs. It is notable that these process-oriented results found in the current study could be related to actual outcomes of social decision-making (e.g., income or other economic outcomes;social well-being). Future work on this would add ecological validity of these process-oriented findings.
The striking high rejection of advantageous unequal offers in the UG in the current study is rare in large-scale societies (but see in refs. 21,38). However, previous work on participants from Western cultures found that when people played the UG as a third party whose personal benefit would not be affected by their decisions, they also displayed a symmetric inequality aversion22, which is very similar to what we find here in ABC Interventions. This suggests that such symmetric inequality aversion is indeed valued by some in Western cultures. Importantly, studies22 including our own have also shown that this social norm is less enforced when self-interest comes into play, which may explain why it is rarely observed. The rejection of advantageous offers also occurs in some small-scale societies where the convention is to repay an unsolicited gift—high value offers are sometimes rejected when the Responder does not want to be indebted to the Proposer39. It raises the possibility that the ABC Interventions in our study might have a stronger belief in reciprocity so that they were more alert to potential cost or repayment caused by advantageous offers in the future, instead of current payoffs. It is just one possible hypothesis; in the future, it should be tested using more elaborate methods because of the obvious difference between these small-scale societies and the culture of our participants.
The longer planning horizon of the ABC Interventions estimated from the MRT suggests their unique norm enforcement behavior during the UG may be motivated by such anticipated future personal costs but also potential social benefits. Consistent with this interpretation, equality enforcement by punishing unequal offers was shown to be positively correlated with altruistic behaviors across cultures40. Indeed, behavior towards equality and fairness despite short-term cost to self has been proposed as important in stabilizing long-term cooperation9,41. Furthermore, ensuring that others comply with social norms can impact others' behaviors to create a more cooperative social environment8, which can produce future societal benefits.
Our approach based on quantitatively prescribed economic games provides a quantitative indicator evaluating possible social benefits of early childhood investment programs. By having participants actively interact with others in social contexts, we provide a more ecologically valid way to measure outcomes on social decision-making compared to self-report questionnaires. Indeed, our findings contribute unique knowledge about long-term correlates of receiving a high-quality early education: changes in social decision-making highlighted by higher social norm enforcing behavior probably motivated by anticipated future benefits related to social cooperation. We should hasten to add that our findings seem to reveal the effects of the educational intervention in the first five years of the ABC participants' life. However, this does not necessarily exclude the possibility that these differences are related to other factors that resulted from the intervention and that occurred during the four decades that followed it. Indeed, since our measures were taken at a single time point (over 40 years after the intervention), known changes on health7, educational level5, social connections as well as other factors induced by the intervention may have also played a role in shaping the participants' social decision-making pattern. Our own analyses did not detect an association between adult educational attainment, for example, and decision-making behavior in this sample. It would be valuable, nonetheless, to further investigate if and how early childhood interventions or any aspects of them can give rise to differences in later experiences that in turn influence social decision-making behaviors. By investing in the early education of highly vulnerable children—with a program that underscores positive adult–child interactions, explicitly teaches about cause-and-effect, permits active learning and early decision-making opportunities, and promotes increasingly complex social cooperation—children realize a brighter future, becoming healthier7, more productive3, and as our results show, stronger promoters of the norms on which our society is built.
As mentioned above, the base sample included 111 children. About four decades after their enrollment, 16 participants attritted with 9 deceased (3 from the intervention group, 6 from the control group), 2 in prison (1 from the intervention group, 1 from the control group), and 5 withdrawal (3 from the intervention group, 2 from the control group; see Table 1 for the pattern of attrition). Among the remaining 95 participants (50 from the intervention group, 45 from the control group), 78 participants, now in their 40s, took part in the current study (participation rate: 82%). Among them, 42 received the standard intervention and the extra educational intervention (ABC Intervention group) while 36 were in the control group during early childhood (ABC Control group) and only received the standard intervention. Several midlife demographic information were also assessed as part of this stage of the ABC project using various self-reported questionnaires (see Supplementary Table 3 and Supplementary Note 5 for midlife demographicinformation).
Additionally, 252 adult participants in Roanoke, Virginia also took part in this study. These participants did not receive any controlled intervention during their childhood (Roanoke Control group).
All participants gave written consent to participate in the experiments, and all procedures were performed in accordance with the Institutional Review Board of the Virginia Tech Carillion Research Institute. Data about gender and age are presented in Supplementary Table 5.
Task procedure
Participants' social behaviors were assessed using the UG and MRT. Stimuli were presented and responses were collected using NEMO (Human Neuroimaging Laboratory, Virginia Tech Carilion Research Institute). The order of the two games was randomized across participants. Details for non-deception and incentivization in the two games are provided in Supplementary Note 6.
In the UG, one player (Proposer) has to decide how to split a $20 endowment between himself and another player (Responder). The Responder has to decide whether to accept or reject the offer. If the offer is accepted, both players get the proposed amounts. However, if the offer is rejected, both players get $0. Participants played 60 rounds as the Responder and were informed that the offers in each round were from different proposers. Abecedarian participants and Roanoke participants were randomly assigned to one of two conditioning type: the medium-high-medium type (MHM) and the medium-low-medium type (MLM) (17 in MLM and 25 in MHM for ABC Interventions; 21 in MLM and 15 in MHM for ABC Controls; 122 in MLM and 130 in MHM for Roanoke Controls). During the first 20 rounds, participants assigned to both types received offers taken from the Medium distribution (Mean = $8, SD = $1.5). During the next 20 rounds, participants assigned to the MHM type received offers from the High distribution (Mean = $12, SD = $1.5) while those assigned to MLM type received offers taken from the Low distribution (Mean = $4, SD = $1.5). For the last 20 rounds, participants from both types received offers from the Medium distribution (Fig. 1a). In three of five rounds after having made their decisions, participants were also asked to rate their emotion towards the current offer on a 1–9 scale, ranging from unhappy to happy using emoticons adapted from the self-assessment manikin42. Participants were informed that they would be paid according to outcomes in one randomly selected round and were encouraged to treat each round as the selected round. The timeline of one round of the UG is illustrated in Supplementary Fig. 3a.
In the MRT, in each round, one player (Investor) received $20 and had to choose to invest any portion of it. This amount of money was tripled and sent to the other player (Trustee) who decided how much of it to repay the Investor. Each participant played 10 consecutive rounds as the Investor in the MRT with the same partner. Unknown to the participants, the Trustee's responses in the MRT were generated using a k-nearest neighbors sampling algorithm on known responses from real players as described in ref. 14. The timeline of one round of the MRT is illustrated in Supplementary Fig. 3b.
Statistics were implemented using SPSS software (IBM SPSS Statistics Version 21.0, IBM Corp.). For all analyses, the significance level was set at 0.05 and Greenhouse–Geisser correction non-sphericity was used when appropriate. Post hoc comparisons were evaluated using two-tailed pairwise tests with Bonferroni correction. Partial eta-squared (η2p) values were provided to demonstrate effect size where appropriate43.
Rejection rate and emotion rating in UG
Our analyses focus on the participants' social behavior towards inequality. Specifically, we tested the interaction between equality and treatment on rejection rate by a (3 × 3) Group (ABC Intervention vs. ABC Control vs. Roanoke Control) × Equality (Disadvantageous Unequal (offers < 10) vs. Equal (offers = 10) vs. Advantageous Unequal (offers > 10)) analyses of covariance (ANCOVA) with Gender as the covariate in the MHM type to take account of the gender unbalance within MHM type. As offers higher than $10 were rarely displayed for in the MLM type, a (3 × 2) Group (ABC Intervention vs. ABC Control vs. Roanoke Control) × Equality (Disadvantageous Unequal (offers < 10) vs. Equal (offers = 10)) ANCOVA with Gender as the covariate was used for the MLM type. Gender was included as a covariate in the analyses to control for the difference in gender composition in the three groups.
Further, the rejection rates for each offer were calculated for each participant—making it possible to precisely describe social behavior from highly disadvantageous, to highly advantageous inequality. Because of the different distribution of offers between MHM and MLM (Fig. 2a and Supplementary Fig. 1a) resulting in small number of offers for certain amounts depending on the conditioning type, different offer sizes were evaluated for MHM and MLM. For each participant with MHM type, offers lower than $8 and offers higher than $12 were respectively pooled together. For each participant with MLM type, offers lower than $5 were pooled together and offers higher than $10—rarely displayed—were not included into the behavioral data analysis in MLM type. For each conditioning type, differences in rejection rates were analyzed using a (3 × 7) Group × Offer Size ANCOVA with Gender as the covariate to take account of the gender unbalance within MLM type.
Differences in emotion ratings were analyzed using a similar approach: a Group × Equality ANCOVA and an Offer Size × Group ANCOVA with Gender as the covariate for each type of conditioning.
Model-based analyses for behaviors in UG. We assumed that the participants' behavior could be modeled by their aversion to offers that deviate from equality and fitted each participant's behaviors to a Fehr–Schmidt inequality aversion model (FS model)17. As previous studies using the UG have shown that people have internal norms (expectations on money allocation) which can be updated based on the history of offers16,44,45, we also fitted the behavioral data to two types of adaptation models, a Bayesian observer model44 and a Rescorla–Wagner model33,46 to test if they outperformed the FS model.
In the FS inequality aversion model, the utility of each offer at each round was represented by the Fehr–Schmidt inequality aversion utility function.
$$U(s_i) = s_i - \alpha \max \left\{ {10 - s_i,0} \right\} - \beta \;{\mathrm {max}}\{ s_i - 10,0\} .$$
Here, U(si) represents the utility of the offer si at round i. This value is discounted by the difference between the amount allocated (offer) to the Responder (si) and an even split ($10). The disutility associated with inequality is controlled by two parameters: α or "envy" (α∈[0,10]) which represents the participant's unwillingness of the participant to accept unequal offers disadvantageous to him/her; β or "guilt" (β∈[0,10]) which represents his/her unwillingness to accept unequal offers advantageous to him/her.
The probability of accepting each offer was modeled using a softmax function:
$$p_{{\mathrm{accept}}} = \frac{{e^{u \ast \gamma }}}{{1 + e^{u \ast \gamma }}}.$$
Here, γ is the softmax inverse temperature parameter where the lower γ is, the more diffuse and variable the choices are (γ∈[0,1]).
Under the Bayesian observer model44 (BO model), we assumed that each participant believed that the offers were sampled from a Gaussian distribution with uncertain mean and variance, and performed Bayesian update after receiving a new offer. Specifically, each participant was assumed to have a prior on the distribution of the offers (s), with mean μ and variance σ2, denoted as s~N(μ,σ2). Since μ and variance σ2 were mixed together, the prior of offers (s) was assumed as p(μ,σ2). The prior was updated following Bayes' rule once the participant received a new offer. The posterior was given by
$$p(\mu \,\sigma ^2|s_i) = \frac{{p(s|\mu ,\sigma ^2)p(\mu |\sigma ^2)}}{{p(s_i)}}.$$
For convenience we assumed a conjugate prior of μ and σ2:
$$p(\mu ,\,{\mathrm{\sigma }}^2) = p(\mu ,|\sigma ^2)\,p({\mathrm{\sigma }}^2),$$
$$p(\mu |\sigma ^2) = {\mathrm {Normal}}(\hat \mu ,\hat \sigma ^2/k)$$
$$p(\sigma ^2) = {\mathrm{Inv}} - \chi ^2(\nu ,\hat \sigma ^2).$$
We set the initial value of the hyperparameters k, ν and \({\hat{\mathrm \sigma }}^2\) as
$$k_0 = 4,\;\nu _0 = 10,\;\hat \sigma _0^2 = 4$$
Two variations of the BO models were tested. The first assumed equality as a fixed initial norm for all participants, \(\hat \mu _0 = 10\). The second assumed that the initial norm could vary between participants, hence \(\hat \mu _0\) was individually fitted using each participant's responses (\(\hat \mu _0 \in \left[ {0,20} \right]\)).
After receiving si, at round i, these values were updated as
$$k_i = k_{i - 1} + 1,v_i = v_{i - 1} + 1,$$
$$\hat \mu _i = \hat \mu _{i - 1} + \frac{1}{{k_i}}(s_i - \hat \mu _{i - 1}),$$
$$\nu _i\hat \sigma _i^2 = \nu _{i - 1}\hat \sigma _{i - 1}^2 + \frac{{k_{i - 1}}}{{k_i}}(s_i - \hat \mu _{i - 1})^2.$$
We define the prevailing norm as μi−1 at round i, and the utility of the offer is given by
$$U(s_i) = s_i - \alpha \max \;\left\{ {\mu _{i - 1} - s_i,0} \right\} - \beta \;\max\{ s_i - \mu _{i - 1},0\} .$$
Here, α represents the unwillingness of the participant to accept offers lower than his/her norm (α∈[0,10]). β represents the unwillingness to accept offers higher than him/her norm (β∈[0,10]).
The probability of accepting each offer was
$$p_{{\mathrm{accept}}} = \frac{{e^{u \ast \gamma }}}{{1 + e^{u \ast \gamma }}},$$
where γ∈[0,1].
The Rescorla–Wagner (RW) model assumed that each participant had internal norms which were updated by the RW rule:33,46
$$x_i = x_{i - 1} + \varepsilon (s_i - x_{i - 1}).$$
Here xi represents the norm at round i and ε is the norm adaptation rate (ε∈[0,1]), which represents the extent to which the norm was influenced by the difference (i.e., norm prediction error) between the current offer si and the preceding norm xi−1. A low ε indicates a lower impact of the norm prediction error on norm updating whereas a high ε indicates a high impact. Similar to the BO model, two variations of the RW model were tested based on the initial norm x0: a fixed initial norm based on equality (x0 = 10) and variable initial norms across participants. The utility of an offer at round i is given by
$$U(s_i) = s_i - \alpha \,{\mathrm{max}} \,\left\{ {x_{i - 1} - s_i,0} \right\} - \beta \,{\mathrm{max}}\{ s_i - x_{i - 1},0\} .$$
Similar to BO model, α here represents the unwillingness of the participant to accept offers lower than his/her norms (α∈[0,10]). β represents the unwillingness to accept offers higher than him/her norms (β∈[0,10]).
All models were then fitted to the behavioral data individually, which estimated the values of α, β, γ, and \(\hat \mu _0\) or x0 for variable starting norm models for each subject by maximizing the log likelihood of choices over 60 trials. Then model comparison was implemented by calculating the Bayesian information criterion score (BIC) for each model for each participant. The model with the lowest mean BIC is considered the winning model since it has the maximal model evidence (Supplementary Table 4). The estimated parameters from the winning model were compared among the three groups of participants with an ANCOVA with Gender as a covariate. Post hoc comparisons were evaluated using Bonferroni correction. The ranges of the free parameters in models presented above were based on previous work44. We tested other ranges that resulted in slightly worse model fitting and did not significantly affect the results presented here.
Fractional investment in MRT
The fractional investment sent from the participant (Investor) in each round was calculated as the amount of investment in each round divided by the resource available to the investor in the current round (i.e., $20). The group difference on the average of investment across 10 rounds was tested with a one-way analysis of variance.
Total earning in MRT
The earning of the participant (Investor) in each round was calculated as the amount from the $20 kept by the Investor plus the amount of repayment sent from the Trustee. The group difference on the sum of earnings across 10 rounds was tested with a one-way analysis of variance.
Model-based analyses for behaviors in MRT
The foundation of players' payoff assessment was also based on the Fehr–Schmidt inequality aversion model17, but the envy term of the equation was omitted for the MRT. To distinguish the guilt parameter in the MRT from the one in the UG, the guilt parameter in the MRT is called inequality aversion, which quantifies the tendency to try and reach a fair outcome with values of {0, 0.4, 1}.
This model of MRT includes six other parameters: (1) planning horizon, which quantifies number of steps to likely plan ahead with values of {1, 2, 3, 4}; (2) theory of mind (ToM), which quantifies the number of mentalization steps with values of {0, 2, 4}; (3) inverse temperature, which quantifies the randomness of the choice preference with values of {1/4, 1/3, 1/2, 1}; (4) risk aversion, which quantifies the value of money kept over money potentially gained with values of {0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8}; (5) irritability, which quantifies tendency to retaliate on repayments worse than expected with values of {0, 0.25, 0.5, 0.75, 1.0}; and (6) irritability Belief, which quantifies the initial belief of likelihood of the partner being irritable with values of {0, 1, 2, 3, 4}. The values for the parameters were selected based on previous work23. The whole collection of parameters that best characterize an individual player are determined by maximizing the likelihood of their choices (over a grid of possible values). See ref. 23 for a detailed description of the model. Since these seven parameters from the model were ordinal variables, the tests of group effects for each parameter were conducted with independent-samples Kruskal–Wallis H tests. Post hoc comparisons were evaluated using two-tailed Mann-Whitney U-tests with Bonferroni correction.
Code availability
The code used to analyze data in the current study is available from the corresponding author on request.
The datasets generated and/or analyzed during the current study are available from the corresponding author on request. A reporting summary for this article is available as a Supplementary Information.
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We thank Laura Bateman and Carrie Bynum for their assistance. Our extreme gratitude also goes out to the families and individuals who have participated in this work over these many years. This work was funded by a Principal Research Fellowship from the Wellcome Trust (to P.R.M.), Virginia Tech (to P.R.M. and S.L.R.), and the FRQ-S and CIHR (to S.H.). P.F. is receipt of a National Institute for Health Research (NIHR) Senior Investigator Award (NF-SI-0514-10157). P.F. was in part supported by the NIHR Collaboration for Leadership in Applied Health Research and Care (CLAHRC) North Thames at Barts Health NHS Trust. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR, or the Department of Health. P.D. was funded by the Gatsby Charitable Foundation; he is currently at Max Planck Institute for Biological Cybernetics, Tuebingen, Germany.
These authors contributed equally: Yi Luo, Sébastien Hétu.
Virginia Tech Carilion Research Institute, Roanoke, VA, 24016, USA
Yi Luo, Sébastien Hétu, Terry Lohrenz, Sharon Landesman Ramey, Libbie Sonnier-Netto, Jonathan Lisinski, Stephen LaConte, P. Read Montague & Craig Ramey
Université de Montréal, Montreal, QC, H3C 3J7, Canada
Sébastien Hétu
Austrian Institute of Technology, 1210, Vienna, Austria
Andreas Hula
Wellcome Trust Centre for Neuroimaging, University College London, 12 Queen Square, London, WC1E 6BT, UK
Peter Dayan, Tobias Nolte & P. Read Montague
Gatsby Computational Neuroscience Unit, University College London, London, WC1E 6BT, UK
Peter Dayan
Anna Freud National Centre for Children and Families, 21 Maresfield Gardens, London, NW3 5SD, UK
Tobias Nolte & Peter Fonagy
Research Department of Clinical, Educational and Health Psychology, University College London, Gower Street, London, WC1E 6BT, UK
Peter Fonagy
Psychiatry Department, Virginia Tech Carilion School of Medicine, Roanoke, VA, 24016, USA
Elham Rahmani
Yi Luo
Terry Lohrenz
Sharon Landesman Ramey
Libbie Sonnier-Netto
Jonathan Lisinski
Stephen LaConte
P. Read Montague
Craig Ramey
Y.L., S.H., T.L., and P.R.M. designed the ultimatum game and the multi-round trust game experiment; C.R. and S.R. designed the ABC experiment and follow-up; Y.L., S.H., S.L.R., C.R., L.S. and P.R.M. performed research; Y.L., S.H., T.L., A.H., P.D., J.L., T.N., P.F., and P.R.M analyzed data; Y.L., S.H., T.L., A.H., P.D., S.L.R., J.L., S.L., C.R., T.N., P.F., E.R. and P.R.M. discussed the results; Y.L., S.H., T.L. and P.R.M. wrote the manuscript. A.H., P.D., S.L., T.N., S.L.R., E.R. and C.R. provided important comments and suggestions that significantly contributed to the manuscript.
Correspondence to P. Read Montague.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Luo, Y., Hétu, S., Lohrenz, T. et al. Early childhood investment impacts social decision-making four decades later. Nat Commun 9, 4705 (2018). https://doi.org/10.1038/s41467-018-07138-5
Exploring Parental Perceptions of Early Childhood Education Among Spanish-Speaking Latinx Families
Julia Rabin
Lisa Vaughn
Farrah Jacquez
Early Childhood Education Journal (2022) | CommonCrawl |
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Luis Mazorra-Aguiar & Philippe Lauret & Mathieu David & Albert Oliver & Gustavo Montero, 2021. "Comparison of Two Solar Probabilistic Forecasting Methodologies for Microgrids Energy Efficiency," Energies, MDPI, vol. 14(6), pages 1-26, March.
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June 2019 , Volume 1 , Issue 2
Accelerating Metropolis-Hastings algorithms by Delayed Acceptance
Marco Banterle, Clara Grazian, Anthony Lee and Christian P. Robert
MCMC algorithms such as Metropolis--Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer a useful generalisation of the Delayed Acceptance approach, devised to reduce such computational costs by a simple and universal divide-and-conquer strategy. The generic acceleration stems from breaking the acceptance step into several parts, aiming at a major gain in computing time that out-ranks a corresponding reduction in acceptance probability. Each component is sequentially compared with a uniform variate, the first rejection terminating this iteration. We develop theoretical bounds for the variance of associated estimators against the standard Metropolis--Hastings and produce results on optimal scaling and general optimisation of the procedure.
Marco Banterle, Clara Grazian, Anthony Lee, Christian P. Robert. Accelerating Metropolis-Hastings algorithms by Delayed Acceptance. Foundations of Data Science, 2019, 1(2): 103-128. doi: 10.3934/fods.2019005.
Flexible online multivariate regression with variational Bayes and the matrix-variate Dirichlet process
Victor Meng Hwee Ong, David J. Nott, Taeryon Choi and Ajay Jasra
2019, 1(2): 129-156 doi: 10.3934/fods.2019006 +[Abstract](2488) +[HTML](899) +[PDF](1020.87KB)
Flexible density regression methods, in which the whole distribution of a response vector changes with the covariates, are very useful in some applications. A recently developed technique of this kind uses the matrix-variate Dirichlet process as a prior for a mixing distribution on a coefficient in a multivariate linear regression model. The method is attractive for the convenient way that it allows borrowing strength across different component regressions and for its computational simplicity and tractability. The purpose of the present article is to develop fast online variational Bayes approaches to fitting this model, and to investigate how they perform compared to MCMC and batch variational methods in a number of scenarios.
Victor Meng Hwee Ong, David J. Nott, Taeryon Choi, Ajay Jasra. Flexible online multivariate regression with variational Bayes and the matrix-variate Dirichlet process. Foundations of Data Science, 2019, 1(2): 129-156. doi: 10.3934/fods.2019006.
Estimation and uncertainty quantification for the output from quantum simulators
Ryan Bennink, Ajay Jasra, Kody J. H. Law and Pavel Lougovski
The problem of estimating certain distributions over {0, 1}d is considered here. The distribution represents a quantum system of d qubits, where there are non-trivial dependencies between the qubits. A maximum entropy approach is adopted to reconstruct the distribution from exact moments or observed empirical moments. The Robbins Monro algorithm is used to solve the intractable maximum entropy problem, by constructing an unbiased estimator of the un-normalized target with a sequential Monte Carlo sampler at each iteration. In the case of empirical moments, this coincides with a maximum likelihood estimator. A Bayesian formulation is also considered in order to quantify uncertainty a posteriori. Several approaches are proposed in order to tackle this challenging problem, based on recently developed methodologies. In particular, unbiased estimators of the gradient of the log posterior are constructed and used within a provably convergent Langevin-based Markov chain Monte Carlo method. The methods are illustrated on classically simulated output from quantum simulators.
Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1(2): 157-176. doi: 10.3934/fods.2019007.
Levels and trends in the sex ratio at birth and missing female births for 29 states and union territories in India 1990–2016: A Bayesian modeling study
Fengqing Chao and Ajit Kumar Yadav
The sex ratio at birth (SRB) has risen in India and reaches well beyond the levels under normal circumstances since the 1970s. The lasting imbalanced SRB has resulted in much more males than females in India. A population with severely distorted sex ratio is more likely to have prolonged struggle for stability and sustainability. It is crucial to estimate SRB and its imbalance for India on state level and assess the uncertainty around estimates. We develop a Bayesian model to estimate SRB in India from 1990 to 2016 for 29 states and union territories. Our analyses are based on a comprehensive database on state-level SRB with data from the sample registration system, census and Demographic and Health Surveys. The SRB varies greatly across Indian states and union territories in 2016: ranging from 1.026 (95% uncertainty interval [0.971; 1.087]) in Mizoram to 1.181 [1.143; 1.128] in Haryana. We identify 18 states and union territories with imbalanced SRB during 1990–2016, resulting in 14.9 [13.2; 16.5] million of missing female births in India. Uttar Pradesh has the largest share of the missing female births among all states and union territories, taking up to 32.8% [29.5%; 36.3%] of the total number.
Fengqing Chao, Ajit Kumar Yadav. Levels and trends in the sex ratio at birth and missing female births for 29 states and union territories in India 1990\u20132016: A Bayesian modeling study. Foundations of Data Science, 2019, 1(2): 177-196. doi: 10.3934/fods.2019008.
On adaptive estimation for dynamic Bernoulli bandits
Xue Lu, Niall Adams and Nikolas Kantas
The multi-armed bandit (MAB) problem is a classic example of the exploration-exploitation dilemma. It is concerned with maximising the total rewards for a gambler by sequentially pulling an arm from a multi-armed slot machine where each arm is associated with a reward distribution. In static MABs, the reward distributions do not change over time, while in dynamic MABs, each arm's reward distribution can change, and the optimal arm can switch over time. Motivated by many real applications where rewards are binary, we focus on dynamic Bernoulli bandits. Standard methods like \begin{document}$ \epsilon $\end{document}-Greedy and Upper Confidence Bound (UCB), which rely on the sample mean estimator, often fail to track changes in the underlying reward for dynamic problems. In this paper, we overcome the shortcoming of slow response to change by deploying adaptive estimation in the standard methods and propose a new family of algorithms, which are adaptive versions of \begin{document}$ \epsilon $\end{document}-Greedy, UCB, and Thompson sampling. These new methods are simple and easy to implement. Moreover, they do not require any prior knowledge about the dynamic reward process, which is important for real applications. We examine the new algorithms numerically in different scenarios and the results show solid improvements of our algorithms in dynamic environments.
Xue Lu, Niall Adams, Nikolas Kantas. On adaptive estimation for dynamic Bernoulli bandits. Foundations of Data Science, 2019, 1(2): 197-225. doi: 10.3934/fods.2019009.
EmT: Locating empty territories of homology group generators in a dataset
Xin Xu and Jessi Cisewski-Kehe
Persistent homology is a tool within topological data analysis to detect different dimensional holes in a dataset. The boundaries of the empty territories (i.e., holes) are not well-defined and each has multiple representations. The proposed method, Empty Territory (EmT), provides representations of different dimensional holes with a specified level of complexity of the territory boundary. EmT is designed for the setting where persistent homology uses a Vietoris-Rips complex filtration, and works as a post-analysis to refine the hole representation of the persistent homology algorithm. In particular, EmT uses alpha shapes to obtain a special class of representations that captures the empty territories with a complexity determined by the size of the alpha balls. With a fixed complexity, EmT returns the representation that contains the most points within the special class of representations. This method is limited to finding 1D holes in 2D data and 2D holes in 3D data, and is illustrated on simulation datasets of a homogeneous Poisson point process in 2D and a uniform sampling in 3D. Furthermore, the method is applied to a 2D cell tower location geography dataset and 3D Sloan Digital Sky Survey (SDSS) galaxy dataset, where it works well in capturing the empty territories.
Xin Xu, Jessi Cisewski-Kehe. EmT: Locating empty territories of homology group generators in a dataset. Foundations of Data Science, 2019, 1(2): 227-247. doi: 10.3934/fods.2019010. | CommonCrawl |
Julius Reichelt
Julius Reichelt (1637–1717) was a German mathematician and astronomer who may have set up the first observatory in the city of Strasbourg.
Biography
Julius Reichelt was born on 5 January 1637 in the city of Strasbourg. In 1644 at the age of 16 he was enrolled as a student and graduated as Doctor of Philosophy in 1660. He was nominated as Professor of Mathematics in 1667. Reichelt traveled through Northern Europe some time after 1666 and met other notable 17th-century scientists and thinkers such as Jan Hudde, Johannes Hevelius, Andreas Concius, Henrik Ruse, Johannes Meyer, Rasmus Bartholin, and Adam Olearius. After his return from his trip Reichelt may have proposed the building of the "turret lantern" as an astronomical observatory at the top of Strasbourg's Hospital Gate based on the Rundetårn (Round Tower) he saw in Copenhagen. Reichelt died in Strasbourg on 19 February 1717.
References
• Strasbourg's "First" Astronomical Observatory by André Heck, Journal of Astronomical History and Heritage (JAH2), published 2010
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Fermat's spiral
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat.[1]
Their applications include curvature continuous blending of curves,[1] modeling plant growth and the shapes of certain spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotrons.
Coordinate representation
Polar
The representation of the Fermat spiral in polar coordinates $(r,\varphi )$ is given by the equation
$r=\pm a{\sqrt {\varphi }}$
for $\varphi \geq 0$.
The two choices of sign give the two branches of the spiral, which meet smoothly at the origin. If the same variables were reinterpreted as Cartesian coordinates, this would be the equation of a parabola with horizontal axis, which again has two branches above and below the axis, meeting at the origin.
Cartesian
The Fermat spiral with polar equation $r=\pm a{\sqrt {\varphi }}$ can be converted to the Cartesian coordinates $(x,y)$ by using the standard conversion formulas $x=r\cos \varphi $ and $y=r\sin \varphi $. Using the polar equation for the spiral to eliminate $r$ from these conversions produces parametric equations for one branche of the curve:
${\begin{cases}x(\varphi )=+a{\sqrt {\varphi }}\cos(\varphi )\\y(\varphi )=+a{\sqrt {\varphi }}\sin(\varphi )\end{cases}}$
and the second one
${\begin{cases}x(\varphi )=-a{\sqrt {\varphi }}\cos(\varphi )\\y(\varphi )=-a{\sqrt {\varphi }}\sin(\varphi )\end{cases}}$
They generate the points of branches of the curve as the parameter $\varphi $ ranges over the positive real numbers.
For any $(x,y)$ generated in this way, dividing $x$ by $y$ cancels the $ a{\sqrt {\varphi }}$ parts of the parametric equations, leaving the simpler equation ${\tfrac {x}{y}}=\cot \left(\varphi \right)$. From this equation, substituting $\varphi $ by $ \varphi =r^{2}/a^{2}$ (a rearranged form of the polar equation for the spiral) and then substituting $r$ by $ r={\sqrt {x^{2}+y^{2}}}$ (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only $x$ and $y$:
${\frac {x}{y}}=\cot \left({\frac {x^{2}+y^{2}}{a^{2}}}\right).$
Because the sign of $a$ is lost when it is squared, this equation covers both branches of the curve.
Geometric properties
Division of the plane
A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. Like a line or circle or parabola, it divides the plane into two connected regions.
Polar slope
From vector calculus in polar coordinates one gets the formula
$\tan \alpha ={\frac {r'}{r}}$
for the polar slope and its angle α between the tangent of a curve and the corresponding polar circle (see diagram).
For Fermat's spiral r = a√φ one gets
$\tan \alpha ={\frac {1}{2\varphi }}.$
Hence the slope angle is monotonely decreasing.
Curvature
From the formula
$\kappa ={\frac {r^{2}+2(r')^{2}-r\,r''}{\left(r^{2}+(r')^{2}\right)^{\frac {3}{2}}}}$
for the curvature of a curve with polar equation r = r(φ) and its derivatives
${\begin{aligned}r'&={\tfrac {a}{2{\sqrt {\varphi }}}}={\tfrac {a^{2}}{2r}}\\r''&=-{\tfrac {a}{4{\sqrt {\varphi }}^{3}}}=-{\tfrac {a^{4}}{4r^{3}}}\end{aligned}}$
one gets the curvature of a Fermat's spiral:
$\kappa (r)={\frac {2r\left(4r^{4}+3a^{4}\right)}{\left(4r^{4}+a^{4}\right)^{\frac {3}{2}}}}.$
At the origin the curvature is 0. Hence the complete curve has at the origin an inflection point and the x-axis is its tangent there.
Area between arcs
The area of a sector of Fermat's spiral between two points (r(φ1), φ1) and (r(φ2), φ2) is
${\begin{aligned}{\underline {A}}&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\,d\varphi \\&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi \,d\varphi \\&={\frac {a^{2}}{4}}\left(\varphi _{2}^{2}-\varphi _{1}^{2}\right)\\&={\frac {a^{2}}{4}}\left(\varphi _{2}+\varphi _{1}\right)\left(\varphi _{2}-\varphi _{1}\right).\end{aligned}}$
After raising both angles by 2π one gets
${\overline {A}}={\frac {a^{2}}{4}}\left(\varphi _{2}+\varphi _{1}+4\pi \right)\left(\varphi _{2}-\varphi _{1}\right)={\underline {A}}+a^{2}\pi \left(\varphi _{2}-\varphi _{1}\right).$
Hence the area A of the region between two neighboring arcs is
$A=a^{2}\pi \left(\varphi _{2}-\varphi _{1}\right).$
A only depends on the difference of the two angles, not on the angles themselves.
For the example shown in the diagram, all neighboring stripes have the same area: A1 = A2 = A3.
This property is used in electrical engineering for the construction of variable capacitors.[2]
Special case due to Fermat
In 1636, Fermat wrote a letter [3] to Marin Mersenne which contains the following special case:
Let φ1 = 0, φ2 = 2π; then the area of the black region (see diagram) is A0 = a2π2, which is half of the area of the circle K0 with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a2π2. Hence:
• The area between two arcs of the spiral after a full turn equals the area of the circle K0.
Arclength
The length of the arc of Fermat's spiral between two points (r(φi), φi) can be calculated by the integral:
${\begin{aligned}L&=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,d\varphi =\cdots \\&={\frac {a}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {1}{\varphi }}+4\varphi }}\,d\varphi .\end{aligned}}$
This integral leads to an elliptical integral, which can be solved numerically.
The arc length of the positive branch of the Fermat's spiral from the origin can also be defined by hypergeometric functions $\operatorname {_{2}F_{1}} \left(a,\,b;\,c;\,z\right)$ and the incomplete beta function $\operatorname {B} \left(z;\,a;\,b\right)$:[4]
${\begin{aligned}L&=a\cdot {\sqrt {\varphi }}\cdot \operatorname {_{2}F_{1}} \left(-{\frac {1}{2}},\,{\frac {1}{4}};\,{\frac {5}{4}};\,-4\cdot \varphi ^{2}\right)\\&=a\cdot {\frac {1-i}{8}}\cdot \operatorname {B} \left(-4\cdot \varphi ^{2};\,{\frac {1}{4}},\,{\frac {3}{2}}\right)\\\end{aligned}}$
Circle inversion
The inversion at the unit circle has in polar coordinates the simple description (r, φ) ↦ (1/r, φ).
• The image of Fermat's spiral r = a√φ under the inversion at the unit circle is a lituus spiral with polar equation
$\;r={\frac {1}{a{\sqrt {\varphi }}}}.$
When φ = 1/a2, both curves intersect at a fixed point on the unit circle.
• The tangent (x-axis) at the inflection point (origin) of Fermat's spiral is mapped onto itself and is the asymptotic line of the lituus spiral.
The golden ratio and the golden angle
In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H. Vogel in 1979[5] is
${\begin{aligned}r&=c{\sqrt {n}},\\\theta &=n\times 137.508^{\circ },\end{aligned}}$
where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.[6]
The pattern of florets produced by Vogel's model (central image). The other two images show the patterns for slightly different values of the angle.
The resulting spiral pattern of unit disks should be distinguished from the Doyle spirals, patterns formed by tangent disks of geometrically increasing radii placed on logarithmic spirals.
Solar plants
Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants.[7]
See also
• List of spirals
• Patterns in nature
• Spiral of Theodorus
References
1. Anastasios M. Lekkas, Andreas R. Dahl, Morten Breivik, Thor I. Fossen: "Continuous-Curvature Path Generation Using Fermat's Spiral" Archived 2020-10-28 at the Wayback Machine. In: Modeling, Identification and Control. Vol. 34, No. 4, 2013, pp. 183–198, ISSN 1890-1328.
2. Fritz Wicke: Einführung in die höhere Mathematik. Springer-Verlag, 2013, ISBN 978-3-662-36804-6, p. 414.
3. Lettre de Fermat à Mersenne du 3 juin 1636, dans Paul Tannery. In: Oeuvres de Fermat. T. III, S. 277, Lire en ligne.
4. Weisstein, Eric W. "Fermat's Spiral". mathworld.wolfram.com. Retrieved 2023-02-04.
5. Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences. 44 (3–4): 179–189. doi:10.1016/0025-5564(79)90080-4.
6. Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8.
7. Noone, Corey J.; Torrilhon, Manuel; Mitsos, Alexander (December 2011). "Heliostat Field Optimization: A New Computationally Efficient Model and Biomimetic Layout". Solar Energy. 86 (2): 792–803. doi:10.1016/j.solener.2011.12.007.
Further reading
• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 31, 186. ISBN 0-486-60288-5.
External links
• "Fermat spiral". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• Online exploration using JSXGraph (JavaScript)
• Fermat's Natural Spirals, in sciencenews.org
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Related
• List of things named after Pierre de Fermat
• Wiles's proof of Fermat's Last Theorem
• Fermat's Last Theorem in fiction
• Fermat Prize
• Fermat's Last Tango (2000 musical)
• Fermat's Last Theorem (popular science book)
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How big a lead can a football team have?
Written by Colin+ in sport.
A reader asks:
What's the biggest lead a football team can have in the table after $n$ games?
In a typical football league, teams get three points for a win, one for a draw, and none for getting beat. After, for example, one game, if one team wins and all of the other games are draws, the winners will have three points, while everyone except the team they beat will have one point — the winners will be two points ahead.
There's not a whole lot more to it — after two games, the biggest possible lead is four points (one team wins both of its games to get six points, and all of the others are draws, leaving everyone else with at most two points). As long as the winning team hasn't played all of the teams, the biggest lead after $n$ games is $2n$ points.
But what if they've played everyone?
In a four-team group, it's possible to have a seven-point lead after three games, rather than just six: if you beat all three of the other teams, you'll have nine points; if they all draw with each other, they each have two points. Assuming you always win and everyone else always draws, once you've played everyone once, you'll have $3n$ points, and the best of the rest will have $n-1$ points - they'll have drawn every game except for the one they lost to you - giving you a margin of $3n - (n-1) = 2n+1$.
In general, if you've played everyone at least $m$ times, your biggest possible margin is $2n + m$. So, when Dunfermline beat the other nine teams in Scottish League One four times, and they all draw with each other, they'll have a lead of $2 \times 36 + 4 = 76$ points.
Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.
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DeepSV: accurate calling of genomic deletions from high-throughput sequencing data using deep convolutional neural network
Lei Cai1,
Yufeng Wu2 &
Jingyang Gao ORCID: orcid.org/0000-0003-1270-62571
Calling genetic variations from sequence reads is an important problem in genomics. There are many existing methods for calling various types of variations. Recently, Google developed a method for calling single nucleotide polymorphisms (SNPs) based on deep learning. Their method visualizes sequence reads in the forms of images. These images are then used to train a deep neural network model, which is used to call SNPs. This raises a research question: can deep learning be used to call more complex genetic variations such as structural variations (SVs) from sequence data?
In this paper, we extend this high-level approach to the problem of calling structural variations. We present DeepSV, an approach based on deep learning for calling long deletions from sequence reads. DeepSV is based on a novel method of visualizing sequence reads. The visualization is designed to capture multiple sources of information in the sequence data that are relevant to long deletions. DeepSV also implements techniques for working with noisy training data. DeepSV trains a model from the visualized sequence reads and calls deletions based on this model. We demonstrate that DeepSV outperforms existing methods in terms of accuracy and efficiency of deletion calling on the data from the 1000 Genomes Project.
Our work shows that deep learning can potentially lead to effective calling of different types of genetic variations that are complex than SNPs.
High-throughput DNA sequencing technologies have generated vast amount of sequence data. These data enable novel approaches for studying many important biological questions. One example is calling genetic variations such as SNPs or SVs from sequence data. There have been many existing computational methods for sequence-based calling of SNPs or SVs. For example, for SNP calling, one popular caller is GATK [1]. On the high level, calling genetic variations from sequence data can be viewed as a classification problem in machine learning. That is, given the sequence data at a candidate variant site, we are to classify the site into one of the two categories: variant or wild-type. Among many existing classification approaches, deep learning based on e.g. convolutional neural network (CNN) is becoming increasingly popular. CNN has outperformed existing approaches in a number of important applications. Among these, the most noticeable application of CNN is image processing, where deep learning has significantly improved the state of the art [2]. A natural research direction is using CNN for genetic variant calling with sequence data. Recently, Google's DeepVariant [3] was developed to call SNPs and short insertion/deletions (indels) from sequence data. The key idea of DeepVariant is viewing the mapped sequence data as images, and treating variant calling as a special kind of image classification. It is reported that DeepVariant can outperform GATK in SNP calling. This demonstrates the potentials of deep learning in the sequence data processing domain. The DeepVariant approach raises a natural research question: can deep learning be applied to call other types of genetic variations from sequence data that are more complex than SNPs and short indels? In this paper, we provide a positive answer for this question: we show that deep learning can be used for accurately calling structural variations from sequence data.
Structural variation refers to relatively long genomic variation, such as deletion, insertion and inversion. Structural variation will lead to complications of many diseases [4], and many cancers are associated with genetic variation [5]. To be specific, we focus on calling long deletions (longer than 50 bp) in this paper. For deletion calling, there exists many approaches including Pindel [6], BreakDancer [7], DELLY [8], CNVnator [9], Breakseq2 [10], Lumpy [11], GenomeStrip2 [12], and SVseq2 [13], among others. Most of these approaches rely on one or multiple information (called signatures) extracted from mapped sequence data: (i) read depth, (ii) discordant read pairs and (iii) split reads. We note that there are also methods performing sequence assembly for deletion calling. While many of the existing methods have been used in large genomics projects such as the 1000 Genomes Project [14], there exists no single method that clearly outperforms other approaches.
In this paper, we present DeepSV, a deep learning based method for long deletion calling from sequence data. DeepSV builds on the general approach of DeepVariant by visualizing mapped sequence reads as images. The key technical aspects of DeepSV are the novel visualization techniques for CNN-based deletion calling and how to work with noisy training data. The visualization procedure combines all major aspects of features with regard to deletions: read depth, split read and discordant pairs. This avoids manual selection of features for classification. We demonstrate that DeepSV outperforms existing methods in calling long deletions on real sequence data from the 1000 Genomes Project. Our work extends the findings of DeepVariant by showing that deep learning can be useful for calling structural variations that are more complex than SNPs and short indels.
The rest of the paper is organized as follows. In Section 2, we survey the existing approaches for calling structural variations from sequencing data, and the application of the machine learning in the this subject. In Section 3, we present our deep leanring based SV calling method. In Section4, we present the research results. In the last section, we provide discussions on the DeepSV approach.
Genomic deletions affect several aspects (called signatures) of the sequence reads mapped onto the given reference genome near the deletion site. (i) Read depth. Mapped read depth within a deletion is likely to be lower than those at wild-type sites. If the deletion is homozygous and the reads are mapped correctly (e.g. not mis-mapped due to repeats), read depth within the deletion should be close to zero. If the deletion is heterozygous, read depth within the deletion should still be lower than expected. Thus, low read depth is a signature of deletions. (ii) Discordant read pair. Consider paired-end reads that are mapped near the deletion with two ends being to the different sides of a deletion. Such read pair is called encompassing pair for the deletion. The mapped insert size (i.e. the outer distance of the two mapped reads of the pair) of an encompassing pair appears to be longer than usual due to the presence of the deletion. This is because the mapped insert size includes the length of the deletion on the reference genome. We say an encompassing read pair is discordant if the difference between its mapped insert size and the known library insert size is at least three times of the standard deviation of the library insert size. Otherwise, we say the read pair is concordant. The longer the deletion is, the more likely an encompassing pair becomes discordant. (iii) Split reads. When a read overlaps the breakpoints of a deletion, the read consists of two parts that are not contiguous on the reference: the part proceeding the left breakpoint and part following the right breakpoint. Such a read is called split read. Here, breakpoint refers to the boundary of the deletion on the reference genome. When a split read is mapped, the read cannot be mapped as a whole. Instead, it is mapped onto two discontinuous regions of the reference. These signatures reveal different aspects of structural variations. A main advantage of using split reads is that split reads can potentially reveal the exact breakpoints of the deletion. In contrast, read depth and discordant pairs cannot lead to exact breakpoints. See Fig. 1 for an illustration.
Above the image is high coverage data, and below is low coverage. The split reads are marked with ① and the discordant reads are marked with ②
Most existing deletion calling methods utilize the above three types of signatures. Earlier methods often use a single signature. For example, BreakDancer uses only discordant read pairs. The original Pindel only relies on split-reads. A number of methods combine multiple signatures and obtained more accurate results. For example, several methods such as DELLY, MATE-CLEVER [15], and SVseq2 combined discordant read pairs and split reads. A main advantage of integrating multiple signatures is better utilizing the information contained in the sequence data. However, it is unclear what is the best way for integrating different signatures for deletion calling. A simple approach is weighting different signatures (e.g. a split read is weighted as 1 and a discordant pair is weighted as 2). Obviously, this leads to the issue of choosing the weights, which is often difficult in practice [16]. Indeed, it is known that parameter settings can affect the results of many existing SV callers [17].
A useful observation is that SV calling can be viewed as a classification problem. That is, for a candidate SV, we want to classify this candidate site to be either a true deletion (denoted as 1) or a non-deletion (denoted as 0) based on the given sequence data near the candidate site. Classification is an important subject of machine learning and there are many existing machine learning methods for classification. Usually classification involves two steps. First, a model is trained from training data. Second, the trained model is used to classify the test data. A main advantage of using a classification model is that there is no need to manually choosing the parameters; parameters are obtained from the training data. There are existing machine learning based approaches for SV calling, including GINDEL [18] and Concod [19]. While these machine learning based methods show promises in accurate calling of SVs, there are also difficulties faced by traditional classification methods. One of the most important issues for traditional classification is feature selection. That is, we need to determine what specific quantities to extract from sequence data to be used in classification. Due to the complex nature of structural variations, it is often unclear what are the best features.
Recently, deep learning is becoming increasingly popular. Deep learning approaches (such as convolutional neural network or CNN) have been applied to several important problems (e.g. image processing, computer vision, natural language processing, to name a few) and led to significant improvements in performance over existing methods. A main advantage of deep learning is that it reduces the need of feature engineering and can potentially better utilize the data. On the other hand, a main disadvantage is that deep learning usually needs more complex models that are more difficult to train than those in traditional approaches. Application of deep learning in sequence data analysis is in its infancy. A pioneering work in this area is Google's DeepVariant. DeepVariant proposes a novel approach for sequence data analysis for the purpose of genetic variant calling: treating mapped sequence data as images and converting genetic variant calling to an image classification problem. As shown in Fig. 2, mapped sequence reads have a natural visualization [20]. Mapped sequence reads near a variant (say a short deletion) or wildtype site can be easily recorded as an image.
The red line is the read-depth value, the green line is split-read value and the insert size spaning the deletion region is larger than library value
This image has different visual appearance at a variant from that from a wildtype. At a short deletion site, the image tends to have a gap. Moreover, read depth tends to be lower than that of the wildtype because short deletion may cause some reads unaligned. DeepVariant relies on the visual difference of images to perform classification for genetic variant calling.
The DeepVariant approach leads to a natural question: can we use deep learning to call more complex genetic variants such as long deletions? SV calling can be somewhat more difficult than SNP or short indel calling. First, SNP calling is more localized: reads relevant to a SNP can be easily fit into a single image. Reads that are relevant for a long deletion can spread out. For example, two ends of a discordant read pair over a long deletion can be mapped to positions that are more than thousands of bases apart. Second, there are more signatures for long deletions than those for SNPs or short indels. For example, discordant read pairs are not associated with SNPs but are important for long deletions. Integrating these diverse set of signatures in visualization needs to be worked out. In this paper, we present DeepSV, a deep learning based method for calling long deletions from sequence reads, which addresses these difficulties.
General description of DeepSV
DeepSV is a deep learning based structural variation calling method. It is based on a new sequence reads visualization approach, which converts mapped sequence reads to images. DeepSV follows the general approach of DeepVariant. Different from DeepVariant, DeepSV aims to calling SVs (especially long deletions that are longer than 50 bp). There are two components in DeepSV: training and variant calling. Both components take mapped reads and the reference genome as input. For model training, DeepSV trains a convolutional neural network (CNN) model from sequence reads with known deletions. Similar to DeepVariant, CNN model training is based on visualizing mapped sequence reads near known deletion sites or at wildtype sites. The key technical aspect of DeepSV is how to train the CNN from sequence images near a deletion. Recall that long deletions have more complex signatures than SNPs or short indels. Note that deletion can be long and different regions of a deletion can be quite different in the visualized reads. For example, near the breakpoint, there is likely a sharp transition from high read depth to low read depth. In the middle of a deletion, there may be no such transition but the read depth can be lower than that near the breakpoint (See Additional file 1: Figure S1).
In order to accurately call deletions with precise breakpoints, it is important to separate these cases. Moreover, to accommodate various signatures of a deletion, DeepSV implements a visualization procedure that takes advantage of the rich information contained in an image to integrate various signals. In a typical color map, there are 8 bits for red, green and blue and so there can be 256 choices for each color. Therefore, one can use various combinations of the three basic colors to represent the configuration of the mapped reads. For example, a pixel corresponding a base of a mapped read can be affected by multiple factors such as whether the read is split read, the quality of the read, whether there is a discordant read pair and so on. When the CNN model is trained, the model is used to call deletions from the sequence images.
DeepSV workflow
Figure 3 shows the overall workflow of DeepSV. DeepSV is composed of three parts. In the first part (Fig. 3a), DeepSV begins by finding candidate deletions in reads aligned to the reference genome using clustering. In the second part (Fig.3b), the deep learning model is trained using a pileup image of the reference and reads around each candidate variant. Pileup image refers to the vertical alignment of bases at each site, rather than the horizontal alignment of bases as reads. The difference between pileup image and tiled image is shown in Additional file 1: Figure S2. In the third part (Fig.3c), the model is used to call the variants.
Overview of the DeepSV. DeepSV extracts candidates training set from the original sequence, and then uses a gradient descent algorithm to train CNN network, and finally generates images according to gene sequence for calling deletions. The gradient descent algorithm is an optimization algorithm commonly used in machine learning and artificial intelligence to recursively approximate the minimum deviation model. a Finding candidate variants and Encoding pileup images. b Training CNN model. c Calling deletions
DeepSV implementation details
To train a classification model, DeepSV takes the aligned sequence reads in binary sequence alignment (BAM) file and a variant call format (VCF) file which containes the known deletions. From sequence data to mapped image, DeepSV goes through three stages. Figure 4 illustrates the DeepSV approach. In the first stage, DeepSV performs filtering operations because the fluctuation of read depths will affect the clustering results. In the second stage, DeepSV eliminates false positives by clustering and then determines precise breakpoints. In the last stage, DeepSV visualizes mapped sequence reads based on sequence characteristics.
The process of DeepSV. DeepSV processing data is divided into three steps: filtering, clustering and visualization
Dealing with noise
Real sequence data tends to have significant noise, which can make the clustering perform poorly. The following lists several such cases.
The read depths fluctuate and some positions have read depths that are either too high or too low than expected. For example, read depths at some sites in the non-deletion region can be very low, while read depths of some sites in a deletion can be very high. Read depth near the breakpoints fluctuates significantly.
The difference on discordant paired-end reads between a deletion and a non-deletion is not obvious when coverage is low.
To address the above two issues, DeepSV uses the following techniques to reduce the effect of noises. First, DeepSV uses a 61 bp long sliding window to filter the read depths. DeepSV uses the following filter formula: \( \frac{\overline{D}}{\sigma}\gamma \). The mapping read depths within the window are d1, d2, ..., d61 (computed by SAMtools [20]), and the D¯ is the average of the read depths of the window. σ is the standard deviation of di and γ represents a coefficient. D¯ can indicate the situation where the read depth is high in the non-deletion region or low in the deletion region. σ reflects the fluctuations of depth. The value of γ is chosen to amplify the trend of the depth values in the window. Our experience indicates that this filtering step reduces the effect of the noise in the data, and improves the performance of the clustering. Now we consider discordant reads and split reads. Since the split read count and the discordant read count can be inversely proportional to the read depth near the breakpoints, we use the negation of the discordant read counts and split read counts (instead of the reads counts themselves). This is to ensure that each feature used in the clustering has the same trend for deletions or non-deletions. This improves the performance of the clustering. Figure 5 shows the clustering details. In the Fig. 5a, the green dots represent the feature points of the deletion regions, and the red and blue colors respectively represent the feature points at upstream and downstream of the deletion region. When there is no filtering, many singular values will appear, and these singular values will be incorrectly assigned to other classes. From Fig. 5b, we can see many singular values are removed by filtering. Figure 5c and d show the comparison before and after filtering. DeepSV successfully excludes the eigenvalues of special sites by filtering.
a, b Represent the clustering result of feature points, in which outliers are drawn in circles, and these boundary outliers are easily clustered into other categories. c, d Show the process of filtering features, where the box is a sliding window
In order to ensure that the boundary of clustering is as close as possible to the breakpoints and improve the accuracy of called deletions, DeepSV uses a modified Euclidean distance formula for the distance measure between two points used in the clustering. Euclidean formula can better reflect the distance relationship of space vectors. It is a commonly used distance formula in clustering algorithm. Considering that each coordinate in multidimensional vector contributes equally to Euclidean distance and they often have random fluctuations of different sizes, we have improved the formula by adding weight adjustment.
$$ Dis\tan ce\left({\mathrm{C}}_{\mathrm{i}},{C}_j\right)=\sqrt{\sum \limits_{k=1}^n\eta {\left(|{C}_{ik}-{C}_{jk}|\right)}^p} $$
CiandCj Ci and Cj represent two points of cluster set. Cik and Cjk CikandCjk are the components of vectors in the form of (normalized read depth, negated split read count, negated discordant read count). When the Cik, Cjk, Cjk values are the normalized depths, η is a fixed number that is greater than one. For discordant read count and split read count, η is set to one. The constant p is a fixed even number.
Clustering process
After dealing with the origin sequence, the k-means clustering algorithm is used on feature sets. We define each point in the sets as a triple, (read depth, discordant read pair count, split-read count). Therefore, we let k = 3 and run k-means clustering to cluster the positions into three categories. The three clusters are denoted as S1, S2,S3 S1, S2, S3 which correspond to the upstream, deletion and downstream regions respectively. Note that at a SV site, read depth tends to decrease while split read count and discordant read count tend to decrease. So we compute a feature value m for each position, where m is equal to the read depth minus split read count and discordant read count. We compute the average feature value of all positions in each of the three clusters S1, S2,S3 S1, S2, S3, which we denote as \( {\overline{\mathrm{m}}}_1,{\overline{m}}_2,{\overline{m}}_3 \) \( {\overline{m}}_1,{\overline{m}}_2,{\overline{m}}_3 \). We let \( \overline{\mathrm{m}} \) \( \overline{m} \) be the minimum of the three mean values. If the positions from the cluster with the minimum mean are largely between the points of the other clusters, then this deletion is considered to be a true deletion. Otherwise, it is a false positive. Figure 6 shows the clustering results. The effect of false positives will be presented in the results section.
shows that DeepSV can filter out the false positives in VCF by clustering the results that do not conform to the deletion characteristics
Sometimes the given breakpoints are not very accurate. This can lead to wrong labeling of the training images, especially near the boundary of the deletion. To find exact breakpoints, we consider the minimum mean cluster S2 S2 with mean ¯2 \( {\overline{m}}_2 \). DeepSV sorts the positions of S2 S2. The minimum and maximum positions, denoted as β1 and \( {\beta}_{\begin{array}{l}2\\ {}\end{array}} \) β1andβ2, are treated as initial breakpoints. There are two cases for the interval [β1, β2] [β1, β2].
[β1, β2] [β1, β2] is close to the given breakpoints.
[β1, β2] [β1, β2] doesn't include the given breakpoints due to the length of the deletion being too long.
We set two pointers ρ1 and ρ1andρ2 ρ2 to β1 β1andβ2 and \( {\beta}_{\begin{array}{l}2\\ {}\end{array}} \) respectively. In the first case if the feature value of ρ1 ρ1 is less than ¯2 \( {\overline{m}}_2 \), then ρ1 ρ1 is moved to the left by one. Similarly, if the feature value of ρ2 ρ1 is less than ¯2 \( {\overline{m}}_2 \), then ρ2 ρ2 is moved to the right by one. In the second case, the two pointers move in the opposite direction. When this process finishes, the two pointers provide the estimate of the two breakpoints of this deletion. Figure 7 shows the breakpoint finding process.
β1and \( {\beta}_{\begin{array}{l}2\\ {}\end{array}} \) represent the initial breakpoints. ρ1 and ρ2 show the moving pointer
Visualizing mapped sequence reads
Model training needs a set of labeled training samples. For image-based deletion calling, we need two sets of images: images from the deletion regions (labeled as 1) and images from the wild-type regions (labeled as 0). In principle, since the deletions are given in the VCF file, creating training sample is straightforward. Mapped reads have the natural pileup form and can be easily converted to images. 1-labeled images are taken within the known deletions and 0-labeled images are from outside the deletions. We partition the reference genome into consecutive non-overlapping windows of 50 bp. The aligned reads in the pileup format are converted into an image. Once the training data is obtained, one may train a CNN.
So far, we have labeled regions along the reference genome to be either deletion or non-deletion. We now describe how to create images for each region. This is a critical step because real sequence data tends to be complex. If the visualization approach is not chosen properly, the CNN may not capture the underlying information about the deletion from the created images.
Recall that an image is composed of pixels, and each pixel has (R, G, B) three-primary colors. The reason for using (R, G, B) image coding sequence is that the single channel graph can only represent gray level image, and the expression ability is limited, while the sequence information is complex. Especially, the CNN has the advantage of natural image processing. DeepSV takes the following simple approach for visualizing the reads: each nucleotide (i.e. A, T, C, or G respectively) is assigned one of these base colors: red (255,0,0), green (0,255,0), blue (0,0,255) and black (0,0,0) respectively; then the base color is slightly modified to integrate the various signatures on deletions. DeepSV considers all the aligned bases (i.e. a column in the image) at a position of the current region. For population sequence reads, a large percentage (95% or higher) of sites are not polymorphisms and the bases from one site tend to have the same color. This gives the visual appearance of column-based images. See Fig. 8 for an illustration. Our experience shows that such column-based images reveal the key aspects of deletions.
Each base is assigned one base color, and each color is composed of three primary colors. According to the characteristics of alignment, the bases at each site are assigned different colors
Integration of deletion signatures. Recall that there are various signatures on deletions (i.e. read depth, discordant pair and split read). Read depth is naturally represented by the pileup images. DeepSV integrates the other two types of signatures by slightly modifying the base colors of the mapped bases based on the signatures. Note that such modification is usually mild and does not destroy the column-based appearance of images. Each read contains multiple aspect of information, e.g., whether it belongs to discordant paired-end reads and whether it is split read. DeepSV uses the following combination of features to determine the color of each mapped base. More specifically, the color of a mapped base is determined by the four quantities, which describe the discordant read pair and split read information at the position. These quantities are explained in Table 1. The sum of these four quantities provide the auxiliary components of the coloring. To see how these four quantities are used to decide the color of a mapped base, we consider the following example. Consider a column of aligned bases (which are say all A's). We first count the number of discordant paired reads (which is say 10), and the number of split reads (which is say 0) that overlap this column. This leads to a color setting (255, 10, 10) for all bases in the column. We then consider each base of the column one by one. For this, we find the four features (is paired, concordant/discordant, mapping quality and map type). Say these features have values (1, 1, 1, 0). This leads to a binary number 13 = 10 + 1 + 1 + 1 + 0.
Table 1 Visualizing an aligned base. Feature: information carried by the base on deletions
As shown in Table 1, each of the four bases has four binary feature values. This gives total 64 combinations. We show an example (See Additional file 1: Table S1) that describes 64 value combinations and color scope. The color range of A base is (255, 0, 0) ~ (255, 235, 235), T base is (0, 255, 0) ~ (235, 255, 235), C base is (0, 0, 255) ~ (235, 235, 255), and the G base is (0, 0, 0) ~ (235, 235, 235).
Because the range of deletion length is from 50 bp to more than 10kbp, a single pileup image cannot cover an entire deletion and discordant pairs may not be contained in a single image. At the same time, because the network requires uniform batch size, we need to use normalized images. DeepSV divides the sequence (including deletion region and non-deletion region) into equal length regions. The length of the region can be customized by users. In this experiment, DeepSV generates the fixed length images in units of 50 bp. Note that the background of each picture is white. This is illustrated in Fig. 9.
These images contain different features. Each vertical colorful bar represents bases aligned to this site. The length of the vertical bar becomes lower in the deletion region but higher in the non-deletion region. a represents the homozygous deletion. b represents the heterozygous deletion. c shows the non deletion. d describes how pictures are sent into a neural network for training and prediction
Training and validating model
DeepSV trains the CNN model with real sequence reads and benchmarked deletions. Tensorflow [21] is used to construct the convolutional neural network and the batch size is set to 128. All genome data analysis is performed on a Linux server with 1080Ti GPUs [22] and a platform of Digits [23]. The parameters setting of model on deletion calling are given in the Additional file 1 (section 3: The parameters setting of model on deletion calling).
Picture labeling and normalization. The nucleotide sequence of the entire region is divided into 50 bp regions. DeepSV generates images of 256 by 256 for these 50 bp and labels each image as 0 or 1. We use all the labeled images in CNN training and use the trained model to call the deletion of the test data. The labeled images are shown in Additional file 1: Figure S4. The deletion calling process is shown in Additional file 1: Figure S5.
We now validate the performance of DeepSV using real data from the 1000 Genomes Project. The called deletions released by the 1000 Genomes Project (phase three) are used as the ground truth for benchmark. The data we use in this paper consist of 40 BAM (binary sequence alignment/map format) files with 20 individuals on chromosomes 1~22. The average insert length is 456 bp, and the standard deviation is between 57 bp~ 78 bp. The average coverage is 10X and 60X. These individuals from three different populations including Yoruba in Ibadan, Nigeria (YRI), Han Chinese in Beijing, China (CHB), and Utah Residents with Northern and Western European Ancestry (CEU). DeepSV needs training data. We use about half of data for the purpose of training, and use the remaining data for testing. The training data set and the test data set are divided according to the following two criteria: (i) ensuring that the training data is sufficient for the model to converge. (ii) ensuring that the test data is sufficient to cover various targets to be detected.
Under the premise of satisfying the above two criteria, the ratio of the training set and the test set can be adjusted according to the actual situation. In this experiment, the training set and the test set are each 50%. For training, we use the data from chromosomes 1 to 11 of these 20 individuals. For testing, we use the data from the chromosomes 12 to 22. Data used in the experiments is given in the Additional file 1: Table S3.
We compare DeepSV with other eight tools including Pindel, BreakDancer, Delly, CNVnator, Breakseq2, Lumpy, GenomeStrip2, and SVseq2. To show the advantage of deep learning, we also compare with an existing machine learning based method, Concod. We examine various aspects of deletion calling by DeepSV and other tools, including the accuracy of calling deletions of different sizes, breakpoint accuracy, impact of sequence data coverage, and the effect of model's activation on precision and loss.
Calling deletions of different sizes
We first evaluate the performance of DeepSV for calling deletions of various sizes. We use the deletions on chromosomes 12 to 22 from 20 individuals from the 1000 Genomes Project as the benchmark. Figure 10 shows the deletion distribution of these benchmarked deletions. We divide the lengths of deletions into five categories. The deletion length roughly follows a normal distribution.
Deletion length distribution for deletions in the chromosomes 12 to 22. The deletion length obeys the normal distribution
We compare the accuracy of deletion calling of different deletion sizes on DeepSV and other calling tools with low or high coverage data. To measure the performance of deletion calling, we use the following statistics: precision, sensitivity, and the F-score. The results on low coverage are shown in Additional file 1: Table S4. The results on high coverage are shown in Additional file 1: Table S5.
DeepSV for long deletions: the effect of complex SV
We now take a closer look at the performance of DeepSV on calling long deletions. Deletion can be classified into homozygous and heterozygous deletion. Moreover, there can exist other types of structural variations (e.g. insertion, translocation and inversion) near the deletion region. This type of structural variation is called complex SV. The importance of complex SV has been recently noticed in the literature [24]. A deletion of longer size is more likely to be a complex SV than shorter deletions. When a deletion is complex, the images created by DeepSV tends to be less clear cut than those from a simple deletion. To evaluate the performance of DeepSV on calling different types of deletions, we study how DeepSV's performance changes when the number of other types of structural variations (e.g. insertions, translocations and inversions) increases. The results are shown in Fig. 11. Our results show that the accuracy of calling homozygous deletions is not affected by the presence of other types of SVs in general. The same roughly holds for heterozygous deletions. For complex deletions, the presence of other SVs reduces the calling accuracy significantly.
Accuracy of calling deletions for different type. Horizontal axis: number of other types of SVs (e.g. insertion/translocation/inversion) near the deletions. Vertical axis: deletion calling accuracy (part d)
Breakpoint accuracy
In this section, we compare the performance of DeepSV with other tools on breakpoint accuracy. Figure 12 shows the distance between the detected and true breakpoints on the all genome data. Once again, breakpoint predictions given by DeepSV are closest to the true breakpoint positions. In many cases, the predicted breakpoint positions of DeepSV are only up to a few base pairs away from the true breakpoint positions. Predictions from other tools are usually further away from the true breakpoint positions. Among them, SVseq2, CNVnator, Lumpy, and BreakSeq2 appear to have the lower breakpoint accuracy than other tools. GenomeStrip2 has the highest resolution in the tools we compared. Our results suggest that DeepSV appears to capture the key characteristics of breakpoints from the images.
Breakpoint accuracy on different detecting tools. The y-axis represents the distance between detected breakpoints and true breakpoints
Impact of reads coverage
Sequence reads coverage can have a large effect on deletion calling. Reads coverage also affects the performance of DeepSV in both training and testing. In order to evaluate the performance of DeepSV on datasets with different coverage, we perform down sampling for the original high coverage data. SAMtools with option "view -s" is used for down sampling, and four datasets are generated with average coverage 48 ×, 36 ×, 24 ×, 12 × respectively. Table 2 shows the precision, sensitivity of DeepSV and the other eight tools on the genome data. In the table, we can see DeepSV has the highest precision compared with all other tools at various coverage. The sensitivity of DeepSV is overall comparable with the best of the other tools.
Table 2 Precision, sensitivity of multiple tools on different coverage data. P indicates precision. S indicates sensitivity
Deletion calling for various frequencies
To see whether DeepSV performs consistently on different population deletion frequencies, we now show the results of DeepSV on various deletion frequencies within the population and compare with the other eight tools. Note that, the precision cannot be calculated because the deletion frequency for the false positive deletions called out by each tool is unknown. Therefore, we only list the sensitivity here. Additional file 1: Table S6 shows the performance of each tool on different deletion frequencies. The results show that DeepSV outperforms the other eight tools for different deletion frequencies. The sensitivity of most tools increases as deletion frequencies increase. GenomeStrp2 has the better sensitivity for medium deletion frequency of 6–10. CNVnator has the lowest sensitivity for medium deletion frequency of 1–5.
Deletion calling for an individual not in training
So far, we use data from 20 individuals where half chromosomes are used for training and the other half for testing. Since training and testing are on different chromosomes, testing is considered to be independent from training. To further validate our method, we now show deletion calling performance for an individual (NA12891) that is not used in the training. The results for this individual with various coverage are shown in Fig. 13. We can see that DeepSV performs well for this new individual with performance similar with those from other individuals.
Performance of deletion calling for NA12891. To verify the DeepSV's validity, we test the individual of NA12891 that is independent from traning and testing
Machine learning and deletion calling
There are existing methods that use other types of machine learning approaches for deletion calling. Concod is one such example. Concod is based on manual feature selection. It performs the consensus-based calling with a support vector machine (SVM) model. We now compare DeepSV with Concod in deletion calling. Both methods need model training. We compare the model training accuracy and loss, as well as the running time of the two methods. Here, model training loss is the model misclassification error on the training data on a trained model. That is, we first use training data to train the model. Then we treat the training data as the test data to see if the model classifies the training data correctly. Note that there is an overfitting issue: a model classifies training data well may not generalize to test data. Nonetheless, a good machine learning model should have small training loss. The results are shown in Fig. 14a, b.
Compared to traditional machine learning method (e.g., Concod), DeepSV shows a more stable state. The training loss of Concord (a) and DeepSV (b). The training accuracy of Concod (c) and DeepSV(d)
As shown in Fig. 14c, d, DeepSV outperforms Concod in training accuracy. This suggests that DeepSV is better in deletion calling than Concod. For running time, Concod has smaller training time when the number of training samples is small. When the number of training samples is large, Concod takes longer time than DeepSV in training. The results are shown in Fig. 15.
The relationship between training time and sample size about Concod and DeepSV
Eliminate false positives
After the data is pre-processed, we can eliminate the false positive deletion in VCF by our clustering appproach. In Table 3, we show the number of false positives detected by DeepSV. Our results show that a significant number of false positives are removed by the clustering approach. We also compare the impact of model training on the removal of false positives and the presence of false positives in the Fig. 16. We can see that removing false positives in training significantly improves the training accuracy of the model.
Table 3 Number of false positives are removed by DeepSV
a Shows the accuracy of the DeepSV training. The blue line represents the training accuracy when the false positive is not removed, and the orange line represents the training accuracy after the false positive is removed. The corresponding (b) shows the training loss
On the high level, calling genetic variations from sequence reads is a process of extracting information in the reads that are relevant to the variations. Google's DeepVariant approach shows that such information can be effectively extracted from color images constructed from the reads. A main advantage for this visualization based approach is that it offers an intuitive way to convert variation calling to image classification. This naturally leads to deep learning, which is currently the leading approach for image classification. Our DeepSV method extends this high-level approach to the case of structural variations. In particular, our results show that the visualization approach can be used for more complex genetic variations. Our results show that the overhead for visualizing sequence reads is low. The images contain useful information on structural variations. Deep learning can outperform other traditional machine learning since it doesn't depend on manual selection of features. This may allow deep learning based methods to better utilize the data.
Usually deletion calling performance is affected by the type of data (e.g. sequence coverage and deletion lengths). A method can perform well for some type of data (say high coverage on short deletions) but doesn't perform well for other types (say low coverage on long deletions). Our results show that DeepSV performs well in almost all the settings. This indicates that DeepSV integrates and effectively uses various sources of information in the sequence data in our simulation.
Similar to other supervised machine learning methods, DeepSV needs labeled training data. The more accurate the training data is, the better DeepSV performs for deletion calling. With large-scale genomics projects such as the 1000 Genomes Project, high-quality training data is becoming available. Most existing methods for SV calling don't use the known genetic variations. This may indicate a potential loss of information and missed chances. DeepSV offers a natural way for using these benchmarked SVs for accurate calling of novel SVs.
This paper focuses on deletion calling. We note that there are other types of structural variations such as long insertions, inversions and copy number variations. A natural research direction is developing methods for calling these types of SVs with deep learning. This will need more thoughts on methodologies. For example, long insertions are different from deletions in that the inserted sequences are not present in the reference genome. This makes the reads visualization more difficult since DeepSV currently visualizes reads on the reference. Such issues need to be resolved in order to apply deep learning to these types of SVs.
We demonstrate a novel method of variation detection, which breaks through the traditional sequence detection method. At the same time, our work shows that deep learning can potentially lead to effective calling of different types of genetic variations that are complex than SNPs.
We use the real data from the phase three of the 1000 Genomes Project. The data are downloaded from: ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/phase3/data/. The software and sample result as part of this project are readily available from GitHub at https://github.com/CSuperlei/DeepSV.
BAM:
Binary sequence alignment
CEU:
Utah Residents with Northern and Western European Ancestry
CHB:
Han Chinese in Beijing, China
CNN:
SNPs:
Single nucleotide polymorphisms
SVM:
Support vector machine
SVs:
Structural variations
VCF:
Variant call format
YRI:
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We would like to thank Kai Ye for useful discussions.
Availability and implementation
DeepSV's source code and sample result as part of this project are readily available from GitHub at https://github.com/CSuperlei/DeepSV/.
Project supported by Beijing Natural Science Foundation (5182018) and the Fundamental Research Funds for the Central Universities & Research projects on biomedical transformation of China-Japan Friendship Hospital (PYBZ1834); YW is partly supported by a grant from US National Science Foundation (III-1526415). The funding bodies did not play any role in the design or development of this study, the analysis and interpretation of data, or in the writing of this manuscript.
Department of Information Science and Technology, Beijing University of Chemical Technology, Beijing, People's Republic of China
Lei Cai & Jingyang Gao
Department of Computer Science and Engineering, University of Connecticut, Storrs, CT, USA
Yufeng Wu
Lei Cai
Jingyang Gao
Conceived and designed the experiments: YW, JG, LC. Performed the experiments: LC. Analyzed the data: LC. Contributed reagents/materials/analysis tools: YW, LC. Wrote the paper: LC, YW, JG. All of authors have read and approved the manuscript.
Correspondence to Jingyang Gao.
Additional file 1.
More specific details about DeepSV.
Cai, L., Wu, Y. & Gao, J. DeepSV: accurate calling of genomic deletions from high-throughput sequencing data using deep convolutional neural network. BMC Bioinformatics 20, 665 (2019). https://doi.org/10.1186/s12859-019-3299-y
Genetic variations
High-throughput sequencing
Imaging, image analysis and data visualization | CommonCrawl |
# Linear regression and its assumptions
Linear regression is a fundamental statistical modeling technique that allows us to predict the value of a variable based on the values of other variables. It is widely used in various fields, including finance, marketing, and healthcare.
To perform linear regression, we need to make several assumptions:
1. Linearity: The relationship between the dependent variable and the independent variables is linear.
2. Independence: The residuals (differences between the predicted values and the actual values) are independent.
3. Homoscedasticity: The variance of the residuals is constant.
4. Normality: The residuals are normally distributed.
Consider the following data set:
| Age | Income |
|-----|--------|
| 25 | 50000 |
| 30 | 60000 |
| 35 | 70000 |
| 40 | 80000 |
We can use linear regression to predict the income based on age.
## Exercise
1. Create a linear regression model using the given data set.
2. Calculate the coefficients of the regression line.
3. Use the model to predict the income for a person of age 45.
# Logistic regression and its use in binary classification
Logistic regression is a statistical modeling technique used for binary classification problems. It is an extension of linear regression and is used to predict the probability of an event occurring.
The logistic regression model is defined by the following equation:
$$\text{Probability} = \frac{1}{1 + e^{-\text{Predictor}}} = \frac{1}{1 + e^{-\beta_0 - \beta_1X}}$$
Where $\text{Predictor}$ is the predicted value, $\beta_0$ is the intercept, $\beta_1$ is the slope, and $X$ is the independent variable.
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use logistic regression to predict the probability of a person being a smoker based on their age.
## Exercise
1. Create a logistic regression model using the given data set.
2. Calculate the coefficients of the logistic regression line.
3. Use the model to predict the probability of a person being a smoker for a person of age 45.
# Support vector machines and their role in classification
Support vector machines (SVM) are a powerful machine learning technique used for classification problems. They are based on the concept of finding the optimal hyperplane that separates the data points of different classes.
The SVM model is defined by the following equation:
$$\text{Predictor} = \text{Sign}(\text{Scalar Product}(\text{Data Point}, \text{Hyperplane}))$$
Where $\text{Data Point}$ is a data point, $\text{Hyperplane}$ is the optimal hyperplane, and $\text{Scalar Product}$ is the dot product of the data point and the hyperplane.
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use SVM to classify the data points into two classes based on their age.
## Exercise
1. Create an SVM model using the given data set.
2. Calculate the coefficients of the SVM hyperplane.
3. Use the model to classify a person of age 45.
# Decision trees and their construction and evaluation
Decision trees are a popular machine learning technique used for classification and regression problems. They are based on the concept of recursively splitting the data into subsets based on the values of a specific feature.
The decision tree model is defined by the following steps:
1. Choose the best feature to split the data.
2. Split the data based on the chosen feature.
3. Repeat steps 1 and 2 for each subset until a stopping criterion is met.
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use a decision tree to classify the data points into two classes based on their age.
## Exercise
1. Create a decision tree model using the given data set.
2. Calculate the depth of the decision tree.
3. Use the model to classify a person of age 45.
# Clustering algorithms and their use in unsupervised learning
Clustering algorithms are a type of machine learning technique used for unsupervised learning problems. They are based on the concept of grouping data points into clusters based on their similarity.
The most common clustering algorithms include:
1. K-means clustering
2. Hierarchical clustering
3. DBSCAN
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use a clustering algorithm to group the data points based on their similarity.
## Exercise
1. Create a K-means clustering model using the given data set.
2. Calculate the centroids of the clusters.
3. Use the model to classify a person of age 45.
# Neural networks and their structure and training
Neural networks are a powerful machine learning technique based on the structure and function of the human brain. They are composed of interconnected nodes called neurons, arranged in layers.
The neural network model is defined by the following steps:
1. Initialize the weights of the neurons.
2. Feed the input data into the neural network.
3. Calculate the output of the neural network.
4. Update the weights based on the error between the predicted output and the actual output.
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use a neural network to classify the data points into two classes based on their age.
## Exercise
1. Create a neural network model using the given data set.
2. Calculate the output of the neural network for a person of age 45.
3. Update the weights of the neural network based on the error between the predicted output and the actual output.
# Advanced techniques in machine learning: deep learning and reinforcement learning
Deep learning and reinforcement learning are advanced machine learning techniques that have revolutionized the field.
Deep learning is based on artificial neural networks with many layers. It is used for tasks such as image and speech recognition, natural language processing, and game playing.
Reinforcement learning is a type of machine learning where an agent learns to make decisions by interacting with its environment. It is used for tasks such as playing games, controlling robots, and autonomous driving.
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use a deep learning model to classify the data points into two classes based on their age.
## Exercise
1. Create a deep learning model using the given data set.
2. Train the model using the data set.
3. Use the model to classify a person of age 45.
# Model evaluation and selection
Model evaluation is the process of comparing the performance of different machine learning models. It is crucial for selecting the best model for a given problem.
The most common evaluation metrics include:
1. Accuracy
2. Precision
3. Recall
4. F1 score
5. Area under the ROC curve
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use the above evaluation metrics to compare the performance of different machine learning models for classifying the data points into two classes based on their age.
## Exercise
1. Create several machine learning models using the given data set.
2. Evaluate the performance of each model using the above evaluation metrics.
3. Select the best model based on the evaluation results.
# Applications of advanced statistical modeling in real-world scenarios
Advanced statistical modeling has numerous applications in real-world scenarios. Some examples include:
1. Financial forecasting
2. Market segmentation
3. Sentiment analysis
4. Recommender systems
5. Fraud detection
6. Image recognition
7. Natural language processing
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use advanced statistical modeling to analyze the data and make predictions or decisions based on the results.
## Exercise
1. Apply advanced statistical modeling to the given data set.
2. Analyze the results and make a prediction or decision based on the analysis.
# Integrating R with other data science tools and platforms
R is a powerful programming language and software environment for statistical computing and graphics. It is widely used in the field of data science.
To integrate R with other data science tools and platforms, we can use the following methods:
1. RStudio: A popular integrated development environment (IDE) for R.
2. Shiny: A web application framework for R.
3. RMarkdown: A document format for R that allows you to create dynamic reports, presentations, and dashboards.
4. R packages: A collection of pre-built functions and data sets that can be easily installed and used in R.
Consider the following data set:
| Age | Smoker |
|-----|--------|
| 25 | 0 |
| 30 | 1 |
| 35 | 1 |
| 40 | 0 |
We can use R to perform advanced statistical modeling on the data set and visualize the results using RStudio, Shiny, or RMarkdown.
## Exercise
1. Install and use RStudio to analyze the given data set.
2. Create a visualization of the results using RStudio, Shiny, or RMarkdown. | Textbooks |
\begin{definition}[Definition:Plato's Geometrical Number]
The actual value of what is generally known as '''Plato's geometrical number''' is uncertain.
The passage in question from {{AuthorRef|Plato}}'s {{BookLink|Republic|Plato}} is obscure and difficult to interpret:
:''But the number of a human creature is the first number in which root and square increases, having received three distances and four limits, of elements that make both like and unlike wax and wane, render all things conversable and rational with one another.''
There are two numbers which it is generally believed it could be:
:$216$
:$12 \, 960, \, 000$
It is believed that the expression:
:''three distances and four limits''
refers to cubing.
It is further believed that the reference is to the area of the $3-4-5$ triangle, which is $6$.
The passage is also deduced to contain a reference to $2 \times 3$.
It is also interpreted by other commentators as being $12 \, 960 \, 000$, which is $60^4$.
\end{definition} | ProofWiki |
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Antarctic ozone hole modifies iodine geochemistry on the Antarctic Plateau
Andrea Spolaor ORCID: orcid.org/0000-0001-8635-91931,2,
François Burgay ORCID: orcid.org/0000-0002-2657-69002,3,
Rafael P. Fernandez4,
Clara Turetta ORCID: orcid.org/0000-0003-3130-29011,2,
Carlos A. Cuevas ORCID: orcid.org/0000-0002-9251-54605,
Kitae Kim6,
Douglas E. Kinnison7,
Jean-François Lamarque ORCID: orcid.org/0000-0002-4225-50747,
Fabrizio de Blasi1,2,
Elena Barbaro1,2,
Juan Pablo Corella5,8,
Paul Vallelonga9,10,
Massimo Frezzotti ORCID: orcid.org/0000-0002-2461-288311,
Carlo Barbante ORCID: orcid.org/0000-0003-4177-22881,2 &
Alfonso Saiz-Lopez ORCID: orcid.org/0000-0002-0060-15815
Nature Communications volume 12, Article number: 5836 (2021) Cite this article
Cryospheric science
Element cycles
Polar stratospheric ozone has decreased since the 1970s due to anthropogenic emissions of chlorofluorocarbons and halons, resulting in the formation of an ozone hole over Antarctica. The effects of the ozone hole and the associated increase in incoming UV radiation on terrestrial and marine ecosystems are well established; however, the impact on geochemical cycles of ice photoactive elements, such as iodine, remains mostly unexplored. Here, we present the first iodine record from the inner Antarctic Plateau (Dome C) that covers approximately the last 212 years (1800-2012 CE). Our results show that the iodine concentration in ice remained constant during the pre-ozone hole period (1800-1974 CE) but has declined twofold since the onset of the ozone hole era (~1975 CE), closely tracking the total ozone evolution over Antarctica. Based on ice core observations, laboratory measurements and chemistry-climate model simulations, we propose that the iodine decrease since ~1975 is caused by enhanced iodine re-emission from snowpack due to the ozone hole-driven increase in UV radiation reaching the Antarctic Plateau. These findings suggest the potential for ice core iodine records from the inner Antarctic Plateau to be as an archive for past stratospheric ozone trends.
The stratospheric ozone layer, which extends from the tropopause (≈17 km in the tropics and ≈11 km in the polar regions) to the upper stratosphere (≈35 km), completely absorbs the 220–290 nm component of the solar electromagnetic spectrum1 and constitutes a natural shielding that reduces the amount of biologically harmful radiation reaching the Earth's surface within the 290–320 nm band2. In the mid-1970s, it was hypothesized that the massive emission of chlorofluorocarbons (CFCs) and halons, commonly used as propellants, solvents and refrigerants, could lead to a decrease in stratospheric ozone concentrations3,4. These compounds, when exposed to UV radiation, release halogen radicals in the stratosphere that lead to ozone destruction through catalytic cycles5,6. The effectiveness of CFCs and halons in depleting stratospheric ozone is maximized at the poles, where the Antarctic ozone hole was identified in the mid-1980s7. The effects of enhanced solar UV radiation, resulting from stratospheric ozone loss, on human health8 and terrestrial5,9 and marine10 ecosystems have been well established. However, a limited number of investigations11,12,13 have been performed to evaluate the impact of ozone hole formation and evolution on the geochemical cycle of photoactive species deposited on polar snow and ice.
In this work, we report the first iodine record from an ice core collected at Dome C (inner East Antarctic Plateau), covering the period of 1800–2012. Among the three available Antarctic iodine records14,15, this is the only one that covers both the pre-ozone hole and the ozone hole periods and enables us to evaluate the influence of increasing incoming solar UV radiation on the snowpack-atmosphere iodine exchange equilibrium during the ozone hole period (1975–2012). Combining field observations, laboratory measurements and chemistry-climate model simulations, we find that the iodine concentration in ice remained relatively stable during the pre-ozone hole period (1800–1974) but has gradually declined by a factor of 2 since the onset of the Antarctic ozone hole in the mid-1970s. We suggest that the enhanced incoming solar UV radiation reaching the Antarctic snowpack due to the formation of the Antarctic ozone hole has significantly amplified the ice-to-atmosphere iodine mass transfer, altering the natural geochemical cycle and mobilization of this essential element on the Antarctic Plateau.
The Dome C iodine records
In 1996, a French-Italian team established the first summer camp at Dome C, the inner Antarctic Plateau, and a new all-year facility, Concordia Station (3233 m a.s.l.; 75°05'59"S, 123°19'56"E, ~1200 km from the Southern Ocean), became winter-over operational during the 2005 season. A 13.72 m shallow ice core was drilled close to Concordia during 2012. This record covers ~212 years, from 1800 to 2012. The ice core chronology of the shallow core, which has an estimated ice age uncertainty of up to 5 years, is based on the annual snow accumulation and on the observed nssSO42− spikes from the most important past volcanic events16,17 (see Supplementary Material S1, hereafter SM). Due to the very dry conditions, low precipitation (25–30 mm water equivalent per year, mm w.e. yr−1) and very thin atmospheric boundary layer (surface pressure of ~650 hPa), Dome C is an ideal site to study snow photochemical processes and snow-atmosphere exchange of reactive elements18,19,20.
The shallow ice core was sampled at 5 (±1) cm resolution, and for each sample, total iodine (I) and sodium (Na) were measured using an inductively coupled plasma sector field mass spectrometer (ICP-SFMS, see the "Methods" section). Both iodine and sodium are mainly emitted by seawater and sea ice within coastal regions and are transported inland21,22. As sodium does not experience photochemical degradation during transport or after deposition, it is widely used as a conservative tracer23. To support the 13.72 m-deep shallow core, a 1.3 m-deep snow pit (Sp2013) and a 4 m-deep snow pit (Sp2017) were dug in December 2013 and December 2017, respectively (see S2 in the SM for further details).
The average iodine concentration for the entire ice core record was 0.056 ± 0.037 (1σ) ng g−1. To detect a possible tipping point discontinuity within the iodine record, we used the Change-Point Analysis test (CPAt24, see S3 in the SM). The results suggest that the iodine concentration record shifted in 1975 (p-value = 0.001) with a confidence interval of 90%. Based on the CPAt test, we identified two main periods for the iodine record, one from 1800 to 1974 and the other from 1975 to 2012. The average iodine concentration for the period of 1800–1974 remains constant at 0.060 ± 0.039 ng g−1, decreasing by almost half to an average concentration of 0.032 ± 0.015 ng g−1 for the period of 1975–2011 (Fig. 1).
Fig. 1: Evolution of iodine concentration and stratospheric ozone over Dome C.
Iodine concentration (grey line, raw data; bold black line, 10 year smoothing) and [I] anomalies, expressed as ng g−1 (blue, positive; red, negative; solid lines with respect to the mean iodine concentration for the period of 1800–2012), were compared with the mean total ozone column above Dome C during the sunlit (1 September to 28 February) period of each year (red dots, TOCDC in Dobson units). The top panel shows the modelled mean sunlit Actinic Flux at 300 nm (purple square, AF300, quanta cm−2 s−1 nm−1) at the Antarctic surface within the closest model grid point to Dome C. The decrease in iodine concentration coincides with the decrease in TOCDC and mirrors the increase in modelled AF300. The pink vertical bar shows the tipping point determined for [I] (1975 CE), which is considered to be the starting date for the ozone hole period.
UV radiation reaching the Antarctic Plateau
To evaluate the hypothesis that the increased UV radiation reaching the Antarctic Plateau due to ozone hole formation has altered iodine concentrations within the ice core, we computed the total ozone column (TOC) and ozone hole evolution during the 1950–2010 period using the CAM-Chem chemistry-climate model25,26 (see "Methods"). Remarkably, the modelled TOC trend above Dome C (TOCDC) matches the iodine record both before and after the observed tipping point discontinuity (Fig. 1c), which coincides with the initial stages of ozone hole formation in ~19755. Furthermore, the modelled mean actinic flux at 300 nm (AF300, Fig. 1a) within the closest model grid point to Dome C (74.84° S; 122.5°E; grid resolution 1.9° × 2.5°, see "Methods") completely mirrors the iodine record and total ozone trend (Fig. 1c), highlighting the large radiative changes affecting the inner Antarctic surface due to ozone hole formation.
We computed the mean modelled TOCDC and AF300 during the sunlit period of each year (i.e., from 1 September to 28 February) before and after the iodine tipping point and considered these values to be representative of the pre-ozone hole (1950–1974) and ozone hole (1975–2011) periods. Within these periods, the modelled TOCDC decreased from 334 ± 16 DU (1950–1974) to 266 ± 41 DU (1975–2017). Consequently, this stratospheric ozone reduction drove an increase in AF300 at Dome C from 1.05 × 1011 to 4.36 × 1011 quanta cm−2 s−1 nm−1. Note that the mean ozone reduction and the actinic flux enhancement reaching the inner Antarctic surface strongly depend on the specific wavelength, location and period (i.e., early spring or mid-summer) considered. A sensitivity analysis of the spatiotemporal variability in these quantities is provided in section S6 in the SM.
During the ozone hole period, the correlation between [I] and sunlit TOCDC has a positive and significant Pearson coefficient, whereas no significant correlation was found for the pre-ozone hole period (pre-1975: r =−0.1, p-value = 0.65; post-1975: r = 0.398, p-value = 0.015; see S3.3 in the SM). Similarly, [I] and UV forcing, expressed as AF300, were uncorrelated before 1975, while they were anti-correlated afterwards, indicating a direct effect induced by the stronger UV radiation reaching the Antarctic Plateau (pre-1975: r = −0.018, p-value = 0.93; post-1975: r = −0.401, p-value = 0.015; S3.4 in the SM). These significant correlations suggest a direct linkage between stratospheric ozone and UV radiation reaching the inner Antarctic Plateau surface, which in turn would alter the postdepositional processes controlling the preservation of iodine within the snowpack, as described below.
Role of ozone hole in the iodine geochemical cycle
To evaluate the physicochemical processes behind the observed negative correlation between UV surface radiation and ice core iodine, we turn to condensed phase iodine photochemistry. Within liquid aerosols and ice crystals, inorganic iodine is mostly in ionic form as iodide (I−(aq)) and iodate (IO3−(aq))27. Although iodate is supposed to be more stable than iodide, several studies highlighted that the actual I−(aq)/IO3−(aq) ratio depends on the sampling location, with I−(aq) being the most abundant species in polar environments28. Indeed, investigations performed on the Talos Dome Antarctic ice core (www.taldice.org 72°49'S, 159°11'E, 2315 m a.s.l., accumulation of 80 mm w.e. yr−1) highlighted that I−(aq) is more stable than IO3−(aq)15. Both iodine species showed strong photoactivity in snow matrices since they are enriched during the polar night and depleted during summer29,30,31. Under laboratory and simulated Antarctic radiation conditions, it was observed that iodate UV absorption and its subsequent photoreduction (IO3−(aq) + hν → IO(g) + O2−(aq)) were enhanced in the 275–400 nm range, with a maximum peak at 295 nm32. Similar conclusions were also reported for the photooxidation of iodide into I2(g) in real snow samples spiked with known amounts of iodide and exposed to Antarctic sunlight (O2(aq) + 4H+(aq) + 6I−(aq) → 2H2O + 2 I3−(aq); I3−(aq) ↔ I2(g) + I−(aq))33. Within this photooxidation mechanism that leads to the release of I2(g) to the atmosphere, a charge-transfer complex with oxygen (I−-O2(aq)) is formed. This complex has a local absorption maximum between 280 and 330 nm, i.e., encompassing the same spectral range where the largest changes in incoming UV radiation were observed between the pre- and post-ozone hole periods (Fig. 2a).
Fig. 2: Iodine photochemical enhancement during the ozone hole period.
Modelled actinic flux (AF) and iodine photoactivation efficiency (J-iodine) at the snow surface in Dome C. a Wavelength-dependent AF reaching the model surface every decade since the 1960s (colour dots, left Y-axis). The right Y-axis shows the absorbance spectra of a frozen iodide solution in equilibrium with oxygen (I-(O2)) and air (I-(air)) environments. b The normalized J-iodine of snow-trapped iodine between 1950 and 2010 relative to the mean J-iodine for the pre-1975 period (dashed-grey horizontal line in (b)). J-iodine wavelength integration was performed within the 280–313 nm band (see dashed-grey vertical lines in (a)) and continuously increased after 1975 due to the ozone hole-driven enhancement in UV radiation during the sunlit period of each year (i.e., from 1 September to 28 February).
Given that iodide abundance in polar ice is higher than iodate, we conducted a timeline simulation of iodine photoactivation within ice and snowpack (J-iodine, Fig. 2b) by computing the wavelength integral of the measured iodide absorption spectra multiplied by the modelled AF reaching the inner Antarctic surface during the 1950–2010 period (see "Methods"). We find that the normalized J-iodine trend shows consistent agreement with the iodine decrease observed for the Dome C ice record. This finding suggests the occurrence of UV-driven re-emission and mobilization of snow-trapped iodide to the gas phase, following the photooxidation of iodine in ice and subsequent release of I2(g) to the atmosphere34. Indeed, J-iodine remained relatively constant during the pre-1975 period, then increased ~1.5-fold for the mean ozone hole era and ~2.0-fold for the 2006–2012 period. This change is indicative of a continuous change in the efficiency of re-emission processes of ice-trapped iodine that altered the steady state equilibrium, assuming that the iodine deposition velocity remains unaltered (see next section). Due to the pronounced changes in AF intensity reaching the inner Antarctic surface above and below 300 nm during different seasons, the post-1975 J-iodine enhancement computed here is very sensitive to the upper bandwidth limit and to the month used to perform the wavelength integration. Indeed, a sensitivity analysis indicates that up to fourfold more efficient iodide photodissociation efficiency is obtained when the narrower 280–307 nm AF band is considered, while an equivalent ~5-fold J-iodine enhancement is reached if only the October mean evolution is used (see Supplementary Figs. 5 and 6). Therefore, regardless of the exact wavelength interval and seasonal period considered, our combined experimental and modelling results indicate that the increase in UV solar radiation observed since the onset of the ozone hole has promoted the release of active iodine from the snowpack to the atmosphere. This validation corroborates our hypothesis that stronger UV radiation led to the enhanced re-emission of iodine deposited on polar ice, thereby explaining the overall decrease in iodine concentration at Dome C during the 1975–2012 period.
Possible competitive processes (CPs)
To explain the observed shift in the iodine trend at Dome C, we explored different alternative hypotheses that involve possible competitive processes (CPs), such as (a) changes in site snow accumulation, (b) dependence on snow pack physical characteristics, (c) snow pack postdepositional processes and spatial variability, (d) changes in the transport mechanism from the coast to the site, and (e) changes in the strength of coastal source emissions. In the SM (S7), we also discuss the possible effects of sample storage on iodine loss from firn samples. All these alternative hypotheses were refuted, as they failed to provide a consistent explanation that connects theory with observations, as discussed below. Additional iodine ice core records exist in both the Arctic and Antarctic and are discussed in the SM (S4 and Supplementary Fig. 1); however, these records do not provide additional information for explaining the Dome C iodine trend.
CP(a) – Change in snow accumulation
The inner Antarctic Plateau is characterized by low accumulation rates and cold temperatures that can affect snow metamorphism. Due to the very low annual snow deposition35, it is possible to estimate snow accumulation at Dome C only at pluriannual scale using snow/firn core analysis. However, the average snow accumulation rate computed during different studies is very similar: for the period of 1955–1998, the rate was 26 ± 1.3 mm w.e. yr−1; for the period of 1816–1998, it was 25 ± 1.3 mm w.e. yr−1 36; and for the 2006–2013 period35, the rate was 27 mm w.e. yr−1. This result leads to the conclusion that the snow accumulation rate at this location has not changed significantly over the last 200 years and that in the worst scenario, it might have affected the iodine signal by no more than 10%. Note that the spatial variability in accumulation might lead to interannual variability but will not affect the recorded trend (Supplementary Fig. 4, lower panel and Supplementary Table 2). For this reason, we assume a constant snow accumulation rate over the entire period covered by the ice core record, and we discard the possibility that snow accumulation changes can explain the large observed reduction in iodine concentration since 1975.
We also investigated the possible role of the precipitation seasonality at Dome C. Combining a Lagrangian trajectory model output and the European Re-Analysis ERA40 data (from 1980 to 2011), it is reported that ~35–40% of the annual precipitation at Dome C occurred during the summer season (December to February) with the remaining 60–65% being homogenously distributed among the remaining months37. We found that the seasonality of snow precipitation is negligible due to the wind redistribution of snow accumulation at the surface38. UV can penetrate the snowpack for the first 20–30 cm, which corresponds to 2–3 years of burial snow due to the low snow accumulation rate at Dome C38. Independently of the seasonal precipitation pattern, all the iodine present in the annual snow layers would experience a similar photochemical activation, thereby greatly minimizing the hypothetical role of a seasonal precipitation pattern change. However, the potential impact of systematic changes in precipitation seasonality over several years remains an uncertainty for the interpretation of future iodine ice core records and might contribute to the observed spatial variability.
CP(b) – Role of snow pack physical characteristics
The snow density can affect the chemical signal both by modifying the extent of light penetration and by changing the gas permeability within the snow pack. The average snow density at Dome C increases with depth, making the snow layers progressively less permeable to gas movement until the close-off depth is reached (≈80 m depth). The gaseous species can move within the snow/ice grains until reaching this depth, but their escape into the atmosphere is potentially facilitated by the higher porosity of the surface layers (i.e., with a lower density). However, the decline in iodine concentration was observed only in the topmost 3.5 m (1.2 m w.e.) of the snow pack (Fig. 1c), and the direct comparison with the Sp2017 iodine and density profiles does not endorse this hypothesis (see Supplementary Fig. 3). Indeed, the Sp2017 density profile presents a typical shallow density increase in the topmost 3.5 m (from 0.32 to 0.38 kg L−1), which cannot justify the observed iodine behaviour.
Another process that can affect the stabilization of iodine species in snow is grain size. The grain size increases with depth40, reducing the specific surface where gas-phase iodine can be bound and/or absorbed. If a relationship between the grain size and iodine concentration existed, we might expect an enhancement of iodine concentration in the topmost part of the record, but we observe the opposite. Snow grain size was measured in Sp2017, and similar to snow density, it cannot explain the iodine trend (Supplementary Fig. 3).
Finally, snow metamorphism is mainly driven by the temperature gradient prevailing in the first metres of snow between the polar summers and winters and occurs on timescales on the order of weeks or months. Continuous snow temperature measurements are routinely performed at Dome C down to 10 m depth41. The largest difference in temperature during the year is measured in the uppermost 2 m, particularly at the surface where the ΔT can reach 44 °C. Note that the largest measured temperature gradient at 2.5 m depth is 8 °C (Supplementary Table 3), which slows snow metamorphosis and possible related processes. In our shallow core, the iodine shift is detected at a depth of 3.5 m. Moreover, when considering iodine postdepositional processes within the snow pack, we need to consider that iodine photochemical activation is fast and can cause a loss of up to 90% of the deposited iodine within a day31,42. The radiative transfer within the snowpack will be further discussed in the next paragraph, but it suggests that the snow metamorphism, the physical characteristics of the snowpack and its sintering process play a negligible role in describing the observed iodine trend.
CP(c) – Inner snow pack postdepositional processes and UV light penetration
We also investigated the effects of UV penetration into snowpack. UV light can penetrate the snowpack for the first 20–30 cm39, corresponding to 2–3 years of burial due to the low snow accumulation rate at Dome C. UV light penetration would have caused the re-emission of iodide as gaseous I2 into the atmosphere for ~2–3 years. However, if this case were true for iodine, we would expect a similar iodine concentration along the entire core or, alternatively, iodine accumulation would occur in the surface layers due to the migration of gaseous iodine from the deeper sections of the core. Evidence that UV penetration in snowpack does not play a significant role also comes from the availability of additional iodine records (i.e., Sp2013 and Sp2017) that showed similar trends and concentrations over the last 40 years (Supplementary Table 2 and Supplementary Fig. 4). Furthermore, our iodine analyses from the Law Dome (LD) ice core, which has an average annual snow accumulation of ≈740 mm w.e. yr−1 (i.e., ~30 times Dome C), showed well-preserved iodine peaks corresponding to the winter periods (polar night)14,29. If light penetration determined re-emission from the snowpack, we would expect a smoother signal in the record compared with the observed sharp peaks. On these grounds, we argue that the iodine trend is influenced by the enhancement of UV radiation and the increase in photoreactivity on the surface snow instead of depending on postdepositional processes occurring within the compacted deep ice. This finding is also supported by the different trend of iodine, which decreases from the surface to 3.5 m (1975 CE), in comparison with other photochemical species analysed in Sp2017 (Supplementary Fig. 4).
CP(d) - Changes in the transport mechanism from the coast to the site
To understand whether changes in the transport mechanisms from the coast to Dome C played a role in explaining the iodine behaviour, we evaluated how sodium concentration has changed over the last two centuries. Overall, the sodium average concentration for the entire record was 41 ± 19 (1σ) ng g−1; however, a slight increase was recorded from 1975 to 2012 (50 ± 23 ng g−1) compared with the period spanning 1800 to 1975 (39 ± 14 ng g−1). Note that the ice core record had some gaps from 1989 to 1997 that were filled with Na concentration data retrieved from Sp2013 and Sp2017 (see S3.5 in the SM), producing a merged sodium (NaM) and iodide (IM) record for direct comparison.
Comparing the NaM and IM profiles, we observe that they have opposite trends during the last few decades (Fig. 3). The increase in sodium concentration recorded since the mid-1970s suggests enhanced transport from the coast to Dome C. This finding is consistent with model simulations and observations that report a poleward shift in the mid-latitude jet stream43,44 and with the enhanced cyclonic activity driven by the phase change in the Interdecadal Pacific Oscillation (IPO) as well as strengthening of westerlies due to global warming45. However, the enhanced poleward transport is not mirrored by the iodine record, which, conversely, has decreased since 1975. The decoupling between NaM and IM is corroborated by the Pearson correlation coefficients calculated for the periods of 1800–1974 and 1975–2017 (see S3.5 in the SM). A significant statistical correlation existed between iodine and sodium during the pre-ozone hole period (1800–1974; r = 0.44; p-value < 0.0001) but not for the ozone hole period (r = −0.1; p-value = 0.54). Thus, the statistical tests support that between 1800 and 1974, iodine and sodium were transported and deposited following similar mechanisms. Once deposited on the snowpack, iodine experienced postdepositional processes whose efficiency remained relatively constant throughout the entire pre-ozone hole period investigated in this study (Figs. 1 and 3). The decoupling observed since 1975 is inconsistent with an iodine decrease driven by changes in atmospheric transport. All of these previous studies provide additional evidence that the different behaviours of Na and I from 1975 to 2017 at Dome C were associated with processes that interfered with iodine deposition and preservation in the snowpack rather than with changes in the snow accumulation rate, coastal source strength or transport efficiency. In fact, enhanced tropospheric transport from the coast to the Antarctic inland45 would have caused an increase in both Na and I concentrations.
Fig. 3: Iodine and sodium merged record from Dome C.
The upper panel shows the merged 1-year smooth records (see S3.5 in the SM for details) using snow pits and ice core data (iodine merged in black— IM - and sodium merged in red—NaM). Light grey and red lines show the raw data, and red and bold curves show the 5-year smoothing. The lower panel shows the iodine enrichment (Ienr = I/(Na*0.00000568, where 0.00000568 is the I/Na mass ratio in seawater), where the solid green line shows the 5-year smoothing. The dashed vertical line indicates the tipping point discontinuity at 1975 CE.
CP(e) - Possible changes in the strength of coastal source emissions
Finally, we explored the possible changes in the strength of iodine emissions from coastal areas. However, this hypothesis is not supported by models or observations. Note that the most relevant iodine source on a global scale is the ocean, providing ~2.3 Tg(I) yr−1 of organic (mostly CH3I) and inorganic (I2 and HOI) iodine46,47. Global iodine emissions show strong latitudinal variability, with stronger emissions in the tropical and mid-latitude oceans and significantly decreasing emissions towards the poles48. In polar regions, iodine sources are mainly connected to first-year sea ice49,50, with both macro- and microalgae living under sea ice51 and abiotic activation of iodine on sea ice and snow surfaces22,52 playing a role. Both the biological and abiotic iodine emissions in the polar regions depend on the thickness and extent of seasonal sea ice, being more prominent for thinner sea ice and large extensions of first-year sea ice. A trend of increasing Antarctic seasonal sea ice extent has been observed for the last decades up to 2014, followed by a decline from 2015 to 201753. Thus, it would be expected that in any case, coastal sea ice emissions of iodine increased during the studied period, contrary to what we observed in the ice core record.
This study suggests that the human-induced stratospheric ozone hole has altered the natural geochemical cycle of iodine. Combining field measurements (ice core, snow pit, surface snow observations) with laboratory (iodine absorption spectra, UV-driven photooxidative mechanisms) and modelling (ozone hole evolution and associated actinic flux changes) results, we conclude that the enhanced incident UV radiation due to the ozone hole has caused a continuous decline in iodine concentration in ice in inner Antarctica since ~1975. This recent phenomenon has modified the previously steady iodine equilibrium between snowpack and the atmosphere in the interior of Antarctica by increasing re-emissions from snowpack. The resulting increase in ice-to-atmosphere iodine mass transfer has relevant implications for polar tropospheric chemistry and for the Earth's radiative budget since iodine catalytic cycles play a crucial role in the destruction of tropospheric ozone26,27,54,55 and can also act as cloud condensation nuclei (CCN)56,57,58. Furthermore, surface re-emission of iodine from central Antarctica and subsequent redistribution by atmospheric transport could potentially be a source of the widespread distribution of iodine observed by satellite measurements over the entire Antarctic continent59. Finally, given the direct link observed between ice core iodine at Dome C and stratospheric ozone hole evolution, we suggest that the ice core iodine present on the Antarctic Plateau may potentially serve as an archive for past stratospheric ozone changes at centennial to millennial time scales.
Shallow core processing and analysis
The shallow core was cut and sampled at 5 ± 1 cm resolution using a ceramic knife and rinsed with ultrapure water (UPW) after each use. Only the central part of the core was used for the analysis, and the outer 2 cm was removed by scraping with a ceramic knife. The core samples were processed at the Institute of Polar Sciences of the National Research Council (ISP-CNR) in a class 1000 inorganic clean room under a class 100 laminar-flow bench. Iodine and sodium analyses in the snow pit and ice core samples were conducted on melted and non-acidified samples. Total sodium (Na) and iodine (I) concentrations were determined by ICP-SFMS following Spolaor et al.60. Each analytical run started and ended with a UPW cleaning session of 3 min to ensure a stable background level throughout the analysis. The external standards that were used to calibrate the analytes were prepared by diluting a 1000 ppm stock IC (ion chromatography) standard solution (TraceCERT® purity grade, Sigma-Aldrich, MO, USA). The standard concentrations ranged between 1 and 200 ng g−1 for sodium and 0.005 and 0.200 ng g−1 for iodine. The relative standard deviation (RSD %) was low for all analytes, ranging from 3 to 4% for sodium and 2 to 5% for iodine. The instrumental limit of detection (LoD), calculated as three times the standard deviation of the blank (n = 10), was 1 ng g−1 for Na and 0.003 ng g−1 for I. We evaluated the stability of iodine by repeating the analysis of a selected sample multiple times. We did not observe any significant iodine loss during the analytical run (≈12 h).
CAM-Chem chemistry-climate model setup: Halogen chemistry and radiative transfer
The stratospheric ozone hole evolution, as well as the radiative transfer of ultraviolet (UV) actinic flux reaching the Antarctic surface, was computed using the global 3-D chemistry-climate model CAM-Chem (Community Atmospheric Model with chemistry, version 4)61, which is the atmospheric component within the CESM framework (Community Earth System Model)62. The model setup follows the CCMI-REFC1 recommendation for 1950–2010 simulations (i.e., we forced the model with prescribed sea surface temperatures and sea ice distributions)63 but incorporates an explicit treatment of very-short lived (VSL) chlorine, bromine and iodine sources and chemistry46,47,64,65,66. The model configuration used here is identical to the one used previously to compare CAM-Chem results with Arctic iodine ice records26, although here, we extracted monthly output with the wavelength-dependent actinic flux at all atmospheric layers and the surface. Additional details of the CAM-Chem configuration and validation are provided in S5 of the SM.
CAM-Chem shows a good ability to reproduce the size and depth of ozone hole evolution within chemistry-climate simulations25 and presents excellent agreement with satellite ozone observations during the modelled period (Supplementary Fig. 2). The total ozone column polar cap (TOCSP) used for the model validation was computed as the cosine-weighted mean within the 63°S–90°S latitudinal band during October. The model mean output during the austral sunlit period (i.e., from 1 September to 28 February of next year) was computed to perform the correlations with the ice core record. The complete sunlit period (spring + summer) was selected because even when the largest changes in TOC and AF300 associated with the ozone hole are observed during spring, the overall intensity of UV radiation reaching the Antarctic surface maximizes during the summer because of the higher solar zenith angle (SZA). The 300 nm wavelength bin used for the actinic flux computation at the model surface (AF300) has an ~3.5 nm width and is centred at 300.5 nm. AF300 values were extracted at the grid point closest to Dome C on the Antarctic Plateau (74.84° S; 122.5°E; grid resolution 1.9° × 2.5°; model mean altitude of 3300 m a.s.l.; see S5 in the SM).
The iodide photolysis rate constant within the snowpack (J-iodine) was determined offline following Eq. (1)
$${{{{{\rm{J}}}}}}-{{{{{\rm{iodine}}}}}}\left({{{{{\mathrm{yr}}}}}}\right)=\int {\sigma }^{{{{{{\mathrm{Abs}}}}}}}\left(\lambda \right)\times {{{{{{\mathrm{AF}}}}}}}^{{{{{{\mathrm{surf}}}}}}}\left(\lambda ,{{{{{\mathrm{yr}}}}}}\right)\times \varPhi \left(\lambda \right)\,{{{{{\mathrm{d}}}}}}\lambda \,$$
where σAbs(λ) is the absorption spectrum of iodide as a function of wavelength (λ) measured in iodine-containing frozen solutions33; AFsurf(λ, yr) is the wavelength-dependent surface CAM-Chem actinic flux reaching Dome C for each year (yr); and Φ(λ) is the quantum yield for the initial UV absorption step leading to the formation of triiodide in the frozen solution, assumed to be unity (Φ(λ) = 1). The wavelength integration used to compute annual J-iodine was performed for the 280–313 nm bandwidth; below 280 nm, the transmittance of the Pyrex filter used for measuring σAbs(λ) was considerably reduced, while above 313 nm, the iodide absorbance of iodide and the I−-O2(aq) complex rapidly dropped to zero33. The resolutions of the CAM-Chem wavelength bins used to compute J-iodine are provided in the SM. Due to the rapid increase in AFsurf intensity within the 290–330 nm range, as well as on the seasonal changes in the net solar radiation reaching the polar regions between spring, summer and fall, the absolute J-iodine computation is very sensitive to the upper bandwidth limit used for wavelength integration and to the monthly period of time considered for each year. A sensitivity analysis is provided in section S6 of the SM (Supplementary Figs. 5 and 6).
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Code availability
The software code for the CESM model is available from http://www.cesm.ucar.edu/models/.
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This project received funding from the European Union's Horizon 2020 Research and Innovation program under grant agreement no. 689443 via project iCUPE (Integrative and Comprehensive Understanding on Polar Environments), part of the European Commission (ERA-PLANET) and by the "Programma Nazionale per la Ricerca in Antartide" (PNRA, project number PNRA16_00295). This study also received funding from the European Research Council Executive Agency under the European Union's Horizon 2020 Research and Innovation programme (Project 'ERC-2016-COG 726349 CLIMAHAL'). NCAR is sponsored by the National Science Foundation under Grant Number 1852977. R.P.F. would like to express thanks for the financial support from CONICET-UNCuyo (SIIP-06/M111) and ANPCyT (PICT 2015-0714). This research was also supported by the Korea Polar Research Institute (KOPRI) project (PE20030) and by the Grant to Department of Science, Roma Tre University (MIUR-Italy Dipartimenti di Eccellenza).
Institute of Polar Sciences, CNR-ISP, Campus Scientifico Via Torino 155, Mestre, 30172, Venice, Italy
Andrea Spolaor, Clara Turetta, Fabrizio de Blasi, Elena Barbaro & Carlo Barbante
Department of Environmental Sciences, Informatics and Statistics, University Ca'Foscari of Venice, via Torino, 155 - 30172, Venice-Mestre, Italy
Andrea Spolaor, François Burgay, Clara Turetta, Fabrizio de Blasi, Elena Barbaro & Carlo Barbante
Paul Scherrer Institute, Laboratory of Environmental Chemistry, 5232, Villigen, PSI, Switzerland
François Burgay
Institute for Interdisciplinary Science, National Research Council (ICB-CONICET), FCEN-UNCuyo, Mendoza, 5501, Argentina
Rafael P. Fernandez
Department of Atmospheric Chemistry and Climate, Institute of Physical Chemistry Rocasolano, CSIC, Madrid, Spain
Carlos A. Cuevas, Juan Pablo Corella & Alfonso Saiz-Lopez
Korea Polar Research Institute, Incheon, 21990, Korea
Kitae Kim
National Center for Atmospheric Research, Boulder, CO, USA
Douglas E. Kinnison & Jean-François Lamarque
CIEMAT, Environmental Department, Av. Complutense 40, 28040, Madrid, Spain
Juan Pablo Corella
Physics of Ice, Climate and Earth, Niels Bohr Institute, University of Copenhagen, Tagensvej 16, Copenhagen, N2200, Denmark
Paul Vallelonga
UWA Oceans Institute, University of Western Australia, Crawley, WA, 6009, Australia
Department of Science, University of Roma Tre, Largo S. Leonardo Murialdo, 1, 00146, Roma, Italy
Massimo Frezzotti
Andrea Spolaor
Clara Turetta
Carlos A. Cuevas
Douglas E. Kinnison
Jean-François Lamarque
Fabrizio de Blasi
Elena Barbaro
Carlo Barbante
Alfonso Saiz-Lopez
A.S. and A.S.-L. devised the research; A.S., F.B., R.P.F. and A.S.-L. wrote the manuscript; F.B., C.T., E.B. and A.S. prepared the ice core and snow pit samples and ran the chemical analysis; F.D.B ran the statistical analysis; R.P.F., C.A.C. and A.S.-L. ran the chemistry-climate model; K.K., D.E. K. and J.-F.L. provided the laboratory data and helped with the model configuration; and C.B., P.V., J-P.C., M.F. and C.A. helped with the data interpretation and manuscript writing.
Correspondence to Andrea Spolaor or Alfonso Saiz-Lopez.
The authors declare no competing interests.
Peer review information Nature Communications thanks Markus Frey, Thorsten Hoffmann and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
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Spolaor, A., Burgay, F., Fernandez, R.P. et al. Antarctic ozone hole modifies iodine geochemistry on the Antarctic Plateau. Nat Commun 12, 5836 (2021). https://doi.org/10.1038/s41467-021-26109-x
DOI: https://doi.org/10.1038/s41467-021-26109-x
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Major advance for measurement based quantum computing?
The Measurement Based Quantum Computing Search Algorithm is Faster than Grover's Algorithm
If this recent paper is true, it seems like a major advance for measurement based quantum computing. The claim is based on numerical evidence, rather than a formal proof. Is this as important/revolutionary as I think it is?
This makes me wonder: Is there a known relation between the speed of gate-based quantum computing and the speed of measurement-based quantum computing?
cc.complexity-theory ds.algorithms quantum-computing
Jim GraberJim Graber
$\begingroup$ MBQC and the circuit model can each simulate the other with constant overhead. Ultimately an MBQC can be viewed as a circuit model computation, so it can never offer a speed up over what can be achieved directly in the circuit model. $\endgroup$ – Joe Fitzsimons Nov 19 '12 at 11:15
A paper which makes strong claims ought to be sufficiently clearly written for readers to check those claims. I don't find the current (http://arxiv.org/pdf/1211.3405v2.pdf) version of this paper expresses its results clearly enough to make a concrete assessment.
But if it were, I'd want to check:
a) whether the quantum part of the algorithm solely consists of Pauli measurements on a cluster state.
Such a measurement-based quantum computation can be simulated using polytime classical processing (due to the Gottesman-Knill theorem), and hence one would have a solely classical superfast search algorithm.
b) whether the oracle used might be of a different type to the oracle in Grover's algorithm.
The oracle in Grover's algorithm has the property that it must recognise the target string, but, crucially, it does not need to know this string in advance, or be able to calculate that target string in polytime (see Nielsen and Chuang chapter 6 for a discussion of this).
Unless P=NP, an oracle which knows the target string needs to be more powerful than one which merely recognises it (consider a search for a solution of an NP-hard problem). I'd want to check whether the oracle here is of the "recognising" or "knowing" kind. If it is the second, then the extra computational power in the oracle might, on its own, explain any exponential improvement reported.
Dan BrowneDan Browne
$\begingroup$ The authors have withdrawn their preprint from the arxiv. arxiv.org/abs/1211.3405v3 $\endgroup$ – Dan Browne Mar 22 '13 at 12:24
I think that this is an example of a preprint on a crank-friendly topic (specifically it asks: "Is the new claimed [revolutionary result] correct?"). But there are specific remarks that can be made about the technical problems of the manuscript in question.
In the second-to-last paragraph on page 1, they indicate that the "measurement angles" are just going to be from the set $\{0,\pi\}$, so that they measure only the states $| \pm \rangle$ — or in the language of observables: only the operators $\pm X$ are measured. Echoing Dan's earlier answer, the cluster state is a stabilizer state. It has been known since at least 1997 (from Daniel Gottesman's introduction of the stabilizer formalism) that Pauli observable measurements on stabilizer states can be simulated efficiently. Indeed, the natural decision problems arising from such simulations belongs not only to $\mathsf P$, but to the rather low class $\mathsf \oplus\mathsf L \subseteq \mathsf{NC^2} \subseteq \mathsf P$, by Aaronson+Gottesman (2004). Any such MBQC algorithm cannot possibly achieve an exponential speedup over a generic classical algorithm, let alone over other quantum algorithms.
Furthermore, they do not even describe how the black box for marking elements (or a suitable substitute) is to be implemented in the MBQC algorithm. They describe something along these lines in Eqns. (4) and (6), but they don't even describe how the angles are chosen, nor how it corresponds to any circuit which marks elements. They show other signs of confusion about Grover's algorithm, as they claim that the "tagging [performed by the oracle] is one to one, i.e. there is a single unique tag for each and every element in the search space". I don't know what that is supposed to mean, but it doesn't sound like the black box in Grover's algorithm to me.
So, with some inspection, I can say that it doesn't seem even to describe how it is meant to represent Grover's algorithm, and there is reason to believe it could never actually do so (nor surpass it) anyway.
Niel de Beaudrap
Not the answer you're looking for? Browse other questions tagged cc.complexity-theory ds.algorithms quantum-computing or ask your own question.
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Finding a basis for quantum measurement with maximum distinguishability | CommonCrawl |
SCAPP: an algorithm for improved plasmid assembly in metagenomes
Software article
David Pellow1,
Alvah Zorea2,
Maraike Probst3,
Ori Furman2,
Arik Segal4,5,
Itzhak Mizrahi2 &
Ron Shamir ORCID: orcid.org/0000-0003-1889-98701
Metagenomic sequencing has led to the identification and assembly of many new bacterial genome sequences. These bacteria often contain plasmids: usually small, circular double-stranded DNA molecules that may transfer across bacterial species and confer antibiotic resistance. These plasmids are generally less studied and understood than their bacterial hosts. Part of the reason for this is insufficient computational tools enabling the analysis of plasmids in metagenomic samples.
We developed SCAPP (Sequence Contents-Aware Plasmid Peeler)—an algorithm and tool to assemble plasmid sequences from metagenomic sequencing. SCAPP builds on some key ideas from the Recycler algorithm while improving plasmid assemblies by integrating biological knowledge about plasmids.
We compared the performance of SCAPP to Recycler and metaplasmidSPAdes on simulated metagenomes, real human gut microbiome samples, and a human gut plasmidome dataset that we generated. We also created plasmidome and metagenome data from the same cow rumen sample and used the parallel sequencing data to create a novel assessment procedure. Overall, SCAPP outperformed Recycler and metaplasmidSPAdes across this wide range of datasets.
SCAPP is an easy to use Python package that enables the assembly of full plasmid sequences from metagenomic samples. It outperformed existing metagenomic plasmid assemblers in most cases and assembled novel and clinically relevant plasmids in samples we generated such as a human gut plasmidome. SCAPP is open-source software available from: https://github.com/Shamir-Lab/SCAPP.
Video abstract
Plasmids play a critical role in microbial adaptation, such as antibiotic resistance or other metabolic capabilities, and genome diversification through horizontal gene transfer. However, plasmid evolution and ecology across different microbial environments and populations are poorly characterized and understood. Thousands of plasmids have been sequenced and assembled directly from isolated bacteria, but constructing complete plasmid sequences from short read data remains a hard challenge. The task of assembling plasmid sequences from shotgun metagenomic sequences, which is our goal here, is even more daunting.
There are several reasons for the difficulty of plasmid assembly. First, plasmids represent a very small fraction of the sample's DNA and thus may not be fully covered by the read data in high-throughput sequencing experiments. Second, they often share sequences with the bacterial genomes and with other plasmids, resulting in tangled assembly graphs. For these reasons, plasmids assembled from bacterial isolates are usually incomplete, fragmented into multiple contigs, and contaminated with sequences from other sources. The challenge is reflected in the title of a recent review on the topic: "On the (im)possibility of reconstructing plasmids from whole-genome short-read sequencing data" [1]. In a metagenomic sample, these problems are amplified since the assembly graphs are much larger, more tangled, and fragmented.
There are a number of tools that can be used to detect plasmid sequences including PlasmidFinder [2], cBar [3], gPlas [4], PlasFlow [5], and others. There is also the plasmidSPAdes assembler for assembling plasmids in isolate samples [6]. However, there are currently only two tools that attempt to reconstruct complete plasmid sequences in metagenomic samples: Recycler [7] and metaplasmidSPAdes [8] (mpSpades). mpSpades iteratively generates smaller and smaller subgraphs of the assembly graph by removing contigs with coverage below a threshold that increases in each iteration. As lower coverage segments of the graph are removed, longer contigs may be constructed in the remaining subgraph. Cyclic contigs are considered as putative plasmids and then verified using the profile of their genetic contents. The main idea behind Recycler is that a single shortest circular path through each node in the assembly graph can be found efficiently. The circular paths that have uniform read coverage are iteratively "peeled" off the graph and reported as possible plasmids. The peeling process reduces the residual coverage of each involved node, or removes it altogether. We note that these tools, as well as our work, focus on circular plasmids and do not assemble linear plasmid sequences.
Here we present SCAPP (Sequence Contents-Aware Plasmid Peeler), a new algorithm that uses the peeling idea of Recycler and also leverages external biological knowledge about plasmid sequences. In SCAPP, the assembly graph is annotated with plasmid-specific genes (PSGs) and nodes are assigned weights reflecting the chance that they are plasmidic based on a plasmid sequence classifier [9]. In the annotated assembly graph, we prioritize peeling off circular paths that include plasmid genes and highly probable plasmid sequences. SCAPP also uses the PSGs and plasmid scores to filter out likely false positives from the set of potential plasmids.
We tested SCAPP on both simulated and diverse real metagenomic data and compared its performance to Recycler and mpSpades. Overall, SCAPP performed better than the other tools across these datasets. SCAPP has higher precision than Recycler in all cases, meaning it more accurately constructs correct plasmids from the sequencing data. SCAPP also has higher recall than mpSpades in most cases, and higher precision in most of the real datasets. We developed and tested a novel strategy given parallel plasmidome and metagenome sequencing of the same sample. We show how to accurately assess the performance of the tools on metagenome data, even in the absence of known reference plasmids.
SCAPP accepts as input a metagenomic assembly graph, with nodes representing the sequences of assembled contigs and edges representing k-long sequence overlaps between contigs, and the paired-end reads from which the graph was assembled. SCAPP processes each component of the assembly graph and iteratively assembles plasmids from them. The output of SCAPP is a set of cyclic sequences representing confident plasmid assemblies.
A high-level overview of SCAPP is provided in Table 1 and depicted graphically in Fig. 1; the full algorithmic details are presented below. For brevity, we describe only default parameters below; see Additional file 1, Section S1 for alternatives.
Graphical overview of the SCAPP algorithm. (A) The metagenomic assembly graph is created from the sample reads. (B) The assembly graph is annotated with read mappings, presence of plasmid specific genes, and node weights based on sequence length, coverage, and plasmid classifier score. (C) Potential plasmids are iteratively peeled from the assembly graph. An efficient algorithm finds cyclic paths in the annotated assembly graph that have low weight and high chance of being plasmids. Cycles with uniform coverage are peeled. (D) Confident plasmid predictions are retained using plasmid sequence classification and plasmid-specific genes to remove likely false positive potential plasmids
Table 1 Overview of SCAPP
SCAPP is available from https://github.com/Shamir-Lab/SCAPP and fully documented there. It was written in Python3 and can be installed as a conda package, directly from Bioconda or from its sources.
The SCAPP algorithm
The full SCAPP algorithm is given in Algorithm 1. The peel function, which defines how cycles are peeled from the graph, is given in Algorithm 2.
Read mapping
The first step in creating the annotated assembly graph (Table 1 step 1a) is to align the reads to the contigs in the graph. The links between paired-end reads aligning across contig junctions are used to evaluate potential plasmid paths in the graph. SCAPP performs read alignment using BWA [10] and the alignments are filtered to retain only primary read mappings, sorted, and indexed using SAMtools [11].
Plasmid-specific gene annotation
We created sets of PSGs by database mining and curation by plasmid microbiology experts from the Mizrahi Lab (Ben-Gurion University). Information about these PSG sets is found in Additional file 1, Section S2. The sequences themselves are available from https://github.com/Shamir-Lab/SCAPP/tree/master/scapp/data.
A node in the assembly graph is annotated as containing a PSG hit (Table 1 step 1b) if there is a BLAST match between one of the PSG sequences and the sequence corresponding to the node (≥75% sequence identity along ≥75% of the length of the gene).
Plasmid sequence score annotation
We use PlasClass [9] to annotate each node in the assembly graph with a plasmid score (Table 1 step 1c). PlasClass uses a set of logistic regression classifiers for sequences of different lengths to assign a classification score reflecting the likelihood of each node to be of plasmid origin.
We re-weight the node scores according to the sequence length as follows. For a given sequence of length L and plasmid probability p assigned by the classifier, the re-weighted plasmid score is: \(s = 0.5 + \frac {p-0.5}{1+e^{-0.001(L-2000)}}\). This tends to pull scores towards 0.5 for short sequences, for which there is lower confidence, while leaving scores of longer sequences practically unchanged.
Long nodes (L>10 kbp) with low plasmid score (s<0.2) are considered probable chromosomal sequences and are removed, simplifying the assembly graph. Similarly, long nodes (L>10 kbp) with high plasmid score (s>0.9) are considered probable plasmid nodes.
Assigning node weights
In order to apply the peeling idea, nodes are assigned weights (Table 1 step 1d) so that lower weights correspond to higher likelihood to be assembled into a plasmid. Plasmid score and PSG annotations are incorporated into the node weights. A node with plasmid score s is assigned a weight w(v)=(1−s)/(C·L) where C is the depth of coverage of the node's sequence and L is the sequence length. This gives lower weight to nodes with higher coverage, longer sequence, and higher plasmid scores. Nodes with PSG hits are assigned a weight of zero, making them more likely to be integrated into any lowest-weight cycle in the graph that can pass through them.
Finding low-weight cycles in the graph
The core of the SCAPP algorithm is to iteratively find a lowest weight ("lightest") cycle going through each node in the graph for consideration as a potential plasmid. We use the bidirectional single-source, single-target shortest path implementation of the NetworkX Python package [12].
The order that nodes are considered matters since in each iteration potential plasmids are peeled from the graph, affecting the cycles that may be found in subsequent iterations. The plasmid annotations are used to decide the order that nodes are considered: first all nodes with PSGs, then all probable plasmid nodes, and then all other nodes in the graph (Table 1 step 2). If the lightest cycle going through a node meets certain criteria described below, it is peeled off, changing the coverage of nodes in the graph. Performing the search for light cycles in this order ensures that the cycles through more likely plasmid nodes will be considered before other cycles.
Assessing coverage uniformity
The lightest cyclic path, weighted as described above, going through each node is found and evaluated. Recycler sought a cycle with near uniform coverage, reasoning that all contigs that form a plasmid should have roughly the same coverage. However, this did not take into account the overlap of the cycle with other paths in the graph (see Fig. 2). To account for this, we instead compute a discounted coverage score for each node in the cycle based on its interaction with other paths as follows:
Evaluating and peeling cycles. Numbers inside nodes indicate coverage. All nodes in the example have equal length. A Cycles (a,e,f) and (c,e,g) have the same average coverage (13.33) and coefficient of variation (CV, 0.35), but their discounted CV values differ: The discounted coverage of node a is 6, and the discounted coverage of node e is 10 in both cycles. The left cycle has discounted CV=0.22 and the right has discounted CV=0. By peeling off the mean discounted coverage of the right cycle (10) one gets the graph in B. Note that nodes g,c were removed from the graph since their coverage was reduced to 0, and the coverage of node e was reduced to 10
The discounted coverage of a node v in the cycle C is its coverage cov(v) times the fraction of the coverage on all its neighbors (both incoming and outgoing), \(\mathcal {N}(v)\), that is on those neighbors that are in the cycle (see Fig. 2):
$${}cov'(v,C) = cov(v) \cdot \left({\sum\limits_{u \in C \land u \in \mathcal{N}(v)}cov(u)}/{\sum\limits_{u \in \mathcal{N}(v)}cov(u)} \right) $$
A node v in cycle C with contig length len(v) is assigned a weight f corresponding to its fraction of the length of the cycle: \(f(v,C) = {len(v)}/{\sum \limits _{u\in C} len(u)}\). These weights are used to compute the weighted mean and standard deviation of the discounted coverage of the nodes in the cycle: \(\mu _{cov'}(C) = \sum \limits _{u \in C} f(u,C)cov'(u,C)\),
$$STD_{cov'}(C) = \sqrt{\sum\limits_{u \in C} f(u,C)(cov'(u,C)-\mu_{cov'}(C))^{2}}$$
The coefficient of variation of C, which evaluates its coverage uniformity, is the ratio of the standard deviation to the mean:
\(CV(C) = \frac {STD_{cov'}(C)}{\mu _{cov'}(C)}\)
Finding potential plasmid cycles
After each lightest cycle has been generated, it is evaluated as a potential plasmid based on its structure in the assembly graph, the PSGs it contains, its plasmid score, paired-end read links, and coverage uniformity. The precise evaluation criteria are described in Additional file 1, Section S3. A cycle that passes them is defined as a potential plasmid (Table 1 steps 3–5). The potential plasmid cycles are peeled from the graph in each iteration as defined in Algorithm 2 (see also Fig. 2).
Filtering confident plasmid assemblies
In the final stage of SCAPP, PSGs and plasmid scores are used to filter out likely false-positive plasmids from the output and create a set of confident plasmid assemblies (Table 1 step 6). All potential plasmids are assigned a length-weighted plasmid score and are annotated with PSGs as was done for the contigs during graph annotation. Those that belong to at least two of the following sets are reported as confident plasmids: (a) potential plasmids containing a match to a PSG, (b) potential plasmids with plasmid score >0.5, (c) self-loop nodes.
We tested SCAPP on simulated metagenomes, human gut metagenomes, a human gut plasmidome dataset that we generated and also on parallel metagenome and plasmidome datasets from the same cow rumen microbiome specimen that we generated. The test settings and evaluation methods are described in Additional file 1, Section S5.
Simulated metagenomes
We created seven read datasets simulating metagenomic communities of bacteria and plasmids and assembled them. Datasets of increasing complexity were created as shown in Table 2. We randomly selected bacterial genomes along with their associated plasmids and used realistic distributions for genome abundance and plasmid copy number. Further details of the simulation can be found in Additional file 1, Section S4, and in Additional file 2. 5M paired-end reads were generated for Sim1 and Sim2, 10M for Sim3 and Sim4, and 20M for Sim5, Sim6, and Sim7.
Table 2 Performance on simulated metagenome datasets. The number of covered plasmids (# covered) reports the number of the simulation plasmids that were covered by reads along at least 95% of their length. The set of covered plasmids is used as the gold standard in calculating the performance metrics. The numbers in parentheses are the median plasmid lengths (in kbp). F1 score is presented as a percent
Table 2 presents features of the simulated datasets and reports the performance of Recycler, mpSpades, and SCAPP on them. For brevity we report only F1 scores; precision and recall scores are reported in Supplementary Table 1, Additional file 1 (Section S6). Here, and throughout, all scores are adjusted to percent. SCAPP had the highest F1 score in all cases, followed by Recycler. SCAPP consistently achieved higher precision than Recycler, allowing it to perform better overall. mpSpades had the highest precision, but assembled far fewer plasmids than the other tools and gained lower recall and F1 scores. In fact, most of the plasmids assembled by mpSpades were also assembled by the other tools (see Figure S1 in Additional file 1), suggesting that these plasmids were easier to capture.
All of the tools assembled mostly shorter plasmids as reflected in the median plasmid lengths. This is likely due to the higher coverage and simplicity in the assembly graph of these plasmids, as also evidenced by the shorter lengths of the covered plasmids. SCAPP assembled many more long plasmids (>10 kbp) than the other tools, achieving much higher recall and higher F1 score for these longer plasmids than the other tools, at the cost of some precision (see Supplementary Table 2 in Additional file 1, Section S6 for results broken down by short and long plasmids).
Human gut microbiomes
We tested the plasmid assembly algorithms on data of twenty publicly available human gut microbiome samples selected from the study of Vrieze et al. [13]. The true set of plasmids in these samples is unknown. Instead, we matched all assembled contigs to PLSDB [14] and considered the set of the database plasmids that were covered by the contigs as the gold standard (see Additional file 1, Section S5 for details). All tools were evaluated according to the same gold standard. We note that this limits the evaluation to known plasmids, potentially over-counting the number of false positive plasmids. We chose the human gut microbiome in this experiment and the next, as it is one of the most widely studied microbiome environments so plasmids in gut microbiome samples are most likely to be represented in the database.
Table 3 presents the results of the three algorithms averaged across all twenty samples. The detailed results on each of the samples are presented in Supplementary Table 2 and Figure S2, Additional file 1 (Section S7). SCAPP performed best in more cases, with mpSpades failing to assemble any gold standard plasmid in over half the samples. We note that all of the cases where SCAPP had recall of 0 occurred when the number of gold standard plasmids was very small and the other tools also failed to assemble them. On the largest samples with the most gold standard plasmids SCAPP performed best, highlighting its superior performance on the types of samples most likely to be of interest in experiments aimed at plasmid assembly. SCAPP consistently outperformed Recycler by achieving higher precision, a result that is consistent with the other experiments.
Table 3 Performance on the human gut metagenomes. Number of plasmids, the median plasmid length (in kbp), and performance measures for all tools are averaged across the twenty samples. The average number of plasmids and median length of the gold standard sets of plasmids were 4.8 and 12.4 respectively
Human gut plasmidome
The protocol developed in Brown Kav et al. [15] enables extraction of DNA from isolate or metagenomic samples with the plasmid content highly enriched. The sequence contents of such a sample is called the plasmidome of the sample. This enrichment for plasmid sequences increases the chance of revealing the plasmids in the sample. The protocol was assessed to achieve samples with at least 65% plasmid contents by Krawczyk et al. [5]. We sequenced the plasmidome of the human gut microbiome from a healthy adult male according to the plasmid enrichment protocol. 18,616,649 paired-end reads were sequenced with the Illumina HiSeq2000 platform, read length 150bp and insert size 1000.
The gold standard set of plasmids, determined as for the gut metagenome samples, consisted of 74 plasmids (median length = 2.1 kbp). Note that the plasmidome extraction process over-amplifies shorter plasmids, as reflected in the shorter median plasmid length. Performance was computed as in the metagenomic samples and is shown in Table 4. SCAPP achieved best overall performance, while mpSpades had lower precision and much lower recall than the other tools.
Table 4 Performance on the human gut plasmidome. Number of plasmids, the median plasmid length (in kbp), and performance measures for all tools
Notably, although the sample was obtained from a healthy donor, some of the plasmids reconstructed by SCAPP matched reference plasmids found in potentially pathogenic hosts such as Klebsiella pneumoniae, pathogenic serovars of Salmonella enterica, and Shigella sonnei. The detection of plasmids previously isolated from pathogenic hosts in the healthy gut indicates potential pathways for transfer of virulence genes.
We used MetaGeneMark [16] to find potential genes in the plasmids assembled by SCAPP. Two hundred ninety-four genes were found, and we annotated them with the NCBI non-redundant (nr) protein database using BLAST. Forty-six of the plasmids contained 170 (58%) genes with matches in the database (>90% sequence identity along >90% of the gene length), of which 77 (45%) had known functional annotations, which we grouped manually in Fig. 3A. There were six antibiotic and toxin (such as heavy metal) resistance genes, all on plasmids that were not in the gold standard set, highlighting SCAPP's ability to find novel resistance carrying plasmids. Sixty of the 77 genes (78%) with functional annotations had plasmid-associated functions: replication, mobilization, recombination, resistance, and toxin-antitoxin systems. Twenty-nine out of the 33 plasmids that contained functionally annotated genes (88%) contained at least one of these plasmid associated functions. This provides a strong indication that SCAPP succeeded in assembling true plasmids of the human gut plasmidome.
Annotation of genes on the plasmids identified by SCAPP in the human gut plasmidome sample. A Functional annotations of the plasmid genes. B Host annotations of the plasmid genes. "Broad-range" plasmids had genes annotated with hosts from more than one phylum
We also examined the hosts that were annotated for the plasmid genes and found that almost all of the plasmids with annotated genes contained genes with annotations from a variety of hosts, which we refer to here as "broad-range" (see Fig. 3B). Of the 40 plasmids with genes from annotated hosts, only 10 (25%) had genes with annotated hosts all within a single phylum. This demonstrates that these plasmids assembled and identified by SCAPP may be involved in one stage of transferring genes, such as the antibiotic resistance genes we detected, across a range of bacteria.
Parallel metagenomic and plasmidome samples
We performed two sequencing assays on the same cow rumen microbiome sample of a four-month old calf. In one subsample total DNA was sequenced. In the other, plasmid-enriched DNA was extracted as described in Brown Kav et al. [15] and sequenced (see Fig. 4). 27,127,784 paired-end reads were sequenced in the plasmidome, and 54,292,256 in the metagenome. Both were sequenced on the Illumina HiSeq2000 platform with read length 150bp and insert size 1000.
Outline of the read-based performance assessment. Plasmidome (I) and metagenome reads (II) are obtained from subsamples of the same sample. (III) The metagenome reads are assembled into a graph. (IV) The graph is used to detect and report plasmids by the algorithm of choice. (V) The plasmidome reads are matched to assembled plasmids. Matched plasmids (red) are used to calculate plasmid read-based precision. (VI) The plasmidome reads are matched to the assembly graph contigs. Covered contigs (red) are considered plasmidic. The fraction of total length of plasmidic contigs included in the detected plasmids gives the plasmidome read-based recall
This parallel data enabled us to assess the plasmids assembled on the metagenome using the plasmidome, without resorting to PLSDB matches as the gold standard. Such assessment is especially useful for samples from non-clinical environments such as the cow rumen, as PLSDB likely under-represents plasmids in them.
Table 5 summarizes the results of the three plasmid discovery algorithms on both subsamples. mpSpades made the fewest predictions and Recycler made the most. To compare the plasmids identified by the different tools, we considered two plasmids to be the same if their sequences matched at >80% identity across >90% of their length. The comparison is shown in Figure S3, Additional file 1 (Section S8). In the plasmidome subsample, 50 plasmids were identified by all three methods. Seventeen were common to the three methods in the metagenome. In both subsamples, the Recycler plasmids included all or almost all of those identified by the other methods plus a large number of additional plasmids. In the plasmidome, SCAPP and Recycler shared many more plasmids than mpSpades and Recycler.
Table 5 Number of plasmids assembled by each tool and their median lengths (in kbp) for the parallel metagenome and plasmidome samples
We also evaluated the results of the plasmidome and metagenome assemblies by comparison to PLSDB as was done for the human gut samples. The metagenome contained only one matching PLSDB reference plasmid, and none of the tools assembled it. The plasmidome had only seven PLSDB matches, and mpSpades, Recycler, and SCAPP had F1 scores of 2.86, 2.67, and 1.74, respectively. The low fraction of PLSDB matches out of the assembled plasmids suggests that the tools can identify novel plasmids that are not in the database.
In order to fully leverage the power of parallel samples, we computed the performance of each tool on the metagenomic sample using the reads of the plasmidomic sample, without doing any contig and plasmid assembly on the latter. The rationale was that the reads of the plasmidome represent the full richness of plasmids in the sample in a way that is not biased by a computational procedure or prior biological knowledge.
We calculated the plasmidome read-based precision by mapping the plasmidomic reads to the plasmids assembled from the metagenomic sample (Fig. 4). A plasmid with >90% of its length covered by more than one plasmidomic read was considered to be a true positive. The precision of an algorithm was defined as the fraction of true positive plasmids out of all reported plasmids. The plasmidome read-based recall was computed by mapping the plasmidomic reads to the contigs of the metagenomic assembly. Contigs with >90% of their length covered by plasmidomic reads at depth >1 were called plasmidic contigs. Plasmidic contigs that were part of the assembled plasmids were counted as true positives, and those that were not were considered false negatives. The recall was defined as the fraction of the plasmidic contigs' length that was integrated in the assembled plasmids. Note that the precision and recall here are measured using different units (plasmids and base pairs, respectively) so they are not directly related. For mpSpades, which does not output a metagenomic assembly, we mapped the contigs from the metaSPAdes assembly to the mpSpades plasmids using BLAST (>80% sequence identity matches along >90% of the length of the contigs).
There were 293 plasmidic contigs in the metagenome assembly graph, with a total length of 146.6 kbp. The plasmidome read-based performance is presented in Fig. 5A. All tools achieved a similar recall of around 12. SCAPP and mpSpades performed similarly, with SCAPP having slightly higher precision (24.0 vs 23.1) but slightly lower recall (11.9 vs 12.2). Recycler had a bit higher recall (13.1), but at the cost of far lower precision (11.7). Hence, a much lower fraction of the plasmids assembled by Recycler in the metagenome were actually supported by the parallel plasmidome sample, adding to the other evidence that the false positive rate of Recycler exceeds that of the other tools.
Performance on the parallel datasets. A Plasmidome read-based performance. B Performance of each tool on the plasmids assembled from the metagenome using as gold standard the plasmids assembled from the plasmidome by the same tool. C Overall performance on the plasmids assembled from the metagenome compared to the union of all plasmids assembled by all tools in the plasmidome
We also compared the plasmids assembled by each tool in the two subsamples. For each tool, we considered the plasmids it assembled from the plasmidome to be the gold standard set, and used it to score the plasmids it assembled in the metagenome. The results are shown in Fig. 5B. SCAPP had the highest precision. Since mpSpades had a much smaller gold standard set, it achieved higher recall and F1. Recycler output many more plasmids than the other tools in both samples, but had much lower precision, suggesting that many of its plasmid predictions may be spurious.
Next, we considered the union of the plasmids assembled across all tools as the gold standard set and recomputed the scores. We refer to them as "overall" scores. Figure 5C shows that overall precision scores were the same as in Fig. 5B, while overall recall was lower for all the tools, as expected. mpSpades underperformed because of its smaller set of plasmids, and SCAPP had the highest overall F1 score. Recycler performed relatively better on recall than the other tools as expected, as it reports many plasmids and has significant overlap with the plasmids reported by the other tools.
We detected potential genes in the plasmids assembled by SCAPP in the plasmidome sample and annotated them as we did for the human gut plasmidome. The gene function and host annotations are shown in Figure S4, Additional file 1 (Section S8). Out of 242 genes, only 34 genes from 17 of the plasmids had annotations, and only 18 of these had known functions, highlighting that many of the plasmids in the cow rumen plasmidome are as yet unknown. The high percentage of genes of plasmid function (15/18) indicates that SCAPP succeeded in assembling novel plasmids. Unlike in the human gut plasmidome, most of the plasmids with known host annotations had hosts from a single phylum.
We summarize the performance of the tools across all the test datasets in Table 6. The performance of two tools was considered similar (denoted ≈) if their scores were within 5% of each other. Performance of one tool was considered to be much higher than the other (≫) if its score was >30% higher (an increase of 5−30% is denoted by >).
Table 6 Summary of performance. Comparison of the performance of the tools on each of the datasets. When multiple samples were tested, the number of samples appears in parentheses, and average performance is reported. For the parallel samples results are for the evaluation of the metagenome based on the plasmidome, and precision and recall are plasmidome read-based. Unless otherwise stated, F1 score is used. Note that in the simulations, SCAPP ≫ mpSpades
We see that in most cases SCAPP was the highest performer. Furthermore, in all other cases SCAPP performed close to the top performing tool.
The runtime and memory usage of the three tools are presented in Table 7. Recycler and SCAPP require assembly by metaSPAdes and pre-processing of the reads and the resulting assembly graph. SCAPP also requires post-processing of the assembled plasmids. mpSpades requires post-processing of the assembled plasmids with the plasmidVerify tool. The reported runtimes are for the full pipelines necessary to run each tool – from reads to assembled plasmids.
Table 7 Resource usage of the three methods. Peak RAM of the assembly step (metaSPAdes for Recycler and SCAPP, metaplasmidSPAdes for mpSpades) in GB. Runtime (wall clock time, in minutes) is reported for the entire pipeline including assembly and any pre-processing and post-processing required. Human metagenome results are an average across the 20 samples
In almost all cases assembly was the most memory intensive step, and so all tools achieved very similar peak memory usage (within 0.01 GB). Therefore, we report the RAM usage for this step.
The assembly step was also the longest step in all cases. SCAPP was slightly slower than Recycler as a result of the additional annotation steps, and mpSpades was 5–40% faster. However, note that mpSpades does not output a metagenomic assembly graph, so users interested in both the plasmid and non-plasmid sequences in a sample would need to run metaSPAdes as well, practically doubling the runtime.
Performance measurements were made on a 44-core, 2.2 GHz server with 792 GB of RAM. Sixteen processes were used where possible. Recycler is single-threaded, so only one process was used for it.
Plasmid assembly from metagenomic sequencing is a very difficult task, akin to finding needles in a haystack. This difficulty is demonstrated by the low numbers of plasmids found in real samples. Even in samples of the human gut microbiome, which is widely studied, relatively few plasmids that have matches in the extensive plasmid database PLSDB were recovered. Despite the challenges, SCAPP was able to assemble plasmids across a number of clinically relevant samples. SCAPP significantly outperformed mpSpades in simulation and on a range of human gut metagenome and plasmidome samples. In simulation mpSpades achieved very high precision at the expense of low recall, and SCAPP had higher combined F1 score. The high precision was not observed in real data, which is more difficult than the simulations. SCAPP was also consistently better than Recycler across almost all tests. Though SCAPP and Recycler share the idea of cycle peeling, SCAPP was shown to have higher precision, due to incorporating additional biological information and better edge weighting.
Another contribution of this study is the joint analysis of the parallel metagenome and plasmidome from the same sample. We show that this enables a novel way to evaluate plasmid assembly algorithms on the metagenome data, by using the coverage information from the plasmidome. This novel approach bypasses the need to rely on known plasmids for evaluation, which is biased due to research focus. We developed several evaluation metrics for such data, and think they can be useful for future plasmid studies, especially in non-clinical and non-human samples where plasmid knowledge is scarce.
A key difficulty in evaluation of performance of plasmid discovery algorithms is the lack of gold standard. The verification of reported plasmids is done either based on prior biological knowledge, which is biased, or by experimental verification, which is slow and expensive. Moreover, such verification evaluates precision but does not give information on the extent of missed plasmids, or recall. While simulations can evaluate both parameters accurately, they are inherently artificial, and necessitate many modeling assumptions that are not fully supported by experimental data. For that reason we chose here to focus primarily on real data, and preferred diversity in the real data types over extensive but artificial simulations. The parallel samples strategy is another partial answer to this problem.
SCAPP has several limitations. Like the other de Bruijn graph-based plasmid assemblers, it may split a cycle into two when a shorter cycle is a sub-path of a longer cycle. It also has difficulties in finding very long plasmids, as these tend to not be completely covered and fragmented into many contigs in the graph. Note however that it produced longer cycles than Recycler. Compared to mpSpades, each algorithm produced longer cycles in different tests. Another limitation is the inherent bias in relying on known plasmid genes and plasmid databases, which tend to under-represent non-clinical samples. With further use of tools like SCAPP, perhaps with databases tailored to specific environments, further improvement is possible.
We introduced SCAPP, a new plasmid discovery tool based on combination of graph theoretical and biological considerations. Overall, SCAPP demonstrated better performance than Recycler and metaplasmidSpades in a wide range of real samples from diverse contexts. By applying SCAPP across large sets of samples, many new plasmid reference sequences can be assembled, enhancing our understanding of plasmid biology and ecology.
The datasets supporting the conclusions of this article are available in the sequence read archive (SRA), accession numbers: ERR1297645, ERR1297651, ERR1297671, ERR1297685, ERR1297697, ERR1297700, ERR1297720, ERR1297738, ERR1297751, ERR1297770, ERR1297785, ERR1297796, ERR1297798, ERR1297810, ERR1297822, ERR1297824, ERR1297834, ERR1297838, ERR1297845, ERR1297852 (for the human gut metagenomes); accession SRR11038083 (for the human gut plasmidome); and accessions SRR11038085 and SRR11038085 (for the cow rumen metagenome and plasmidome samples, respectively). Project name: SCAPP Project homepage: https://github.com/Shamir-Lab/SCAPP Operating system: Platform independent (tested on Linux) Programming language (Python3) License: MIT
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We thank members of the Shamir Lab for their help and advice—Roye Rozov, Lianrong Pu, Hagai Levi, and Nimrod Rappoport.
PhD fellowships from the Edmond J. Safra Center for Bioinformatics at Tel-Aviv University and Israel Ministry of Immigrant Absorption (to DP). Israel Science Foundation (ISF) grant 1339/18, US - Israel Binational Science Foundation (BSF) and US National Science Foundation (NSF) grant 2016694 (to RS), ISF grant 1947/19 and ERC Horizon 2020 research and innovation program grant 640384 (to IM).
Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 6997801, Israel
David Pellow & Ron Shamir
Department of Life Sciences, Ben-Gurion University of the Negev and the National Institute for Biotechnology in the Negev, Beer-Sheva, 8410501, Israel
Alvah Zorea, Ori Furman & Itzhak Mizrahi
Institute of Microbiology, University of Innsbruck, Innsbruck, A-6020, Austria
Maraike Probst
Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, 8410501, Israel
Arik Segal
Soroka University Medical Center, Beer-Sheva, 8410501, Israel
David Pellow
Alvah Zorea
Ori Furman
Itzhak Mizrahi
Ron Shamir
DP developed and implemented the SCAPP algorithm and benchmark experiments, performed analysis, and wrote the manuscript. AZ assisted with analysis and plasmid annotations and wrote the manuscript. MP curated plasmid-specific genes, assisted with gene annotations and wrote the manuscript. OF oversaw the parallel cow rumen metagenome-plasmidome experiment. AS oversaw the human gut plasmidome experiment. IM oversaw experimental and analysis aspects of the project and edited the manuscript. RS oversaw the computational and analysis aspects of project and edited the manuscript. All authors edited and approved the final manuscript.
Correspondence to David Pellow.
Sequencing of the human gut plasmidome was approved by the local ethics committee of Clalit HMO, approval number 0266-15-SOR. Extraction and sequencing of the cow rumen microbiome was approved by the local ethics committee of the Volcani Center, approval numbers 412/12IL and 566/15IL.
Additional file 1 — Supplementary information
Supplementary methods, experimental settings information, and results supporting the main text of this paper, including Figures S1-S4, Supplementary Tables 1 and 2. PDF file.
Additional file 2 — Simulation reference genomes
Tab-separated list of the human gut-specific reference genomes used in the simulations.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
Pellow, D., Zorea, A., Probst, M. et al. SCAPP: an algorithm for improved plasmid assembly in metagenomes. Microbiome 9, 144 (2021). https://doi.org/10.1186/s40168-021-01068-z | CommonCrawl |
February 2019 , Volume 26, Issue 3, pp 1895–1908 | Cite as
Dexamethasone-containing bioactive dressing for possible application in post-operative keloid therapy
Agnieszka Rojewska
Anna Karewicz
Marta Baster
Mateusz Zając
Karol Wolski
Mariusz Kępczyński
Szczepan Zapotoczny
Krzysztof Szczubiałka
Maria Nowakowska
First Online: 10 December 2018
Bioactive dressing based on bacterial cellulose modified with carboxymethyl groups (mBC) was successfully prepared and studied. The surface of mBC was activated using carbodiimide chemistry and decorated with alginate/hydroxypropyl cellulose submicroparticles containing dexamethasone phosphate (DEX-P). Prior to their deposition particles were coated with chitosan in order to facilitate their binding to mBC, and to increase the control over the release process. The detailed physicochemical characterization of the particles and the bioactive dressing was performed, including the determination of the particles' size and size distribution, DEX-P encapsulation efficiency and loading, particles' distribution on the surface of the mBC membrane, as well as DEX-P release profiles from free and mBC-bound particles. Finally, the preliminary cytotoxicity studies were performed. The fabricated bioactive material releases DEX-P in a controlled manner for as long as 25 h. Biological tests in vitro indicated that the dexamethasone-containing submicroparticles are not toxic toward fibroblasts, while effectively inhibiting their proliferation. The prepared bioactive dressing may be applied in the treatment of the post-operative wounds in the therapy of keloids and in other fibrosis-related therapies.
Alginate Hydroxypropyl cellulose Dexamethasone phosphate Modified bacterial cellulose Wound healing Drug release
Dexamethasone sodium phosphate (DEX-P) is a water-soluble derivative of dexamethasone (DEX), a potent corticosteroid used widely to treat various inflammatory and autoimmune conditions. Although many pharmaceutical applications of DEX-P are based on its anti-inflammatory activity (Hickey et al. 2002; Zhang et al. 2016; Hu et al. 2017), its use as a bioactive component of the implants, stents and wound dressings can also benefit from the anti-proliferative (Bao et al. 2016) and anti-apoptotic properties of the drug toward fibroblasts (Nieuwenhuis et al. 2010). DEX and DEX-P were shown to have a negative influence on wound healing process, but only when administered systemically for longer time periods, in particular when the patient was treated with that drug prior to injury (Wang et al. 2013). No significant influence of the short-term post injury/surgery application of DEX and DEX-P was, however, confirmed so far. On the contrary, there are reports on the positive effects of DEX in healing process. DEX was applied with success in repair of the mucous membrane defects in oral submucous fibrosis (Raghavendra Reddy et al. 2012). In that study Reddy et al. have used DEX-impregnated collagen membranes to cover the raw wound created by surgical excision of fibrous bands. The presence of the drug was shown to decrease the inflammatory reaction and extent of the fibrosis process, most probably due to the reduced proliferation and deposition of fibroblasts. Beule et al. (2009) have shown the ability of DEX to decrease postoperative osteogenesis in a standardized animal wound model for endoscopic surgery of sinus. Li et al. (2014) have recently proposed the electrospun polymer fiber meshes based on poly(lactide-co-glycolide) (PLGA) as a delivery vehicle for DEX and green tea polyphenols. The obtained, bioactive dressing was proposed for the post-operative therapy of keloids. Hydrophilic green tea polyphenols were introduced as necessary permeation enhancers for hydrophobic DEX, simultaneously providing antibacterial activity.
The long-term systemic treatment with DEX may lead to various adverse effects, including swelling, insomnia, bleeding of the stomach or intestines, Cushing's Syndrome, diabetes, or osteoporosis (Vardy et al. 2006; Ren et al. 2015). The systemic absorption of the topically administered DEX is low, but not negligible (Weijtens et al. 2002). Thus, there is a need for a controlled delivery system for DEX-P. In response to this challenge, we have developed a nanoparticulate delivery system for DEX-P based on the alginate-hydroxypropylcellulose (ALG/HPC) composite. Such system can be utilized in various topical applications to increase the duration, efficiency and safety of therapy. In a current paper we used that system as a component of a bioactive dressing. For that purpose the obtained ALG/HPC particles containing DEX-P were coated with a thin layer of chitosan to increase the control over the drug release and enable their binding to bacterial cellulose (Bionanocellulose®, BC). They were then covalently bound to the modified Bionanocellulose® membrane (mBC), which was obtained by functionalization of BC with carboxyl groups. BC was chosen because it constitutes an excellent wound dressing (Ul-Islam 2013; Liu 2016). It is a natural biopolymer material consisting of the interconnected network of cellulose fibrils. BC has high surface area and high ability for water retention. Due to its high sorption ability it allows to remove exudates from the wound, while providing the moist environment which promotes healing processes and prevents the formation of scars (Moritz et al. 2014; Napavichayanun et al. 2016). Ionically-modified BC was shown to have a better stability in water and higher water retention (Spaic 2014).
The studies presented in the current paper were carried out based on the hypothesis that one can fabricate the bioactive wound dressing material by immobilization on the surface of BC the polysaccharide submicroparticles containing dexamethasone phosphate (ensuring prolonged and controlled release profile of that drug) which can be particularly useful in the therapy of post-operative wounds, especially those resulting from the surgical treatment of the pathologies related to the uncontrolled proliferation of the fibrous tissue (fibrosis, keloid).
Experimental part
The sheets of Bionanocellulose® modified with carboxymethyl groups (mBC) were kindly provided by Biovico company [(6.07 ± 0.42) nmol of carboxylic groups per 1 mm2 of mBC surface (Guzdek et al. 2018)]. Dexamethasone 21-phosphate disodium salt (DEX-P, ≥ 98%, Sigma-Aldrich), hydroxypropyl cellulose (HPC, MW = 100,000 g/mol, Sigma-Aldrich), alginic acid sodium salt (ALG, medium molecular weight, from brown algae, Sigma-Aldrich; Mv = 260,000 g/mol, M/G ratio = 1.20), chitosan (low molecular weight, Sigma-Aldrich, Mv = 120,000 g/mol, degree of deacetylation DDA = 79%), calcium chloride (p.a., Fluka, Poland), 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC, Sigma-Aldrich, commercial grade, powder), N-hydroxysuccinimide (NHS, 98%, Sigma-Aldrich), 2-propanol (99.9% pure, Sigma Aldrich), acetonitrile (gradient grade for liquid chromatography, Sigma-Aldrich), acetic acid (≥ 99.5%, Chempur), fluorescein sodium salt (powder, Sigma Aldrich), rhodamine B isothiocyanate (powder, Sigma-Aldrich), sodium phosphate monobasic dihydrate (≥ 99.0%, Sigma-Aldrich) were used as received. Dulbecco's Modified Eagle's Medium–high glucose (DMEM), phosphate buffered saline tablets (PBS) and Cell Proliferation Kit II (XTT) were purchased from Sigma-Aldrich; HyClone Research Grade Fetal Bovine Serum, South American Origin (FBS), HyClone trypsin–EDTA and HyClone Penicillin–Streptomycin solution were purchased from Symbios.
Synthesis of ALG/HPC submicroparticles containing DEX-P
To encapsulate DEX-P into the nanospheres, 20 mg of the drug was dissolved in 5 mL of water. Then 50 mg of sodium alginate (ALG) and 12.5 mg of hydroxypropyl cellulose (HPC) were dissolved in this solution. After stirring for 1 h at room temperature, the resulting solution was injected at 0.05 mL/min with a syringe pump (Aladdin-1000, World Precision Instruments) to a solution of cross-linking agent (0.2 M calcium chloride, 10 mL) under continuous stirring. The obtained ALG/HPC particles were filtered off using a fritted glass funnel (11-G4), washed with distilled water and isopropanol and dried at room temperature.
Coating ALG/HPC particles with chitosan
20 mg of nanospheres was added to 1 mL of 1% (w/v) solution of chitosan in 1% (w/v) acetic acid and stirred at room temperature for 2 h. The excess of the chitosan was then removed by filtration on a Büchner funnel (11-G4) and the obtained ALG/HPC-Ch particles were dried at room temperature.
Deposition of ALG/HPC-Ch on BC modified with carboxyl groups (mBC)
To attach the chitosan-coated nanospheres on the surface of nanocellulose modified with carboxyl groups, a 4 cm2 sheet of mBC was immersed in an acetate buffer (pH 5.5) at room temperature. NHS (0.025 M) and EDC (0.008 M) were then added sequentially and the sample was continuously stirred at 150 rpm for 120 min. The activated BC was rinsed with deionized water, incubated with the suspension of the nanospheres (100 mg) in phosphate buffer (pH 7.4) for 30 min, and washed thoroughly with deionized water. The deposition reaction was carried out in the ultrasound bath to prevent aggregation of the particles. The obtained samples of mBC with attached particles (ALG/HPC-Ch-mBC) were dried in the air.
Release profiles: entrapment efficiency and loading capacity determination
20 mg of the ALG/HPC particles loaded with DEX-P was placed in a 15 mL centrifuge tube and 5 mL of 10 mM PBS (pH 7.4) was added. The sample was incubated at 37 °C (IKA, a KS 3000 incubator) with continuous agitation (140 rpm). After defined time intervals the sample was centrifugated at 10,000 rpm for 5 min and then the supernatant was collected. The new portion (5 mL) of PBS was added to the nanospheres and the system was placed back in the incubator. The experiment was performed in four repetitions.
To study the release profiles of DEX-P from ALG/HPC-Ch-mBC, a 4 cm2 sample of ALG/HPC-Ch-mBC was incubated in 5 mL of 10 mM PBS (pH 7.4). The sample was incubated at 37 °C with continuous agitation (140 rpm). After defined time intervals the supernatant was collected, replaced with the new portion (5 mL) of PBS and the system was placed back in the incubator. The experiment was performed in four repetitions.
The concentration of DEX-P in the collected samples was determined using an HPLC system (Waters) consisting of a 515 pump, a rheodyne-type dosing system and a 2996 Photodiode Array Detector. The separation was performed on a C18 column (3.9 mm × 150 mm, 5 µm), using a 30:70 (v/v) mixture of acetonitrile and 10 mM phosphate buffer (pH 7.4) as the mobile phase at a flow rate of 0.5 mL/min. The drug signal was detected at 241 nm (absorption maximum of DEX-P). All the measurements were done in triplicate. DEX-P concentration was calculated based on the calibration curve obtained for the standard solutions (R2 = 0.999). Entrapment efficiency (EE [%]) and loading capacity (LC [%]) were calculated based on the total amount of the drug released from the obtained particles, using the following equations:
$$EE \left[ \% \right] = \frac{Total\, weight\, of\,DEX - P \,in \,the\, obtained \,submicroparticles}{Weight\, of\,DEX - P \,used \,in\, the\, synthesis} \times 100$$
$$LC \,\left[ \% \right] = \frac{Weight\, of\, DEX - P \,encapsulated \,in\, the\, sample}{Weight\, of\, the\, sample} \times 100$$
Microscopic and spectroscopic characterization of the unbound and mBC-bound particles
SEM analysis was carried out using a PhenomWorld Pro scanning electron microscope. ALG/HPC-Ch spheres were dried at room temperature on a watch glass, and then the obtained material was placed on a carbon tape. The mBC membrane with the attached particles was stretched flat on a glass slide, dried in vacuum and placed on a carbon tape. Atomic force microscopic (AFM) images were obtained using a Dimension Icon AFM microscope (Bruker, Santa Barbara, CA) working in the PeakForce Tapping (PFT) and QNM® modes with standard silicon cantilevers for measurements in the air (nominal spring constant of 0.4 N/m).
For confocal laser scanning microscopy (CLSM) studies the mBC membranes were stained with the aqueous solution of fluorescein sodium salt (0.1 mg/mL) for 72 h before the particle deposition. Chitosan-coated particles were also labeled with rhodamine B isothiocyanate (0.1 mg/mL solution) in 0.1 M phosphate buffer (pH 9.0) 24 h prior to the deposition on mBC. The deposition was carried out as previously described. Images were acquired using an A1-Si Nikon (Japan) confocal laser scanning system built on a Nikon inverted microscope Ti-E using a Plan Apo 100×/1.4 Oil DIC objective. Diode lasers (488 nm and 561 nm) were used for excitation.
FTIR spectra were recorded using a Nicolet iS10 FT-IR spectrometer equipped with an ATR accessory (SMART iTX).
Cytotoxicity studies
Mouse Embryonic Fibroblasts MEF ATCC SCRC-1008 (MEFs) were maintained in a cell culture dish containing DMEM with streptomycin (100 µg/mL) and penicillin (100 U/mL) supplemented with 5% (v/v) FBS. The cells were incubated at 37 °C, 90% humidity with 5% CO2. Before toxicity and proliferation assessment, the cells (at approximately 70% confluence) were washed twice with PBS solution and subsequently harvested after 3 min incubation with 1 mL of 0.25% trypsin with 0.1% EDTA. After adding 3 mL of DMEM [with 5% (v/v) FBS] the cell suspension was centrifugated at 1250 g for 5 min, the supernatant was removed, and the pellet was resuspended in DMEM and 5% (v/v) FBS.
Toxicity and proliferation
MEFs suspended in DMEM supplemented with 5% (v/v) FBS were seeded into a 48-well cell culture plate (0.5 mL) at 5.2 × 104 (cytotoxicity) or 2.5 × 104 (proliferation) cells/well and incubated (37 °C, 5% CO2, 90% humidity). After 8 h (proliferation) or 29 h (cytotoxicity) the medium was replaced with 0.5 mL of fresh DMEM (supplemented with 5% (v/v) FBS in the case of proliferation test) containing different nanospheres concentration. After 23.5 h (cytotoxicity) or 42 h (proliferation) XTT assay was performed. It is based on reduction of tetrazolium salt XTT to formazan salt which occurs only in metabolically active (live) cells. The medium was removed and 200 µL of fresh DMEM with 100 µL of the activated XTT mixture was added to each well [with 5% (v/v) FBS in a final solution in the case of proliferation test]. After 2.5 h of incubation the plate was analyzed using a microplate reader (EPOCH2, Biotek Instruments, Inc) by measuring the absorbance at 460 nm. The results are normalized to an untreated control (without the nanospheres). All the data were presented as the mean of three replicates with standard deviation of the mean.
Low solubility of DEX in water (85 mg/L) (Messner and Loftsson 2010) limits its application as a component of the wound-healing dressings, as these are supposed to provide moist environment and are often based on hydrogels. On the other hand, DEX-P has similar anti-inflammatory and immunosuppressive properties, while showing higher solubility in aqueous media, thus it can be successfully introduced into the hydrogel matrix. To ensure the control over the release of DEX-P we have encapsulated it into the nanoparticulate system based on natural polysaccharides, and then used these particles to modify the surface of mBC.
Encapsulation of DEX-P in ALG/HPC particles
The initial experiments allowed us to select the optimal composition of the ALG/HPC hydrogel matrix of the particles, which in the case of DEX-P was found to be 4:1 w/w. We have shown previously that this particular hydrogel composition can be used to encapsulate bioactive macromolecules [heparin (Karewicz et al. 2010)] and alkaline phosphatase (Karewicz et al. 2014). The DEX-P containing submicroparticles were obtained using the extrusion technique. Their diameter was maintained at an average value of 170 nm by adjusting the injection rate of the polymeric mixture containing the active agent into the solution of cross-linking agent (0.2 M CaCl2) and the flow of the inert gas, which was applied parallel to the injection flow.
Figure 1a shows a typical SEM image of the particles produced using the method described above. They were spherical in shape and showed moderate tendency to aggregate. The dispersity of the obtained particles was significant, which is a typical outcome of the extrusion method. However, the diameter of 96% of particles did not exceed 500 nm, with the average diameter of 175 nm. The histogram showing the distribution of sizes of the obtained particles is presented in Fig. 1b.
a SEM analysis of the uncoated ALG/HPC particles with DEX-P, b histogram showing size distribution of ALG/HPC particles
Coating ALP/HPC particles with chitosan
To facilitate the attachment of the DEX-P-loaded particles to the surface of mBC via EDC/NHS chemistry, as well as to increase the control over the release profile, the nanospheres were coated with a thin layer of chitosan using polycation-polyanion electrostatic interactions. The chitosan-coated particles (ALG/HPC-Ch) retained the spherical shape and did not significantly change their average size as illustrated by the SEM image (Fig. 2). The diameter of 96% of the chitosan-coated particles did not exceed 600 nm, with the average diameter of 190 nm. Due to the fact that the coating process was carried out in an aqueous solution, the drug loss was unavoidable and it was estimated that ca. 18% of the initially loaded drug was lost. The concentration of the drug released from the ALG/HPC-Ch particles was, however, still at the desired therapeutic level (Dayanarayana et al. 2014).
a SEM image of chitosan-coated particles (ALG/HPC-Ch) containing encapsulated DEX-P, b histogram showing size distribution of ALG/HPC-Ch with DEX-P
Release profiles of DEX-P from ALG/HPC and ALG/HPC-Ch particles
Release profiles of DEX-P from the submicroparticles were studied under physiological conditions (PBS, pH 7.4, 37 °C). Each experiment was conducted fourfold, and each collected supernatant was measured in triplicate. Figure 3 shows a typical chromatogram (panel A) and the release profiles (panel B). 50% of DEX-P was released from the uncoated submicroparticles within the first 45 min and from the particles coated with chitosan within the first 75 min. In both systems the drug was still being released after 24 h.
a Typical chromatogram of DEX-P released under physiological conditions (37 °C, pH 7.4) from the uncoated ALG/HPC particles containing DEX-P, b The release profiles of DEX-P from the uncoated ALG/HPC particles (squares) and particles coated with chitosan (circles)
Based on the total amount of DEX-P released from the submicroparticles, the encapsulation efficiency (EE) and loading capacity (LC) were calculated for both uncoated and chitosan-coated particles. The obtained EE and LC values for uncoated particles were found to be (65.1 ± 1.1)% and (14.9 ± 0.6)%, respectively. For the chitosan coated particles these values were slightly lower: (53.0 ± 1.9)% and (12.2 ± 1.7)%, respectively. EE was satisfactory for both systems, and LC was comparable or higher than that obtained for DEX-P in other micro/nanoparticulate systems described in literature (Jaraswekin et al. 2007).
Deposition of ALG/HPC-Ch particles on mBC
The ALG/HPC/Ch submicroparticles containing DEX-P were covalently attached to the surface of Bionanocellulose® modified with carboxymethyl groups (mBC) using EDC/NHS methodology. The coupling reaction led to the formation of amide bonds between the amino groups of chitosan and the carboxyl groups of mBC. To avoid the use of an excess of the coupling agents, the concentrations of EDC and NHS necessary for activation of carboxymethyl groups were first optimized. In order to minimize the release of DEX-P from the nanospheres during their binding to mBC, the shortest required exposure time to the coupling agents solution was also established. The optimal concentrations and exposure time were found based on the changes in the amount of the particles bound to mBC and DEX-P released from the mBC modified with nanospheres (Table 1). To estimate the weight of attached particles, the weight of particles remaining in the solution after the deposition process was determined. The concentration of DEX-P was measured with HPLC in the same conditions as in the release profile experiments. In all experiments EDC:NHS ratio was 1:3.
Optimization of the mBC activation procedure with EDC and NHS as coupling agents
Time of exposure to EDC/NHS (h)
Concentration of EDC (mol/L)
Concentration of NHS (mol/L)
DEX-P released from attached particles (mg/cm2)
Weight of attached particles (mg/cm2)
The optimal time of mBC exposure to coupling agents was established first and was found to be 2 h. It enabled effective attachment of particles (19.93 ± 0.10 mg of particles/cm2) and the highest amount of drug released from mBC (2.15 ± 0.15 mg/cm2). The amount of attached particles decreased only slightly with decreasing concentration of EDC and NHS therefore the amount of both compounds could be significantly reduced while maintaining an effective attachment of the nanospheres to mBC. A fourfold reduction of EDC and NHS concentrations (to 0.008 M and 0.025 M, respectively) allowed to remove any detectable (based on HPLC analysis) traces of coupling agents from the material by washing with small amount of water, without any significant reduction in the amount of entrapped DEX-P.
To confirm the formation of amide bonds, the FTIR-ATR spectra of the unmodified mBC sheets, ALG/HPC-Ch and ALG/HPC-Ch-mBC were measured (Fig. 4). The covalent attachment of the particles to the mBC surface was confirmed by the presence of the strong amide band I (at 1644 cm−1, stretching vibrations of C=O group of amide bond) and amide band II (1561 cm−1, deforming vibrations of N–H group of amide bond) in the sample of ALG/HPC-Ch-mBC
FTIR-ATR spectra of (a) mBC, (b) ALG/HPC-Ch, and (c) ALG/HPC-Ch-mBC
a SEM image of mBC, b SEM image of ALG/HPC-Ch-mBC, c histogram showing size distribution of ALG/HPC-Ch bound to the surface of mBC
The morphology of the ALG/HPC-Ch-mBC material was visualized using SEM, AFM and confocal microscopy. SEM images revealed the presence of a large amount of nanometric spherical structures at the surface of mBC (Fig. 5). SEM images of the surface of pristine mBC and the surface of mBC with attached ALG/HPC-Ch particles are shown in Fig. 5a, b. The comparison of the histograms presented on Figs. 2b and 5c leads to the conclusion that the smaller submicroparticles are preferentially attached to mBC.
This observation was confirmed using the AFM visualization. The results of the AFM measurements are presented in Fig. 6. The strands of submicroparticles decorating nanocellulose fibrils are clearly visible. The analysis of the AFM images allows to define the average size of the attached particles as being around 100–120 nm, which is in a good agreement with the SEM analysis. It should be stressed, that due to the significant pressure of the tip exerted on the soft hydrogel particle, the AFM images do not represent well the shape of the particle, thus the diameter can be reasonably estimated only in the horizontal direction.
a AFM image of mBC, b AFM image of ALG/HPC-Ch-mBC material containing DEX-P, c AFM cross-section profiles marked as 1 and 2 in the image b
The confocal microscopy was used to study the extent to which the particles penetrate the 3D structure of the mBC sheet. The mBC fibrils and DEX-P-loaded particles were fluorescently labeled with fluorescein sodium salt and rhodamine B isothiocyanate (green and red fluorescence), respectively. The 3D image (Fig. 7a) shows a spatial distribution of the ALG/HPC-Ch particles in mBC. Most of the submicroparticles occupy the surface or the space close to the surface of mBC, however, small number of particles was distributed evenly in the whole volume of the mBC sheet. Therefore, although most of the drug would be released form the surface, a small amount of DEX-P will diffuse slowly through the nanocellulose hydrogel to reach the contact with the wound much later, thus possibly prolonging the healing effect of the material. The 2D image (Fig. 7b) shows the surface of mBC with particles distributed alongside the cellulose nanofibrils.
Confocal microscopy images of the mBC labelled with fluorescein after the deposition of rhodamine B-labelled ALG/HPC-Ch submicroparticles containing DEX-P: a 3D image in the green channel (FITC), b 3D image in the red channel (rhodamine B), c 3D image showing a merge of green and red channels, d 2D image of the sample—merge of green and red channels. (Color figure online)
Release of DEX-P from ALG/HPC-Ch-mBC hydrogel material
The release of DEX-P from the mBC-attached submicroparticles was studied. The drug was released from the material in a controlled manner, as shown in Fig. 8. The release profile is advantageous, with fast release during the first few hours, and slow but still significant delivery of DEX-P for up to 2 days. A separate experiment was also designed to confirm the beneficial impact of the encapsulation of DEX-P in the submicroparticles attached to the mBC hydrogel matrix on the resulting release profile. For this purpose mBC was incubated in 0.75 mg/mL solution of DEX-P in PBS for 1 h in order to allow its diffusion into the material. The release profile of the free drug entrapped physically in mBC was then studied. The obtained profile (Fig. 8) was characterized by undesirable "burst release" and all the entrapped drug was released within 2 h. That confirms that our approach involving entrapment of DEX-P in the mBC hydrogel matrix is reasonable.
Release profiles of DEX-P from ALG/HPC-Ch-mBC (squares) and from mBC pre-incubated in a PBS solution of free DEX-P (0.75 mg/mL) (circles)
We attempted to fit different kinetic models, frequently applied to the drug release from the nano/microparticulate systems, to our data in order to obtain some insight into the possible release mechanism. The fitting parameters obtained for the three models used (Higuchi, Peppas and Weibull) are presented in Table 2. The Higuchi model resulted in the relatively poor approximation for all of the obtained systems, thus it was not taken into consideration in further analysis. Fitting the experimental data to the Peppas model gave the highest R2 values for both: unbound and m-BC-bound chitosan-coated particles, therefore it was used as the best model for the release from the proposed dressing. It also gave a relatively good fit for uncoated particles, as illustrated in Fig. 9. This model is a short time approximation, so the fitting procedure has to be limited to the first 60% of the release profile. In this semi-empirical equation a is the kinetic constant and k is an exponent characterizing the diffusion mechanism. For uncoated particles the release exponent k suggested that the drug release was driven by a Fickian diffusion from the spherical matrix, whereas for coated particles and mBC attached particles a non-Fickian diffusion is the most possible release mechanism. The empirical Weibull model allows to take into consideration the whole data set. Depending on the correlation between the d value in the Weibull equation and the type of diffusional mechanism of drug release one can specify the type of release from the obtained particles (Papadopoulou et al. 2006). For the uncoated particles d was found to be 0.75, which suggests that Fickian diffusion in either fractal or Euclidian spaces is dominant. For mBC with attached submicroparticles d is in the range of 0.75–1, which is typical for a combined (Fickian diffusion and Case II transport) mechanism. For chitosan-coated particles d is higher than 1, suggesting more complex release mechanism. In all cases the Peppas model analysis leads to the conclusions which are in agreement with these obtained based on the Weibull model.
Parameters obtained by fitting different release kinetic models to experimental release profiles for uncoated, chitosan-coated and mBC-bound particles
Uncoated submicroparticles
Unbound chitosan-coated submicroparticles
mBC with attached chitosan-coated submicroparticles
Higuchi \(y = a\sqrt x\)
Peppas \(y = ax^{k}\)
Weibull \(y = a - \left( {a - b} \right)e^{{\left( { - kx} \right)^{d} }}\)
− 2.57
Release data fits to Peppas model for ALG/HPC particles (squares), ALG/HPC-Ch particles (circles) and mBC bound ALG/HPC-Ch particles (triangles)
Biological studies—cell viability and proliferation
Nanocellulose is well known to be highly biocompatible and non-toxic (Lin and Dufresne 2014). To verify whether the ALG/HPC-Ch submicroparticles generate any toxic effect to fibroblasts, the cell viability assay was performed. Figure 10a shows that at the concentrations up to 20 mg/mL the empty ALG/HPC-Ch submicroparticles (carrier) caused no toxicity to the mouse embryonic fibroblasts (MEFs), while for the DEX-P-loaded particles only slight toxicity was observed, with cell viability in the range of 80–95%. At higher ALG/HPC-Ch concentrations (40 and 80 mg/mL) some toxicity of the carrier was registered for empty particles (still the viability was above 70%), whereas the protective effect of DEX-P was revealed. This is in agreement with some previous reports where the protective effect of DEX on various cell lines, including endothelial cells (Zakkar et al. 2011) and fibroblasts (Mendoza-Milla et al. 2005) was observed. These results and no significant fibroblast necrosis suggest the DEX-containing dressing is safe. This is essential, as the healthy tissue on the edges of the wound may be in contact with the dressing. It is also important for the wound itself, as necrosis may increase inflammatory response (Davidovich et al. 2014), resulting in the elevated pro-fibrotic activity (White and Mantovani 2013). In attempt to assess the influence of the DEX-P-loaded ALG/HPC-Ch on the proliferation of fibroblasts the suitable proliferation test was also performed (Fig. 10b). The proliferation was not hindered by neither empty nor loaded submicroparticles at the concentration below 10 mg/mL. Although DEX-P has positively influenced the proliferation observed for the loaded ALG/HPC-Ch in the concentration range of 10–20 mg/mL, at higher concentrations the inhibition effect of both: the carrier and the drug on the proliferation rate is clearly visible. The inhibitory effect was more pronounced for DEX-P containing submicroparticles, as expected (Wu et al. 2006). One can conclude that the DEX-P-loaded submicroparticles at concentrations equal or higher than 40 mg/mL can effectively inhibit fibroblast proliferation. As shown before, 40 mg of DEX-P-loaded submicroparticles is deposited on ca. 5 cm2 of the ALG/HPC-Ch-mBC surface, although it can be also observed, that the material will contain relatively limited amount of water and will be in direct contact with the wound. This may lead to higher DEX-P concentrations and a satisfactory decrease in proliferation rate even at considerably lower surface of contact.
a MEF cell viability test results, b MEF cell proliferation test results (black—empty particles, grey—DEX-P-loaded particles)
DEX-P was successfully encapsulated in the ALG/HPC submicroparticles. These particles were then surface-modified with chitosan to obtain ALG/HPC-Ch system. Both ALG/HPC and ALG/HPC-Ch particles showed spherical morphology, high DEX-P encapsulation efficiency (65.1% and 53%, respectively) and drug loading values, which were comparable to other delivery systems for DEX-P. A thin layer of chitosan coating increased the control over the DEX-P release profile and allowed to covalently attach ALG/HPC-Ch particles to mBC. DEX-P was released from the particles attached to mBC in a controlled manner for up to 2 days. Additional experiments confirmed the advantage of DEX-P encapsulation in submicroparticles prior to introduction into mBC matrix over the direct introduction of the drug into mBC matrix. Preliminary biological studies showed that no toxicity is induced by both empty and DEX-P-loaded submicroparticles up to 20 mg/mL and only a slight decrease in fibroblasts viability at the concentration up to 80 mg/mL could be observed. At the concentration above 40 mg/mL the DEX-P-loaded particles have effectively inhibited proliferation of fibroblasts.
Based on the results of our studies we can conclude that we have successfully fabricated the novel bioactive wound dressing material combining the advantageous properties of BC and these of dexamethasone. Due to the decoration of BC surface with submicroparticles containing encapsulated DEX-P the drug release from the material could be controlled, ensuring its local concentration at the required therapeutic level (e.g. that needed to inhibit the fibroblasts proliferation).
In view of the negligible cytotoxicity combined with anti-inflammatory properties and ability to inhibit fibroblast proliferation the proposed system may constitute a promising bioactive dressing useful for the treatment of the wound fibrosis.
Authors would like to thank The National Centre for Research and Development (NCBiR) for the financial support in the form of grant no. K/NCB/000013 obtained in the frame of the INNOTECH Programme. The research was carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Programme (Contract No. POIG.02.01.00-12-023/08). Karol Wolski would like to thank the Fundation for Polish Science for the financial support (START 96.2018).
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1.Faculty of ChemistryJagiellonian University in KrakówKrakówPoland
Rojewska, A., Karewicz, A., Baster, M. et al. Cellulose (2019) 26: 1895. https://doi.org/10.1007/s10570-018-2182-8
Received 17 August 2018
Accepted 06 December 2018
First Online 10 December 2018 | CommonCrawl |
A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium
DCDS-S Home
Existence and decay of solutions of the 2D QG equation in the presence of an obstacle
October 2014, 7(5): 1045-1063. doi: 10.3934/dcdss.2014.7.1045
Stokes and Navier-Stokes equations with perfect slip on wedge type domains
Siegfried Maier 1, and Jürgen Saal 1,
Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut, 40204 Düsseldorf, Germany, Germany
Received March 2013 Revised June 2013 Published May 2014
Well-posedness of the Stokes and Navier-Stokes equations subject to perfect slip boundary conditions on wedge type domains is studied. Applying the operator sum method we derive an $\mathcal{H}^\infty$-calculus for the Stokes operator in weighted $L^p_\gamma$ spaces (Kondrat'ev spaces) which yields maximal regularity for the linear Stokes system. This in turn implies mild well-posedness for the Navier-Stokes equations, locally-in-time for arbitrary and globally-in-time for small data in $L^p$.
Keywords: Kondrat'ev spaces, perfect slip, $\mathcal{H}^\infty$-calculus., Stokes equations, wedge domains.
Mathematics Subject Classification: Primary: 76D035, 35K65; Secondary: 76D0.
Citation: Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045
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\begin{document}
\title{Almost isometric constants for partial unconditionality} \author{R. M. Causey}
\address{Department of Mathematics\\ University of South Carolina\\ Columbia, SC 29208\\ U.S.A.} \email{[email protected]}
\author{S. J. Dilworth}
\address{Department of Mathematics\\ University of South Carolina\\ Columbia, SC 29208\\ U.S.A.} \email{[email protected]}
\begin{abstract} We discuss optimal constants of certain projections on subsequences of weakly null sequences. Positive results yield constants arbitrarily close to $1$ for Schreier type projections, and arbitrarily close to $1$ for Elton type projections under the assumption that the weakly null sequence admits no subsequence generating a $c_0$ spreading model. As an application, we prove that a weakly null sequence admitting a spreading model not equivalent to the $c_0$ basis has a quasi-greedy subsequence with quasi-greedy constant arbitrarily close to $1$.
\end{abstract}
\subjclass[2010]{46B15, 41A65} \thanks{The second author was supported by the National Science Foundation under Grant Number DMS-1361461}
\maketitle
\section{Introduction}
A standard result in Banach space theory, due to Bessaga and Pe\l czy\'nski, \cite{BP}, is that every seminormalized, weakly null sequence admits a basic subsequence with basis constant arbitrarily close to $1$. Maurey and Rosenthal \cite{MR} famously gave an example of a normalized, weakly null sequence admitting no unconditional subsequence. Since then, significant attention has been paid to notions of partial unconditionality of subsequences of weakly null sequences. Two examples of such notions, examples which will be the focus of this work, are as follows: Given a seminormalized, weakly null sequence $(x_n)$, for any $\varepsilon>0$ and $q\in \mathbb{N}$, there exists a subsequence $(x_{n_i})$ of $(x_n)$ so that for any scalars $(a_i)\in c_{00}$, the scalar sequences with finite support, and any set $E\subset \mathbb{N}$ with $|E|\leqslant q$, $$\|\sum_{i\in E} a_i x_{n_i}\|\leqslant (1+\varepsilon)\|\sum a_ix_{n_i}\|.$$ The first proof of this fact was given by Odell \cite{O}, wherein he applied a standard diagonalization procedure to obtain a subsequence $(x_{n_i})$ of $(x_n)$ so that for any scalar sequence $(a_i)\in c_{00}$ and any set $E\subset \mathbb{N}$ with $|E|\leqslant \min E$, $$\|\sum_{i\in E}a_i x_{n_i}\|\leqslant (2+\varepsilon)\|\sum a_i x_{n_i}\|.$$ The latter property of the sequence $(x_{n_i})$ is called \emph{Schreier unconditionality}.
Another mode of unconditionality was introduced by Elton \cite{E}, and it is known as \emph{Elton unconditionality} or \emph{near unconditionality}. For $0<\delta <1$, a sequence $(x_n)$ is $\delta$- Elton unconditional provided that there exists a constant $K\geqslant 1$ such that for any $(a_n)\in c_{00}$ with $\sup_n |a_n|\leqslant 1$ and any $E\subset \{n: |a_n|\geqslant \delta\}$, $$\|\sum_{i\in E} a_i x_i\|\leqslant K\|\sum a_ix_i\|.$$ Elton \cite{E} showed that for each $\delta\in (0,1)$, every seminormalized, weakly null sequence admits a subsequence which is $\delta$-Elton unconditional with constant $K=K(\delta)$ depending only on $\delta$.
The focus of this work is to combine Schreier unconditionality with monotonicity to show that every semi-normalized weakly null sequence admits a subsequence for which certain projections have norm arbitrarily close to $1$. An analogous result which combines Elton unconditionality and monotonicity is proved for a certain class of weakly null sequences.
As an application we show that the members of this class of sequences admit quasi-greedy subsequences with quasi-greedy constant arbitrarily close to $1$, which improves \cite[Theorem 5.4]{DKK}.
We are ready to state the main theorem.
\begin{theorem} Let $(x_n)$ be a seminormalized, weakly null sequence. \begin{enumerate}[(i)]\item For any $\varepsilon>0$ and any sequence $(q_i)_{i\geqslant 0}$ of natural numbers, there exists a subsequence $(y_n)$ of $(x_n)$ so that for any $n\in \mathbb{N}$, any set $E\subset \mathbb{N}$ with $n<\min E$ and $|E|\leqslant q_n$, and any $(a_i)\in c_{00}$, $$\|\sum_{i=1}^n a_i y_i+ \sum_{i\in E} a_i y_i\|\leqslant (1+\varepsilon) \|\sum a_iy_i\|.$$ \item Suppose that no subsequence of $(x_n)$ generates a spreading model equivalent to the canonical $c_0$ basis. For any $\varepsilon>0$ and sequence $(\delta_n)_{n\geqslant 0}\subset (0,1)$, there exists a subsequence $(y_n)$ of $(x_n)$ so that for any scalars $(a_i)\in c_{00}$ such that $\sup_{i\in \mathbb{N}}|a_i|\leqslant 1$, any $n\in \mathbb{N}$, and any set $E\subset \{i\in \mathbb{N}: n<i, |a_i|\geqslant \delta_n\}$, $$\|\sum_{i=1}^n a_iy_i + \sum_{i\in E} a_iy_i\|\leqslant (1+\varepsilon)\|\sum a_iy_i\|.$$ \end{enumerate}
\label{main theorem} \end{theorem}
In this work, $X$ will denote a Banach space over the real or complex scalars and $B_X$ will denote its closed unit ball. We let $\mathbb{B}$ denote $[-1,1]$ in the real case or the closed unit disk in the complex plane in the complex case. For each $n\in \mathbb{N}$, $\mathbb{B}^n$ will be endowed with the $\ell_\infty^n$ metric. If $N$ is an infinite subset of $\mathbb{N}$, we let $[N]$ (resp. $[N]^n$) denote the infinite (resp. cardinality $n$) subsets of $N$. We identify subsets of $\mathbb{N}$ with strictly increasing sequences in $\mathbb{N}$ in the natural way and let $\mathbb{N}_0:=\mathbb{N}\cup \{0\}$.
\section{Extensions of Schreier and Elton unconditionality}
Recall that a sequence $(x_n)$ in a Banach space has the sequence $(s_n)$ in a (possibly different) Banach space as a \emph{spreading model} provided that for each $k\in \mathbb{N}$ and $\varepsilon>0$, there exists $m=m(k, \varepsilon)$ so that for any $m\leqslant n_1<\ldots < n_k$ and any scalars $(a_i)_{i=1}^k$, $$\Bigl|\|\sum_{i=1}^k a_i s_i\| - \|\sum_{i=1}^k a_i x_{n_i}\|\Bigr|<\varepsilon.$$ We say $(x_n)$ \emph{generates} the spreading model $(s_n)$. Recall also that every seminormalized, weakly null sequence has a subsequence which generates a $1$-suppression unconditional spreading model.
Theorem \ref{main theorem}$(i)$ will follow from the next lemma.
\begin{lemma} Let $X$ be a Banach space. Given a weakly null sequence $(x_n)\subset B_X$, $(q_n)_{n\geqslant 0} \subset \mathbb{N}$, and $(\varepsilon_n)_{n\geqslant 0}\subset (0,\infty)$, there exist $1\leqslant n_1<n_2<\ldots$ with the following property: For any $k\in \mathbb{N}_0$, $x^*_0\in B_{X^*}$, and any $\varnothing \neq E\subset \mathbb{N}$ with $k<\min E$ and $|E|\leqslant q_k$, there exists a functional $x^*\in B_{X^*}$ so that \begin{enumerate}[(i)]\item for $i\in \{1, \ldots, k\}\cup E$, $|x^*_0(x_{n_i})- x^*(x_{n_i})|\leqslant \varepsilon_k$, \item for $1\leqslant i\leqslant \max E$, $i\notin \{1, \ldots, k\}\cup E$, $|x^*(x_{n_i})|\leqslant \varepsilon_i.$ \end{enumerate}
\label{lemma1} \end{lemma}
For clarity, we isolate the following step of the proof of Lemma \ref{lemma1}.
\begin{lemma} Suppose $k\in \mathbb{N}_0$, $(y_i)_{i=1}^k\subset B_X$, and $(z_i)\subset B_X$ is weakly null. Fix $l\in \mathbb{N}$ and $\varepsilon>0$. If $A\subset \mathbb{B}^k$ and $B\subset \mathbb{B}^l$ are any sets, then there exists a subsequence $(z_i)_{i\in M}$ of $(z_i)$ so that if $x^*\in B_{X^*}$ and $r_0<\ldots < r_l$, $r_i\in M$, are such that $(x^*(y_i))_{i=1}^k\in A$ and $(x^*(z_{r_i}))_{i=1}^l\in B$, then there exists $y^*\in B_{X^*}$ so that $(y^*(y_i))_{i=1}^k\in A$, $(y^*(z_{r_i}))_{i=1}^l\in B$, and $|y^*(z_{r_0})|\leqslant \varepsilon$. \label{lemma2} \end{lemma}
Here it should be understood that if $k=0$, the hypotheses and conclusions involving $(y_i)_{i=1}^k$ and $A$ should be omitted.
The next lemma will easily yield Theorem \ref{main theorem}$(ii)$ after a standard diagonalization.
\begin{lemma} Fix $n\in \mathbb{N}_0$, $\varepsilon>0$, and $\delta>0$. If $(x_i)_{i=1}^n\subset X$ and $(y_i)\subset X$ is a semi-normalized, weakly null sequence generating a spreading model not equivalent to the canonical $c_0$ basis, then there exists a subsequence $(z_i)$ of $(y_i)$ so that for all scalars $(a_i)_{i=1}^n$ and $(b_i)\in c_{00}$ with $|a_i|, |b_i|\leqslant 1$ for all appropriate $i$, and for all sets $E\subset \{i\in \mathbb{N}: |b_i|\geqslant \delta\}$, $$ \|\sum_{i=1}^n a_ix_i + \sum_{i\in E} b_i z_i\|\leqslant (1+\varepsilon) \|\sum_{i=1}^n a_ix_i + \sum b_iz_i\|.$$
\label{lemma3} \end{lemma}
We use these lemmas to prove Theorem \ref{main theorem}, and then return to the proofs of the lemmas.
\begin{proof}[Proof of Theorem \ref{main theorem}]$(i)$ Let us assume $\varepsilon<1$ and $(x_i)\subset B_X$ is $(1+\varepsilon/2)$-basic. Fix $\eta>0$ such that $0<\eta^{-1}< \inf_n \|x_n\|$. By \cite{O}, by passing to a subsequence of $(x_i)$, we may assume that for any $(a_i)\in c_{00}$, $\|(a_i)\|_{c_0}\leqslant \eta \|\sum a_ix_i\|$. Fix $(\varepsilon_k)_{k\geqslant 0}$ so that for each $k\in \mathbb{N}_0$, $$ (k+q_k)\varepsilon_k + \sum_{i=k+1}^\infty \varepsilon_i < \varepsilon/2\eta.$$ Without loss of generality, by Lemma \ref{lemma1} we may pass to a subsequence, relabel, and assume $(x_i)$ satisfies the conclusions of Lemma \ref{lemma1} with this choice of $(\varepsilon_k)_{k\geqslant 0}$ and $(q_k)_{k\geqslant 0}$. Choose $k\in \mathbb{N}_0$, $E\subset \mathbb{N}$ with $k<\min E$ and $|E|\leqslant q_k$, and $(a_i)\in c_{00}$. If $E=\varnothing$, the fact that $(x_i)$ is $(1+\varepsilon/2)$-basic gives the desired inequality. So assume $E\neq \varnothing$. Fix $x^*_0\in B_{X^*}$ so that $$\|\sum_{i=1}^k a_ix_i + \sum_{i\in E}a_ix_i\|=x^*_0\Bigl(\sum_{i=1}^k a_ix_i + \sum_{i\in E}a_ix_i\Bigr).$$ Choose $x^*$ as in the conclusion of Lemma \ref{lemma1}. Then \begin{align*} (1+\varepsilon/2)\|\sum a_ix_i\| & \geqslant \|\sum_{i=1}^{\max E}a_ix_i\| \geqslant x^*\Bigl(\sum_{i=1}^{\max E}a_ix_i\Bigr) \\ & \geqslant x^*_0\Bigl(\sum_{i=1}^k a_ix_i + \sum_{i\in E}a_ix_i\Bigr) - \sum_{i=1}^k|a_i||(x^*_0-x^*)(x_i)| \\ & - \sum_{i\in E}|a_i||(x^*_0-x^*)(x_i)| - \sum_{\underset{i\notin \{1, \ldots, k\}\cup E}{i=1}}^{\max E} |a_i||x^*(x_i)| \\ & \geqslant \|\sum_{i=1}^k a_ix_i + \sum_{i\in E}a_ix_i\| - \|(a_i)\|_{c_0}\Bigl((k + q_k)\varepsilon_k - \sum_{i=k+1}^\infty \varepsilon_i\Bigr) \\ & \geqslant \|\sum_{i=1}^k a_ix_i + \sum_{i\in E}a_ix_i\| - \eta\Bigl( (k+q_k)\varepsilon_k + \sum_{i=k+1}^\infty \varepsilon_i\Bigr)\|\sum a_ix_i\| \\ & \geqslant \|\sum_{i=1}^k a_ix_i + \sum_{i\in E}a_ix_i\| - (\varepsilon/2)\|\sum a_ix_i\|. \end{align*}
Adding $(\varepsilon/2)\|\sum a_ix_i\|$ to both sides finishes the proof.
$(ii)$ This follows easily from recursive applications of Lemma \ref{lemma3}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemma2}]
Let $\mathcal{U}$ consist of all $E=(j_i)_{i=1}^l\in [\mathbb{N}]^l$ such that there exists $x^*\in B_{X^*}$ so that $(x^*(y_i))_{i=1}^k\in A$ and $(x^*(z_{j_i}))_{i=1}^l\in B$. By the finite Ramsey theorem, there exists $N\in [\mathbb{N}]$ such that either $\mathcal{U}\cap [N]^l=\varnothing$ or $[N]^l\subset \mathcal{U}$. In the first case, we let $M=N$ and note that the conclusion is vacuously satisfied. In the second case, let $\mathcal{V}$ consist of those $E=(j_i)_{i=0}^l\in [N]^{l+1}$ such that there exists $x^*\in B_{X^*}$ such that $(x^*(y_i))_{i=1}^k\in A$, $(x^*(z_{j_i}))_{i=1}^l\in B$, and $|x^*(z_{j_0})|\leqslant \varepsilon$. Applying the finite Ramsey theorem again, there exists $M\in [N]$ such that either $[M]^{l+1}\cap \mathcal{V}=\varnothing$ or $[M]^{l+1}\subset \mathcal{V}$. We show that the first alternative cannot hold, so that this $M$ satisfies the conclusion. Assume that the first alternative holds. Let $M=(m_i)_{i=1}^\infty$, $m_1<m_2<\ldots$. Note that for each $1\leqslant j\in \mathbb{N}$, $(m_{j+r})_{r=1}^l\in \mathcal{U}$. Therefore there exists $x^*_j\in B_{X^*}$ so that $(x^*_j(y_i))_{i=1}^k\in A$ and $(x^*_j(z_{m_{j+r}}))_{r=1}^l\in B$. Since for each $1\leqslant i\leqslant j$, $(m_i, m_{j+1}, \ldots, m_{j+l})\notin \mathcal{V}$, it must be that $|x^*_j(x_{m_i})|\geqslant \varepsilon$. Then if $x^*$ is any $w^*$ cluster point of $(x_j^*)$, $|x^*(z_{m_i})|\geqslant \varepsilon$ for all $i\in \mathbb{N}$, contradicting the weak nullity of $(z_i)$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemma1}] Fix $(\delta_n)_{n=0}^\infty \subset (0,\infty)$ decreasing to zero such that $2\sum_{i=k}^\infty\delta_i \leqslant \varepsilon_k$ for each $k\in \mathbb{N}_0$. Use Lemma \ref{lemma2} recursively as follows: Assume $1\leqslant n_1<\ldots <n_i$ and $\mathbb{N}=M_0\supset \ldots \supset M_i\in [\mathbb{N}]$ have been chosen. For fixed $1\leqslant j\leqslant q_{i+1}$, let $(A_p, B_p)_{p=1}^r$ be such that $(A_p)_{p=1}^r$ is a cover of $\mathbb{B}^i$ by sets of diameter less than $\delta_{i+1}$ and $(B_p)_{p=1}^r$ is a cover of $\mathbb{B}^j$ by sets of diameter less than $\delta_{i+1}$. By applying Lemma \ref{lemma2} successively to $A=A_p$ and $B=B_q$ for each $1\leqslant p,q\leqslant r$, we may pass to a subsequence $M^j_{i+1}$ of $M_i$ satisfying the conclusion, with $\varepsilon=\delta_{i+1}$, of Lemma \ref{lemma2} for each of these pairs. Of course, we may do this successively for each $1\leqslant j\leqslant q_{i+1}$. We let $M$ be the sequence obtained by repeating this process for each $1\leqslant j\leqslant q_{i+1}$, $n_{i+1}=\min M$, and $M_{i+1}=M\setminus \{n_{i+1}\}$. We will show that the subsequence $(x_{n_i})$ resulting from this recursive definition satisfies the conclusion of Lemma \ref{lemma1}.
Fix $k\in \mathbb{N}_0$, $\varnothing \neq E\subset \mathbb{N}$ with $k<\min E$ and $|E|\leqslant q_k$. Fix $x^*_0\in B_{X^*}$. We choose $(x_i^*)_{i=1}^{\max E}$ recursively as follows: If $x_i^*\in B_{X^*}$ has been chosen, and if $i+1\in \{1, \ldots k\}\cup E$, let $x_{i+1}^*=x_i^*$. Otherwise, we let $A= A_p\subset \mathbb{B}^i$ and $B=B_q\subset \mathbb{B}^j$ be members of the covers of $\mathbb{B}^i$ and $\mathbb{B}^j$, respectively, from the previous paragraph such that $(x_i^*(x_{n_r}))_{r=1}^i\in A$ and $(x^*_i(x_{n_r}))_{i<r\in E}\in B$, where $j=|E\cap \{i+1, i+2, \ldots\}|$. Then there exists $x_{i+1}^*\in B_{X^*}$ so that $(x^*_{i+1}(x_{n_r}))_{i=1}^i\in A$, $(x^*_{i+1}(x_{n_r}))_{i<r\in E}\in B$, and $|x^*_{i+1}(x_{n_{i+1}})|\leqslant \delta_{i+1}$. This completes the recursive definition. Observe that with this definition, for each $1\leqslant i<\max E$ and $j\in \{1, \ldots, i\}\cup (E\cap \{i+1, i+2, \ldots\})$, $|x^*_i(x_{n_j})- x^*_{i+1}(x_{n_j})|\leqslant \delta_{i+1}$. Then for $j\in \{1, \ldots, k\}\cup E$, $$|(x^*_{\max E}- x^*_0)(x_{n_j})| = |(x^*_{\max E}- x_k^*)(x_{n_j})|\leqslant \sum_{i=k}^{\max E-1}|(x^*_{i+1}- x^*_i)(x_{n_j})|\leqslant \sum_{i=k}^\infty\delta_i \leqslant \varepsilon_k.$$ For $j\leqslant \max E$ with $j\notin \{1, \ldots, k\}\cup E$, $$|x^*_{\max E}(x_{n_j})|\leqslant |x^*_j(x_{n_j})| + \sum_{i=j+1}^{\max E-1} |(x^*_{i+1}-x_i^*)(x_{n_j})|\leqslant \delta_j+\sum_{i=j}^\infty \delta_i \leqslant \varepsilon_j.$$
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemma3}] Assume $\varepsilon<1$. Fix $\alpha\in (0, 1)$ so that $(1+2\alpha)/(1-\alpha)<1+\varepsilon$. Let $M=\max_{1\leqslant i\leqslant n} \|x_i\|$. By passing to a subsequence, we may assume $(y_i)$ generates the spreading model $(s_i)$, and recall that $(s_i)$ is not equivalent to the canonical $c_0$ basis. Let $g(n)=\|\sum_{i=1}^n s_i\|$. Note that the properties of spreading models generated by weakly null sequences imply that $g(n)=\|\sum_{i\in A}s_i\|$ for any $A$ with $|A|=n$ and $g(1)\leqslant g(2)\leqslant \ldots$. Since $(s_i)$ is not equivalent to the $c_0$ basis, $g(n)\to \infty$. Choose $q\in \mathbb{N}$ so that $\delta g(q)/8 > nM/\alpha$. By passing to a subsequence of $(y_i)$ and relabelling, we may assume that for any set $B\subset \mathbb{N}$ with $|B|\leqslant q$, $(y_i)_{i\in B}$ is $2$-equivalent to $(s_i)_{i=1}^{|B|}$. By \cite[Theorem 3.3]{DOSZ}, we may assume by passing to a further subsequence that
$(y_i)$ is $\delta$-Elton unconditional with constant $1+\alpha$. By passing to yet a further subsequence and arguing as in Theorem~ \ref{main theorem}(i), we may assume that for any set $B$ with $|B|\leqslant q$ and any scalars $(a_i)_{i=1}^n$, $(b_i)\in c_{00}$, $\|\sum_{i=1}^n a_ix_i +\sum_{i\in B} b_iy_i\|\leqslant (1+\varepsilon)\|\sum_{i=1}^n a_ix_i+ \sum_i b_iy_i\|$.
Fix scalars $(a_i)_{i=1}^n$ and $(b_i)\in c_{00}$ with $\max_i |a_i|, \max_i |b_i|\leqslant 1$. We consider two cases. For the first case, assume that $\| \sum_i b_iy_i\|\leqslant nM/\alpha$, and let $A=\{i: |b_i|\geqslant \delta\}$. We claim that $|A|<q$. If it were not so, we could choose $B\subset A$ with $|B|=q$. Then by our choice of $(y_i)$ and the $1$-suppression unconditionality of $(s_i)$, \begin{align*} nM/\alpha & \geqslant \|\sum b_i y_i\| \geqslant (1/2) \|\sum_{i\in B} b_i y_i \| \\ & \geqslant (1/4)\|\sum_{i\in B} b_i s_i\| \geqslant (\delta/8) g(q),\end{align*} a contradiction. Since $|A|<q$, $\|\sum_{i=1}^n a_ix_i+ \sum_{i\in A} b_iy_i\|\leqslant (1+\varepsilon)\|\sum_{i=1}^n a_ix_i+\sum b_iy_i\|$ by the last sentence of the previous paragraph.
For the second case, assume that $\| \sum_i b_iy_i\|\geqslant nM/\alpha$, and note that then $$\|\sum_{i=1}^n a_ix_i + \sum_i b_iy_i\| \geqslant \|\sum b_iy_i\| - nM \geqslant (1-\alpha)\|\sum_i b_iy_i\|.$$ Hence \begin{align*} \|\sum_{i=1}^n a_ix_i + \sum_{i\in E} b_i y_i\| & \leqslant nM + \|\sum_{i\in E} b_i y_i\| \leqslant (1+2\alpha) \|\sum_i b_i y_i\| \\ & \leqslant (1+2\alpha)/(1-\alpha)\|\sum_{i=1}^n a_ix_i + \sum_i b_iy_i\| \\ & \leqslant (1+\varepsilon)\|\sum_{i=1}^n a_ix_i + \sum_i b_iy_i\|. \end{align*}
\end{proof}
\section{Applications to quasi-greedy bases}
Fix a seminormalized basis $(e_i)$ for a Banach space $X$. For $n\in \mathbb{N}$, $x=\sum a_ie_i$, and $n\in \mathbb{N}$, we let $G_n(x)=\sum_{i\in A} a_ie_i$, where $|A|=n$ and $\min_{i\in A} |a_i|\geqslant \max_{i\in \mathbb{N}\setminus A}|a_i|$. Of course, such a set $A$ will not be uniquely defined in general. Recall that the basis $(e_i)$ is said to be \emph{quasi-greedy} provided there exists a constant $C_{qg}$ such that for all $x\in X$ and $n\in \mathbb{N}$, $\|G_n(x)\|\leqslant C_{qg} \|x\|$. This definition was introduced by Konyagin and Temlyakov \cite{KT}
and was shown to be equivalent to norm convergence of $(G_n(x))$ to $x$, for each $x \in X$, by Wojtaszczyk \cite{W}. In order to check that $(e_i)$ is quasi-greedy with constant $C$ it clearly suffices to prove that for each $a>0$ and $x\in X$, $\|\mathcal{G}_a(x)\|\leqslant C\|x\|$, where $\mathcal{G}_a(x)=\sum_{i:|a_i|\geqslant a} a_ie_i$ and $x=\sum a_ie_i$.
\begin{theorem} Let $(x_n)$ be a seminormalized, weakly null sequence so that no subsequence generates a spreading model equivalent to the canonical $c_0$ basis. Then for any $\varepsilon>0$, there exists a subsequence $(y_n)$ of $(x_n)$ which is a quasi-greedy basis for its span with constant $(1+\varepsilon)$.
\end{theorem}
\begin{remarks} (a) The weaker result with $1+\varepsilon$ replaced by $3+\varepsilon$ was shown in \cite[Theorem 5.4]{DKK} (cf.\ \cite[Corollary 3.4]{DOSZ}).
(b) The result fails if we omit the hypothesis concerning $c_0$ spreading models. In fact, in \cite[p. 59]{DOSZ}, an equivalent norm on $c_0$ is given so that the unit vector basis has no quasi-greedy subsequence with constant less than $8/7$.
(c) The constant of $1 + \varepsilon$ is optimal, even in a Hilbert space. Indeed, it was proved in \cite[Corollary 2.3]{GW} that a normalized basis $(x_n)$ of a Hilbert space is quasi-greedy with $C_{qg}=1$ if and only if $(x_n)$ is orthonormal. Hence, if $\langle x_n, x_m \rangle > 0$ for all $n,m$, it follows that $(x_n)$ does not admit a subsequence with $C_{qg}=1$. Clearly, such an $(x_n)$ will also satisfy the spreading model assumption of the theorem.
(d) It was proved recently \cite{AA} that a semi-normalized basis $(x_n)$ of a Banach space is quasi-greedy with $C_{qg}=1$ if and only if $(x_n)$ is $1$-suppression unconditional.
\end{remarks}
\begin{proof} Assume $\varepsilon<1$. By passing to a subsequence, scaling, and passing to a closely equivalent norm, we may assume $(x_n)$ is normalized and monotone. As above, we may assume $(x_n)$ generates a spreading model $(s_n)$, and we let $g(n)=\|\sum_{i=1}^n s_i\|$, $g(0)=0$. Choose $(q_n)_{n=0}^\infty \subset \mathbb{N}$ so that $g(q_k)\varepsilon >32 (k+1)$ for all $k\in \mathbb{N}_0$. By passing to a subsequence, we may assume that for any $k\in \mathbb{N}_0$ and any $\varnothing \neq C\subset \mathbb{N}$ with $k< \min C$ and $|C|\leqslant q_k$, $(x_n)_{n\in C}$ is $2$-equivalent to $(s_n)_{n\in B}$. Let $(y_i)=(x_{n_i})$, where $(x_{n_i})$ satisfies the conclusion of Theorem \ref{main theorem}$(i)$ with this choice of $(q_n)_{n=0}^\infty$ and $\varepsilon$ replaced by $\varepsilon/2$.
Fix $x=\sum a_iy_i$ with $\|x\|=1$ and fix $a>0$. Choose $k\in \mathbb{N}_0$ so that $2a\in [\varepsilon/(k+1), \varepsilon/k)$, where $\varepsilon/0:=\infty$. Write $G_a(x)= \sum_{i\in A}a_iy_i + \sum_{i\in B} a_iy_i$, where $A=\{i:i\leqslant k, |a_i|\geqslant a\}$ and $B= \{i: i> k, |a_i|\geqslant a\}$. We claim that $|B|\leqslant q_k$. If it were not so, we could choose $C\subset B$ with $|C|=q_k$. Then \begin{align*} 1 & = \|x\| \geqslant \frac{1}{2} \|\sum_{i=1}^k a_i y_i + \sum_{i\in C}a_i y_i\| \\ & \geqslant \frac{1}{4} \|\sum_{i\in C} a_i y_i\| \geqslant \frac{1}{8} \|\sum_{i\in C} a_i s_i\| \\ & \geqslant \frac{a g(q_k)}{16} \geqslant \frac{g(q_k) \varepsilon}{32 (k+1)}>1, \end{align*} a contradiction. Thus $|B|\leqslant q_k$, as claimed. Then \begin{align*} \|G_a(x)\| & = \|\sum_{i=1}^k a_i y_i + \sum_{i\in B} a_i y_i - \sum_{i\in \{1, \ldots, k\}\setminus A} a_i y_i\| \leqslant (1+\varepsilon/2) \|x\| + ka \\ & \leqslant 1+\varepsilon/2 +\varepsilon/2 = 1+\varepsilon. \end{align*}
\end{proof}
\begin{remark}
Recall that for $\Delta\geqslant 1$, a basic sequence $(e_i)$ is \emph{democratic} if for any finite sets $A, B\subset \mathbb{N}$ with $|A|\leqslant |B|$, $\|\sum_{i\in A} e_i\|\leqslant \Delta \|\sum_{i\in B}e_i\|$. It was shown in \cite[Proposition 5.3]{DKK} that any seminormalized, weakly null sequence having no subsequence generating a spreading model equivalent to the canonical $c_0$ basis has a $(1+\varepsilon)$ democratic subsequence. Recall also that a basic sequence $(e_i)$ is said to be \emph{almost greedy} provided there exists a constant $C_a$ so that for any $x\in X$, $\|x-G_n(x)\|\leqslant C_a \tilde{\sigma}_n(x)$, where $$\tilde{\sigma}_n(x)= \inf_{|A|\leqslant n} \|x-\sum_{i\in A} e_i^*(x)e_i\|.$$ It was also shown in \cite[Theorem 3.3]{DKKT} that a basic sequence is almost greedy if and only if it is quasi-greedy and democratic, and the constant $C_a$ of $(e_i)$ does not exceed $8C^4 \Delta + C+1$ if $(e_i)$ is quasi-greedy with constant $C$ and democratic with constant $\Delta$. Thus we have shown that any seminormalized, weakly null sequence having no subsequence generating a spreading model equivalent to the canonical $c_0$ basis has a subsequence which is almost greedy with constant $10+\varepsilon$.
\end{remark}
\end{document} | arXiv |
Research article | Open | Open Peer Review | Published: 12 July 2017
Expanding co-payment for methadone maintenance services in Vietnam: the importance of addressing health and socioeconomic inequalities
Bach Xuan Tran1,2,
Quyen Le Nguyen4,
Long Hoang Nguyen5,
Huong Thu Thi Phan3,
Huong Thi Le1,
Tho Dinh Tran6,
Thuc Thi Minh Vu7 &
Carl A. Latkin2
Ensuring high enrollment while mobilizing resources through co-payment services is critical to the success of the methadone maintenance treatment (MMT) program in Vietnam. This study assessed the willingness of patients to pay (WTP) for different MMT services delivery models and determined its associated factors.
A facility based survey was conducted among 1016 MMT patients (98.7% male, 42% aged 35 or less, and 67% living with spouse) in five MMT clinics in Hanoi and Nam Dinh province in 2013. Socioeconomic, HIV and health status, history of drug use and rehabilitation, and MMT experience were interviewed. WTP was assessed using contingent valuation method, including a set of double-bounded binary questions and a follow-up open-ended question. Point and interval data models were used to estimate maximum willingness to pay.
95.5% patients were willing to pay for MMT at the monthly mean price of US$ 32 (95%CI = 28–35). Higher WTP was associated with higher level of educational attainment, higher income, male sex, and had high expenses on opiates prior to MMT. Patients who reported having any problem in Pain/ Discomfort, and who did not have outpatient care last year were willing to pay less for MMT than others.
High level of WTP supports the co-payment policies as a strategy to mobilize resources for the MMT program in Vietnam. However, it is necessary to ensure equalities across patient groups by acknowledging socioeconomic status of different settings and providing financial supports for disadvantaged patients with severe health status.
As the home of more than half of drug use population in the world, Asian countries have been inordinately hard hit by the twin epidemics of HIV and substance use [1]. Since most people who are opioid dependent use heroin, opium, or pharmaceutical opioids, mainly through the injecting route, it becomes the major driver of the spread of HIV epidemics in Asia [2,3,4]. Injecting drug use (IDU) together with high stigma and discrimination by community and the lack of enabling policies are major social and structural barriers to scaling up comprehensive care and treatment services for patients in large injection-driven HIV epidemics [3]. Over the past decade, there have been substantial efforts by Asian governments to expand the coverage of methadone maintenance treatment (MMT) services. In China, Malaysia, Indonesia, and Vietnam, empirical evidence has demonstrated that MMT has brought about significant changes in health, social, and economic well-being of MMT patients and their families [5,6,7,8,9,10,11]. Therefore, MMT has become an essential component in national HIV/AIDS plans in many Asian settings.
Among Asian countries, Vietnam is in a precarious situation given a commitment to provide MMT for 80,000 people who inject drug (PWID). To date, MMT has been provided to more than 46.000 drug users at 251 free-standing MMT clinics and Provincial/District Health Centres in 58/63 provinces of Vietnam [12]. Moreover, it has been offered free-of-charge. However, while a substantial number of drug users is remained to reach the target of this program, these foreign aids are rapidly decreasing in recent years, causing the financial burden for the Vietnam Government [13, 14]. Without the financial aids from foreign donors, subsidies from the Government could only contribute up to 50% of total operational cost of HIV-related services, including MMT program, until 2020 [15]. In this case, Vietnam Ministry of Health has prioritized several resource optimization and mobilization policies to ensure the sustainability of HIV program. For instance, in MMT clinics, several HIV-related services such as Harm reduction programs, HIV testing and counselling and antiretroviral therapy (ART) are integrated in order to reduce the operational cost. In addition, the Vietnam Government requires the involvement of provincial budgets and co-payment by users in all HIV-related services [16]. These policies, so far, have been effective in the short run and contributed to the expansion of the service for over 24,000 MMT patients.
Generally, the government policy encourages local authorities to cover investment costs, including facilities, human resources, and training for MMT clinics; meanwhile, patients are supposed to co-pay the fee that covers the medication. Co-payment for treatment is popular in Vietnam health care system, contributing to over 50% of total health expenditure [17]. In terms of MMT, the current fee that has been applied in such socialized service models is US$0.5 that approximates 50% of the unit cost for MMT [14, 18]. However, the application of a universal fee might impede some patient groups from accessing and adhering to the treatment. In other words, patients with different characteristics have different decisions in co-payment. Literature in Vietnam, Taiwan and the United States found that patients with higher education, being employed, having higher income, receiving treatment at clinics in high level of health system were willing to pay for MMT than other patients [19,20,21,22]. Meanwhile, patients with higher age, having health problems or not believing in the effectiveness of treatment had lower amount of willingness to pay for drug rehabilitation [19,20,21]. Evidently, when the co-payment mechanism is implemented, the differences in the capacity to pay for MMT among patients with different characteristics may raise the socio inequalities as generating the barriers for accessing MMT (e.g. patients having lower socio-economic status are unaffordable to pay the service with high fee). Therefore, understanding factors associated with patients' willingness-to-pay (WTP) for the service is necessary for developing contextualized policies on co-payment MMT services.
A previous study in Vietnam demonstrated a high level of WTP for MMT by current MMT services users, with the mean amount of WTP being US$ 15.9 per month [20]. However, at the price of 50% of unit cost, there was only a half of target population willing to use the service. Moreover, there were several limitations in the previous analysis. First, its sample includes only HIV positive drug users who were approached at antiretroviral treatment clinics, while the majority of current drug users at MMT clinics in Vietnam are HIV-negative. Those patients with HIV/AIDS were more likely to suffer from catastrophic health expenditure; meanwhile HIV-negative drug users might have more hope to prevent HIV infection that affect their WTP for the services [8, 14, 20, 23, 24]. Second, This prior study was conducted at HIV outpatient clinics instead of MMT clinics across levels of health administration of the Vietnamese health system [25, 26]. Consequently, the implication of previous research findings is limited. Thus, the purpose of this study was to assess the WTP for MMT among MMT patients attending different services delivery models and examine its associated factors with focuses on socio-economic status, health and drug use-related characteristics.
Survey design and sampling
During June to August 2013, a cross-sectional study was conducted in Ha Noi and Nam Dinh province, involving five MMT clinics. We selected the two provinces in consultation with program managers at the Vietnam Authority of HIV/AIDS for a purposive comparison of an experienced setting—Hanoi and a new setting—Nam Dinh Province. There were four clinics located at district health centers, namely Tu Liem, Ha Dong, Long Bien, and Xuan Truong, and one clinic located at Nam Dinh Provincial AIDS Center. Criteria for selecting these clinics included 1) delivering MMT services; 2) representing both urban and rural areas, and 3) covering various levels of health system such as provincial- and district- levels. The characteristics of study sites are listed in Table 1.
Table 1 Study settings and sample size
A convenience sample of 1016 patients was enrolled in the study, accounting for 80–90% of the sample frame. Interviewers were master students in Public Health at Hanoi Medical University who were working in the field of HIV and who were not affiliated with the clinics where they invited patients to participate.
Survey participants comprised clients who met following inclusion criteria: 1) were taking or initiating MMT at the selected sites' 2) visited the clinics during the; and 3) aged 18 years and above. First, we assessed the selection criteria among patients and then invited eligible patients to a designated counselling room for the interviews. After that, we introduced the survey to patients and asked them to provide written informed consent if they agreed to participate. Finally, we interviewed them with structured questionnaires.
Measures and instruments
Face-to-face interviews were conducted by well-trained interviewers using structured questionnaires to collect data on socioeconomic characteristics, health status, history of drug use and rehabilitation, and experience with current MMT. Monthly per capita household income was self-reported by patients and included all sources of income for each household member. Household expenditures were estimated on the basis of recurring expenses (e.g. food, utilities, rent and education) in the past month and non-recurring expenses (e.g. construction, health care, furniture, travel and entertainment) in the past year [27, 28]. Equivalent costs of these goods in 2013 were estimated. Health status in five dimensions (mobility, self-care, usual activities, pain/discomfort and anxiety/depression) was measured using the five-level EQ-5D (EQ-5D-5 L) instrument (EuroQol Group, Rotterdam, Netherlands) [29].
Measurement of willingness to pay
WTP for MMT was assessed using contingent valuation (CV) method [20], by which all patients were clearly presented "the scenario" that they would value:
The problem: Patients were reminded the negative impacts of opioid use on HIV prevention, care, and treatment. This included an increased risks of transmitting HIV to others if they shared needles and syringes, a sub-optimal adherence and poorer outcomes of HIV/AIDS treatment, and a deteriorated health status and quality of life [30,31,32,33]. Also, other socioeconomic impacts of opioid dependence were discussed, for instance, stigma and discrimination, economic burden, and poverty risk of households. Traditional drug rehabilitation services available for opioid users showed limited long-term efficacy, and a large proportion of patients relapsed to drug abuse after several rehabilitation periods.
The attributes of MMT services: The patients were then introduced into the effectiveness of MMT as a substitution therapy for opioid dependants. MMT is cost-effective in reducing the frequency of opioid use, improving health status and quality of life of patients, supporting adherence to ART [9, 34, 35]. In addition, drug users taking MMT can have earlier access to health care services, receive adequate health information, counselling, and referrals, which in turn, may reduce other risky health behaviours, for example, alcohol use disorders [31]. Furthermore, patients can continue to be productive, released from stigma, discrimination, and financial burden of opioid as well as health care expenses associated with opioid abuse [34].
The market: The patients were presented the scale-up plan of MMT services that would be offered to them or their family members. Patients would be required to visit the MMT clinics once a day to take MMT under direct supervisions of health care workers. Currently, MMT services are delivered by the public health care system, and offered free-of-charge, however, its coverage is very low as of 20% by 2013 [36]. Since the international financial support is decreasing, it would be difficult for the Government to expand the coverage of this service. The patients were then asked the maximum they were willing to pay out-of-pocket for the MMT services.
The CV method: has been widely used as a valid method for eliciting patients' preference and WTP and has been applied in the previous study in Vietnam [20]. Double-bounded dichotomous choice questions backed by an open-ended question were used for eliciting patients' WTP for MMT services. Initially, patients were asked if they were willing to pay for a monthly fee of 1,000,000 Vietnam Dong (~US$ 50) to take MMT. This starting value was the most-update estimation of unit cost for one-month MMT. If the respondent indicated a WTP the first amount offered, then interviewers asked a follow-up question with the new threshold at double the first one. If the respondent was unwilling to pay, interviewers then halved the price. The question was repeated until the amount to be offered was four times or one-fourth of the starting value. Finally, patients were asked an open-ended question follows the double-bounded binary questions: "What is the maximum price you would be willing to pay per month for the MMT?" Fig. 1. presents the details of CV questions.
Contingent valuation questions
Student-t and Chi-square tests were used to examine the differences in opioid use behaviours among MMT patients in different sites. Since WTP was interviewed using both double-bounded and open-ended questions, it included a mixture of censored and uncensored data. The point and interval data models, which consist of interval data model and simple Tobit model, were used to estimate the average WTP for MMT services by different patients groups [37].
In the typical interval model, the WTP is supposed to have lognormal distribution, which is calculated by formula as below [37]:
$$ {logWTP}_i^{\ast }={x\prime}_i\beta +{\varepsilon}_i $$
Where WTPi represents the true value of WTP of patient i; x' i denotes a vector of independent variables and ε i represents a random element (with normal distribution, mean zero and standard deviation σ). In the interval model, the interval censoring means that the value of WTP is between the selected bid (denoted t li ) and the next bid in the scale (denoted t ui ). Thus, the logWTP* also lies in (logt li ; logt ui ). The right censoring means that logWTP* is higher than logt ui ; and the left censoring refers that logWTP* is lower than logt li . As a result, the log-likelihood function is constituted by three components (interval censoring, right censoring and left censoring) as below [37]:
$$logL=\sum\limits^{N}_{i=1} \left\{ \begin{array}{ll} I^a_i\ \text{log}\ \left(\upphi\left(\frac{logt_{ui}-x^{\prime}{~}_{i}\beta}{\sigma}\right)-\upphi\left(\frac{logt_{li}-x^{\prime}{~}_{i}\beta}{\sigma}\right)\right)\\ +I^b_i\ \text{log}\ \left(1-\upphi\left(\frac{logt_{ui}-x^{\prime}{~}_{i}\beta_2}{\sigma}\right)\right)+I^c_i\ \text{log}\ \left(\upphi\left(\frac{logt_{ui}-x^{\prime}{~}_{i}\beta}{\sigma}\right)\right) \end{array} \right\}$$
Where logL means Log-likelihood function; σ is the scale parameter; ɸ denotes the cumulative standard normal density function; I a i, I b i and I c i are binary variables (value 0/1 options) that have value 1 if the data are treated as (a) interval censored (I a i = 1 , I b i = 0 and I c i = 0); or (b) right-censored (I a i = 0 , I b i = 1 and I c i = 0); or (c) left-censored (I a i = 0 , I b i = 0 and I c i = 1) [37].
For example, a patient selected the highest bid in WTP scale (i.e. US$ 200) and he states US$ 220 as a maximum point. In this case, the data are coded as interval censored if we assume WTP to be between t li = 200 and t ui = 220. The observation is treated as right-censored if we assume WTP to be more than t li = 220. Otherwise, if this patient is not willing to pay even the lowest bid in WTP scale (i.e. US$ 12.5), the observation is considered left-censored with WTP being inferior to t ui = 12.5. Zero responses from patients who are not willing to pay are considered left-censored data [38].
Meanwhile, in the point and interval models, if the patients state the points (below the lowest bid or above the highest bid), these observations are coded as uncensored data; while the others are considered censored data. In this case, the model includes four components: three components (censored data) from the interval data model and one component (uncensored data) from the simple Tobit model:
$$ logL=\sum\limits^{N}_{i=1} \left\{ \begin{array}{ll} I^a_i\ \text{log}\ \left(\upphi\left(\frac{logt_{ui}-x^{\prime}{~}_{i}\beta}{\sigma}\right)-\upphi\left(\frac{logt_{li}-x^{\prime}{~}_{i}\beta}{\sigma}\right)\right)\\ +I^b_i\ \text{log}\ \left(1-\upphi\left(\frac{logt_{ui}-x^{\prime}{~}_{i}\beta_2}{\sigma}\right)\right)+I^c_i\ \text{log}\ \left(\upphi\left(\frac{logt_{ui}-x^{\prime}_{i}\beta}{\sigma}\right)\right)\\ +I^{d}_i \left(- log{\sigma}_{i} + log\phi\left(\upphi\left(\frac{logWTP^{*}_{i}-x^{\prime}_{i}\beta}{\sigma}\right)\right)\right) \end{array} \right\} $$
Where φ denotes the probability density function of the standard normal distribution. When we ask: "What is the maximum price you would be willing to pay per month for the MMT?", if a patient states his WTP of US$ 220 after selecting the highest bid, this observation is considered uncensored (WTP* = 220, I a i = 0 , I b i = 0, I c i = 0 and I d i = 1).
In the point and interval data models, because we know the threshold (t li ; t ui ), we can estimate the coefficient (β) and the scale parameter (σ). In addition, the marginal effects of independent variables on increasing or decreasing WTP value can also be computed [38].
The mean WTP is calculated by using formula with the intercept of the models:
$$ Mean\ WTP= \exp \left({\beta}_0+\frac{\sigma^2}{2}\right) $$
In multivariable analysis, determinants of patients' WTP were examined, including an "a priori" defined set of candidate variables: 1) socio-demographics: sex, age, education, marital status, employment, 2) economic status: household's income and capacity-to-pay, 3) opioid use behaviours: current use, experienced drug rehabilitation, length since first opioid use, frequency of opioid use, opioid expenses, 2) clinical characteristics: HIV stages, CD4 cell count, length of ART, and currently in MMT 3) health status: reported having problems in each EQ-5D dimension. The reduced model was constructed using a stepwise forward selection strategy, which included variables based on the log-likelihood ratio test at a p-value <0.1, and excluded variables at p-values >0.2 [39].
Characteristics of respondents in each site are presented in Table 2. The majority the study sample was male (98.7%), about two-thirds (67.5%) were living with spouse or partners; 44.7% completed high school or above, and 53.4% were self-employed. The proportion of MMT patients living with spouse was lowest at Nam Dinh PAC (54.8%), and unemployment was highest at Tu Liem DHC (31.8%). The differences among clinics were found in marital status, education attainment, employment and religion (p < 0.05).
Table 2 Demographic characteristics of respondents by MMT sites
In Table 3, we compared drug use history of patient across settings. In general, 8.6% patients were HIV-positive and 73.4% ever injected drug. The types of previous drug rehabilitation were diverse across settings. At Tu Liem, Long Bien and Ha Dong clinics, more than half of respondents had experience with private voluntary centers, and this was higher than at Nam Dinh and Xuan Truong clinics (p < 0.05). In Xuan Truong DHC, the rural setting, the proportion of having drug rehabilitation at voluntary and compulsory centers was the lowest as of 36.8% and 9.6%, respectively compared to other clinics (p < 0.05). In rural and suburban areas, Xuan Truong DHC and Ha Dong PRC, patients had fewer number of previous drug rehabilitation episodes in comparison with other clinics (p < 0.05). On average, patients reported spending 295,000 Vietnamese Dong (15 USD) per day for opiates use.
Table 3 History of drug use and treatment among respondents across MMT sites
Figure 2 reveals the proportion of participants willing to pay for different bids. There was willingness to pay for MMT services data on 95.5% patients. They reported a WTP average price of 639,000 Vietnamese Dong (32 USD) per month. Patients with HIV/AIDS or not-yet-on MMT or never experienced other drug rehabilitation reported lower price of WTP. Patients in Hanoi were willing to pay more for MMT than those in Nam Dinh Province (Table 4).
Proportion of participants willing to pay for different bids
Table 4 Willingness to pay for MMT by different patient groups
In Table 5, we determined factors associated with WTP for MMT with focus on different service delivery models. We found that the WTP of patients taking MMT at urban DHC or suburban RPC was significantly lower than those attending rural DHC or in facility without comprehensive HIV or general health care. Socioeconomic status, health status and drug-related characteristics were found to be predictors of WTP for MMT. Higher WTP was associated with higher level of educational attainment, higher income, male sex, and had high expenses on opiates prior to MMT. HIV status did not remain significant in the reduced model, rather, those patients who reported having any problem in Pain/Discomfort, and who did not have outpatient care last year were willing to pay less for MMT than others. Duration on MMT, number of years of addiction, and times of previous drug rehabilitation episodes were excluded in the reduced model.
Table 5 Factors associated with patients' willingness to pay for MMT services. (Unit: 1000 Vietnam Dong)
This study assessed the willingness of drug users to pay for MMT services. Involving a large number of patients in two epicenters of injection-driven HIV epidemics in Vietnam, we were interested to examine if the integration of MMT with other HIV/AIDS or general health care services may influence patients' WTP. We found that almost all patients were willing to pay for MMT service at an average price of 31 USD per month, approximately the unit cost for providing the service [20]. The availability of other health care services did not increase the WTP of patients. However, those with better physical health status were willing to pay less for MMT. Better socioeconomic status and great level of drug addiction, measured by reported money spent on opiates, significantly predicted higher price that patients were willing to pay for MMT.
To date, this is the largest health facility survey to examine the WTP for MMT services. In the literature, there were few studies in developed countries that showed the willingness of patients to pay for drug rehabilitations [19, 21, 22]. For example, Bishai et al. estimated a WTP a greater monthly amount for drug rehabilitation in Baltimore, Maryland (US$ 29–64) [22]. Zarkin estimated a WTP US$37 for substance abuse treatment among drug users in North Carolina and New York [19]. This is an advancement of previous research in assessing the WTP for MMT services in Vietnam that only involved a small number of HIV positive drug users in HIV outpatient clinics [20]. In the current study, we conducted the survey at various MMT service models including both HIV-negative and HIV-positive patients. Comparing to a prior assessment, respondents in this study sample reported less expenses related to opiates use and more WTP for the MMT services [20]. However, those patients with physical health problems had a lower WTP for MMT similar to patients with HIV/AIDS in the previous study. Findings of this study also support previous work that income, educational attainment, and better health status predicted a lower amount of WTP for MMT [20]. In addition, we further explored that expenses on opiates use previously and having outpatient care predicted higher amount of WTP. It appeared that those patients who had better socioeconomic status or more sufferings from addiction in the past have greater demand for the service than others.
Notably, this study found that patients attending to the MMT clinics with comprehensive care services (comprises MMT, ART, VCT, GH) were willing to pay less amount than those in the MMT clinic with only VCT. The reason for this phenomenon is still unclear and should be elucidated in further studies. However, we assume several explanations based on previous literatures. First, the clinic providing MMT and VCT is located in provincial level, while other clinics are placed in district level. A previous study in Vietnam suggested that the amount of WTP for MMT among patients in provincial clinics were higher than patients in district clinics and even central clinics [20]. Second, patients in the comprehensive clinics had to experience a higher out-of-pocket payment for health services than those in their counterpart, especially patients in the rural clinic [28]. Another analysis in general Vietnamese population from 2002 to 2010 shows that people in rural area were more likely to suffer catastrophic expenditure and impoverishment than those in urban setting [40].
While both availability of comprehensive health care services and duration on MMT did not clearly predict WTP for MMT, implications of this study's findings mainly focus on individual factors. First, given the high WTP, scaling up co-payment MMT clinics is feasible and can be an effective strategy to mobilize resources to sustain the MMT program as well as the HIV/AIDS system in Vietnam. Second, policies on co-payment MMT services should acknowledge the differences in socioeconomic status of target population; using the average income per capita as a reliable basis for justifying the user fee in each setting. Finally, there should be continuing financial supports for those patients living with HIV/AIDS and who had poor health status. These patient groups are not only economically vulnerable to health care costs but also less motivated to take MMT. It is surprising that those with Pain/ Discomfort were less willing to pay for MMT. Perhaps the MMT was not at a sufficiently high dose or that they didn't perceive that MMT is adequately helping them. These results suggest the need for pain management for some MMT patients. As it has been known that WTP is associated with medical care retention and compliance, therefore, better case management and support for severe patients is critical, especially in integrative MMT models where patients had complicated health care demand [41]. In a longitudinal study in Vietnam, HIV disease stage and drug interaction between antiretroviral or TB drugs and MMT predict patients' ongoing drug use during MMT [42] and should be taken into account in any programs that required co-payments.
The strengths of this study include a large sample size of MMT patients in various settings in two epicenters of Vietnam. However, there are several limitations should be acknowledged. First, convenient sampling technique might limit the representativeness of the sample and the capacity to generalize the findings to all MMT patients [43, 44]. In addition, our limitation is that we only selected one clinic at the provincial level; therefore, further research should be warranted in a larger scale with more provincial sites to increase the representativeness of this level of health system. Second, we were not able to collect the MMT dose and cost data. Nonetheless, as the first assessment in different MMT facilities, findings of this study are helpful for developing co-payment policies for MMT services in Vietnam.
In conclusion, co-payment policies can be applied to MMT services as a strategy to mobilize resources for the program. Also, it is necessary to ensure equalities across patient groups by acknowledging socioeconomic status of different settings and providing financial supports for disadvantaged patients with poor health status. Given the economic vulnerability of drug users, future research may focus on household's capacity- and willingness-to pay, and interventions to economically empower them.
ART :
Antiretroviral therapy
CV :
Contingent valuation
DHC :
District Health Center
EQ-5D-5 L :
EuroQol – 5 dimensions – 5 levels
GH :
HIV :
IDU :
Injecting drug use
MMT :
Methadone maintenance treatment
PAC :
Provincial AIDS Center
PWID :
People who inject drugs
VAS :
VCT :
Voluntary counselling and testing services
WTP :
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The authors would like to acknowledge supports by the Vietnam Authority of HIV/AIDS Control for the implementation of the study.
There was no funding for this analysis.
The data that support the findings of this study are available from the Vietnam Authority of HIV/AIDS Control but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of the Vietnam Authority of HIV/AIDS Control.
Institute for Preventive Medicine and Public Health, Hanoi Medical University, Hanoi, Vietnam
Bach Xuan Tran
& Huong Thi Le
Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA
& Carl A. Latkin
Authority of HIV/AIDS Control, Ministry of Health, Hanoi, Vietnam
Huong Thu Thi Phan
Institute for Global Health Innovations, Duy Tan University, Da Nang, Vietnam
Quyen Le Nguyen
School of Medicine and Pharmacy, Vietnam National University, Hanoi, Vietnam
Long Hoang Nguyen
Department of Hepatobiliary Surgery, Viet-Duc Hospital, Hanoi, Vietnam
Tho Dinh Tran
Department of Immunology and Allergy, National Otolaryngology Hospital, Hanoi, Vietnam
Thuc Thi Minh Vu
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BXT, QLN, LHN, HTTP, HTL, TDT, VTMT, CAL conceived of the study, and participatedin its design and implementation and wrote the manuscript. BXT and QLNanalyzed the data. All authors read and approved the final manuscript.
Correspondence to Quyen Le Nguyen.
This study's protocol was approved by the IRB of Vietnam Authority of HIV/AIDS Control. Data collection procedures were also approved by the directors of the MMT clinics. Written informed consent was obtained from all participants.
https://doi.org/10.1186/s12913-017-2405-y
Methadone maintenance
Contingent valuations
Integrative models | CommonCrawl |
\begin{document}
\title{The embedding theorem in Hurwitz--Brill--Noether Theory}
\begin{abstract} We generalize the Embedding Theorem of Eisenbud--Harris from classical Brill--Noether theory to the setting of Hurwitz--Brill--Noether theory.
More precisely, in classical Brill--Noether theory, the embedding theorem states that a general linear series of degree \(d\) and rank \(r\) on a general curve of genus \(g\) is an embedding if \(r \geq 3\). If \(f \colon C \to \mathbb{P}^1\) is a general cover of degree \(k\), and \(\mathcal{L}\) is a line bundle on \(C\), recent work of the authors shows that the splitting type of \(f_* \mathcal{L}\) provides the appropriate generalization of the pair \((r, d)\) in classical Brill--Noether theory \cite{CPJ1, CPJ2, LLV, HannahRefined}.
In the context of Hurwitz--Brill--Noether theory, the condition \(r \geq 3\) is no longer sufficient to guarantee that a general such linear series is an embedding. We show that the additional condition needed to guarantee that a general linear series \(|\mathcal{L}|\) is an embedding is that the splitting type of \(f_* \mathcal{L}\) has at least three nonnegative parts. This new extra condition reflects the unique geometry of \(k\)-gonal curves, which lie on scrolls in \(\mathbb{P}^r\). \end{abstract}
\section{Introduction}
Brill--Noether theory provides the bridge between the classical perspective on curves as subsets of projective space and the modern theory of abstract curves. Given an algebraic curve $C$, the line bundles that could give rise to a degree \(d\) explicit realization of \(C\) in \(\mathbb{P}^r\) are parameterized by its \defi{Brill-Noether locus} \[ W^r_d (C) \colonequals \{ \mathcal{L} \in \operatorname{Pic}^d (C) \mbox{ } \vert \mbox{ } h^0 (\mathcal{L}) \geq r+1 \} . \] For \(C\) a general curve, the geometry of \(W^r_d(C)\) is well-understood by the main theorems of classical Brill--Noether theory, established in a series of papers from the 1980s \cite{ im, fl, gp, bn, kempf, kl}. In particular, \(W^r_d(C)\) is nonempty if and only if \(\rho(g,r,d) \colonequals g-(r+1)(g-d+r)\) satisfies \(\rho \geq 0\). In this case, the universal $\mathcal{W}^r_d$ has a unique irreducible component $\mathcal{W}^r_{d, \text{BN}}$ dominating the moduli space $\mathcal{M}_g$.
Equipped with a good understanding of the moduli space $\mathcal{W}^r_{d, \text{BN}}$, one can return to the original motivation and ask finer questions about the map to projective space corresponding to a general line bundle in $\mathcal{W}^r_{d, \text{BN}}$. The first natural such question is whether a general such line bundle defines an \emph{embedding} into projective space, that is, when is it very ample? In their \emph{Embedding Theorem} \cite[Theorem 1]{EH83}, Eisenbud--Harris show that a general line bundle in $\mathcal{W}^r_{d, \text{BN}}$ is very ample if $r \geq 3$.
\begin{rem} In addition to being sufficient, the condition $r \geq 3$ is almost necessary. More precisely, there are only four triples $(g, r, d)$ with $r < 3$ where the general $\mathcal{L} \in \mathcal{W}^r_{d, \text{BN}}$ is very ample, namely \((g, r, d) \in \{ (0, 1,1), (0, 2, 2), (1, 2, 3), (3, 2, 4) \} \). \end{rem}
In this article, we consider the analogous problem when the curve $C$ is equipped with a fixed degree \(k\) map $f \colon C \to \mathbb{P}^1$. In this setting, the analogues of Brill--Noether loci are the Brill--Noether splitting loci, whose definition we now recall. Given a vector $\vec{e} = (e_1 , \ldots , e_k)$ with \(e_1 \leq e_2 \leq \cdots \leq e_k\), we define the vector bundle $\mathcal{O} (\vec{e}) \colonequals \bigoplus_{i=1}^k \mathcal{O}_{\mathbb{P}^1} (e_i)$. We then define the \defi{Brill--Noether splitting locus} \[ W^{\vec{e}} (C,f) \colonequals \left\{ \mathcal{L} \in \operatorname{Pic} (C) \mbox{ } \vert \mbox{ } f_* \mathcal{L} \simeq \mathcal{O} (\vec{e}) \text{ or a specialization thereof}\right\} . \]
The expected dimension of $W^{\vec{e}} (C,f)$ is given by \[ \rho' (g, \vec{e}) \colonequals g- \sum_{i,j} \max \{ 0 , e_j - e_i - 1 \} . \] When \((C, f)\) is general, \(W^{\vec{e}}(C, f)\) is nonempty if and only if $\rho'(g, \vec{e}) \geq 0$ \cite{CPJ1, CPJ2, HannahRefined}. In this case, when the characteristic of the ground field is \(0\) or greater than \(k\), the universal $\mathcal{W}^{\vec{e}}$ has a unique irreducible component $\mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ dominating the Hurwitz space $\mathcal{H}_{g,k}$ \cite{LLV}. Our main results determine when a general line bundle in this component is basepoint free or very ample. (See Remark \ref{char} for the situation when the characteristic is positive but less than or equal to $k$.)
\begin{ithm}\label{Thm:bpf} A general line bundle $(f, \mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ is basepoint free if and only if either \(e_{k-1} \geq 0\), or $\mathcal{L} \simeq f^* \mathcal{O}_{\mathbb{P}^1}(n)$ for $n \geq 0$. \end{ithm}
\begin{ithm} \label{Thm:VA-intro} A general line bundle in $\mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ is very ample if $e_{k-2} \geq 0$ and $r = h^0(\mathcal{O}(\vec{e})) - 1 \geq 3$. \end{ithm}
\begin{rem} As in the original Embedding Theorem, the sufficient condition in Theorem \ref{Thm:VA-intro} is almost necessary. We make this precise by giving the necessary and sufficient conditions for very ampleness in Theorem \ref{Thm:VA}. \end{rem}
Taking $k$ sufficiently large, the condition $r \geq 3$ in Theorem \ref{Thm:VA-intro} implies the original Embedding Theorem of Eisenbud--Harris. In the regime of small $k$, another condition is needed to capture the unique geometry of \(k\)-gonal curves: The condition involving the number of nonnegative parts of \(\vec{e}\) reflects the fact that such maps necessarily factor through projective bundles over \(\mathbb{P}^1\).
Correspondingly, our approach to this problem splits into two parts: One involving the ``ampleness of the map from the curve to the projective bundle'', and a second involving ``ampleness of the map from the projective bundle to the projective space along the curve''.
More generally, we can completely understand the ``degree of ampleness'' of the first of these maps. Recall that a line bundle $\mathcal{L}$ on a curve $C$ is called \defi{$p$-very ample} if, for every effective divisor $D$ on $C$ of degree $p+1$, we have $h^0 (\mathcal{L}(-D)) = h^0 (\mathcal{L}) - (p+1)$. Note that a line bundle is 0-very ample if and only if it is basepoint free, and it is 1-very ample if and only if it is very ample. In this way, the notion of $p$-very ampleness is a natural generalization of both basepoint freeness and very ampleness.
In the setting of curves with a fixed map to $\mathbb{P}^1$, it is simpler to study a variant of $p$-very ampleness that is relative to the map to $\mathbb{P}^1$. Given a cover $f \colon C \to \mathbb{P}^1$, we say that a divisor $D$ on $C$ is \defi{fibral} if \(D\) is effective and supported in a fiber of \(f\). We make the following definition.
\begin{defin} \label{def:rel-va} Let $f\colon C \to \mathbb{P}^1$ be a cover. We say that a line bundle $\mathcal{L}$ on $C$ is \defi{relatively $p$-very ample} if, for every fibral divisor $D$ of degree $p+1$ on $C$, we have \[ h^0 (\mathcal{L}(-D)) = h^0 (\mathcal{L}) - (p+1) . \] Similarly, we say that $\mathcal{L}$ is \defi{birationally relatively $p$-very ample} if the above holds for all but finitely many fibral divisors $D$ of degree $p+1$ on $C$. \end{defin} \noindent This definition is only useful when $p \leq k - 1$, since for $p \geq k$, there are no degree $p+1$ fibral divisors.
\begin{ithm} \label{bi-rel-va-intro} Assume $p \leq k - 1$. A general line bundle in \(\mathcal{W}^{\vec{e}}_{\mathrm{BN}}\) is birationally relatively \(p\)-very ample if and only if $e_{k-p} \geq 0$. \end{ithm} \begin{ithm} \label{rel-va-intro} Assume $p \leq k - 1$. A general line bundle in \(\mathcal{W}^{\vec{e}}_{\mathrm{BN}}\) is relatively \(p\)-very ample if $e_{k-p-1} \geq 0$. \end{ithm}
\begin{rem} Again, the condition in Theorem \ref{rel-va-intro} is almost necessary. We make this precise in Theorem \ref{rel-va}, where we give the necessary and sufficient conditions for relative $p$-very ampleness. \end{rem}
Our proofs proceed by embedded degeneration. We begin in Section~\ref{Sec:Prelim} by recalling the correspondence between line bundles \(\mathcal{L} \in W^{\vec{e}}(C, f)\) and maps \(C \to \mathbb{P}\mathcal{O}(\vec{e})^\vee\); this extends the correspondence in classical Brill--Noether theory between line bundles \(\mathcal{L} \in W^r_d(C)\) and maps \(C \to \mathbb{P} H^0(\mathcal{L})^\vee\). Our theorems can therefore be approached by constructing suitable curves \(C \subset \mathbb{P}\mathcal{O}(\vec{e})^\vee\). This we do inductively: In Section~\ref{sec:our_degeneration}, we explain how to start with a suitable curve \[C_{k - 1} \subset \mathbb{P}\mathcal{O}(-e_1, -e_2, \ldots, -e_{k-1}) \subset \mathbb{P}\mathcal{O}(\vec{e})^\vee,\] and attach lines and a section to form a certain reducible curve \(X \subset \mathbb{P}\mathcal{O}(\vec{e})^\vee\). Even though the curve \(X\) does not satisfy the necessary properties, we are able to analyze the deformations of \(X\) in Sections~\ref{Sec:Smooth}--\ref{Sec:VA} to show that a suitably general deformation does.
Simply by considering the curves we construct in this manner, we obtain a 6-page ``constructive'' proof of Hurwitz--Brill--Noether existence (see Remark~\ref{existence}). Even just this result is highly nontrivial: The original proof by Jensen--Ranganathan of a special case \cite{JR} was quite involved, and subsequent complete proofs have relied on deep results about either tropical geometry \cite{CPJ1}, intersection theory on the moduli stack of vector bundles \cite{P1}, or analogs of the regeneration theorem \cite{LLV}. Moreover, the constructive nature of this proof opens up the door to probing many other aspects of the geometry of $k$-gonal curves; its usefulness is unlikely to be limited to only the study of ampleness given here.
\begin{rem}[Characteristic hypotheses] \label{char} When the characteristic of the ground field is zero or greater than \(k\), the Hurwitz space $\mathcal{H}_{k,g}$ is irreducible. By contrast, when the characteristic of the ground field is positive but less than or equal to $k$, the Hurwitz space $\mathcal{H}_{k,g}$ is not known to be irreducible. Nevertheless, our proofs show that there exists a component of $\mathcal{W}^{\vec{e}}$ dominating \emph{some} component of the Hurwitz space, in which the statements of the theorems hold. When we say that $f\colon C \to \mathbb{P}^1$ is a general cover, we shall mean that it is general in the appropriate component of the Hurwitz space.
When the characteristic of the ground field is positive but less than or equal to $k$, the main theorems of \cite{LLV} also apply to some component of the Hurwitz space (see \cite[Remark 1]{LLV}). A priori, this component need not be the same as any of our components in the present paper. Moreover, the components here need not be the same for different $\vec{e}$. \end{rem}
\section{Preliminaries on projective bundles and splitting types} \label{Sec:Prelim}
Throughout, we denote our splitting type by $\vec{e} = (e_1, \ldots, e_k)$ with $e_1 \leq \cdots \leq e_k$. We write \(\deg(\vec{e}) \colonequals e_1 + \cdots + e_k\).
We define
\[u(\vec{e})\colonequals h^1(\mathbb{P}^1, \operatorname{End}(\mathcal{O}(\vec{e}))) = \sum_{i < j} \max\{0, e_j - e_i - 1\}.\]
Note that $\rho'(g, \vec{e}) = g - u(\vec{e}).$
Let $f \colon C \to \mathbb{P}^1$ be a degree $k$, genus $g$ cover, and suppose
\[
f_*\mathcal{L} = E \simeq \mathcal{O}(\vec{e}) .
\] Pulling back to $C$, there is a natural surjection
\[f^*E = f^*f_*\mathcal{L} \to \mathcal{L},\]
which corresponds to evaluation of sections on a fiber at a point of $C$. Dualizing, we obtain an injection
\[\mathcal{L}^\vee \hookrightarrow f^*E^\vee \] with locally free cokernel. This defines an embedding \begin{equation} \label{CintoPE} \begin{tikzcd} C \arrow[hookrightarrow]{r}{\iota} \arrow{rd}[swap]{f} &\mathbb{P} E^\vee \arrow{d}{\pi} \\ & \mathbb{P}^1 \end{tikzcd} \end{equation} such that $\iota^*\mathcal{O}_{\mathbb{P} E^\vee}(1) = \mathcal{L}$. Throughout, we use the subspace convention for projective bundles, so that $\pi_* \mathcal{O}_{\mathbb{P} E^\vee}(1) = E$.
\begin{lem} \label{good-embeddings} For every $t \in \mathbb{P}^1$, the fiber $f^{-1}(t)$ is a basis for $\pi^{-1}(t) \simeq \mathbb{P}^{k-1}$. Conversely, suppose we have an embedding $\iota\colon C \to \mathbb{P} E^\vee$ factoring $f$, as in \eqref{CintoPE}. If $f^{-1}(t)$ is a basis for $\pi^{-1}(t)$ for all $t \in \mathbb{P}^1$, then \(f_* \iota^*\mathcal{O}_{\mathbb{P} E^\vee}(1) \simeq E\). \end{lem} \begin{proof} To say $f^{-1}(t)$ is a basis for $\pi^{-1}(t) \simeq \mathbb{P}^{k-1}$ is to say that the evaluation map
\[ H^0(\mathcal{O}_{\mathbb{P} E^\vee}(1)|_{\pi^{-1}(t)}) \to \mathcal{O}_{\mathbb{P} E^\vee}(1)|_{f^{-1}(t)} \] is an isomorphism. This in turn is equivalent to saying that the map \[ E = \pi_* \mathcal{O}_{\mathbb{P} E^\vee}(1) \to f_* \iota^*\mathcal{O}_{\mathbb{P} E^\vee}(1) \] is an isomorphism at $t$. \end{proof}
The bundle \(E^\vee\) admits a filtration \begin{equation} \label{filt} 0 \subset \mathcal{O}(-e_1) \subset \mathcal{O}(-e_1, -e_2) \subset \cdots \subset \mathcal{O}(-e_1, -e_2, \ldots, -e_{k-1}) \subset \mathcal{O}(-e_1, -e_2, \ldots , -e_{k-1}, -e_k) = E^\vee. \end{equation} If the splitting type has distinct parts, this filtration is unique (it is the Harder--Narasimhan filtration); otherwise, we choose such a filtration. We define $F_i \colonequals \mathcal{O}(-e_1, -e_2, \ldots, -e_i)$ to be the $i$th bundle above, so \eqref{filt} becomes \[ 0 \subset F_1 \subset \cdots \subset F_{k-1} \subset F_k = E^\vee . \] For ease of notation, write \(\Sigma_i \colonequals \mathbb{P} F_i\). The above filtration of \(E^\vee\) gives a sequence of inclusions \[\Sigma_1 \subset \Sigma_2 \subset \cdots \subset \Sigma_{k - 1} \subset \Sigma_k = \mathbb{P} E^\vee.\] We also define \[F_- \colonequals F_{\#\{i : e_i < 0\}} \simeq \bigoplus_{i : e_i < 0} \mathcal{O}(-e_i) \quad \text{and} \quad F_0 \colonequals F_{\#\{i : e_i \leq 0\}} \simeq \bigoplus_{i : e_i \leq 0} \mathcal{O}(-e_i). \]
Projection away from $\mathbb{P} F_- \subset \mathbb{P} E^\vee$ defines a rational map which we denote $\operatorname{pr}_-\colon \mathbb{P} E^\vee \dashedrightarrow \mathbb{P}(E^\vee/F_-)$. The line bundle $\mathcal{O}_{\mathbb{P} E^\vee}(1)$ on $\mathbb{P} E^\vee$ defines a rational map $\mathbb{P} E^\vee \dashedrightarrow \mathbb{P}^r$, where \[ r + 1 = h^0(\mathbb{P}^1, E) = h^0(\mathbb{P} E^\vee, \mathcal{O}_{\mathbb{P} E^\vee}(1)) = \sum_{i=1}^k \max\{0, e_i + 1\}. \]
This map factors through $\operatorname{pr}_-$. In particular, the complete linear series $|\mathcal{L}| \colon C \to \mathbb{P}^r$ factors as \begin{center} \begin{tikzcd} C \arrow[hook]{r}{\iota} & \mathbb{P} E^\vee \arrow[dashed]{r}{\operatorname{pr}_-} & \mathbb{P}(E^\vee/F_-) \arrow{r} &\mathbb{P}^r. \end{tikzcd} \end{center} Thus, we break our problem into two parts: First we study the map $(\operatorname{pr}_- \circ \iota) \colon C \to \mathbb{P}(E^\vee/F_-)$. Then we study $\mathbb{P}(E^\vee/F_-) \to \mathbb{P}^r$ along the image of the first map. The image of $ \mathbb{P}(E^\vee/F_-) \rightarrow \mathbb{P}^r$ is a cone over $\mathbb{P}(E^\vee/F_0) \hookrightarrow \mathbb{P}^r$\eledit{,} with vertex a linear space which is the image of $\mathbb{P}(F_0/F_-)$.
The definition of (birational) relative very ampleness given in Definition~\ref{def:rel-va} is the analog of (birational) very ampleness for the first map \(\operatorname{pr}_- \circ \iota\). Our first task is, thus, to determine exactly when \(\mathcal{L}\) is (birationally) relatively very ample.
\section{Overview of inductive strategy} \label{sec:our_degeneration}
The following theorem is the basis for our inductive strategy to prove the main theorems.
\begin{thm}\label{thm:main_inductive} Let \(k \geq 1\) and let \(E = \mathcal{O}(\vec{e})\). There exists a smooth curve \(C \subset \mathbb{P} E^\vee\) of any genus \(g\) such that $\rho'(g, \vec{e}) \geq 0$ (with \(g=u(\vec{e}) = 0\) when \(k=1\)) satisfying all of the following conditions: \begin{enumerate}
\item\label{main_inductive:pushforward} \((\pi|_C)_*\mathcal{O}(1)|_C \simeq E\), \item\label{main_inductive:h1vanishes} \(h^1(N_{C/ \mathbb{P} E^\vee}) = 0\),
\item \label{main_inductive:general} the map \(\pi|_C \colon C \to \mathbb{P}^1\) is general in the Hurwitz space, and \item \label{main_inductive:subscroll} the curve \(C\) meets \(\Sigma_{k-1}\) at finitely many points, and does not meet \(\Sigma_{k-2}\). \end{enumerate} \end{thm}
\begin{rem} \label{existence} Combining Lemma \ref{good-embeddings} with Theorem \ref{thm:main_inductive} parts \eqref{main_inductive:pushforward} and \eqref{main_inductive:general} gives a short alternative proof of the existence theorem in Hurwitz--Brill--Noether theory, i.e., that $W^{\vec{e}}(C, f)$ is non-empty when $\rho'(g, \vec{e}) \geq 0$. \end{rem}
Our proof of Theorem \ref{thm:main_inductive} will be by induction on the rank \(k\) of \(E\). For the base case, $k=1$, the curve $C$ is the unique section of the $\mathbb{P}^0$-bundle $\mathbb{P} F_1$; in this case, the genus of $C$ is zero. For $k \geq 2$, we will construct a curve satisfying Theorem~\ref{thm:main_inductive} of arbitrary genus \(g \geq u(\vec{e})\).
Write \(\vec{e}_{<k} = (e_1, e_2, \ldots, e_{k - 1})\) for the splitting type of \(F_{k-1}^\vee\), and choose a direct sum decomposition \(E^\vee \simeq F_{k-1} \oplus \mathcal{O}(-e_k)\). (As mentioned in Section~\ref{Sec:Prelim}, if \(e_k > e_{k-1}\), then the bundle \(F_{k-1}\) is canonical; otherwise we make a choice.) We have an inclusion \(\Sigma_{k-1} = \mathbb{P} F_{k-1} \hookrightarrow \Sigma_k = \mathbb{P} E^\vee\) as a divisor. By induction on \(k\), there exists a smooth curve \(C_{k-1}\) in \(\Sigma_{k-1}\) of genus \(u(\vec{e}_{<k})\) satisfying Theorem~\ref{thm:main_inductive}.
For the inductive step, for any genus \(g \geq u(\vec{e})\), we construct a nodal curve \(X\) of arithmetic genus \(g\) as the union of such a curve \(C_{k-1} \subset \Sigma_{k-1}\), a section \(S_k\) corresponding to a choice of splitting \(\mathcal{O}(-e_k) \hookrightarrow E^\vee\), and a collection of lines connecting \(S_k\) to \(C_{k - 1}\). More precisely, define \begin{equation} \label{mdef} m \colonequals g+1 - u(\vec{e}_{<k}). \end{equation}
Choose \(m\) general points \(p_1, \dots, p_m\) on \(C_{k-1}\) over general points \(t_1, \dots, t_m\) in \(\mathbb{P}^1\). Write \(q_i\) for the unique point on \(S_k\) over \(t_i\), and let \( \Gamma \colonequals p_1+ \cdots + p_m\). We will write \(L_i\) for the line in the fiber \(\mathbb{P} E^\vee|_{t_i}\) joining \(p_i\) and \(q_i\). Define \[X \colonequals C_{k-1} \cup S_k \cup L_1 \cup \cdots \cup L_m.\] By construction, \(X\) has arithmetic genus \(g\). The following diagram illustrates \(X\):
\begin{center} \begin{tikzpicture}[scale=1.5] \draw (-3,2.5) node {\(X \colonequals C_{k-1} \cup S_k \cup (L_1 \cup \cdots \cup L_m)\)}; \draw[thick] (1.3,2.15) -- (1.25,3.3); \draw[thick] (3.5,1.865) -- (3.75,3.2); \draw[thick] (4.5,1.895) -- (4.5,3.1); \draw[thick, below] (1.3,2.15) node {\(L_1\)}; \draw[thick, below] (3.5,1.865) node {\(L_{m-1}\)}; \draw[thick, below] (4.5,1.895) node {\(L_m\)}; \filldraw[white] (3.7,2.9) circle (2.5pt); \filldraw[white] (1.25, 3.13) circle (1.25pt); \filldraw[white] (4.5, 2.9) circle (1.25pt); \draw[ left] (0,2) node {\( S_k\)}; \draw[ thick] (0, 2) .. controls (1,2.2) .. (2.5, 2); \draw[ thick] (2.5, 2) .. controls (4,1.8) .. (5, 2); \draw (5, 2) -- (5.5, 2.5) -- (5.5, 3.5); \draw (0, 2) -- (0, 3); \draw (5, 2) -- (5, 3); \draw (0, 2+1) .. controls (1,2.2+1) .. (2.5, 2+1); \draw (2.5, 2+1) .. controls (4,1.8+1) .. (5, 2+1); \draw (0+.5, 2+1+.5) .. controls (1+.5,2.2+1+.5) .. (2.5+.5, 2+1+.5); \draw (2.5+.5, 2+1+.5) .. controls (4+.5,1.8+1+.5) .. (5+.5, 2+1+.5); \draw (5+.5, 2+1+.5) -- (5, 2+1); \draw (0, 2+1) -- (0+.5, 2+1+.5); \draw (0, 2-.75) .. controls (1,2.2-.75) .. (2.5, 2-.75); \draw (2.5, 2-.75) .. controls (4,1.8-.75) .. (5, 2-.75); \draw[left] (-2.75,1.5) node {\(\mathbb{P}^1\)}; \draw[->] (-3,2.25) -- (-3,1.75); \draw (-3,2) node[left]{$f_0$}; \draw[ thick] (1, 3.5) .. controls (1.1, 3.5+.1*.2) and (1.4, 3.55) ..(1.5, 3.55); \draw[ thick] (1.5, 3.55) .. controls (2, 3.55) and (2.5, 3.4 + .5*.2) .. (3, 3.4); \draw[ thick] (3, 3.4) .. controls (3.5, 3.4 - .5*.2) and (4, 3.25 - .5*.15) .. (4.5, 3.25); \draw[ thick] (4.5, 3.25) .. controls (4.75, 3.25 + .25*.15) and (5, 3.15 + .5*.18) .. (4.5, 3.15); \draw[ thick] (4.5, 3.15) .. controls (4, 3.15 - .5*.18) and (3.5, 3.2 - .5*.2).. (3, 3.2); \draw[ thick] (3, 3.2) .. controls (2.75, 3.2 + .25*.2) and (2, 3.4 + .5*.05).. (1.5, 3.4); \draw[thick] (1.5, 3.4) .. controls (.5, 3.5 - .05) and (.75, 3.2+.75*.05) .. (1.5,3.2); \draw[ thick] (1.5,3.2) .. controls (2, 3.2-.5*.05) and (2.5, 3 +.5*.2) .. (3,3); \draw[ thick] (3,3) .. controls (3.5, 3 -.5*.2) and (4, 3 - .5*.18) .. (4.5,3); \draw[ thick] (4.5,3) .. controls (4.55, 3 + .05*.18) and (4.7, 3.1 - .05*.5) .. (4.75, 3.1); \draw[thick] (2.5, 2.5) node {...}; \draw[thick] (.63, 3.27) node{\(C_{k-1}\)}; \draw[thick] (-0.3, 3.05) node{\(\Sigma_{k-1}\)}; \filldraw (1.252,3.227) circle[radius=0.03]; \filldraw (3.731,3.11) circle[radius=0.03]; \filldraw (4.5,3) circle[radius=0.03]; \draw (1.1,3) node{\(p_1\)}; \draw (4.7,2.8) node{\(p_m\)}; \filldraw (1.3,2.14) circle[radius=0.03]; \filldraw (3.5,1.87) circle[radius=0.03]; \filldraw (4.5,1.9) circle[radius=0.03]; \draw (1.15,2.3) node{\(q_1\)}; \draw (4.7,2.1) node{\(q_m\)}; \filldraw (1.3,2.14-.75) circle[radius=0.02]; \filldraw (3.5,1.87-.75) circle[radius=0.02]; \filldraw (4.5,1.9-.75) circle[radius=0.02]; \draw[thick, below] (1.3,2.15-.75) node {\(t_1\)}; \draw[thick, below] (3.5,1.865-.75) node {\(t_{m-1}\)}; \draw[thick, below] (4.5,1.895-.75) node {\(t_m\)}; \draw[thick] (2.5, 1.1) node {...}; \end{tikzpicture} \end{center} In subsequent sections, we prove that a general deformation of $X$ satisfies Theorem \ref{thm:main_inductive}.
\section{The normal bundle of \(X\): smoothing the nodal curve} \label{Sec:Smooth}
In this section, we show that there exists a deformation of $X$ smoothing all the nodes. As a consequence, we obtain Theorem~\ref{thm:main_inductive} parts~\eqref{main_inductive:pushforward} and~\eqref{main_inductive:h1vanishes}. Recall that the deformation theory of $X$ in $\Sigma_k$ is controlled by its normal bundle $N_X \colonequals N_{X/\Sigma_k}$. For this reason, we now prove several results on the normal bundle of \(X\) restricted to various components.
\subsection{\boldmath The normal bundles of \(\Sigma_{k-1}\) and \(S_k\)} We first record some elementary results about the normal bundles of the two projective subbundles \(\Sigma_{k-1}\) and \(S_k\) in \(\Sigma_k\). Since these subbundles correspond to the direct sum decomposition \(E^\vee \simeq F_{k - 1} \oplus \mathcal{O}_{\mathbb{P}^1}(-e_k)\), we have \begin{align}
N_{\Sigma_{k-1}} &\simeq \mathcal{O}_{\mathbb{P} E^\vee}(1)|_{\Sigma_{k-1}} \otimes \pi^* \mathcal{O}_{\mathbb{P}^1}(-e_k) \label{eq:normal_bundle_sigma} \\
N_{S_k} &\simeq \mathcal{O}_{\mathbb{P} E^\vee}(1)|_{S_k} \otimes \pi^* F_{k-1} \simeq \bigoplus _{i < k} \mathcal{O}_{\mathbb{P}^1}(e_k - e_i). \label{nsk} \end{align} In particular, since \(e_k \geq e_i\) for all \(i\leq k\), we obtain \begin{equation}\label{eq:h1NS} H^1(N_{S_k}) = 0. \end{equation}
\subsection{\boldmath The normal bundle of \(C_{k-1}\)}\label{sec:N_C} The normal bundle exact sequence \begin{equation} \label{eq:standard}
0 \to N_{C_{k-1}/\Sigma_{k-1}} \to N_{C_{k-1}} \to N_{\Sigma_{k-1}}|_{C_{k-1}} \to 0, \end{equation} for \(C_{k-1} \subset \Sigma_{k-1} \subset \mathbb{P} E^\vee\), allows us to combine results about \(N_{\Sigma_{k-1}}\) with our inductive hypotheses about \(C_{k-1} \subset \Sigma_{k-1}\) to deduce information about \(N_{C_{k-1}}\). By induction we have \(H^1(N_{C_{k-1}/\Sigma_{k-1}}) = 0\) and
\((\pi|_{C_{k-1}})_* \mathcal{O}_{\mathbb{P} E^\vee}(1)|_{C_{k-1}} \simeq F_{k-1}^\vee\). Hence combining \eqref{eq:normal_bundle_sigma} with the push-pull formula, we have \begin{equation*}
(\pi|_{C_{k-1}})_*N_{\Sigma_{k-1}}|_{C_{k-1}} \simeq F_{k-1}^\vee(-e_k) \simeq \bigoplus_{i < k} \mathcal{O}_{\mathbb{P}^1}(e_i - e_k) . \end{equation*} In particular, we have \begin{equation}\label{eq:h1_NF}
h^1(N_{\Sigma_{k-1}}|_{C_{k-1}}) = \sum_{i < k} \max \{0, e_k - e_i - 1\}. \end{equation}
The restricted normal bundle \(N_X|_{C_{k-1}}\) is a positive elementary modification of \(N_{C_{k-1}}\) at the points \(\Gamma = p_1+ \dots + p_m\). See \cite[Sections 3.1--3.2]{interpolation} for a brief introduction to elementary modifications of normal bundles. These modifications are described in the following lemma.
\begin{lem}\label{lem:Nx_restC}
The restricted normal bundle \(N_X|_{C_{k-1}}\) sits in the exact sequence \begin{equation}\label{eq:NX_restricted_C}
0 \to N_{C_{k-1}/\Sigma_{k-1}} \to N_{X}|_{C_{k-1}} \to N_{\Sigma_{k-1}}|_{C_{k-1}}(\Gamma) \to 0. \end{equation} \end{lem} \begin{proof} This follows from the normal bundle exact sequence \eqref{eq:standard} and the fact that each line \(L_i\) meets \(\Sigma_{k-1}\) transversely at \(p_i\) as in \cite[Equation (7)]{interpolation}. \end{proof}
\subsection{\boldmath The normal bundles of the \(L_i\)}\label{sec:N_lines} The normal bundle of each component \(L_i \subset X\) is also simple to describe. Write \(\mathbb{P} E_{t_i}^\vee\) for the fiber of \(\mathbb{P} E^\vee\) over \(t_i \in \mathbb{P}^1\). Then the restricted normal bundle sits in an exact sequence \begin{equation}\label{eq:normal_line_unmod}
0 \to [N_{L_i/\mathbb{P} E_{t_i}^\vee} \simeq \mathcal{O}(1)^{\oplus (k - 2)}] \to N_{L_i} \to [N_{\mathbb{P} E_{t_i}^\vee}|_{L_i} \simeq \mathcal{O}] \to 0. \end{equation} Again, the restriction of \(N_X\) to one of the line components \(L_i\) is a positive modification of \(N_{L_i}\). Since \(C_{k-1} \cup S_k\) meets \(\mathbb{P} E_{t_i}^\vee\) transversely at \(p_i\) and \(q_i\), we obtain the exact sequence \begin{equation} \label{lem:NX_restL}
0 \to [N_{L_i/\mathbb{P} E_{t_i}^\vee} \simeq \mathcal{O}(1)^{\oplus k-2}] \to N_{X}|_{L_i} \to [N_{\mathbb{P} E_{t_i}^\vee}|_{L_i}(p_i + q_i) \simeq \mathcal{O}(2)] \to 0. \end{equation}
\subsection{\boldmath Deformations of \(C_{k-1}\) transverse to \(\Sigma_{k-1}\)} In this section we prove the following lemma.
\begin{lem} \label{Cor:Surj} The following map is surjective: \[ H^0 (N_X) \to H^0 (N_{\Sigma_{k-1}} \vert_{C_{k-1}} (\Gamma)). \] \end{lem} \begin{proof} The map factors as
\[H^0(N_X) \to H^0(N_X|_{C_{k-1}}) \to H^0(N_{\Sigma_{k-1}}|_{C_{k-1}}(\Gamma)).\] We consider each of the above maps in turn. The second map appears in the long exact sequence associated to the exact sequence~\eqref{eq:NX_restricted_C}. By induction, we have $H^1 (N_{C_{k-1}/\Sigma_{k-1}}) = 0$, and surjectivity follows.
Next we show that $H^0(N_X) \to H^0(N_X|_{C_{k-1}})$ is surjective. By considering the exact sequence for restriction to \(C_{k-1}\): \[ 0 \to N_X \vert_{S_k \cup L_1 \cup \cdots \cup L_m} (- \Gamma) \to N_X \to N_X \vert_{C_{k-1}} \to 0 , \] it suffices to show that $H^1 (N_X \vert_{S_k \cup L_1 \cup \cdots \cup L_m} (- \Gamma)) = 0$. To see this, restrict further to \(S_k\): \begin{equation} \label{lns} 0 \to \bigoplus_{i=1}^m N_X \vert_{L_i} (-p_i - q_i ) \to N_X \vert_{S_k \cup L_1 \cup \cdots \cup L_m}(-\Gamma) \to N_X \vert_{S_k} \to 0 . \end{equation} By \eqref{lem:NX_restL}, we have $h^1 (N_X \vert_{L_i} (-p_i -q_i )) = 0$.
Finally, \(N_X|_{S_k}\) is a positive modification of \(N_{S_k}\). By \eqref{eq:h1NS}, we have \(H^1(N_{S_k}) = 0\); hence \(H^1(N_X|_{S_k}) = 0\). \end{proof}
\subsection{\boldmath Smoothing the nodes of \(X\)} Recall that if \(X\) is a nodal curve and \(p\) is a node of \(X\), we have the exact sequence of sheaves \[0 \to N_X^p \hookrightarrow N_X \xrightarrow{\operatorname{ev}_p} T_p^1 \to 0,\] which plays a crucial role in the deformation theory of \(X\). The sheaf \(T_p^1\) is a skyscraper sheaf supported at \(p\). The subsheaf \(N_X^p\) of $N_X$ corresponds to deformations of \(X\) failing to smooth the node \(p\). Since \(T_p^1\) is supported at \(p\), the sheaves \(N_X\) and \(N_X^p\) are isomorphic away from \(p\). Likewise, for any finite subset \(\Delta\) of the nodes of \(X\), we can define \(N_X^\Delta\). (When \(\Delta\) is the entire singular locus of \(X\), this sheaf is known as the \defi{equisingular normal sheaf}, cf.\ \cite[Section 4.7.1]{sernesi}.)
\begin{lem}\label{lem:smooth_defs_exist} For any points \(p = p_i\) and \(q = q_i\), we have \[H^1(N_X^{p,q}) = 0.\] In particular, there exists a deformation of \(X\) smoothing all of the nodes. \end{lem} \begin{proof} Restriction to the (disconnected) curve \(C_{k-1} \cup S_k\) gives an exact sequence
\[ 0 \to \bigoplus_{j=1}^m N_X|_{L_j}(-p_j - q_j) \to N_X^{p,q} \to N_X^p|_{C_{k-1}} \oplus N_X^q|_{S_k} \to 0.\] Making modifications at each of the points \(\{p_1, \dots, p_m\}\) except \(p\), the exact sequence of Lemma \ref{lem:Nx_restC} induces the exact sequence
\[0 \to N_{C_{k-1}/\Sigma_{k-1}} \to N_X^p|_{C_{k-1}} \to N_{\Sigma_{k-1}}|_{C_{k-1}}(p_1 + \cdots + \hat{p} + \cdots + p_m) \to 0.\] The subbundle has no higher cohomology by induction (Theorem \ref{thm:main_inductive}\eqref{main_inductive:h1vanishes}). By \eqref{eq:h1_NF},
\[ h^1(N_{\Sigma_{k-1}}|_{C_{k-1}}) = \sum_{i \neq k} \max \{ 0, e_k - e_i - 1 \} \leq m - 1.\] Since twisting up by a general point \(p_i\) decreases the \(h^1\) by \(1\), the quotient has no higher cohomology as well. Thus \(H^1(N_X^{p,q}) = 0\) as desired.
The vanishing of \(H^1(N_X^{p,q})\) implies that each of the evaluation maps \[H^0(N_X) \xrightarrow{\operatorname{ev}_{p,q}} T^1_{p,q}\] is surjective. A general section of \(N_X\) therefore smooths all of the nodes of \(X\). \end{proof}
\subsection{Proof of Theorem~\ref{thm:main_inductive} parts~\eqref{main_inductive:pushforward} and~\eqref{main_inductive:h1vanishes}} By Lemma \ref{lem:smooth_defs_exist}, smooth deformations of the curve \(X\) exist. We will show that a general such deformation \(C\) has all of the desired properties in Theorem \ref{thm:main_inductive}.
\begin{lem} \label{Lem:2PointsOnALine} Under specialization to $f_0\colon X \to \mathbb{P}^1$, the limit of a fiber of \(f \colon C \to \mathbb{P}^1\) is either \begin{itemize} \item a fiber of $f_0 \colon X \to \mathbb{P}^1$ over a point $t \notin \{t_1, \ldots, t_m\}$, or \item contained in the fiber of $f_0$ over $t_i$ and contains exactly two points on $L_i$. \end{itemize} In particular, the limit of a fibral divisor on $C$ contains at most two points on any $L_i$. \end{lem}
\begin{proof} The limit of a fiber is necessarily supported on a fiber, say over \(t \in \mathbb{P}^1\). If \(f_0^{-1}(t)\) does not contain a line, there is nothing to prove. Otherwise, the limit of a fiber contains the points on \(C_{k-1}\) over \(t = t_i\) that are in the smooth locus of \(X\), which accounts for \(k-2\) of the \(k\) points of the limit. Thus the limit of a fiber must contain exactly \(2\) points on $L_i$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main_inductive} parts~\eqref{main_inductive:pushforward} and~\eqref{main_inductive:h1vanishes}] Applying Lemma~\ref{Lem:2PointsOnALine}, the limit of any fiber of \(f \colon C \to \mathbb{P}^1\) spans the corresponding fiber of \(\mathbb{P} E^\vee\); therefore, the same is true for any fiber of \(f\). Part \eqref{main_inductive:pushforward} therefore follows from
Lemma \ref{good-embeddings}.
Next, the bundle \(N_X\) is a positive modification of \(N_{X}^{p,q}\):
\begin{equation}
\label{eq:Npq}
0 \to N_X^{p, q} \hookrightarrow N_X \xrightarrow{\operatorname{ev}_{p,q}} T_{p,q}^1 \to 0.
\end{equation} By considering the long exact sequence in cohomology, and using Lemma~\ref{lem:smooth_defs_exist} and the fact that \(T_{p,q}^1\) is punctual, we see that \(H^1(N_X) = 0\). By semicontinuity, the same is true for a deformation \(C\) of \(X\), which completes the proof of Theorem \ref{thm:main_inductive}\eqref{main_inductive:h1vanishes}. \end{proof}
Let $\operatorname{Hilb}_{k,g}(\mathbb{P} E^\vee)$ be the Hilbert scheme of curves in $\mathbb{P} E^\vee$ having arithmetic genus $g$ and relative degree $k$ over $\mathbb{P}^1$. Since \(H^1(N_X) = 0\) by Theorem~\ref{thm:main_inductive}\eqref{main_inductive:h1vanishes}, we obtain the following.
\begin{cor} \label{Cor:SmoothPoint} The curve $X \subset \mathbb{P} E^\vee$ is a smooth point of $\operatorname{Hilb}_{k,g}(\mathbb{P} E^\vee)$, and hence lives in a unique component of $\operatorname{Hilb}_{k,g}(\mathbb{P} E^\vee)$. \end{cor}
\section{The nodal curve is general} \label{sec:degeneration_general}
We saw in Corollary~\ref{Cor:SmoothPoint} above that $X$ is contained in a unique component of $\operatorname{Hilb}_{k,g}(\mathbb{P} E^\vee)$. In this section, we will establish Theorem~\ref{thm:main_inductive}\eqref{main_inductive:general}, i.e., we will show that this component dominates some component of the Hurwitz space $\mathcal{H}_{k,g}$. (Recall that when the characteristic exceeds \(k\), the Hurwitz space is irreducible and thus this component dominates the entire Hurwitz space.) Since covers are determined by their ramification data, this can be done by analyzing the ramification behavior of a general deformation of \(X\).
Recall that, by the Riemann--Hurwitz formula, a simply branched cover \(C \to \mathbb{P}^1\) has \(2g + 2k - 2\) branch points when the characteristic is zero or odd. However, when the characteristic is \(2\), there is necessarily wild ramification. The generic behavior is that the cover looks locally in a neighborhood of a ramification point like an Artin-Schreier cover \[y^2 + y = \frac{\omega}{x},\] for some constant \(\omega\) (depending on the choice of local coordinate \(x\) at the branch point). In this case the cover has \(g + k - 1\) branch points.
Our goal is thus to show that a general deformation of \(X\) is simply branched with general branch points, and moreover, if the characteristic is \(2\), then the corresponding constants \(\omega\) are also general.
By induction, these conditions hold for \(C_{k-1} \to \mathbb{P}^1\). We therefore have to check that a general deformation of \(X\) creates the claimed general branching behavior near each \(t_i\).
Recall from \eqref{lem:NX_restL} that \(N_X|_{L_i}\) has a quotient \(N_{\mathbb{P} E_{t_i}^\vee}|_{L_i}(p_i+q_i)\). Let \(x\) be a local coordinate for \(\mathbb{P}^1\) at \(t_i\), inducing an isomorphism
\[N_{\mathbb{P} E_{t_i}^\vee}|_{L_i}(p_i+q_i) \simeq \mathcal{O}_{L_i}(p_i + q_i).\] Let \(s\) be a global coordinate on \(L_i\), so that \(p_i\) is at \(s = \infty\) and \(q_i\) is at \(s = 0\). Then if \(\sigma\) is a section of \(N_X\) whose image in \(H^0(\mathcal{O}_{L_i}(p_i + q_i))\) is \(as + b/s + c\), the geometry of the first-order deformation of \(X\) corresponding to \(\sigma\) is given by the graph of the image of \(\sigma\): \begin{equation} \label{locgeom} x = (as + b/s + c) \cdot \epsilon. \end{equation}
\begin{lem} \label{lem:deform_p_eqs} The branch points of \eqref{locgeom} occur at \[x = (c \pm 2\sqrt{ab}) \cdot \epsilon.\] Furthermore, if the characteristic is \(2\), then after a change of coordinates, \eqref{locgeom} becomes: \[y^2 + y = \frac{\sqrt{ab} \cdot \epsilon}{x - c\epsilon}.\] \end{lem} \begin{proof} The ramification points satisfy \(dx/ds = (a - b/s^2) \cdot \epsilon = 0\). Solving for \(s\), we find \(s = \pm \sqrt{b/a}\); substituting this into \eqref{locgeom}, we conclude that \eqref{locgeom} is branched at the claimed values of \(x\).
Note that, when the characteristic is \(2\), there is only one branch point, at \(x = c\epsilon\). To obtain the desired equation, we make the change of coordinates: \[s = \sqrt{\frac{b}{a}} \cdot \left(1 + \frac{1}{y}\right). \qedhere\] \end{proof}
The key point is that the branching behavior of the corresponding deformation of \(X\) is determined by the products \(ab\). (Since the \(t_i\) were already general, the value of \(c\) can be absorbed into a shift of the \(t_i\).) The smoothing parameters \(a\) and \(b\) appearing in \eqref{locgeom} are the images of the section \(\sigma\) in the deformation space \(T_{p_i}^1 \oplus T_{q_i}^1\) of the singularities at \(p_i\) and \(q_i\). The parameter \(ab\) appearing in Lemma~\ref{lem:deform_p_eqs} is therefore the image of \(\sigma\) under the quadratic map \begin{equation} \label{quadmap} H^0(N_X) \to \bigoplus_{i=1}^m T_{p_i}^1 \oplus T_{q_i}^1 \to \bigoplus_{i=1}^m T_{p_i}^1 \otimes T_{q_i}^1, \end{equation} which multiplies the smoothing parameters of the two nodes \(p_i\) and \(q_i\) together. Our goal is thus to show the surjectivity of the quadratic map \eqref{quadmap}.
The first step is to reduce this problem to the surjectivity of a linear map. By \eqref{eq:Npq}, together with Lemma~\ref{lem:smooth_defs_exist}, there is a section $s \in H^0(N_X)$ whose image under \eqref{quadmap} is nonzero in each component on the right-hand side. Now suppose
\[s' \in H^0(N_X|_{S_k \cup L_1 \cup \cdots \cup L_m}(-\Gamma) ) = \ker[H^0(N_X) \to H^0(N_X|_{C_{k-1}})] \subset H^0(N_X).\] Then $s' + s$ has fixed nonzero evaluation in each factor $T_{p_i}^1$ (independent of choice of $s'$). To show that \eqref{quadmap} is surjective, it therefore suffices to show that such sections $s'$ can attain any collection of values in the $T_{q_i}^1$ factors, i.e., to establish the surjectivity of the linear map \[ H^0(N_X \vert_{S_k \cup L_1 \cup \cdots \cup L_m}(-\Gamma)) \to \bigoplus_{i=1}^m T_{q_i}^1. \]
By \eqref{lem:NX_restL}, we have $H^1(N_X \vert_{L_i}(-p_i - q_i)) =0$, so considering the long exact sequence in cohomology associated to \eqref{lns}, we see that \(N_X \vert_{S_k \cup L_1 \cup \cdots \cup L_m}(-\Gamma) \to N_X \vert_{S_k}\) is surjective on global sections. It thus suffices to show that $H^0(N_X \vert_{S_k}) \to \bigoplus_{i=1}^m T_{q_i}^1$ is surjective. To see this, recall that we have an exact sequence \[ 0 \rightarrow N_{S_k} \rightarrow N_X \vert_{S_k} \rightarrow \bigoplus_{i=1}^m T_{q_i}^1 \rightarrow 0, \] and $H^1(N_{S_k}) = 0$ by (\ref{eq:h1NS}).
\section{Intersections with subscrolls} \label{Sec:Subscroll}
In this section, we prove Theorem~\ref{thm:main_inductive}\eqref{main_inductive:subscroll}. To do so, we first prove the following general fact about evaluation maps of line bundles.
\begin{lem} \label{Lem:Eval}
Let $C$ be a smooth curve, and $\mathcal{L}$ a line bundle on $C$, and $D$ an effective divisor on $C$, and $\Delta \subset \mathcal{L}|_D$ a proper linear subspace. Let $\Gamma$ be a general set of points on $C$ with $\# \Gamma \geq h^1 (\mathcal{L}) + 1$. Then the image of the evaluation map \[
H^0 (\mathcal{L} (\Gamma)) \to \mathcal{L}|_D \] is not contained in $\Delta$. \end{lem}
\begin{proof} Let $p \in C$ be a general point. If $h^0 (\mathcal{L}) \neq 0$ and the statement holds for $\mathcal{L} (-p)$, then it holds for $\mathcal{L}$ as well. Similarly, if $h^1 (\mathcal{L}) \neq 0$ and the statement holds for $\mathcal{L}(p)$, then it holds for $\mathcal{L}$ as well. We may therefore reduce to the case where $h^0 (\mathcal{L}) = h^1 (\mathcal{L}) = 0$.
Let $A = a_1 + \cdots + a_n$ be a general effective divisor on $C$ of the same degree as $D$. We have an exact sequence \[
H^0 (\mathcal{L} (A)) \to \mathcal{L}|_D \to H^1 (\mathcal{L}(A - D )) . \] We claim that $h^1 (\mathcal{L}(A - D )) = 0$. Indeed, specializing to the case $A = D$, we have $h^1(\mathcal{L}) = 0$, so since $A$ is general, the claim follows by semicontinuity.
In particular, the evaluation map $ H^0 (\mathcal{L} (A)) \to \mathcal{L}|_D $ is surjective. For each $i$, there is a nonzero section of $H^0 (\mathcal{L}(A))$ that vanishes on $A -a_i$, and these sections span $H^0 (\mathcal{L} (A))$. It follows that there exists an $i$ such that the image of $H^0 (\mathcal{L}(a_i))$ is not contained in $\Delta$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main_inductive}\eqref{main_inductive:subscroll}]
By induction on $k$, the curve $C_{k-1}$ meets $\Sigma_{k-2}$ at finitely many points, and does not meet $\Sigma_{k-3}$. Let $p \in C_{k-1}$ be one of the finitely many points in the intersection $C_{k-1} \cap \Sigma_{k-2}$. We show that there is a deformation of \(X\) that does not meet $\Sigma_{k-2}$ in a neighborhood of $p$. Since both \(\Sigma_{k-2}\) and $C_{k-1}$ are contained in \(\Sigma_{k-1}\), it suffices to show that there exists a section of \(N_X\) whose image in \(N_{\Sigma_{k-1}}|_p\) is nonzero. By Lemma~\ref{Cor:Surj}, the map \[ H^0 (N_X) \to H^0 (N_{\Sigma_{k-1}} \vert_{C_{k-1}}(\Gamma)) \] is surjective. Now consider the evaluation map \begin{equation}\label{eq:eval_p}
H^0 (N_{\Sigma_{k-1}} \vert_{C_{k-1}}(\Gamma)) \to N_{\Sigma_{k-1}}|_p. \end{equation} By construction, \begin{align*} \#\Gamma = m &\geq 1 + \sum_{i < k} \max \{ 0, e_k - e_i - 1 \} && \text{ by \eqref{mdef}} \\ &= 1 + h^1(N_{\Sigma_{k-1}} \vert_{C_{k-1}}) && \text{ by \eqref{eq:h1_NF}}. \end{align*}
Hence, applying Lemma \ref{Lem:Eval}, we see that \eqref{eq:eval_p} is nonzero. The preimage of the zero subspace of \(N_{\Sigma_{k-1}}|_p\) in \(H^0(N_X)\) is therefore a hyperplane, and a general section of \(N_X\) separates $C_{k-1}$ and \(\Sigma_{k-2}\). This shows that a general deformation of $X$ does not meet $\Sigma_{k-2}$.
Since a general deformation of $X$ is irreducible and $X$ is not contained in $\Sigma_{k-1}$, a general deformation of $X$ meets $\Sigma_{k-1}$ in finitely many points. \end{proof}
\section{``Relative" ampleness: maps to the nonnegative scroll}
In this section, we prove the necessary and sufficient conditions for (birational) relative very ampleness. We start by proving the statements in the introduction, and then give the full statement in Section \ref{stronger}.
\begin{proof}[Proof of Theorem \ref{bi-rel-va-intro}] The dimension of the span of a fibral divisor under $C \to \mathbb{P}^r$ is the dimension of its span in $C \to \mathbb{P} E^\vee \to \mathbb{P} \left( E^\vee / F_-\right)$. It is therefore necessary that the fibers of $\mathbb{P} \left( E^\vee / F_-\right) \to \mathbb{P}^1$ have dimension at least $p$, equivalently that $e_{k-p} \geq 0$.
To prove that $e_{k-p} \geq 0$ is sufficient, we induct on \(k\). We assume the statement for all \(k' < k\). Let \[X = C_{k-1} \cup L_1 \cup \cdots L_m \cup S_k\] be the degenerate curve constructed in Section~\ref{sec:our_degeneration}, and let \(C\) be a general deformation of \(X\) in \(\mathbb{P} E^\vee\).
Assume that \(\vec{e}\) has at least \(p+1\) nonnegative parts. Let $(f,\mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ be general. We will show that only finitely many fibral divisors \(D\) of degree \(p+1\) satisfy $h^0 (C,\mathcal{L}(-D)) \geq h^0 (C,\mathcal{L}) - p$. Let \(x\) be a general point on \(C\). Every component of \(C \times_{\mathbb{P}^1} \cdots \times_{\mathbb{P}^1} C\) dominates \(C\), and hence contains a divisor whose support contains \(x\). By upper-semicontinuity, it therefore suffices to show \(h^0 (C,\mathcal{L}(-D)) = h^0(C, \mathcal{L}) - (p+1)\) for every divisor \(D\) of the form \(D' + x\), where \(D'\) is supported in the same fiber as \(x\).
To establish this, let $x_0$ be a general point on the section $S_k$, and suppose $x$ specializes to $x_0$. Every point of $X$ distinct from $x_0$ in the same fiber lies on the component $C_{k-1}$, and by induction, the restriction of $\mathcal{L}$ to $C_{k-1}$ is birationally relatively $(p-1)$-very ample. Since \(x_0\) is general, it follows that all divisors $D'$ of degree $p$ on $C_{k-1}$ in the fiber \(f_0^{-1}(f_0(x_0))\) have linear span in \(\mathbb{P} \left( F_{k-1} / F_-\right) \) that is $(p-1)$-dimensional. Since \(x_0\) does not lie in $\mathbb{P} \left( F_{k-1} / F_-\right) $, the linear span of $D'+x_0$ in \(\mathbb{P} \left( E^\vee / F_-\right) \) is $p$-dimensional, and the result follows. \end{proof}
The remainder of this subsection is devoted to proving Theorem \ref{rel-va-intro}. If \( \vec{e}\) has at least \(p+2\) nonnegative parts, then \(\vec{e}_{<k}\) has at least \(p+1\) nonnegative parts. Therefore \(\mathcal{L}|_{C_{k-1}}\) is birationally relatively \(p\)-very ample by Theorem~\ref{bi-rel-va-intro}. There are finitely many ``bad'' fibers containing either a line \(L_i\) or a fibral divisor \(D'\) of degree $p+1$ on \(C_{k-1}\) with \(h^0(C_{k-1}, \mathcal{L}(-D')) \geq h^0(C_{k-1}, \mathcal{L}) - p\). Any fibral divisor \(D\) of degree $p+1$ on \(X\) that is not supported in a bad fiber has linear span in \(\mathbb{P} \left( E^\vee / F_{-} \right)\) of dimension \(p\). It therefore suffices to show that no divisor of degree \(p+1\) supported in a bad fiber is a limit of a divisor from \(C\) failing to impose independent conditions on \(\mathcal{L}\). We split this into two cases, first considering the fibers that contain a line \(L_i\).
\begin{lem} \label{Lem:FibralSupport} Suppose that $e_{k-p-1} \geq 0$. Let $D$ be the limit on $X$ of a fibral divisor of degree $p+1$ on $C$ whose span in $\mathbb{P} \left( E^\vee / F_{-} \right)$ has dimension $p-1$ or less. Then $D$ is supported on $C_{k-1}$ in a fiber not containing a line and has span of dimension \(p-1\). \end{lem}
\begin{proof} Because $D$ contains at most two points on a line by Lemma~\ref{Lem:2PointsOnALine}, the span of $D$ is equal to the span of $D' + x_0$ where \(x_0\) is some point on \(X\) and $D'$ is a degree $p$ divisor on $C_{k-1}$ (if two points limit to a line $L_i$, replacing one of them with $p_i = C_{k-1} \cap L_i$ does not change the span and brings our divisor to this form). By the inductive hypothesis, the span of $D'$ in $\mathbb{P}(F_{k-1}/F_-)$ has dimension $p-1$. Thus, $x_0$ must already lie in the span of $D'$. Hence, $D$ is supported on $C_{k-1}$.
By Theorem \ref{bi-rel-va-intro}, a general fibral divisor of degree \(p+1\) supported on \(C_{k-1}\) has span in \(\mathbb{P} (F_{k-1}/F_-)\) of dimension \(p\). Because the lines are attached in general fibers, $D$ lies in a fiber not containing a line. \end{proof}
\begin{proof}[Proof of Theorem~\ref{rel-va-intro}]
Let $D$ be the limit on $X$ of a fibral divisor of degree $p+1$ on $C$ whose span in $\mathbb{P} \left( E^\vee / F_{-} \right)$ has dimension $p-1$ or less. By Lemma~\ref{Lem:FibralSupport}, we may assume that $D$ is contained in a fiber of $C_{k-1}$ and has a span of dimension $p-1$. By Theorem \ref{bi-rel-va-intro}, there are finitely many such divisors, so it suffices to show that for each one, we can find a deformation of $X$ so that any corresponding deformation of $D$ has image in $\mathbb{P}(E^\vee/F_-)$ of dimension $p$.
Let $V \subseteq H^0(N_X) $ be the subspace of sections that preserve the property that some corresponding deformation of \(D\) lives in a linear space of dimension $p-1$ in $\mathbb{P}(E^\vee/F_-)$. We shall show that the image of $V$ under
\begin{equation} \label{123} H^0(N_{X}) \to H^0(N_{\Sigma_{k-1}}|_{C_{k-1}}(\Gamma)) \to N_{\Sigma_{k-1}}|_D \end{equation}
is contained in a proper linear subspace $\Delta \subset N_{\Sigma_{k-1}}|_D$.
Since the first map is surjective (Lemma \ref{Cor:Surj}), it will then suffice to show that there is a section of $N_{\Sigma_{k-1}}|_{C_{k-1}}(\Gamma)$ whose image under the second map does not lie in $\Delta$ (for which we will use Lemma \ref{Lem:Eval}).
Because $D$ does not meet $\mathbb{P} F_-$ (by Theorem \ref{thm:main_inductive}\eqref{main_inductive:subscroll}), we know $N_{\Sigma_{k-1}}|_D \simeq N_{\mathbb{P}(F_{k-1}/F_-)/\mathbb{P}(E^\vee/F_-)}|_{D}$. Let $\Lambda \subset \mathbb{P}(E^\vee/F_-)$ be the span of the image of $D$ under projection from $\mathbb{P} F_-$. If a deformation of $D$ continues to have span in $\mathbb{P}(E^\vee/F_-)$ of dimension $p-1$, then there would exist a deformation of $\Lambda$ that contains it. Therefore the image of $V$ under the composition \eqref{123} is necessarily contained in the image $\Delta$ of
\begin{equation} \label{s} H^0(N_{\mathbb{P}(F_{k-1}/F_-)/\mathbb{P}(E^\vee/F_-)}|_{\Lambda}) \rightarrow N_{\mathbb{P}(F_{k-1}/F_-)/\mathbb{P}(E^\vee/F_-)}|_{D} = N_{\Sigma_{k-1}}|_D. \end{equation}
Since $\Lambda$ is contained in a fiber and $\mathbb{P}(F_{k-1}/F_-) \subset \mathbb{P}(E^\vee/F_-)$ is a hyperplane in each fiber, we have $N_{\mathbb{P}(F_{k-1}/F_-)/\mathbb{P}(E^\vee/F_-)}|_{\Lambda} \simeq \mathcal{O}_{\Lambda}(1)$. Hence, the source of \eqref{s} has dimension
\[h^0(N_{\mathbb{P}(F_{k-1}/F_-)/\mathbb{P}(E^\vee/F_-)}|_{\Lambda}) = \dim \Lambda + 1 = p.\]
On the other hand, $D$ has degree $p+1$, so the target of \eqref{s} has dimension $p+1$. Hence, the image $\Delta \subset N_{\Sigma_{k-1}}|_D$ of \eqref{s} is a proper subspace.
Finally, by Lemma \ref{Lem:Eval}, since $\Gamma$ is general and contains $m \geq h^1(N_{\Sigma_{k-1}}|_{C_{k-1}})+1$ points, there exists a section in
$H^0(N_{\Sigma_{k-1}}|_{C_{k-1}}(\Gamma))$ that misses
$\Delta \subset N_{\Sigma_{k-1}}|_D$. \end{proof}
\subsection{Necessary and sufficient conditions for relative very ampleness} \label{stronger}
The following theorem is a stronger version of Theorem \ref{rel-va-intro} from the introduction. \begin{thm} \label{rel-va} Assume $p \leq k - 1$. A general line bundle in \(\mathcal{W}^{\vec{e}}_{\mathrm{BN}}\) is relatively \(p\)-very ample if and only if either
\begin{enumerate} \item\label{rel-va-1} $e_{k-p-1} \geq 0$, \item\label{rel-va-2} $p = 0$ and $e_k \geq 0$ and $e_{k-1} - e_1 \leq 1$ and $\rho'(g,\vec{e}) = 0$, \item\label{rel-va-3} $p = k-2$ and $e_{2} \geq 0$ and $e_k - e_2 \leq 1$ and $\rho'(g,\vec{e}) = 0$, \item\label{rel-va-4} $p = k-1$ and $e_1 \geq 0$, or \item\label{rel-va-5} $g = 0$ and $e_{k-p} \geq 0$. \end{enumerate} \end{thm}
By Theorem \ref{bi-rel-va-intro}, the condition $e_{k-p} \geq 0$ is necessary, and by Theorem~\ref{rel-va-intro}, the condition $e_{k-p-1} \geq 0$ is sufficient. We therefore suppose $e_{k-p-1} < 0 \leq e_{k-p}$.
By Theorem \ref{bi-rel-va-intro}, there are finitely many fibral divisors of degree $p+1$ that fail to be independent in their fiber of $\mathbb{P}(E^\vee/F_-)$ over $\mathbb{P}^1$. We shall call these \defi{dependent fibral divisors}. Below, we use intersection theory to compute the number $N$ of dependent fibral divisors of degree $p+1$, counted with multiplicity. In particular, when $N = 0$, there are no dependent fibral divisors of degree $p+1$, so a general $\mathcal{L}$ will be relatively $p$-very ample (instead of just birationally relatively $p$-very ample). In other words, in these cases, $\mathcal{L}$ is more ample than expected because of a numerical coincidence.
On the other hand, of course, if $N \neq 0$, then $\mathcal{L}$ cannot be relatively $p$-very ample. This will prove the only if direction in Theorem \ref{rel-va}.
\subsubsection{Number of dependent fibral divisors via intersection theory} To perform our intersection theory calculation, we realize the locus of dependent fibral divisors of degree $p+1$ as a degeneracy locus where a map of vector bundles drops rank. Let $A$ be the closure of the complement of the diagonals in the $(p+1)$-fold fiber product $C \times_{\mathbb{P}^1} \times \cdots \times_{\mathbb{P}^1} C$. Over a point $t \in \mathbb{P}^1$ that is not a branch point of $f$, the fiber of $h\colon A \to \mathbb{P}^1$ consists of ordered tuples $(a_1, \ldots, a_{p+1})$ of $p+1$ distinct points in the fiber of $f$ over $t$. The map $h\colon A \to \mathbb{P}^1$ is therefore finite of degree $k!/(k - p-1)!$.
If $t \in \mathbb{P}^1$ is a branch point of $f$, the fiber of $h\colon A \to \mathbb{P}^1$ over $t$ consists of tuples $(a_1, \ldots, a_{p+1})$ where $a_i = a_j$ is a ramification point, for some pair $(i,j)$. Let $Z \subset A$ be the locus of points $(a_1, \ldots, a_{p+1}) \in A$ where the $a_i$ fail to be distinct. By Riemann--Hurwitz, the degree of the ramification divisor of $f$ is $2g -2 + 2k$. It follows that \[\deg Z = (2g - 2 + 2k)\cdot {p+1 \choose 2} \cdot \frac{(k-2)!}{(k - p - 1)!}. \]
Let $\pi_j\colon A \to C$ be the projection onto the $j$th factor. Note that $\deg \pi_j = (k-1)!/(k-p-1)!$. For each $j$ there is an evaluation map \begin{equation} \label{evh} h^*E = \pi_j^*f^*(f_*\mathcal{L}) \to \pi_j^*\mathcal{L}. \end{equation} The fiber of the kernel of \eqref{evh} over a point $(a_1, \ldots, a_{p+1}) \in A$ is the space of linear forms on the fiber of $\mathbb{P} E^\vee$ that vanish on $a_j \in C \subset \mathbb{P} E^\vee$. Taking the sum of \eqref{evh} over all $j$, we obtain a map of vector bundles \[h^*E \rightarrow \bigoplus_{j=1}^{p+1} \pi_j^*\mathcal{L} \] on $A$ whose kernel is the bundle of linear forms on the fibers of $\mathbb{P} E^\vee$ that vanish on $a_1, \ldots, a_{p+1}$.
Recall that we assume $E$ has exactly $p+1$ non-negative parts, which form a canonical rank $p+1$ subbundle. (This subbundle of $E$ is the same as $(E^\vee/F_-)^\vee$, which is the space of linear forms on our projective bundle $\mathbb{P}(E^\vee/F_-)$.)
The composition \[\phi\colon h^*\left(\bigoplus_{i \geq k-p} \mathcal{O}(e_i) \right) \rightarrow h^*E \to \bigoplus_{j=1}^{p+1}\pi_j^*\mathcal{L}\] drops rank when there exists a linear form on $\mathbb{P}(E^\vee/F_-)$ that vanishes on all of $a_1, \ldots, a_{p+1}$. In other words, $\det \phi$ vanishes when $a_1, \ldots, a_{p+1}$ are linearly dependent in $\mathbb{P}(E^\vee/F_-)$. If $a_i = a_j$ (so $(a_1, \ldots, a_{p+1}) \in Z$), then $a_1, \ldots, a_{p+1}$ are automatically dependent. Hence, $Z \subset V(\det \phi)$. Meanwhile, away from $Z$, the determinant $\det \phi$ vanishes precisely when $a_1 + \ldots + a_{p+1}$ is a dependent fibral divisor. By Theorem \ref{bi-rel-va-intro}, there are finitely many dependent fibral divisors, so $\det \phi$ vanishes in the expected codimension. Note also that Theorem~\ref{rel-va-intro} ensures a general $(f, \mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\text{BN}}$ is relatively $(p+1)$-very ample, so there are no dependent fibral divisors of degree $p$. Thus, the degree of the scheme of dependent fibral divisors of degree $p+1$ is \begin{align} N &= \frac{1}{(p+1)!}(\deg [V(\det \phi)] - \deg Z) \\ & = \frac{1}{(p+1)!} \left( \sum_{j=1}^{p+1} \deg \pi_j^*\mathcal{L} - \sum_{i = k-p}^k \deg h^*\mathcal{O}(e_i) - \deg Z \right) \notag \\ &=\frac{1}{(p+1)!} \left( (p+1) (\deg \pi_i)(\deg \mathcal{L}) - (\deg h)(e_{k-p} + \ldots + e_k) - \deg Z \right) \notag \\ & = {k-1 \choose p} (e_1 + \ldots + e_k + g + k - 1) - {k \choose p+1}(e_{k-p} + \ldots + e_k) - (g - 1 + k){k-2 \choose p-1}. \label{hi} \end{align}
\subsubsection{The case of no dependent fibral divisors} The edge cases when $\mathcal{L}$ is more ample than expected occur when $N =0$. Multiplying \eqref{hi} by $\frac{(p+1)!(k - 1 - p)!}{(k-2)!}$, the condition $N = 0$ is equivalent to \[ (p+1)(k-1)(e_1 + \ldots + e_k ) + (p+1)(k-1 - p)(g + k - 1) = k(k-1)(e_{k-p} + \ldots + e_k), \] or equivalently, \begin{align*} (p+1)(k-1 - p)(g + k - 1) &= (k-1)\left((k - p - 1)(e_{k-p} + \ldots + e_k) - (p+1)(e_1 + \ldots + e_{k-p-1})\right) \\ &=(k-1)\sum_{i = 1}^{k-p-1} \left(\sum_{j=k-p}^{k} e_j - e_i\right). \end{align*} This in turn means \begin{equation} \label{meq} (p+1)(k-1-p)g = (k-1)\sum_{i = 1}^{k-p-1} \left(\sum_{i=k-p}^{k} e_j - e_i - 1\right) \leq (k - 1) u(\vec{e}) \leq (k - 1)g. \end{equation} If $g = 0$, then all inequalities in \eqref{meq} are equalities and hence $N = 0$. Therefore, we have relative $p$-very ampleness whenever there are $p+1$ nonnegative parts, which is Theorem \ref{rel-va}\eqref{rel-va-5}.
Assume for the remainder that $g \neq 0$. Dividing \eqref{meq} by $g$, we see that \begin{equation} \label{last} (p+1)(k-1-p) \leq k - 1. \end{equation} There are only three possible values for $p$ where this inequality holds, which translate into Theorem~\ref{rel-va}\eqref{rel-va-2}--\eqref{rel-va-4}. \begin{itemize} \item $p = 0$. Since equality holds in \eqref{last}, all inequalities in \eqref{meq} are actually equalities. Hence, $g = u(\vec{e})$, equivalently $\rho'(g,\vec{e}) =0$. Furthermore, $\sum_{i = 1}^{k-1} e_k - e_i - 1= u(\vec{e})$, which implies that the other parts are balanced, i.e., $e_{k-1} - e_1 \leq 1$. \item $p = k-2$. Again, equality holds in \eqref{last}, so all inequalities in \eqref{meq} are actually equalities. This gives $g = u(\vec{e})$, equivalently $\rho'(g,\vec{e}) = 0$. Furthermore, $\sum_{i=2}^{k} e_j - e_1 - 1 = u(\vec{e})$, which implies that the other parts are balanced, i.e., $e_k - e_2 \leq 1$. \item $p=k-1$. Our assumption that $\vec{e}$ has exactly $p+1$ nonnegative parts means $e_1 \geq 0$. \end{itemize}
\section{Very ampleness: map to projective space\label{Sec:VA}} In this section, we use the complete characterization in Theorem~\ref{rel-va} of relative \(p\)-very ampleness to prove Theorem~\ref{Thm:bpf} and a refined version of Theorem \ref{Thm:VA-intro} giving necessary and sufficient conditions for very ampleness.
\begin{proof}[{Proof of Theorem~\ref{Thm:bpf}}] When \(p = 0\), relative \(0\)-very ampleness is equivalent to \(0\)-very ampleness. It therefore suffices to show that conditions \eqref{rel-va-1}--\eqref{rel-va-5} in Theorem~\ref{rel-va} are equivalent to Theorem~\ref{Thm:bpf}. Conditions~\eqref{rel-va-1}--\eqref{rel-va-5} in Theorem~\ref{rel-va} when \(p=0\) simplify to \begin{enumerate}[(a)] \item \label{a} $e_{k-1} \geq 0$, or \item \label{b} $e_k \geq 0$ and $e_{k-1} - e_1 \leq 1$ and $\rho'(g, \vec{e}) = 0$. \end{enumerate} (To show that Theorem~\ref{rel-va}\eqref{rel-va-5} reduces to \eqref{b} above, note that the pushforward of a line bundle under any map of genus \(0\) curves is balanced.)
Finally, suppose that \eqref{b} holds and \eqref{a} does not, so $e_{k-1} < 0$. In this case, we have \[h^0(\mathcal{L}\otimes f^*\mathcal{O}_{\mathbb{P}^1}(-e_k)) = 1 \qquad \text{and} \qquad h^1(\mathcal{L} \otimes f^*\mathcal{O}_{\mathbb{P}^1}(-e_k)) = \sum_{i < k} e_k - e_i - 1 = u(\vec{e}) = g. \] Hence, $\mathcal{L} \otimes f^*\mathcal{O}_{\mathbb{P}^1}(-e_k) \cong \mathcal{O}_C$, or equivalently, $\mathcal{L} \cong f^*\mathcal{O}_{\mathbb{P}^1}(e_k)$. \end{proof}
Ordinary \(p\)-very ampleness implies relative \(p\)-very ampleness, but the converse implication does not hold for $p \geq 1$. Nevertheless, we have the following inductive tool:
\begin{lem} \label{lem:it} For a general $(f, \mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\mathrm{BN}}$, the line bundle $\mathcal{L}$ is \(p\)-very ample (resp.\ birationally \(p\)-very ample) if both: \begin{itemize} \item It is relatively \(p\)-very ample (resp.\ birationally relatively \(p\)-very ample), and \item For a general \((f',\mathcal{L}') \in \mathcal{W}^{\vec{e}'}_{\mathrm{BN}}\) where \(\vec{e}' = (e_1 - 1, e_2 - 1, \ldots, e_k - 1)\), the line bundle $\mathcal{L}'$ is \((p - 1)\)-very ample (resp.\ birationally \((p - 1)\)-very ample). \end{itemize} \end{lem}
\begin{proof} Take any effective divisor \(D\) of degree \(p + 1\), and suppose \(D\) nontrivially intersects a fiber \(F\) of the map \(C \to \mathbb{P}^1\). Because \(\mathcal{L}\) is relatively \(p\)-very ample, \(D \cap F\) is in linear general position. It thus suffices to show that the projection from \(F\) of \(D - (D \cap F)\) is in linear general position. This follows from the \((p - 1)\)-very ampleness of \(\mathcal{L}' = \mathcal{L} \otimes f^* \mathcal{O}_{\mathbb{P}^1}(-1)\). \end{proof}
\begin{rem} \label{rem:dumb} As an immediate consequence of Theorem~\ref{Thm:bpf} and Lemma~\ref{lem:it}, we see that for a general $(f, \mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\mathrm{BN}}$, the line bundle $\mathcal{L}$ is $p$-very ample if: \begin{align*} e_{k} & \geq p \\ e_{k-1} &\geq p \\ e_{k-2} &\geq p-1 \\ \vdots \\ e_{k-p-1} &\geq 0. \end{align*} Similarly, for a general \((f, \mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\mathrm{BN}}\), the line bundle $\mathcal{L}$ is birationally \(p\)-very ample if: \begin{align*} e_k &\geq p \\ e_{k-1} &\geq p-1 \\ \vdots \\ e_{k-p} &\geq 0. \end{align*} \end{rem}
The result in Remark~\ref{rem:dumb} is likely far from sharp. Farkas proved \cite{farkas} that a general line bundle in the classical Brill--Noether locus \(\mathcal{W}^r_d\) is \(p\)-very ample if \(r \geq 2p+1\). We have already shown that a general line bundle in \(\mathcal{W}^{\vec{e}}_{\mathrm{BN}}\) is \textit{relatively} \(p\)-very ample if \(e_{k-p-1}\geq 0\), and so the natural conjecture would be as follows.
\begin{conj}\label{p-very-conj} A general line bundle in $\mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ is \(p\)-very ample if \[e_{k-p-1} \geq 0 \quad \text{and} \quad r = h^0(\mathcal{O}(\vec{e})) - 1 \geq 2p+1.\] \end{conj}
When \(p=1\), we prove this conjecture in Theorem~\ref{Thm:VA-intro}; in other words, the remainder of the paper deals with the discrepancy between what we expect by Conjecture~\ref{p-very-conj} and what is covered by Remark~\ref{rem:dumb} when \(p=1\). Note that this discrepancy increases as \(p\) increases.
We now give a characterization of birational very ampleness, which will feed into our proof of Theorem \ref{Thm:VA-intro}.
\begin{lem} \label{lem:3parts} For $(f, \mathcal{L}) \in \mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ general, $\mathcal{L}$ is birationally very ample if and only if \begin{enumerate} \item \label{pt1} \(e_{k-2} \geq 0\), \item \(e_{k-1} \geq 0\) and \(e_{k}\geq 1\), or \item \(g=0\) and \(e_{k-1} \geq 0\). \end{enumerate} \end{lem} \begin{proof}
By Theorem~\ref{bi-rel-va-intro}, it is necessary that \(e_{k-1} \geq 0\). If in addition \(e_{k} \geq 1\), then a generic line bundle in \(\mathcal{W}^{\vec{e}}_{\text{BN}}\) is birationally very ample by Remark \ref{rem:dumb}. On the other hand, if \(e_{k-1} = e_k = 0\) and \(e_{k-2} < 0\), then \(|\mathcal{L}|\) maps \(C\) to \( \mathbb{P}^1\), and so \(\mathcal{L}\) is birationally very ample if and only if \(g=0\).
It remains to consider the case \(e_{k-2} \geq 0\). We will use our degeneration introduced in Section~\ref{sec:our_degeneration}. The map $X \to \mathbb{P}^r$ sends $C_{k-1}$ into a proper linear space (with non-zero degree) and sends $S_k$ to a rational normal curve in a complementary linear space (possibly contracting it to a point if \(e_k = 0\)). Since the lines $L_i$ are attached at general points of $C_{k-1}$, their images in $\mathbb{P}^r$ are distinct. In particular, the map $X \to \mathbb{P}^r$ is birationally very ample along the lines $L_i$. \end{proof}
\subsection{Proof of Theorem \ref{Thm:VA-intro}} \label{mc}
We must prove very ampleness in the following two cases: \begin{enumerate} \item When \(\vec{e}\) has at least four nonnegative parts, none of which are positive. \item When \(\vec{e}\) has at least three nonnegative parts, at least one of which is positive. \end{enumerate}
\subsubsection{At least four parts of degree $0$ and no positive parts} \label{edge04} Here, we consider splitting types with $e_k = e_{k-1} = e_{k-2}= e_{k-3} = 0$.
We argue by degeneration to the reducible curve $X$ constructed in Section \ref{sec:our_degeneration}. By our hypothesis on $\vec{e}$, we know $\mathcal{O}_{\mathbb{P} E^\vee}(1)|_{C_{k-1}}$ is a line bundle whose pushforward has splitting type with at least three parts of degree $0$ (and no positive parts). Thus, by Lemma~\ref{lem:3parts}\eqref{pt1}, we have that
$\mathcal{O}_{\mathbb{P} E^\vee}(1)|_{C_{k-1}}$ is birationally very ample on $C_{k-1}$.
The complete linear system of $\mathcal{O}_{\mathbb{P} E^\vee}(1)$ maps $X$ to $\mathbb{P}^r$ with $r \geq 3$. This map contracts $S_k$ to a point and sends
$C_{k-1}$ birationally onto its image in a complementary hyperplane $\alpha\colon C_{k-1} \to \mathbb{P}^{r-1} \subset \mathbb{P}^r$.
We will show that for a deformation $C$ of $X$, every degree \(2\) effective divisor $D$ on $C$ embeds into $\mathbb{P}^r$. We have two cases to consider: \begin{enumerate} \item \label{ck} $D$ limits to a divisor on $C_{k-1}$ which is collapsed to a point in $\mathbb{P}^{r-1} \subset \mathbb{P}^r$. \item \label{sk} $D$ limits to a divisor on $S_{k}$ (which is necessarily collapsed to a point in $\mathbb{P}^r$). \end{enumerate}
Case \eqref{ck}: By inducting on the number of nonnegative parts of $\vec{e}$,
we may assume $\vec{e}$ has exactly four parts of degree $0$, i.e., $r = 3$.
By Lemma \ref{lem:3parts}, there are finitely many bad degree \(2\) effective divisors $D$ on $C_{k-1}$ that are collapsed to a point $\alpha(D) \in \mathbb{P}^2 \subset \mathbb{P}^3$, so it suffices to show that a general deformation of $X$ will separate any given bad $D$. Let $M = N_{\Sigma_{k-1}}|_{C_{k-1}} = \mathcal{O}_{\mathbb{P} E^\vee}(1)|_{C_{k-1}}$ (see \eqref{eq:normal_bundle_sigma} in case $e_k = 0$). We have an identification
\[M|_D \simeq \alpha^*N_{\mathbb{P}^2/\mathbb{P}^3}|_{\alpha(D)}. \] A deformation of $X$ separates $D$ in the map to $\mathbb{P}^3$ if it does not take values in the subspace
\[\Delta = \alpha^*H^0(N_{\mathbb{P}^2/\mathbb{P}^3}|_{\alpha(D)}) \subset H^0(\alpha^*N_{\mathbb{P}^2/\mathbb{P}^3}|_{\alpha(D)}) = M|_D. \] Thus it suffices to show that the image of
\[H^0(N_X) \rightarrow H^0(M(\Gamma)) \rightarrow M|_{D}\]
does not lie in the subspace $\Delta$. By Lemma~\ref{Cor:Surj}, the first map above is surjective. Since $\Gamma$ is general with $\# \Gamma \geq h^1( M) + 1$, the image of $H^0(M(\Gamma)) \rightarrow M|_{D}$ does not lie in $\Delta$ by Lemma~\ref{Lem:Eval}.
Case \eqref{sk}: Every degree \(2\) divisor $D$ on $S_k$ is collapsed to a point in $\mathbb{P}^r$. By \eqref{nsk} we have a filtration \begin{equation} \label{eq:filt} 0 \rightarrow \bigoplus_{i \leq k-4} \mathcal{O}(-e_i) \rightarrow N_{S_k} \rightarrow \mathcal{O}^{\oplus 3} \rightarrow 0. \end{equation}
We have that $N_X|_{S_k}$ is a positive modification of $N_{S_k}$. Since $C_{k-1}$ does not meet $\Sigma_{k-4}$ by Theorem \ref{thm:main_inductive}\eqref{main_inductive:subscroll}, these modifications do not meet the subbundle in \eqref{eq:filt}, i.e., \(k=0\) in \cite[Equation (7)]{interpolation}. Hence, we obtain a filtration \begin{equation} \label{filtN}
0 \rightarrow \bigoplus_{i \leq k-4} \mathcal{O}(-e_i) \rightarrow N_X|_{S_k} \rightarrow Q \rightarrow 0, \end{equation}
where $Q$ is a positive modification of $\mathcal{O}^{\oplus 3}$ at the $q_i$. For $i \leq k-4$, we have $e_i \leq 0$ by assumption, so $h^1( \bigoplus_{i \leq k-4} \mathcal{O}(-e_i))= 0$. Hence, we have a surjection $H^0(N_X|_{S_k}) \to H^0(Q)$.
Since a general section in $H^0(N_X)$ smooths all the $q_i$, and the images of the lines \(L_i\) are distinct, it suffices to show that it separates any degree
$2$ effective divisor $D$ in $S_k \smallsetminus \{q_1, \ldots, q_m\}$. For such $D$, we have a natural trivialization of $Q|_D$. If a section in $H^0(N_X)$ has non-constant image under
\[H^0(N_X) \to H^0(N_X|_{S_k}) \to H^0(Q) \to Q|_{D},\]
then the corresponding deformation separates $D$. We know $H^0(N_X) \to H^0(N_X|_{S_k})$ is surjective and $H^0(N_X|_{S_k}) \to H^0(Q)$ is surjective. Therefore, we wish to show that a general section of $Q$
misses the constant subspace when evaluated in $Q|_{D}$ for all degree $2$ effective divisors $D$ supported on \(S_k \smallsetminus \{q_1, \dots, q_m\}\).
There is a $2$-dimensional family of degree $2$ effective divisors $D$ in $S_k \smallsetminus \{q_1, \ldots, q_m\}$. For each such $D$, we get a ``bad subspace" of $H^0(Q)$ defined as the preimage of the constant sections $\Delta \subset Q|_{D}$ under
$H^0(Q) \to Q|_{D}$. If $m \geq 3$, then we claim that all summands of $Q$ are positive. Indeed, because the $p_i$ are general, and the image of \(C_{k - 1}\) in \(\mathbb{P}^{r - 1}\) is nondegenerate, the modifications cannot all lie in a proper trivial subbundle. Then, it is a codimension $3$ condition on the space of global sections $H^0(Q)$ to lie in the constant subspace along $D$. The union of a $2$-dimensional family of codimension $3$ ``bad subspaces" of $H^0(Q)$ cannot be all of \(H^0(Q)\), so a general section will not lie in any bad subspace. If $m = 1$ or $m = 2$, then $N_X|_{S_k}$ has an $\mathcal{O}(1)$ summand, and every section that is non-constant in the $\mathcal{O}(1)$ component is non-constant along every $D$.
\subsubsection{At least three nonnegative parts at least one of which is positive} \label{edge00a} If $e_{k-1} > 0$, the result follows from Remark~\ref{rem:dumb}; we therefore suppose $e_{k-2} = e_{k-1} = 0$ and $b \colonequals e_k > 0$.
Let $X = C_{k-1} \cup L_1 \cup \cdots \cup L_m \cup S_k \subset \mathbb{P} E^\vee$ be our degenerate curve. The complete linear series for $\mathcal{O}_{\mathbb{P} E^\vee}(1)$ sends $X$ to $\mathbb{P}^{a+b+1}$ by sending $C_{k-1}$ onto a $\mathbb{P}^a$ and $S_k$ onto a degree $b$ rational normal curve in a complementary $\mathbb{P}^{b}$. Write $\alpha \colon C_{k-1} \to \mathbb{P}^a$ for the restriction of $|\mathcal{O}_{\mathbb{P} E^\vee}(1)|$ to $C_{k-1}$. Our goal is to show that for a general deformation $C$ of $X$, every degree $2$ effective divisor on $C$ embeds into $\mathbb{P}^{a+b+1}$. The only degree \(2\) effective divisors on $X$ that are collapsed to a point in $\mathbb{P}^{a+b+1}$ are divisors on $C_{k-1}$ that live in a fiber of $\alpha \colon C_{k-1} \to \mathbb{P}^a$.
Let $\Gamma = \{p_1, \ldots, p_m\}$ be the points on $C_{k-1}$ where the lines $L_1, \ldots, L_m$ are attached. Write \begin{equation} \label{Ldef}
\mathcal{L} \colonequals N_{\Sigma_{k-1}}|_{C_{k-1}} = \mathcal{O}_{\mathbb{P} E^\vee}(1)|_{C_{k-1}} \otimes f^* \mathcal{O}_{\mathbb{P}^1}(-e_{k}) = \alpha^* \mathcal{O}_{\mathbb{P}^1}(1) \otimes f^*\mathcal{O}_{\mathbb{P}^1}(-b), \end{equation}
where the second equality follows from \eqref{eq:normal_bundle_sigma}. By Lemma \ref{lem:Nx_restC} we have $N_{X}|_{C_{k-1}}/(N_{C_{k-1}/\Sigma_{k-1}}) = \mathcal{L}(\Gamma)$.
Because $b> 0$, the map $\mathbb{P}(E^\vee/F_{k-3}) \to \mathbb{P}^{a+b+1}$ is an embedding away from the codimension $1$ subscroll $\mathbb{P}(F_{k-1}/F_{k-3}) \subset \mathbb{P}(E^\vee/F_{k-3})$. If a section of $N_{X}$ has image in $\mathcal{L}(\Gamma)$ that does not vanish along a degree \(2\) effective divisor $D$ on $C_{k-1}$, then the corresponding deformation of $X$ separates $D$. Since $H^0(N_X) \to H^0(\mathcal{L}(\Gamma))$ is surjective by Lemma~\ref{Cor:Surj}, it suffices to show that, for general $\Gamma$, a general section of $\mathcal{L}(\Gamma)$ has a vanishing locus which contains no degree $2$ effective divisor contained in a fiber of $\alpha$.
If $e_{k-3} = 0$, then by Lemma \ref{lem:3parts}, we know that $\alpha\colon C_{k-1} \to \mathbb{P}^a$ is birational onto its image. Thus, there are finitely many degree $2$ effective divisors $D$ on $C_{k-1}$ that are collapsed by $\alpha$. For each such $D$, Lemma \ref{Lem:Eval} applied to $\Delta = 0 \subset \mathcal{L}|_D$ shows that a general section of $\mathcal{L}(\Gamma)$ is nonzero on $D$.
For the remainder, we therefore assume $e_{k-3} < 0$, and hence $a = 1$. If $\deg \alpha = 1$, then no degree \(2\) effective divisor of $X$ is collapsed in the map to $\mathbb{P}^{b+2}$. We therefore assume for the remainder of this section that $\deg \alpha \geq 2$. In this case, $\alpha\colon C_{k-1} \to \mathbb{P}^1$ collapses infinitely many degree $2$ effective divisors $D$, making this case more difficult than the previous paragraph.
Throughout, we write $g' \colonequals u(\vec{e}_{<k})$ for the genus of $C_{k-1}$. If $\vec{e} = (-1, \ldots, -1, 0, 0, b)$, then we have
$g' = u(\vec{e}_{<k}) = u(-1, \ldots, -1, 0, 0) = 0$, and so $\deg \alpha = g' - 1 + (k-1) - \deg(\vec{e}_{<k}) = 1$. Thus, we may also assume that $e_1 < -1$, and consequently, the quantity $m$ defined in \eqref{mdef} satisfies \begin{equation} \label{mgeq3} m \geq 1+ \sum_{j < k} (e_k - e_j - 1) \geq 3. \end{equation}
If $m \geq g'$, then $\mathcal{L}(\Gamma)$ is a general line bundle, so it has a section with the desired property. We can therefore assume $m < g'$ for the remainder.
Since we are assuming $b > 0$, we have \[h^0(\mathcal{L}) = h^0(f_*\mathcal{L}) = h^0\left(\mathcal{O}(\vec{e}_{<k}) \otimes \mathcal{O}(-b)\right)= 0.\] We claim that it suffices to consider the case of minimal $m$, namely $m = h^1(\mathcal{L}) + 1$. Indeed, if $m > h^1(\mathcal{L}) + 1$, let $\Gamma' \subset \Gamma$ be a subset of $h^1(\mathcal{L}) + 1$ points. Then, if we have treated the case of minimal $m$, it follows that a general section in $H^0(\mathcal{L}(\Gamma')) \subset H^0(\mathcal{L}(\Gamma))$ has the desired property. We therefore assume $m = h^1(\mathcal{L}) + 1$ for the remainder of this section. From \eqref{mdef} and our assumptions $e_1 < -1$ and $e_i \leq -1$ for $i \leq k-3$, we see that \[ m = 1+\sum_{i=1}^{k-1} (b - e_i - 1) > 1+ (k-3)b + 2(b-1) = (k-1)b - 1 \geq (k-2)b.\]
\begin{lem} Let $x_1, \ldots, x_{m - (k-2)b}$ be general points on $C_{k-1}$. Then there exists $\Gamma \in \operatorname{Sym}^{m}C_{k-1}$ such that $h^0(\mathcal{L}(\Gamma)) = 1$, and the unique section of $\mathcal{L}(\Gamma)$ vanishes along the $x_i$ but at no other points in the fibers $\alpha^{-1}(\alpha(x_i))$. \end{lem} \begin{proof} Let $d = \deg \alpha$ and let $\{h_1, \ldots, h_d\}$ be the points of a general fiber of $\alpha$. Define \[\Gamma\colonequals \{x_1, \ldots, x_{m-(k-2)b}\} \cup \bigcup_{i \leq b} \{f^{-1}(f(h_{i})) \smallsetminus h_{i} \}.\] Now, using \eqref{Ldef}, we see \[\mathcal{L}(\Gamma) = \mathcal{O}\left(\Gamma + h_1 + \ldots + h_d - \sum_{i = 1}^b f^{-1}(f(h_i))\right) = \mathcal{O}(h_{b+1} + \ldots + h_d + x_1 + \ldots + x_{m-(k-2)b}).\] In particular, $\mathcal{L}(\Gamma)$ has a section that vanishes along the $x_i$ but at no other points in the fibers $\alpha^{-1}(\alpha(x_i))$. It remains to see that $h^0(\mathcal{L}(\Gamma)) = 1$.
First, we observe that $h^0(\mathcal{O}(h_{b+1} + \ldots + h_d)) = 1$ since
$h^0(\mathcal{O}(h_1 + \ldots + h_d)) = 2$ and $\mathcal{O}(h_1 + \ldots + h_d)$ is basepoint free. By Riemann--Roch, \[\chi(\mathcal{O}(h_{b+1} + \ldots + h_d)) = \chi(\mathcal{O}(h_1 + \ldots + h_d)) - b = \chi(\mathcal{L}) + b(k-1) - b = [0 - (m+1)] + b(k-2).\] Hence, $h^1(\mathcal{O}(h_{b+1} + \ldots + h_d)) = m - b(k-2)$, so twisting up by this many general points kills the $h^1$ without increasing $h^0$. \end{proof}
We now show that for $\Gamma$ general, \emph{every} zero of a general section of $\mathcal{L}(\Gamma)$ is isolated in its fiber under $\alpha$. For this, consider the incidence correspondence \[\Psi = \{(x, \Gamma) \in C_{k-1} \times \operatorname{Sym}^m C_{k-1}: h^0(\mathcal{L}(\Gamma)(- x)) \neq 0 \}.\] Because $\chi(\mathcal{L}(\Gamma)(-x)) = 0$, the locus $\Psi \subset C_{k-1} \times \operatorname{Sym}^m C_{k-1}$ is pure codimension $1$.
By construction, if $h^0(\mathcal{L}(\Gamma)) = 1$, then the fiber of $\Psi$ over $\Gamma \in \operatorname{Sym}^m C_{k-1}$ is the vanishing locus of the unique section. We have already found one such $\Gamma$ where there exists a point $x$ that is the only point of $\alpha^{-1}(\alpha(x))$ in the fiber of $\Psi \to \operatorname{Sym}^m C_{k-1}$ over $\Gamma$. It therefore suffices to show that $\Psi$ is irreducible, which is Lemma \ref{irr} below. To prove Lemma \ref{irr}, we first need the following.
\begin{lem} \label{lem:h0andh1} Suppose $L$ is a line bundle on a curve $C$ with $h^0(L) \geq 2$ and $h^1(L) = 0$. Then, for general $x \in C$, the line bundle $L(x)$
is birationally very ample.
\end{lem}
\begin{proof}
Consider $P = \{(p, q) \in \operatorname{Sym}^2 C : h^0(L(-p - q)) \geq h^0(L) - 1\}$. Since it is defined by a determinantal condition, $P$ is either empty or pure dimension $1$. If $(p, q) \notin P$, then $p$ and $q$ already impose independent conditions on $L$, so they impose independent conditions on $L(x)$ for all $x \in C$. Now fix some component of $P$ and a general $(p, q)$ in that component.
It suffices to show that, for general $x \in C$, the points $p$ and $q$ impose independent conditions on $L(x)$. Since every component of $P$ has dimension $1$, we may assume that $p$ and $q$ are not both basepoints of $L$, so $h^0(L(-p-q)) = h^0(L) - 1$, and therefore $h^1(L(-p-q)) = 1$. Since $x$ is general, we have $h^1(L(x)(-p-q)) = 0$. Hence, $h^0(L(x)(-p-q))= h^0(L(x)) - 2$, as desired.
\end{proof}
\begin{lem} \label{irr} If $m < g'$, then the variety $\Psi$ is irreducible. \end{lem} \begin{proof} First we show that no fiber of $\Psi$ over $C_{k-1}$ is the entire space $\operatorname{Sym}^m C_{k-1}$. Let $g'$ be the genus of $C_{k-1}$. Recall that $\chi(\mathcal{L}(\Gamma)(-x)) = 0$ for any $x\in C_{k-1}$. If $h^0(\mathcal{L}(\Gamma)(-x)) > 0$ for all $\Gamma$, then we would have $h^1(\mathcal{L}(\Gamma)(-x)) > 0$ for all $\Gamma$, so $h^1(\mathcal{L}(-x)) \geq m + 1$. But this is impossible since we are assuming $h^1(\mathcal{L}) = m-1$. Since $\Psi$ is pure codimension $1$, every component must dominate $C_{k-1}$.
It remains to show that for a general $x \in C_{k-1}$, the fiber of $\Psi$ over $x$ is irreducible. Let $K$ denote the canonical bundle on $C_{k-1}$. Because $\chi(\mathcal{L}(\Gamma)(-x)) = 0$, we have \begin{equation}\label{cgs} h^0(\mathcal{L}(\Gamma)(-x)) = h^1(\mathcal{L}(\Gamma)(-x)) = h^0(K \otimes \mathcal{L}^{-1}(x)(-\Gamma)). \end{equation} Note that $h^0(K \otimes \mathcal{L}^{-1}) = h^1(\mathcal{L}) = m - 1 \geq 2$ by \eqref{mgeq3} and $h^1(K \otimes \mathcal{L}^{-1}) = h^0(\mathcal{L}) = 0$. Therefore, Lemma \ref{lem:h0andh1} says that $K \otimes \mathcal{L}^{-1}(x)$ is birationally very ample.
Let $s = h^0(K \otimes \mathcal{L}^{-1}(x)) - 1$. The cohomology group in \eqref{cgs} is non-zero if and only if the image of $\Gamma$ under the map $C_{k-1} \to \mathbb{P}^s$ given by $|K \otimes \mathcal{L}^{-1}(x)|$
is contained in a hyperplane. Hence, the fiber of $\Psi$ over $x$ consists of those $\Gamma$ that are contained in hyperplane sections of $C_{k-1} \to \mathbb{P}^s$. Since $C_{k-1} \to \mathbb{P}^{s}$ is birationally very ample, we may use the Uniform Position Principle if the characteristic is $0$ or if $s \geq 4$ for any characteristic \cite{Borys}. In these cases, the Uniform Position Principle says that the collection of $\Gamma$ contained in hyperplane sections of the image is irreducible.
The cases where $s \leq 3$ occur when $m = 3$ or $4$. If $m = 3$, then $\vec{e} = (-2,0,0,1)$, which has $g' = 2$. If $m = 4$, then $\vec{e} = (-2, 0, 0, 2), (-2,-1,0,0,1),$ or $(-3, 0, 0,1)$, which have $g' = 2, 2,$ and $4$ respectively. In each of these cases $m \geq g'$. \end{proof}
\subsection{Necessary and sufficient conditions for very ampleness}
The remainder of the paper is devoted to proving the following theorem.
\begin{thm} \label{Thm:VA} A general line bundle in $\mathcal{W}^{\vec{e}}_{\mathrm{BN}}$ is very ample if and only if either
\begin{enumerate} \item\label{va-1} $e_{k-2} \geq 0$ and $r = h^0(\mathcal{O}(\vec{e})) - 1 \geq 3$, \item\label{va-2} $k = 3$ and $e_2 \geq 1$ and $e_3 - e_2 \leq 1$ and $\rho'(g,\vec{e}) = 0$, \item\label{va-3} $k = 2$ and $e_{1} \geq 1$, \item\label{va-4} $k = 2$ and $(e_1,e_2) = (0,g)$ or $(e_1,e_2) = (0, g+1)$ (these are degree $2g +1$ and $2g+2$), \item\label{va-5} $g = 0$ and $e_{k-1} \geq 0$, \item\label{va-6} $g = 1$ and $\vec{e} = (-1, 0, 1)$ or $\vec{e} = (-1,\ldots,-1,0,0,0)$ (smooth plane cubic), or \item\label{va-7} $g = 3$ and $\vec{e} = (-2, 0, 1)$ or $\vec{e} = (-2,-1,\ldots,-1,0,0,0)$ (smooth plane quartic). \end{enumerate} \end{thm}
The main content of the above theorem is the sufficiency of part \eqref{va-1}, which is Theorem \ref{Thm:VA-intro} from the introduction. The proof of this more precise theorem relies on our precise characterization of relative very ampleness given in Theorem~\ref{rel-va}.
Since a very ample line bundle is relatively very ample, we consider each of the cases in Theorem~\ref{rel-va} in the next five subsections.
\subsubsection{Case \eqref{va-1} of Theorem \ref{rel-va}: when \(e_{k-2}\geq 0\)} If $e_{k-2} \geq 0$ and \(r \geq 3\), then a generic line bundle in \(\mathcal{W}^{\vec{e}}_{\text{BN}}\) is very ample by Theorem \ref{Thm:VA-intro}. So it remains to consider the case that \(e_{k-2} = e_{k-1} = e_k = 0\) and \(e_{k-3} < 0\). By assumption, \(\deg \vec{e} \leq -k + 3\).
In this case, \(|\mathcal{L}|\) is birationally very ample by Lemma~\ref{lem:3parts}\eqref{pt1} and maps \(C\) to \(\mathbb{P}^2\). This map is an embedding if and only if the genus of \(C\) is equal to the arithmetic genus of a plane curve of degree \(g - 1 + k + \deg(\vec{e})\). That is, we must have the equality \begin{equation}\label{eq:genus_plane}
g = \frac{(g + k + \deg \vec{e} - 2)(g + k + \deg \vec{e} - 3)}{2}.
\end{equation}
On the other hand, we know that \(\rho'(g, \vec{e}) \geq 0\) and hence
\begin{equation}\label{eq:rhop}
g \geq u(\vec{e}) \geq \sum_{j < k-2 \leq i} e_i-e_j-1 = 3(-k+3 - \deg \vec{e}).
\end{equation} Combining \eqref{eq:genus_plane} and \eqref{eq:rhop} we obtain \[ g \geq \frac{(g -g/3 + 1)(g -g/3)}{2} = \frac{(2g + 3)(2g)}{18}, \] and hence \(g \leq 3\). Those genera that satisfy \eqref{eq:genus_plane} are \(g=0\), which falls into Theorem \ref{Thm:VA}\eqref{va-5}, and \(g=1\), which falls into Theorem \ref{Thm:VA}\eqref{va-6}, and \(g=3\), which falls into Theorem \ref{Thm:VA}\eqref{va-7}.
Having dealt with this case, we will assume for the remainder of the proof that \(e_{k-2} < 0\).
\subsubsection{Case \eqref{va-2} of Theorem \ref{rel-va}: when $p = 0$} This case is not relevant, since we assume $p = 1$.
\subsubsection{Case \eqref{va-3} of Theorem \ref{rel-va}: when \(k=3\) and \(e_2 \geq 0\) and \(e_3 - e_2 \leq 1\) and \(\rho'(g, \vec{e}) = 0\)}
If \(e_2 \geq 1\), then the map from the nonnegative scroll \(\mathbb{P}(E^\vee/F_-)\) to \(\mathbb{P}^r\) is an embedding, and so the composition with the map from \(C\) to the nonnegative scroll (which we already know is an embedding by Theorem~\ref{rel-va}) is also an embedding. This is Theorem~\ref{Thm:VA}\eqref{va-2}.
It remains to consider the case that \(e_2 = 0\). First assume that \(e_2 = e_3 = 0\). Then \(|\mathcal{L}|\) maps \(C\) to \(\mathbb{P}^1\), and hence is very ample if and only if \(g=0\). This therefore falls into Theorem~\ref{Thm:VA}\eqref{va-5}.
The only remaining case is \(e_2=0\) and \(e_3 = 1\). In this case \(C\) must again be a smooth plane curve. A similar calculation as above
shows that \(g=1\) or \(g=3\), which fall into cases~\eqref{va-6} and \eqref{va-7} of Theorem~\ref{Thm:VA}.
\subsubsection{Case \eqref{va-4} of Theorem \ref{rel-va}: when \(k=2\) and \(e_1 \geq 0\)}
If \(e_1 \geq 1\), then as in the previous case, the map from the nonnegative scroll \(\mathbb{P}(E^\vee/F_-) = \mathbb{P} E^\vee\) to \(\mathbb{P}^r\) is an embedding, and so the map from \(C\) to \(\mathbb{P}^r\) is also an embedding. This is Theorem~\ref{Thm:VA}\eqref{va-3}.
If \(e_1 = 0\), then since the directrix is contracted by the complete linear system of \(\mathcal{O}_{\mathbb{P} E^\vee}(1)\), it suffices to determine when the curve meets the directrix at most once. This is an intersection theory calculation on the Hirzebruch surface \(\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-e_2))\). If \(F\) denotes the class of a fiber and \(D\) denotes the class of the directrix, then \([C] = (e_2 + g + 1)F + 2D\). Hence, \(C \cdot D = 0\) if and only if \(e_2 = g+1\), and \(C \cdot D = 1\) if and only if \(e_2 = g\). This is Theorem~\ref{Thm:VA}\eqref{va-4}.
\subsubsection{Case \eqref{va-5} of Theorem \ref{rel-va}: when \(g=0\) and \(e_{k-1} \geq 0\)}
In all of these cases the line bundle on \(\mathbb{P}^1\) is of degree at least \(1\), and is hence very ample. This is Theorem~\ref{Thm:VA}\eqref{va-5}.
\end{document} | arXiv |
Birch's theorem
In mathematics, Birch's theorem,[1] named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.
Statement of Birch's theorem
Let K be an algebraic number field, k, l and n be natural numbers, r1, ..., rk be odd natural numbers, and f1, ..., fk be homogeneous polynomials with coefficients in K of degrees r1, ..., rk respectively in n variables. Then there exists a number ψ(r1, ..., rk, l, K) such that if
$n\geq \psi (r_{1},\ldots ,r_{k},l,K)$
then there exists an l-dimensional vector subspace V of Kn such that
$f_{1}(x)=\cdots =f_{k}(x)=0{\text{ for all }}x\in V.$
Remarks
The proof of the theorem is by induction over the maximal degree of the forms f1, ..., fk. Essential to the proof is a special case, which can be proved by an application of the Hardy–Littlewood circle method, of the theorem which states that if n is sufficiently large and r is odd, then the equation
$c_{1}x_{1}^{r}+\cdots +c_{n}x_{n}^{r}=0,\quad c_{i}\in \mathbb {Z} ,\ i=1,\ldots ,n$
has a solution in integers x1, ..., xn, not all of which are 0.
The restriction to odd r is necessary, since even degree forms, such as positive definite quadratic forms, may take the value 0 only at the origin.
References
1. Birch, B. J. (1957). "Homogeneous forms of odd degree in a large number of variables". Mathematika. 4: 102–105. doi:10.1112/S0025579300001145.
| Wikipedia |
Volume 82, Numbers 1-2, 2016
An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices
Abstract. For a bounded lattice $L$, the principal congruences of $L$ form a bounded ordered set $\princ L$. G. Grätzer proved in 2013 that every bounded ordered set can be represented in this way. Also, G.Birkhoff proved in 1946 that every group is isomorphic to the group of automorphisms of an appropriate lattice. Here, for an arbitrary bounded ordered set $P$ with at least two elements and an arbitrary group $G$, we construct a selfdual lattice $L$ of length sixteen such that $\princ L$ is isomorphic to $P$ and the automorphism group of $L$ is isomorphic to $G$.
DOI: 10.14232/actasm-015-817-8
Keyword(s): principal congruence, lattice congruence, lattice automorphism, ordered set, bounded poset, quasi-colored lattice, preordering, quasiordering, monotone map, simultaneous representation, independence, automorphism group
Received September 6, 2015, and in revised form September 28, 2015. (Registered under 67/2015.)
Permutations assigned to slim rectangular lattices
Tamás Dékány, Gergő Gyenizse, Júlia Kulin
Abstract. Slim rectangular lattices were introduced by G. Grätzer and E. Knapp in Acta Sci. Math. 75, 29--48, 2009. They are finite semimodular lattices $L$ such that the poset $\jir L$ of join-irreducible elements of $L$ is the cardinal sum of two nontrivial chains. Using deep tools and involved considerations, a 2013 paper by G. Czédli and the present authors proved that a slim semimodular lattice is rectangular iff so is the Jordan--Hölder permutation associated with it. Here, we give an easier and more elementary proof.
DOI: 10.14232/actasm-015-271-y
AMS Subject Classification (1991): 06C10
Keyword(s): rectangular lattice, semimodularity, slim lattice, planar lattice, combinatorics of permutations
Received June 26, 2014, and in revised form March 2, 2015. (Registered under 21/2015.)
A note on the structure of $\mathbb{F}_{p^{k}}A_{5}{/}J(\mathbb{F}_{p^{k}}A_{5})$
Neha Makhijani, R. K. Sharma, J. B. Srivastava
Abstract. Let $\mathbb{F}_{p^{k}}A_{5}$ be the group algebra of $A_{5}$, the alternating group of degree $5$, over $\mathbb{F}_{p^{k}}=GF(p^{k})$, where $p$ is a prime. Using the theory developed by Ferraz in [Ferraz08], we give an explicit description for the Wedderburn decomposition of $\mathbb{F}_{p^{k}}A_{5}$ modulo its Jacobson radical.
AMS Subject Classification (1991): 16S34; 16U60
Keyword(s): group algebra, Wedderburn decomposition, unit group
Received August 19, 2014. (Registered under 61/2014.)
Finite loop algebras of $RA2$ loops
Swati Sidana, R. K. Sharma
Abstract. Let $L=M(G,2)$ be a $RA2$ loop and $F[L]$ be its loop algebra over a field $F$. In this article, we obtain the unit loop of $F[L]/J(F[L]),$ where $L=M(D_{2p},2)$ is obtained from the dihedral group of order $2p$ ($p$ odd prime), $J(F[L])$ is the Jacobson radical of $F[L]$ and $F$ is a finite field of characteristic $2$. The structure of $1+J(F[L])$ is also determined.
AMS Subject Classification (1991): 20N05, 17D05
Keyword(s): loop algebra, Moufang loop, Zorn's algebra, general linear loop, loops $M(G, 2)$
Received February 4, 2014, and in revised form August 27, 2015. (Registered under 6/2015.)
Generating sets of infinite full transformation semigroups with restricted range
K. Tinpun, J. Koppitz
Abstract. In the present paper, we consider minimal generating sets of infinite full transformation semigroups with restricted range modulo specific subsets. In particular, we determine relative ranks.
AMS Subject Classification (1991): 20M20, 54H15
Keyword(s): generating sets, transformation semigroups, restricted range, relative ranks
Received January 7, 2015, and in revised form September 15, 2015. (Registered under 2/2015.)
Essential normality of automorphic composition operators
Liangying Jiang, Caiheng Ouyang, Ruhan Zhao
Abstract. We first characterize those composition operators that are essentially normal on the weighted Bergman space $A^2_s(D)$ for any real $s>-1$, where induced symbols are automorphisms of the unit disk $D$. Using the same technique, we investigate automorphic composition operators on the Hardy space $H^2(B_N)$ and the weighted Bergman spaces $A^2_s(B_N)$ ($s>-1$). Furthermore, we give some composition operators induced by linear fractional self-maps of the unit ball $B_N$ that are not essentially normal.
DOI: 10.14232/actasm-014-060-x
AMS Subject Classification (1991): 47B33; 32A35, 32A36
Keyword(s): composition operator, essentially normal, automorphism, linear fractional maps
Received August 19, 2014, and in revised form October 19, 2014. (Registered under 60/2014.)
Additive solvability and linear independence of the solutions of a system of functional equations
Eszter Gselmann, Zsolt Páles
Abstract. The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations \Eq{*}{ d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \qquad(x,y\in{\msbm R}, k\in\{0,\ldots,n\}) } is studied, where $\Delta_n:=\big\{(i,j)\in\Z \times\Z \mid0\leq i,j\mbox{ and }i+j\leq n\big\}$ and $\Gamma\colon \Delta_n\to{\msbm R} $ is a symmetric function such that $\Gamma(i,j)=1$ whenever $i\cdot j=0$. On the other hand, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n}\colon{\msbm R} \to{\msbm R} $ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\colon{\msbm R} \to{\msbm R} $, the iterates $d^0,d^1,\dots,d^n$ of $d$ are shown to be linearly independent, and the graph of the mapping $x\mapsto(x,d^1(x),\dots,d^n(x))$ to be dense in ${\msbm R} ^{n+1}$.
AMS Subject Classification (1991): 16W25, 39B50
Keyword(s): derivation, higher order derivation, iterates, linear dependence
Received April 23, 2014, and in revised form August 19, 2014. (Registered under 34/2014.)
Generalized monotonicity of sequences and functions of bounded $p$-variation
S. S. Volosivets, A. A. Tyuleneva
Abstract. It is well known that for a non-negative sequence $\{a_n\}_{n=1}^\infty $ the continuity of the sum $\sum ^\infty_{n=1}a_n\cos nx$ is equivalent to the convergence of the series $\sum ^\infty_{n=1}a_n$. We prove that for generalized monotone $\{a_n\}_{n=1}^\infty $ the last condition implies the so-called $p$-absolute continuity in the sense of L. C. Young and E. R. Love, where $1< p< \infty $. In this case we give estimates for the $p$-variation moduli of continuity and best approximations in terms of Fourier coefficients of a function. As a corollary of the above results some Konyushkov-type theorems on the equivalence of $O$- and $\asymp $-relations are established.
AMS Subject Classification (1991): 42A32, 42A10, 42A16, 41A25
Keyword(s): $p$-variation, $L^p$, best approximation, fractional moduli of continuity, Fourier coefficients, equivalence of $O$- and $\asymp $-relations
Received November 12, 2014, and in revised form August 15, 2015. (Registered under 74/2014.)
Almost everywhere and norm convergence of the inverse continuous wavelet transform in Pringsheim's sense
Kristóf Szarvas, Ferenc Weisz
Abstract. The inverse wavelet transform is studied with the help of the summability means of Fourier transforms. Norm and almost everywhere convergence of the inversion formula is obtained for $L_p$ functions. The points of the set of the almost everywhere convergence are characterized as the Lebesgue points.
AMS Subject Classification (1991): 42C40; 42C15, 42B08, 42A38, 46B15
Keyword(s): continuous wavelet transform, $\theta $-summability, inversion formula
Received June 10, 2014, and in revised form January 6, 2015. (Registered under 45/2014.)
On the essential minimum modulus of linear operators in Banach spaces
Ha?kel Skhiri
Abstract. We shall show in this paper a quite useful formula connecting the essential minimum modulus and minimum modulus of any linear bounded operator defined on a separable Hilbert space. Since this formula does not depend on the structure of Hilbert space, this result enables us to define the essential minimum modulus of linear operators in the more general context of Banach spaces. The connection between our definition and that given by Zemánek [Geometric interpretation of the essential minimum modulus, Operator Theory: Adv. Appl., 6, (1982), 225-227] is discussed. Moreover, the notion of the essential surjectivity modulus and left (resp. right) essential minimum modulus on Banach spaces are also defined and will be studied in this paper. The asymptotic formula for the essential spectrum of a semi-Fredholm operator with index zero in terms of the left and right essential minimum moduli is proved.
AMS Subject Classification (1991): 47A53, 47A55, 47A60, 46B04
Keyword(s): Banach space, minimum modulus, surjectivity modulus, essential minimum modulus, Calkin algebra, essential spectrum, semi-Fredholm, index
Received May 14, 2014, and in final form January 17, 2016. (Registered under 38/2014.)
The class of b-limited operators
Abdelmonaim El Kaddouri, Kamal El Fahri, Mohammed Moussa
Abstract. We introduce a new class of operators that we call b-limited operators. Properties of b-limited operators, the relationship between the b-limited operators and various classes of operators are studied.
AMS Subject Classification (1991): 46B42, 47B60, 47B65
Keyword(s): order limited operator, b-AM-compact operator, limited operator, (b)-property, discrete Banach lattice
Received October 13, 2014, and in final form January 10, 2015. (Registered under 70/2014.)
Adjoint of sums and products of operators in Hilbert spaces
Zoltán Sebestyén, Zsigmond Tarcsay
Abstract. We provide sufficient and necessary conditions guaranteeing equations $(A+B)^*=A^*+B^*$ and $(AB)^*=B^*A^*$ concerning densely defined unbounded operators $A,B$ between Hilbert spaces. We also improve the perturbation theory of selfadjoint and essentially selfadjoint operators due to Nelson, Kato, Rellich, and Wüst. Our method involves the range of two-by-two matrices of the form $M_{S,T}=\soperator{-T}{S}$ that makes it possible to treat real and complex Hilbert spaces jointly.
AMS Subject Classification (1991): 47A05, 47A55, 47B25
Keyword(s): closed operator, adjoint, selfadjoint operator, operator product, operator sum, perturbation
The isometric equivalence problem on various Banach spaces
Sara Botelho-Andrade
Abstract. In this paper we solve the isometric equivalence problem for composition operators on Hardy spaces of the bi-disk, for generalized composition operators on the Bloch space and for elementary operators on the symmetric subspace of $\mathcal{B}(\mathcal{H})$.
AMS Subject Classification (1991): 47A06, 47B25, 47B33
Keyword(s): isometric equivalence, Hardy space, Bloch space, spaces of symmetric operators, composition operators, elementary operators
Received May 31, 2014, and in revised form October 20, 2014. (Registered under 44/2014.)
Some perturbation results through localized SVEP
Pietro Aiena, Salvatore Triolo
Abstract. Some classical perturbation results on Fredholm theory are proved and extended by using the stability of the localized single-valued extension property under Riesz commuting perturbations. In the last part, we give some results concerning the stability of property $(gR)$ and property $(gb)$.
AMS Subject Classification (1991): 47A10, 47A11; 47A53, 47A55
Keyword(s): localized SVEP, operators with topological uniform descent, Riesz operators, property $(gR)$ and property $(gb)$
Received April 17, 2014, and in revised form December 5, 2014. (Registered under 35/2014.)
Spectra of some weighted composition operators on $H^{2}$
Carl C. Cowen, Eungil Ko, Derek Thompson, Feng Tian
Abstract. We completely characterize the spectrum of a weighted composition operator \W on \HtD when \ph has Denjoy--Wolff point $a$ with $0<|\ph '(a)|< 1$, the iterates, $\ph_n$, converge uniformly to $a$, and $\psi $ is in \Hi(the space of bounded analytic functions on \D ) and continuous at $a$. We also give bounds and some computations when $|a|=1$ and $\ph '(a)=1$ and, in addition, show that these symbols include all linear fractional \ph that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights $\psi $ so that \W is seminormal.
AMS Subject Classification (1991): 47B33, 47B35, 47A10, 47B20, 47B38
Keyword(s): weighted composition operator, spectrum of an operator, hyponormal operator
Received May 21, 2014, and in revised form September 18, 2014. (Registered under 42/2014.)
Intersection theory and the Horn inequalities for invariant subspaces
H. Bercovici, W. S. Li
Abstract. We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class $C_{0}$, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model.
DOI: 10.14232/actasm-015-538-z
Keyword(s): invariant subspaces, Horn inequalities
Received May 19, 2015. (Registered under 38/2015.)
Perturbations of invariant subspaces of compact operators
Michael Gil'
Abstract. The paper deals with Schatten--von Neumann operators in a Hilbert space. A sharp perturbation bound for invariant subspaces is established. It can be considered as a particular generalization of the Davis--Kahan theorem.
Keyword(s): Hilbert space, compact operators, invariant subspaces, perturbations
Received February 22, 2015, and in revised form April 27, 2015. (Registered under 14/2015.)
On a generalization of $\rho $-contractions
B. Chevreau, A. Crăciunescu
Abstract. A result of Eckstein asserts that for any $\rho $-contraction $T$ on a Hilbert space $\mathcal{H}$ the sequence $(||T^{n}h||)_{n}$ is convergent for any $h\in\mathcal {H}$. We show that this remains true for a natural generalization of the class of $\rho $-contractions, which we call the class of $(\rho,N)$-contractions (notation: $\mathcal{C}_{\rho,N}(\mathcal{H})$). Our argument follows the lines of Mlak's proof of Eckstein's result, but is somewhat simplified by a study of coisometric $(\rho,N)$-dilations of these operators, which seems to be of independent interest. Along the way we also point out that Gavruta's example extends to the class of $(\rho,N)$-contractions. Namely, let $\mathcal{C}_{\infty,\infty }(\mathcal{H}) :=\cup_{\rho,N}\mathcal{C}_{\rho,N}(\mathcal{H})$; then, for any integer $p>1$, there exists an operator $T$ such that $T^{p}=I$ and $T\notin\mathcal {C}_{\infty,\infty }(\mathcal{H})$.
Keyword(s): contractions, coisometric dilations, similarity
Received July 18, 2014, and in final form January 5, 2015. (Registered under 4/2014.)
Chebyshev center, best proximity point theorems and fixed point theorems
S. Rajesh, P. Veeramani
Abstract. Brodskii and Milman proved that there exists a point in $C(A)$, the set of all Chebyshev centers of $A$, which is fixed by every surjective isometry from $A$ into $A$ whenever $A$ is a nonempty weakly compact convex set having normal structure in a Banach space. Motivated by this result, Lim et al. proved that every isometry from $A$ into $A$ has a fixed point in $C(A)$ whenever $A$ is a nonempty weakly compact convex set having normal structure in a Banach space. In this paper, we prove that every relatively isometry map $T\colon A\cup B \rightarrow A\cup B$, satisfying $T(A) \subseteq B$ and $T(B) \subseteq A$, has a best proximity point in $C_{A}(B)$, the set of all Chebyshev centers of $B$ relative to $A$, whenever the nonempty weakly compact convex proximal pair $(A, B)$ has proximal normal structure and rectangle property. Also, we prove that, under suitable assumptions, an analogous result of Brodskii and Milman for relatively isometry mappings holds. In case of $A = B$, we obtain the results of Brodskii and Milman, and Lim et al. as a particular case of our results.
AMS Subject Classification (1991): 47H09, 47H10
Keyword(s): asymptotic center, Chebyshev center, best proximity points, proximal pairs, relatively nonexpansive maps, rectangle property
Received December 30, 2014. (Registered under 83/2014.)
On the convexity of a hitting distribution for discrete random walks
Gábor V. Nagy, Attila Szalai
Abstract. We examine the convexity of the hitting distribution of the real axis for symmetric random walks on $\duz ^2$. We prove that for a random walk starting at $(0,h)$, the hitting distribution is convex on $[h-2,\infty )\cap\duz $ if $h\ge2$. We also show an analogous fact for higher-dimensional discrete random walks. This paper extends the results of a recent paper [NT].
AMS Subject Classification (1991): 60G50; 05A20
Keyword(s): discrete random walk, integer lattice, convexity
Received April 2, 2014. (Registered under 26/2014.)
Parameter estimation for the subcritical Heston model based on discrete time observations
Mátyás Barczy, Gyula Pap, Tamás T. Szabó
Abstract. We study asymptotic properties of some (essentially conditional least squares) parameter estimators for the subcritical Heston model based on discrete time observations derived from conditional least squares estimators of some modified parameters.
AMS Subject Classification (1991): 91G70, 60H10, 62F12, 60F05
Keyword(s): Heston model, conditional least squares estimation
Received February 23, 2015, and in revised form October 24, 2015. (Registered under 16/2015.) | CommonCrawl |
Proceedings of the American Mathematical Society
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The Jordan decomposition of vector-valued measures
by B. Faires and T. J. Morrison PDF
Proc. Amer. Math. Soc. 60 (1976), 139-143 Request permission
This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and $\mathcal {A}$ is a Boolean algebra. Then an additive set function $\mu :\mathcal {A} \to L$ can be expressed as the difference of two positive additive measures if and only if $\mu (\mathcal {A})$ is order bounded. A sufficient condition for a countably additive set function with values in ${c_0}(\Gamma )$, for any set $\Gamma$, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set.
J. Diestel, The Radon-Nikodym property and the coincidence of integral and nuclear operators, Rev. Roumaine Math. Pures Appl. 17 (1972), 1611–1620. MR 333728
J. Diestel, Applications of weak compactness and bases to vector measures and vectorial integration, Rev. Roumaine Math. Pures Appl. 18 (1973), 211–224. MR 317042
J. Diestel and B. Faires, On vector measures, Trans. Amer. Math. Soc. 198 (1974), 253–271. MR 350420, DOI 10.1090/S0002-9947-1974-0350420-8
B. Faires, Grothendieck spaces and vector measures, Ph.D. Dissertation, Kent State Univ., 1974.
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Ulrich Krengel, Über den Absolutbetrag stetiger linearer Operatoren und seine Anwendung auf ergodische Zerlegungen, Math. Scand. 13 (1963), 151–187 (German). MR 176034, DOI 10.7146/math.scand.a-10697
D. R. Lewis, A vector measure with no derivative, Proc. Amer. Math. Soc. 32 (1972), 535–536. MR 296248, DOI 10.1090/S0002-9939-1972-0296248-2
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J. J. Uhl Jr., Orlicz spaces of finitely additive set functions, Studia Math. 29 (1967), 19–58. MR 226395, DOI 10.4064/sm-29-1-19-58
J. J. Uhl Jr., Extensions and decompositions of vector measures, J. London Math. Soc. (2) 3 (1971), 672–676. MR 286974, DOI 10.1112/jlms/s2-3.4.672
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Journal: Proc. Amer. Math. Soc. 60 (1976), 139-143
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DOI: https://doi.org/10.1090/S0002-9939-1976-0419723-X | CommonCrawl |
ISI B.Stat & B.Math 2014 Objective Paper| Problems & Solutions
Here, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
The system of inequalities
a-b^{2} \geq \frac{1}{4}, b-c^{2} \geq \frac{1}{4}, c-d^{2} \geq \frac{1}{4}, d-a^{2} \geq \frac{1}{4} \quad \text { has }
(A) no solutions
(B) exactly one solution
(C) exactly two solutions
(D) infinitely many solutions.
Let $\log _{12} 18=a$. Then $\log {24} 16$ is equal to
(A) $\frac{8-4 a}{5-a}$
(B) $\frac{1}{3+a}$
(C) $\frac{4 a-1}{2+3 a}$
(D) $\frac{8-4 a}{5+a}$
The number of solutions of the equation $\tan x+\sec x=2 \cos x,$ where $0 \leq x \leq \pi$ is
(A) $0$;
(B) $1$;
(C) $2$;
(D) $3$.
Using only the digits $2,3$ and $9,$ how many six digit numbers can be formed which are divisible by $6$ ?
(A) $41$;
(B) $80$;
(C) $81$;
(D) $161$.
What is the value of the following integral?
\int_{\frac{1}{2014}}^{2014} \frac{\tan ^{-1} x}{x} d x
(A) $\frac{\pi}{4} \log 2014$;
(B) $\frac{\pi}{2} \log 2014$;
(C) $\pi \log 2014$;
(D) $\frac{1}{2} \log 2014$.
A light ray travelling along the line $y=1,$ is refiected by a mirror placed along the line $x=2 y .$ The reflected ray travels along the line
(A) $4 x-3 y=5$;
(B) $3 x-4 y=2$;
(C) $x-y=1$;
(D) $2 x-3 y=1$.
For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Then the number of real solutions of $|2 x-[x]|=4$ is
(D) $4$ .
What is the ratio of the areas of the regular pentagons inscribed inside and circumscribed around a given circle?
(A) $\cos 36^{\circ}$;
(B) $\cos ^{2} 36^{\circ}$;
(C) $\cos ^{2} 54^{\circ}$;
(D) $\cos ^{2} 72^{\circ}$.
Let $z_{1}, z_{2}$ be nonzero complex numbers satisfying $\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right| .$ The circumcentre of the triangle with the points $z_{1}, z_{2},$ and the origin as its vertices is given by
(A) $\frac{1}{2}\left(z_{1}-z_{2}\right)$;
(B) $\frac{1}{3}\left(z_{1}+z_{2}\right)$;
(C) $\frac{1}{2}\left(z_{1}+z_{2}\right)$;
(D) $\frac{1}{3}\left(z_{1}-z_{2}\right)$.
In how many ways can 20 identical chocolates be distributed among 8 students so that each student gets at least one chocolate and exactly two students get at least two chocolates each?
(A) $308$;
(B) $364$;
(C) $616$;
(D) $\left(\begin{array}{c} 8\\2 \end{array} \right) \left(\begin{array}{c} 17 \\ 7 \end{array}\right)$.
Two vertices of a square lie on a circle of radius $r,$ and the other two vertices lie on a tangent to this circle. Then, each side of the square is
(A) $\frac{3 r}{2}$;
(B) $\frac{4 r}{3}$;
(C) $\frac{6 r}{5}$;
(D) $\frac{8 r}{5}$.
Let $P$ be the set of all numbers obtained by multiplying five distinct integers between 1 and $100 .$ What is the largest integer $n$ such that $2^{n}$ divides at least one element of $P ?$
(D) $25$.
Consider the function $f(x)=a x^{3}+b x^{2}+c x+d,$ where $a, b, c$ and $d$ are real numbers with $a>0$. If $f$ is strictly increasing, then the function $g(x)=$ $f^{\prime}(x)-f^{\prime \prime}(x)+f^{\prime \prime \prime}(x)$ is
(A) zero for some $x \in \mathbb{R}$;
(B) positive for all $x \in \mathbb{R}$;
(C) negative for all $x \in \mathbb{R}$;
(D) strictly increasing.
Let $A$ be the set of all points $(h, k)$ such that the area of the triangle formed by $(h, k),(5,6)$ and (3,2) is 12 square units. What is the least possible length of a line segment joining (0,0) to a point in $A ?$
(A) $\frac{4}{\sqrt{5}}$;
(B) $\frac{8}{\sqrt{5}}$;
(C) $\frac{12}{\sqrt{5}}$;
(D) $\frac{16}{\sqrt{5}}$.
Let $P$=$\{a b c: a, b, c \text{ positive integers }, a^{2}+b^{2}=c^{2},\text { and }3 \text { divides } c\}$ . What is the largest integer $n$ such that $3^{n}$ divides every element of $P$?
Let $A_{0}=\emptyset$ (the empty set). For each $i=1,2,3, \ldots,$ define the set $A_{i}=$ $A_{i-1} \cup\{A_{i-1}\} .$ The set $A_{3}$ is
(A) $\emptyset$;
(B) $\{\emptyset\}$;
(C) ${\emptyset,\{\emptyset}\}$;
(D) $\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$.
Let $f(x)=\frac{1}{x-2} \cdot$ The graphs of the functions $f$ and $f^{-1}$ intersect at
(A) $(1+\sqrt{2}, 1+\sqrt{2})$ and $(1-\sqrt{2}, 1-\sqrt{2})$;
(B) $(1+\sqrt{2}, 1+\sqrt{2})$ and $\left(\sqrt{2},-1-\frac{1}{\sqrt{2}}\right)$;
(C) $(1-\sqrt{2}, 1-\sqrt{2})$ and $\left(-\sqrt{2},-1+\frac{1}{\sqrt{2}}\right)$;
(D) $\left(\sqrt{2},-1-\frac{1}{\sqrt{2}}\right)$ and $\left(-\sqrt{2},-1+\frac{1}{\sqrt{2}}\right)$.
Let $N$ be a number such that whenever you take $N$ consecutive positive integers, at least one of them is coprime to $374 .$ What is the smallest possible value of $N ?$
Let $A_{1}, A_{2}, \ldots, A_{18}$ be the vertices of a regular polygon with 18 sides. How many of the triangles $\Delta A_{i} A_{j} A_{k}, 1 \leq i<j<k \leq 18,$ are isosceles but not equilateral?
The limit $\lim _{x \rightarrow 0} \frac{\sin ^{\alpha} x}{x}$ exists only when
(A) $\alpha \geq 1$;
(B) $\alpha=1$;
(C) $|\alpha| \leq 1$;
(D) $\alpha$ is a positive integer.
Consider the region $R=\{(x, y): x^{2}+y^{2} \leq 100, \sin (x+y)>0\} .$ What is the area of $R ?$
(A) $25 \pi$;
(B) $50 \pi$;
(D) $100 \pi-50$.
Considcr a cyclic trapezium whose circumcentre is on one of the sides. If the ratio of the two parallel sides is $1: 4,$ what is the ratio of the sum of the two oblique sides to the longer parallel side?
(A) $\sqrt{3}: \sqrt{2}$;
(B) $3: 2$;
(C) $\sqrt{2}: 1$;
(D) $\sqrt{5}: \sqrt{3}$.
Consider the function $f(x)=\{\log _{e}\left(\frac{4+\sqrt{2 x}}{x}\right)\}^{2}$ for $x>0 .$ Then
(A) $f$ decreases upto some point and increases after that;
(B) $f$ increases upto some point and decreases after that;
(C) $f$ increases initially, then decreases and then again increases;
(D) $f$ decreases initially, then increases and then again decreases.
What is the number of ordered triplets $(a, b, c),$ where $a, b, c$ are positive integers (not necessarily distinct), such that $a b c=1000 ?$
Let $f:(0, \infty) \rightarrow(0, \infty)$ be a function differentiable at $3,$ and satisfying $f(3)=$ $3 f^{\prime}(3)>0 .$ Then the limit
\lim _{x \rightarrow \infty}\left(\frac{f\left(3+\frac{3}{x}\right)}{f(3)}\right)^{x}
(A) exists and is equal to $3$;
(B) exists and is equal to $e$;
(C) exists and is always equal to $f(3)$;
(D) need not always exist.
Let $z$ be a non-zero complex number such that $\left|z-\frac{1}{z}\right|=2$. What is the maximum value of $|z| ?$
(B) $\sqrt{2}$;
(D) $1+\sqrt{2}$.
The minimum value of
|\sin x+\cos x+\tan x+cosec x+\sec x+\cot x| \text { is }
(B) $2 \sqrt{2}-1$;
(C) $2 \sqrt{2}+1$;
For any function $f: X \rightarrow Y$ and any subset $A$ of $Y$, define
f^{-1}(A)={x \in X: f(x) \in A}
Let $A^{c}$ denote the complement of $A$ in $Y$. For subsets $A_{1}, A_{2}$ of $Y$, consider the following statements:
(i) $f^{-1}\left(A_{1}^{c} \cap A_{2}^{c}\right)=\left(f^{-1}\left(A_{1}\right)\right)^{c} \cup\left(f^{-1}\left(A_{2}\right)\right)^{c}$
(ii) If $f^{-1}\left(A_{1}\right)=f^{-1}\left(A_{2}\right)$ then $A_{1}=A_{2}$.
(A) both (i) and (ii) are always true;
(B) (i) is always true, but (ii) may not always be true;
(C) (ii) is always true, but (i) may not always be true;
(D) neither (i) nor (ii) is always true.
Let $f$ be a function such that $f^{\prime \prime}(x)$ exists, and $f^{\prime \prime}(x)>0$ for all $x \in[a, b] .$ For any point $c \in[a, b],$ let $A(c)$ denote the area of the region bounded by $y=f(x)$ the tangent to the graph of $f$ at $x=c$ and the lines $x=a$ and $x=b .$ Then
(A) $A(c)$ attains its minimum at $c=\frac{1}{2}(a+b)$ for any such $f$;
(B) $A(c)$ attains its maximum at $c=\frac{1}{2}(a+b)$ for any such $f$;
(C) $A(c)$ attains its minimum at both $c=a$ and $c=b$ for any such $f$;
(D) the points $c$ where $A(c)$ attains its minimum depend on $f$.
In $\triangle A B C,$ the lines $B P, B Q$ trisect $\angle A B C$ and the lines $C M, C N$ trisect $\angle A C B .$ Let $B P$ and $C M$ intersect at $X$ and $B Q$ and $C N$ intersect at $Y .$ If $\angle A B C=45^{\circ}$ and $\angle A C B=75^{\circ},$ then $\angle B X Y$ is
(A) $45^{\circ}$;
(B) $47 \frac{1^{\circ}}{2}$;
(C) $50^{\circ}$;
(D) $55^{\circ}$.
Some useful link
ISI B.stat & B.math 2015 problems & solutions
Our ISI & CMI Entrance program
Barycenter I.S.I 2014 Problem 2- watch & learn | CommonCrawl |
\begin{document}
\title{Parametric down-conversion from a wave-equations approach: geometry and absolute brightness.} \author{Morgan W. Mitchell} \affiliation{ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain} \date{26 September 2008}
\begin{abstract} Using the approach of coupled wave equations, we consider spontaneous parametric down-conversion (SPDC) in the narrow-band regime and its relationship to classical nonlinear processes such as sum-frequency generation. We find simple expressions in terms of mode overlap integrals for the absolute pair production rate into single spatial modes, and simple relationships between the efficiencies of the classical and quantum processes. The results, obtained with Green function techniques, are not specific to any geometry or nonlinear crystal. The theory is applied to both degenerate and non-degenerate SPDC. We also find a time-domain expression for the correlation function between filtered signal and idler fields.
\end{abstract}
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\section{Introduction}
Spontaneous parametric down-conversion (SPDC) has become a workhorse technique for generation of photon pairs and related states in quantum optics. Improvements in both nonlinear materials \cite{Fejer:1992} and down-conversion geometries have led to a steady growth in the brightness of these sources \cite{Kwiat:1995,Kwiat:1999,Giorgi:2003,Fiorentino:2004, Pelton:2004,Kuklewicz:2004,Fiorentino:2005,Wolfgramm:2008}.
Applications of the bright sources include fundamental tests of quantum mechanics, quantum communications, quantum information processing, and quantum metrology \cite{O'brien:2003,Groblacher:2007,Higgins:2007,Sauge:2007}.
Although down-conversion sources typically have bandwidths of order $10^{11}$ Hz, for the brightest sources even the output in a few-MHz window can be useful for experiments. This permits a new application, the interaction of down-conversion pairs with atoms, ions, or molecules. Indeed, sources for this purpose have been demonstrated \cite{Haase:2009}. Many modern applications use single-spatial-mode collection, either for improved spatial coherence, to take advantage of fiber-based technologies, or to separate the source and target for experimental convenience.
Remarkably, despite the importance of bright, single-spatial-mode sources, general methods for calculating the absolute brightness of such a source are not found in the literature. By absolute brightness, we mean the number of pairs per second that are collected, for specified beam shapes, pump power, filters and crystal characteristics. A number of calculations study the dependence of brightness on parameters such as beam widths or collection angles, but these typically give only relative brightness: the final results contain an unknown multiplicative constant \cite{Ljunggren:2005,Kurtsiefer:2001a}. While useful for optimizing a given source, they are less helpful when designing new sources. A recent paper computes the absolute brightness for a specific geometry: gaussian beams in the thin-crystal limit \cite{Ling:2008}.
In this paper, we calculate the absolute brightness for narrow-band, paraxial sources. The results are quite general, for example they apply equally well to crystals with spatial or temporal walk-off, for non-gaussian beams, etc. The Green-function approach we use is well suited to describing the temporal features of the down-conversion pairs, and we are able to predict the time correlations in a particularly simple way. To our knowledge, this method of deriving the time-correlations is also novel.
Perhaps of greatest practical importance, we derive very simple relationships between the efficiency of classical parametric processes and their corresponding quantum parametric processes. For example, in any given geometry the efficiency of sum-frequency generation and spontaneous parametric down-conversion are proportional. This allows the use of existing classical calculations and/or experiments with classical nonlinear optics to predict the brightness of quantum sources.
The paper is organized as follows: In section \ref{sec:Precedents}, we describe briefly the variety of theoretical treatments that have been applied to parametric down-conversion, and our reasons for making a new calculation. In Section \ref{sec:Formalism} we describe the formalism we use, based on an abstract paraxial wave equation and Green function solutions. In Section \ref{sec:Results} we calculate the absolute brightness and efficiencies for non-degenerate and degenerate parametric down-conversion and the corresponding classical processes. In section \ref{sec:Conclusions} we summarize the results.
\section{Background} \label{sec:Precedents}
The characteristics of parametric down-conversion light have been calculated in a number of different ways. Kleinman \cite{Kleinman:1968} used a Hamiltonian of the form \begin{equation} H' = -\frac{1}{3} \int d^3 x \, {\bf E} \cdot \chi : {\bf E} \end{equation} and the Fermi ``golden rule'' to derive emission rates as a function of frequency and angle. Zel'dovich and Klyshko \cite{Zeldovich:1969} proposed to use a mode expansion and calculate pair rates treating the quantum process as a classical parametric amplifier seeded by vacuum noise . Detailed treatment along these lines is given in \cite{Shen:1984,Klyshko:1988}. The problem of collection into defined spatial modes was not considered, indeed the works emphasize that the {\em total} rate of emission is {\em independent} of pump focusing.
After the observation of SPDC temporal correlations by Burnham and Weinberg \cite{Burnham:1970}, Mollow \cite{Mollow:1973} described detectable field correlation functions (coincidence distributions) in terms of source-current correlations and Green functions of the wave equation. This Heisenberg-picture calculation derived absolute brightness for multi-mode collection, e.g., for detectors of defined area at defined positions. It did not give brightness for single-mode collection, nor a connection to classical nonlinear processes. Hong and Mandel \cite{Hong:1985} used a mode-expansion to compute correlation functions based on the Heisenberg-picture evolution and an interaction Hamiltonian of the form \begin{equation} H_I = \frac{1}{2} \int d^3x \, \chi^{(2)}_{ijk} E_iE_jE_k.\end{equation} As with Mollow's calculation, they find singles and pair detection rates, but only for multi-mode detection \footnote{We note that Hamiltonian-based treatments often do not agree about the constant preceding the integral $\int d^3 x \chi^{(2)} E^3$, not even its sign. This question does not present a problem for the present calculation, which does not employ a Hamiltonian.} .
Ghosh, {\em et al.} \cite{Ghosh:1986} used the same Hamiltonian in a Schr\"{o}dinger-picture description, truncating the time evolution at first order to derive a ``two-photon wave-function." This last method has become the most popular description of SPDC, including work on efficient collection into single spatial modes\cite{Ljunggren:2005,Kurtsiefer:2001a}. Many works along these lines are cited in reference \cite{Ljunggren:2005}. Recently, Ling, et al.~\cite{Ling:2008} calculated the absolute emission rate based on a similar interaction Hamiltonian and a gaussian-beam mode expansion. In this way, they are able to calculate the absolute pair rate for non-degenerate SPDC in a uniform, thin crystal into gaussian collection modes. As described in Section \ref{Sec:Ling}, our calculation agrees with that of Ling et al. while also treating other crystal geometries, general beam shapes, and degenerate SPDC.
Notable differences among the calculations include Heisenberg {vs.} Schr\"{o}dinger picture and calculating in direct space { vs.} inverse space via a mode expansion. While they are of course equivalent, Heisenberg picture calculations are easier to compare to classical optics, while Schr\"{o}dinger picture calculations are more similar to the state representations in quantum information. As our goal is in part to connect classical and quantum efficiencies, we use the Heisenberg picture. Also, we note that the Schr\"{o}dinger-picture ``two-photon wave-function''
has a particular pathology: the first-order treatment of time evolution means the Schr\"{o}dinger picture state is not normalized and never contains more than two down-conversion photons. While this is not a problem for calculation of relative brightness or pair distributions\cite{Valencia:2007,Mosley:2008}, it does prevent calculation of absolute brightness. The choice of inverse { vs.} real space calculation is also one of convenience: for large angles in birefringent media or detection in momentum space, plane waves are the ``natural'' basis for the calculation. However, most bright sources use paraxial geometries and collection into defined spatial modes, e.g., the gaussian modes of optical fibers. In these situations, the advantages of a mode expansion disappear, while the local nature of the $\chi^{(2)}$ interaction makes real-space more ``natural.'' Thus we opt for a real-space calculation.
Our treatment of SPDC is based on coupled wave equations, a standard approach for multi-wave mixing in non-linear optics \cite{Boyd:2008}. The calculations are done in the Heisenberg picture, so that the evolution of the quantum fields is exactly parallel to that of the classical fields described by nonlinear optics. This allows the re-use of well-known classical calculations such as those by Boyd and Kleinman \cite{Boyd:1968}. As in the approach of Mollow, we use Green functions to describe the propagation, and find results that are not specific to any particular crystal or beam geometry. Unlike Mollow's calculation, we work with a paraxial wave equation (PWE). This allows us to simply relate the classical and quantum processes through momentum-reversal, which takes the form of complex conjugation in the PWE.
We focus on narrow-band parametric down-conversion, for which the results are particularly simple. By narrow-band, we mean that the bandwidths of the pump and of the collected light are much less than the bandwidth of the SPDC process, as set by the phase-matching conditions. This includes recent experiments with very narrow filters \cite{Haase:2009}, but also a common configuration in SPDC, in which the down-conversion bandwidth is $\sim 10$ nm while the filter bandwidths are $< 1$ nm.
\section{Formalism} \label{sec:Formalism}
\subsection{description of propagation}
We are interested in the envelopes ${\bf\cal E}_\pm$ for forward- and backward-directed of parts of the quantum field $E^{(+)}(t,{\bf x}) = ({\bf\cal E}_{+}\exp[+ i k z] + {\bf\cal E}_{-}\exp[- i k z ] )\exp[- i \omega t]$ where $k$ is the average wave-number and $\omega$ is the carrier frequency. These propagate according to a paraxial wave equation \begin{eqnarray} \label{Eq:DiffEq}{\cal D}_{\pm} {\bf\cal E}_{\pm} &=& {\cal S}_\pm,\end{eqnarray} where ${\cal D}_{\pm}$ is a differential operator and $ {\cal S}_\pm$ is a source term (later due to a $\chi^{(2)}$ non-linearity).
The formal (retarded) solution to equation (\ref{Eq:DiffEq}) is \begin{equation} {\bf\cal E}_{\pm}(x) = {\bf\cal E}_{0\pm}(x)+\int d^4 x' {\cal G}_\pm(x;x') {\cal S}_\pm(x') \end{equation} where $x$ is the four-vector $(t,{\bf x})$, ${\bf\cal E}_{0\pm}(x)$ is a solution to the source-free (${\cal S} = 0$) equation, and ${\cal G}_\pm$ are the time-forward Green functions, defined by \begin{eqnarray} {\cal D}_\pm {\cal G}_\pm(x;x') &=& \delta^4(x-x') \nonumber \\ {\cal G}_\pm(x;x') & = & 0 ~~~~~ t < t'. \end{eqnarray}
\newtext{For illustration, we consider the paraxial wave equation (PWE), for which
\begin{eqnarray} \label{Eq:PWE} {\cal D}_\pm &\equiv& \nabla_T^2 \pm 2ik(\partial_z \pm v_{g}^{-1} \partial_t ) \\ {\cal S}_\pm &=& \frac{\omega^2}{c^2 \varepsilon_0} {\cal P}^{(\rm
NL)}_\pm. \end{eqnarray}
Here $\nabla_T^2$ is the transverse Laplacian, $k = n(\omega) \omega /c$ is the wave-number, $v_g \equiv \partial \omega / \partial k_z$ is the group velocity, and $\cal P^{(\rm NL)}$ is the envelope for the nonlinear polarization.}
We note that ${\cal D}_\pm$ is invariant under translations of $x$, and that time reversal $t\rightarrow -t$ is equivalent to direction-reversal and complex conjugation, i.e., ${\cal D}_\pm \rightarrow {\cal D}_\mp^*$ .
The results we obtain will be valid for any equation obeying these symmetries. In particular, the results will also apply to propagation with dispersion and/or spatial walk-off, which can be included by adding other time and/or spatial derivatives to ${\cal D}$.
From the symmetries of ${\cal D}_\pm$, it follows that the Green functions depend only on the difference $x-x'$, and that ${\cal G}_+(t,{\bf x};t',{\bf x}') = {\cal G}^*_-(t,{\bf x}',t',{\bf x})$. Also, the time-backward (or ``advanced'') Green functions ${\cal H}_\pm$, defined by \begin{eqnarray} {\cal D}_\pm {\cal H}_\pm(x;x') &=& \delta^4(x-x') \nonumber \\ {\cal H}_\pm(x;x') & = & 0 ~~~~~ t > t'\end{eqnarray} obey ${\cal H}_\pm(x;x') = {\cal G}_\pm^*(x',x)$.
\subsection{boundary and initial value problems}
If the value of the field is known on a plane $z=z_{\rm src}$, the field downstream of that plane is \begin{equation} \label{Eq:BoundaryValueProblem} {\bf\cal E}_\pm(x) = {\beta}_z \int d^4x' {\cal G}_\pm(x;x') {\bf\cal E}_{\pm}(x') \delta(z'-z_{\rm src})\end{equation} where ${\beta}_z \equiv \pm 2ik$. Similarly, if the field is known at an initial time $t=t_0$, the field later is \begin{equation}\label{Eq:InitialValueProblem} {\bf\cal E}_\pm(x) = {\beta}_t \int d^4x' {\cal G}_\pm(x;x'){\bf\cal E}_\pm(x')\delta(t'-t_0)\end{equation} where ${\beta}_t = 2ik/v_g$. Similar relationships hold for the advanced Green functions. If the field is known in some plane $z=z_0$ downstream, then \begin{eqnarray} \label{Eq:BVProblem2} {\bf\cal E}_{\pm}(x) &=& {\beta}_{z}^* \int d^4 x'\, {\cal H}_\pm(x;x'){\bf\cal E}_{\pm}(x')\delta(z'-z_0)
\nonumber \\ & = & {\beta}_{z}^* \int d^4 x'\, {\bf\cal E}_{\pm}(x')\delta(z'-z_0){\cal G}^*_\pm(x';x) \end{eqnarray} while if the field is known at some time $t_f$ in the future, \begin{eqnarray} \label{Eq:BackFromTheFuture} {\bf\cal E}_{\pm}(x) &=& {\beta}_{t}^* \int d^4 x'\, {\cal H}_\pm(x;x'){\bf\cal E}_{\pm}(x')\delta(t'-t_f)
\nonumber \\ & = & {\beta}_{t}^* \int d^4 x'\, {\bf\cal E}_{\pm}(x')\delta(t'-t_f){\cal G}^*_\pm(x';x) \end{eqnarray}
\subsection{quantization}
The field envelopes are operators which obey the equal-time commutation relation \begin{equation} \label{Eq:EqualTimeCommutator} [{\bf\cal E}({\bf x},t),{\bf\cal E}^\dagger({\bf x}',t)] = A_{\gamma}^2 \delta^3({\bf x}'-{\bf x})\end{equation} where $A_{\gamma} \equiv \sqrt{\hbar \omega /2 n n_g \varepsilon_0}$ is a photon units scaling factor and $n_g \equiv c/v_g$ is the group index. For narrow-band fields, $A_{\gamma}^{-2}\expect{{\bf\cal E}^\dagger{\bf\cal E}} $ describes a photon number density, and $v_g A_{\gamma}^{-2}\expect{{\bf\cal E}^\dagger{\bf\cal E}}$ and $v_{gs}v_{gi} A_{\gamma i}^{-2}A_{\gamma s}^{-2} \expect{{\bf\cal E}_s^\dagger{\bf\cal E}_i^\dagger{\bf\cal E}_i{\bf\cal E}_s}$ describe single and pair fluxes. We find the unequal-time commutation relation from equation (\ref{Eq:InitialValueProblem}) \begin{equation}\left. [{\bf\cal E}(x),{\bf\cal E}^\dagger(x')]\right._{t>t'} = {\beta}_t A_{\gamma}^2 {\cal G}(x;x')\end{equation} so that $\bra{0}{{\bf\cal E}(x){\bf\cal E}^\dagger(x')}\ket{0} = {\beta}_t A_{\gamma}^2 {\cal G}(x;x')$ for $t>t'$. For the PWE, $A_{\gamma}^{-2} v_g = 2 n c \varepsilon_0/\hbar \omega$ and ${\beta}_t A_{\gamma}^2 = i \hbar \omega^2 /c^2\varepsilon_0$.
To calculate singles rates, we will need to evaluate expressions of the form $\expect{{\bf\cal E}{\bf\cal E}^\dagger}$. For this, a useful expression is derived in the Appendix: Equation (\ref{Eq:AlternatePropagator})
\begin{eqnarray} \expect{{\bf\cal E}_{}(x){\bf\cal E}_{}^\dagger(x') }&=&
\frac{2\hbar n \omega^3 }{c^3 \varepsilon_0} \int d^4 x'' \delta(z''-z_0) \nonumber \\ & & \times {\cal G}^*(x'';x) {\cal G}(x'';x'). \end{eqnarray} Here $z_0$ is any plane down-stream of $x$ and $x'$.
\subsection{single spatial modes}
A single spatial mode $M_\pm({\bf x})$ is a time-independent solution to the source-free wave equation ${\cal D}_\pm M_\pm({\bf x})=0$. $M_\pm^*({\bf x})$ is the corresponding momentum-reversed solution ${\cal D}_\mp M_\pm^*({\bf x})=0$. We assume the normalization $\int d^3x
|M_\pm({\bf x})|^2 \delta(z) = 1$. For single-mode collection, it will be convenient to define the projection of a field ${\cal E}(x)$ onto the mode $M$ as \begin{equation} {\cal E}_M(t) \equiv \int d^3x M^*({\bf x})\delta(z-z_0) {\cal E}(x)\end{equation} (here and below, the $+/-$ propagation direction is the same for ${\cal E},M$). Here $z_0$ is some plane of interest, and ${\cal E}_M(t)$ describes the magnitude of the field component in this plane. Similarly, if the envelope is constant, the field distribution is \begin{equation} {\cal E}(x) = {\cal E}_M(t) M({\bf x})
.\end{equation} The optical power is (MKS units) $P_M(t) = 2 n c\varepsilon_0 \int d^3x |{\bf\cal E}(t,x)|^2\delta(z-z_0) = 2 n c\varepsilon_0 |{\bf\cal E}_M(t)|^2$.
Given an upstream source ${\cal S}(x)$, the $M$ component of the generated field is \begin{eqnarray} {\cal E}_M(t) &=& \int d^3x d^4x' M^*({\bf x})\delta(z-z_0) \nonumber \\ & & \times {\cal G}(x;x') {\cal S}(x').\end{eqnarray}
If the source is time-independent, then Equation (\ref{Eq:BVProblem2}) and the time-translation symmetry of ${\cal G}$ imply \begin{equation} {\cal E}_M(t) = \frac{1}{{\beta}_z} \int d^3x' M^*({\bf x}') {\cal S}(x').\end{equation}
Similarly, if a product ${\cal E}_1(x_1){\cal E}_2(x_2)$ is given by a constant pair source ${\cal S}^{(2)}(x)$ as \begin{eqnarray} {\cal E}_1(x_1){\cal E}_2(x_2) &=& \int d^4x' {\cal G}_1(x_1;x'){\cal G}_2(x_2;x') \nonumber \\ & & \times {\cal S}^{(2)}(x').\end{eqnarray} then the time-integrated mode-projected component is \begin{eqnarray} \int dt_1 {\cal E}_{1M_1}(t_1){\cal E}_{2M_2}(t_2) &=& \frac{1}{{\beta}_{1z}{\beta}_{2z}} \int d^3x' M_1^*({\bf x}')\nonumber \\ & & \times M_2^*({\bf x}') {\cal S}^{(2)}(x').\end{eqnarray}
\subsection{Coupled wave equations}
We now introduce a $\chi^{(2)}$ nonlinearity, which produces a nonlinear polarization that appears as a source term in the propagation equations. We consider three fields, ``signal,'' ``idler'' and ``pump'' with carrier frequencies $\omega_s, \omega_i,\omega_p$ and wave-numbers $k_s,k_i,k_p$, respectively. The respective field envelopes ${\bf\cal E}_s,{\bf\cal E}_i,{\bf\cal E}_p$ evolve according to \begin{eqnarray} {\cal D}_p {\bf\cal E}_p &=& \omega_p^2 g {\bf\cal E}_s {\bf\cal E}_i \exp[i \Delta k z]\nonumber \\ {\cal D}_s {\bf\cal E}_s &=& \omega_s^2 g {\bf\cal E}_p {\bf\cal E}_i^\dagger \exp[-i \Delta k z] \nonumber \\ {\cal D}_i {\bf\cal E}_i &=& \omega_i^2 g {\bf\cal E}_p {\bf\cal E}_s^\dagger \exp[-i \Delta k z]\end{eqnarray} where $g = -4m({\bf x})d /c^2 $, $d$ is the effective nonlinearity, equal to half the relevant projection of $\chi^{(2)}$, and $\Delta k \equiv k_p-k_s-k_i$ is the wave-number mismatch. The dimensionless function $m({\bf x})$ describes the distribution of $\chi^{(2)}$. For example in a periodically-poled material it alternates between $\pm 1$. We can take $\Delta k=0$ without loss of generality, as the phase oscillation can be incorporated directly in the envelopes. The propagation directions ($\pm$) will be omitted unless needed for clarity. Note that for transparent materials $\chi^{(2)}$ is real, and $\chi^{(2)}(\omega_p;\omega_s+\omega_i)= \chi^{(2)}(\omega_s;\omega_p-\omega_i)= \chi^{(2)}(\omega_i;\omega_p-\omega_s)$.
First-order perturbation theory is sufficient to describe situations in which pairs are produced. For example, if ${\bf\cal E}_{0s},{\bf\cal E}_{0i},{\bf\cal E}_{0p}$ are source-free solutions,
then \begin{eqnarray} \label{Eq:EsToFirstOrder} {\bf\cal E}_s & = & {\bf\cal E}_{0s} + \omega_s^2 \int d^4x' {\cal G}_{s}(x;x') \nonumber \\ & & \times g(x') {\bf\cal E}_{0p}(x'){\bf\cal E}_{0i}^\dagger(x') + O(g^2).\end{eqnarray}
and similar expressions for ${\cal E}_i,{\cal E}_p$ are sufficient to give the lowest-order contribution to the pair-detection rate $W^{(2)}\propto \expect{{\bf\cal E}^\dagger_{s} {\bf\cal E}^\dagger_{i} {\bf\cal E}_{i}{\bf\cal E}_{s}}$. Higher-order expansions would be necessary for double-pair production, etc.
\subsection{narrow-band frequency filters}
In most down-conversion experiments, some sort of frequency filter is used.
Assuming this filter is linear and stationary, the field reaching the detector is \begin{equation} {\bf\cal E}^{(F)}(t) = \int dt' F(t-t'){\bf\cal E}(t') + G(t-t'){\bf\cal E}_{\rm res}(t').\end{equation} Here ${\bf\cal E}_{\rm res}$ is a reservoir field required to maintain the field commutation relations. Assuming the reservoir is in the vacuum state, it will not produce detections and can be ignored. Defining
${H}_F(t_i,t_s) \equiv \expect{ {\bf\cal E}^{(F_i)}_i(t_i) {\bf\cal E}^{(F_s)}_s(t_s) },$ the fields that leave the filter obey \begin{eqnarray} {H}_F(t_i,t_s) &=& \int dt' dt'' F_i(t_i-t') F_s(t_s-t'') \nonumber \\ & & \times \expect{ {\bf\cal E}_i(t') {\bf\cal E}_s(t'') }.\end{eqnarray} In the narrowband case, i.e., when the correlation time between signal and idler is much less than the time-scale of the impulse response functions, we can take $\expect{ {\bf\cal E}_i(t') {\bf\cal E}_s(t'') } \approx {\cal A} \delta(t'-t'')$ where the constant ${\cal A} \equiv \int dt_i \expect{{\bf\cal E}_i(t_i){\bf\cal E}_s(t_s)}$. We find \begin{eqnarray} {H}_F(t_i,t_s) &\approx& {\cal A}\int dt' F_i(t_i-t') F_s(t_s-t') \nonumber \\ & \equiv & {\cal A} f(t_s-t_i) .\end{eqnarray} With this, we see that the flux of pairs is \begin{eqnarray} W^{(2)}(t_s-t_i) &=&
\frac{4 n_s n_i c^2 \varepsilon_0^2}{\hbar^2 \omega_s \omega_i }
|{\cal A}f(t_s-t_i)|^2 \end{eqnarray} with a total coincidence rate of \begin{eqnarray} W^{(2)} &=& \int dt_i W^{(2)}(t_s-t_i) \nonumber \\ & = & \frac{n_s n_i c^2
\varepsilon_0^2}{\hbar^2 \omega_s \omega_i } |{\cal A}|^2 \int dt_i |2 f(t_s-t_i)|^2 \nonumber \\ & \equiv & \frac{n_s n_i c^2
\varepsilon_0^2}{\hbar^2 \omega_s \omega_i } |{\cal A}|^2 \Gamma_{\rm eff}.\end{eqnarray} We note that \begin{equation} \label{Eq:GammaArea}
\Gamma_{\rm eff} = \frac{2}{\pi} \int d\Omega T_s(\Omega)T_i(-\Omega)\end{equation} where $T_{s,i}(\Omega) \equiv |\int dt
\exp[i \Omega t] F_{s,i}(t)|^2$ are the signal and idler filter transmission spectra, respectively. For this reason we refer to $\Gamma_{\rm eff}$ as the effective line-width (in angular frequency) for the combined filters. Also important will be the singles rate \begin{eqnarray} \label{Eq:SinglesRateGeneral}W^{(1)} &=& A_{\gamma s}^{-2}v_{gs} \expect{[{\bf\cal E}_s^{(F_S)}(t_s)]^\dagger{\bf\cal E}_s^{(F_S)}(t_s)}\nonumber \\ & = & \frac{2 n_s c \varepsilon_0 }{\hbar \omega_s} \int dt' dt'' F_s^*(t_s-t') F_s(t_s-t'') \nonumber \\ & & \times \expect{ {\bf\cal E}_s^\dagger(t') {\bf\cal E}_s(t'')} \nonumber \\ & \approx & \frac{2 n_s c \varepsilon_0 }{\hbar
\omega_s} {\cal C} \int dt' |F_s(t_s-t')|^2 \nonumber \\ & \equiv & \frac{n_s c \varepsilon_0 }{2 \hbar \omega_s} {\cal C} \Gamma_{{\rm eff},s} \end{eqnarray} where ${\cal C} \equiv \int dt' \expect{ {\bf\cal E}_s^\dagger(t') {\bf\cal E}_s(t'')}$. $\Gamma_{{\rm eff},s}$ is the effective line-width for the signal filter.
\section{Results} \label{sec:Results}
With the calculational tools described above, we now demonstrate the central results of this paper. We first express the efficiency of continuous-wave sum-frequency generation (SFG) in terms of a mode-overlap integral. This effectively reduces the non-linear optical problem to three uncoupled propagation problems. We then show that the efficiency of parametric down-conversion in the same medium is proportional to the SFG efficiency, for modes with the same shapes but opposite propagation direction. The constant of proportionality is found, allowing calculations of absolute efficiency based either on material properties such as $\chi^{(2)}$ or measured SHG efficiencies. Similarly, the singles production efficiency is related to difference-frequency generation (DFG) and the collection efficiency is calculated. The same quantities for the degenerate case are also found.
\subsection{sum-frequency generation}
We consider first the process of SFG, for un-depleted signal and idler and no input pump. Signal and idler are constant and come from single-modes, \begin{eqnarray} \label{Eq:SingleModeSFGBrightness} {\bf\cal E}_{M_p}(t_p) & = & -{\bf\cal E}_{M_i}{\bf\cal E}_{M_s} \frac{4\omega_p^2 d }{c^2{\beta}_{z,p}} \nonumber \\ & & \times \int d^3x' M_p^*(x')m(x') M_i(x')M_s(x') \nonumber \\ & \equiv & -{\bf\cal E}_{M_i}{\bf\cal E}_{M_s}\frac{4\omega_p^2 d}{c^2{\beta}_{z,p}} I_{SFG}.\end{eqnarray}
The conversion efficiency is \begin{eqnarray} \label{Eq:SingleModeSFGEff} {Q}_{SFG}^{} &\equiv& \frac{P_{M_p}}{P_{M_s}P_{M_i}} =
\frac{8\omega_p^4 n_p d^2|I_{SFG}|^2}{ n_s n_i c^5 \varepsilon_0
|{\beta}_{z,p}|^2} \nonumber \\ & = & \frac{2 \omega_p^2 d^2}{c^3 \varepsilon_0 n_p n_s n_i }|I_{SFG}|^2\end{eqnarray}
The efficiency of a cw, single-mode source is thus proportional to the spatial overlap of the pump, signal, and idler modes, weighted by the nonlinear coupling $g$.
\subsection{non-degenerate parametric down-conversion}
Next we consider the process of parametric down-conversion.
Using Equation (\ref{Eq:EsToFirstOrder}), we can calculate to first order in $g$ the correlation function \begin{eqnarray} \label{Eq:NDPDCCorrFun} \expect{{\bf\cal E}_{i}(x_i){\bf\cal E}_{s}(x_s)} &=& \omega_s^2 \int d^4 x' {\cal G}_s(x_s,x') \nonumber \\ & & \times\expect{{\bf\cal E}_{0,i}(x_i){\bf\cal E}^\dagger_{0,i}(x')}\nonumber \\ & & \times g(x'){\bf\cal E}_{0,p}(x') \nonumber \\ & = & i \frac{\hbar \omega_i^2 \omega_s^2}{c^2\varepsilon_0} \int d^4 x' {\cal G}_s(x_s,x')\nonumber \\ & & \times{\cal G}_i(x_i,x') g(x'){\bf\cal E}_{0,p}(x') \end{eqnarray} For constant pump and single-mode collection we have \begin{eqnarray} \label{Eq:SingleModeSPDCA} {\cal A}_{M_i M_s} &\equiv& \int dt_s \expect{{\bf\cal E}_{M_i}(t_i){\bf\cal E}_{M_s}(t_s)} \nonumber \\ &=& \frac{i \hbar\omega_s \omega_i d }{c^2\varepsilon_0 n_s n_i}{\bf\cal E}_{M_p} \nonumber \\ & & \times \int d^3x' M_s^*(x') M_i^*(x') m(x') M_p(x') \nonumber \\ & \equiv & \frac{i \hbar\omega_s \omega_i d }{ c^2\varepsilon_0 n_s n_i}{\bf\cal E}_{M_p} I_{DC} . \end{eqnarray}
We note that $I_{DC} = I_{SFG}^*$. Also, the conjugate modes describe backward-propagating fields, as if the source fields were sent through the nonlinear medium in the opposite direction. Thus if we want to know the brightness of down-conversion when all beams are propagating to the left, it is sufficient to calculate (or measure) the efficiency of up-conversion when all beams are propagating to the right. Using equations (\ref{Eq:SingleModeSFGEff}) and (\ref{Eq:SingleModeSPDCA}) we find
\begin{eqnarray} \label{Eq:SingleModeSPDCBrightness} \left|{\cal A}_{M_i M_s}\right|^2 &=& \frac{\hbar^2 \omega_i^2\omega_s^2}{4 c^2 \varepsilon_0^2 n_s n_i \omega_p^2 }P_p {Q}_{SFG}^{} . \end{eqnarray}
\subsection{brightness}
We can now consider the brightness of the filtered, single-mode source. The rate of detection of pairs is \begin{eqnarray} \label{Eq:PDCPairRate}W^{(2)} &=& \frac{n_s n_i c^2
\varepsilon_0^2}{\hbar^2 \omega_s \omega_i } |{\cal A}|^2 \Gamma_{\rm eff} \nonumber \\ & = & \Gamma_{\rm eff} \frac{\omega_i\omega_s}{4\omega_p^2 }P_p {Q}_{SFG}^{} \end{eqnarray}
This simple expression is the first main result: The rate of pairs is simply the joint collection bandwidth $\Gamma_{\rm eff}$, times the ratio of frequencies, times the pump power, times the up-conversion efficiency ${Q}_{SFG}^{}$. Note that the last quantity can be calculated if the mode shapes and $\chi^{(2)}({\bf x})$ are known, for example in the paper of Boyd and Kleinman, or simulated for more complicated situations. Most importantly, it is directly measurable.
\subsection{difference-frequency generation}
We now consider the classical situation in which pump and signal beam are injected into the crystal and idler is generated. We will see that this directly measurable process is related to the singles generation rate by parametric down-conversion. The generated idler is \begin{eqnarray} {\bf\cal E}_i(x) & = & \omega_i^2 \int d^4x' {\cal G}_{i}(x;x') g(x') {\bf\cal E}_{0p}(x'){\bf\cal E}_{0s}^*(x').\nonumber \end{eqnarray} If pump and signal are from modes $M_P,M_S$, respectively, we find \begin{eqnarray} {\bf\cal E}_i(x) & = & -\frac{4\omega_i^2 d}{c^2} {\bf\cal E}_{M_p}(t_p){\bf\cal E}_{M_s}^*(t_s) \int d^4x' {\cal G}_{i}(x;x') \nonumber \\ & & \times m(x') M_{p}(x')M_s^*(x')
.\end{eqnarray} The total power generated is $P_{i} = 2 c n_i \varepsilon_0
\int d^3 x_i \delta(z_i-z_0) |{\bf\cal E}_i(x_i)|^2$ where $z_0$ indicates a plane downstream of the generation. We find \begin{eqnarray} P_{i} & = & \frac{2\omega_i^2 d^2 }{c^3 \varepsilon_0 n_s n_i n_p } {P_p}{P_s} \int d^3 x_i \delta(z_i-z_0) \nonumber \\ & & \times
\left|{\beta}_{z,i}^{} \int d^4x' {\cal G}_{i}(x_i;x') m(x')
M_{p}(x')M_s^*(x') \right|^2 \nonumber \\ & \equiv & {P_p}{P_s} \frac{ 2\omega_i^2 d^2}{c^3 \varepsilon_0 n_s n_i n_p }
|I_{DFG}^{(s)}|^2 \nonumber \\ & \equiv & {P_p}{P_s} Q_{DFG}
. \label{Eq:PDFG}\end{eqnarray}
\subsection{singles rates in PDC}
We can find the rate of detection of singles in the mode $M_S$ by equation (\ref{Eq:SinglesRateGeneral}) and using Equation (\ref{Eq:AlternatePropagator}) \begin{eqnarray} {\cal C} &=& \int dt_s \expect{{\bf\cal E}_{M_S}^\dagger(x_s){\bf\cal E}_{M_S}(x_s')} \nonumber \\ & = & \int dt_s d^3x_s d^3x_s' M_{s}({\bf x}_s) M_{s}^*({\bf x}_s') \nonumber \\ & & \times \delta(z_s-z_0)\delta(z_s'-z_0) \expect{{\bf\cal E}_{s}^\dagger(x_s){\bf\cal E}_{s}(x_s') }\nonumber \\ & = &
\frac{|{\bf\cal E}_{Mp}|^2\omega_s^4}{|{\beta}_{z,s}|^2}\int d^3x d^3x' M_{s}({\bf x}) g({\bf x}) M_{p}^*({\bf x}) \nonumber \\ & & \times \expect{{\bf\cal E}_{0i}(x){\bf\cal E}_{0i}^\dagger(x') }M_{s}^*({\bf x}') g({\bf x}')
M_{p}({\bf x}')\nonumber \\ & = & \frac{2\hbar \omega_i \omega_s^2 d^2}{c^3 n_s^2 n_i \varepsilon_0}{|{\bf\cal E}_{Mp}|^2} \int d^4 x'' \delta(z''-z_0)
\nonumber \\ & & \times \left|{\beta}_{i,z} \int d^3x G_i(x'';x) M_{s}^*({\bf x}) m({\bf x})
M_{p}({\bf x})\right|^2\end{eqnarray}
so that
\begin{eqnarray} \label{Eq:SinglesRateSignal}W^{(1)} &=& \frac{ c \omega_i
\omega_s}{32 n_p n_s n_i \varepsilon_0} {P_p} |I_{DFG}^{(s)}|^2 \Gamma_{{\rm eff},s} \nonumber \\ & = & \frac{\omega_s}{4 \omega_i}\Gamma_{{\rm eff},s} {P_p} Q_{DFG}^{(s)} \end{eqnarray}
\subsection{conditional efficiency}
The conditional efficiency for the idler (probability of collecting the idler, given that the signal was collected) is \begin{eqnarray} \label{Eq:PDCSignalEfficiency} \eta_s &\equiv&
\frac{W^{(2)}}{{W_s^{(1)}}} = \frac{\Gamma_{\rm eff}}{\Gamma_{s}}\frac{|I_{SFG}|^2}{|I_{DFG}^{(s)}|^2} \end{eqnarray}
\subsection{degenerate processes}
Up to this point, we have discussed only non-degenerate processes, i.e., those in which the signal and idler fields are distinct and do not interfere. This is always the case for type-II down-conversion, and will be the case for type-I down-conversion if the frequencies and/or directions of propagation are significantly different. We now consider degenerate processes, in which there is only one down-converted field (signal).
The above discussion is modified only slightly. The signal and pump evolve by \begin{eqnarray} {\cal D}_p {\bf\cal E}_p &=& \frac{1}{2}\omega_p^2 g {\bf\cal E}_s {\bf\cal E}_s \nonumber \\ {\cal D}_s {\bf\cal E}_s &=& \omega_s^2 g {\bf\cal E}_p {\bf\cal E}_s^\dagger.\end{eqnarray}
\subsection{second harmonic generation}
The calculation of second-harmonic generation (SHG) proceeds exactly as in sum-frequency generation, except for the factor of one half and with all ``idler'' variables replaced by ``signal'' variables. Thus we find \begin{eqnarray} P_p = P^2_s {Q}_{SHG} \end{eqnarray} where \begin{eqnarray} \label{Eq:SHGEfficiency} {Q}_{SHG} &=& \frac{ \omega_p^2 d^2
}{2 c^3\varepsilon_0 n_p n_s^2 }|I_{SHG}|^2\end{eqnarray} and \begin{eqnarray} \label{Eq:SHGOverlap} I_{SHG} & \equiv & \int d^3x M_p^*({\bf x})m({\bf x}) M_s({\bf x})M_s({\bf x}) .\end{eqnarray}
\subsection{average parametric gain}
The other classical process of interest is parametric amplification of the signal by the pump. The first-order solution for the signal field is \begin{eqnarray} {\bf\cal E}_s & = & {\bf\cal E}_{0s} + \omega_s^2 \int d^4x' {\cal G}_{s}(x;x') \nonumber \\ & & \times g(x') {\bf\cal E}_{0p}(x'){\bf\cal E}_{0s}^\dagger(x')\nonumber \\ & \equiv & {\bf\cal E}_{0s} + {\bf\cal E}_{1s} . \end{eqnarray} The signal power at the output is \begin{eqnarray} P_s &=& 2 n_s c
\varepsilon_0 \int d^3 x_s \delta(z_s-z_0)|{\bf\cal E}_s(x_s)|^2 \nonumber \\ & = & 2 n_s c \varepsilon_0 \int d^3 x_s \delta(z_s-z_0) \left(
|{\bf\cal E}_{0,s}(x_s)|^2\right. \nonumber \\ & & + 2 Re[{\bf\cal E}_{0,s}(x_s){\bf\cal E}_{1,s}^*(x_s)] + \left.
|{\bf\cal E}_{1,s}(x_s)|^2\right).\end{eqnarray} The first term is the input signal power $P_{0,s}$, the second term depends on the relative phase $\phi_p - 2 \phi_s$, and the last term is the phase-independent contribution to the gain, an experimentally accessible quantity. We have \begin{eqnarray} \overline{\delta P} &\equiv&\expect{P_s - P_{0,s}}_{\phi_s} \nonumber \\ & = & 2 n_s c \varepsilon_0
\int \int d^3 x_s \delta(z_s-z_0) |{\bf\cal E}_{1,s}(x_s)|^2 \nonumber \\ & = & \frac{8
\omega_s^2 \varepsilon_0 d^2}{cn_s} |{\bf\cal E}_{0s}|^2 |{\bf\cal E}_{p}|^2
\int d^4 x_s \delta(z_s-z_0) \nonumber \\ & & \times \left|{\beta}_{z,s} \int d^3x' {\cal G}_{s}(x_s;x') \right. \nonumber \\ & & \times \left. m({\bf x}')M_p({\bf x}')M_s^*({\bf x}')
\right|^2 \nonumber \\ & \equiv & P_{0s} P_{p}\frac{2 \omega_s^2 d^2}{c^3 n_s^2 n_p \varepsilon_0} |I_{APG}|^2 \nonumber \\ & \equiv & P_{0s} P_{p} Q_{APG}.\end{eqnarray}
\subsection{degenerate PDC}
Next we consider the process of degenerate parametric down-conversion, for which \begin{eqnarray} {\bf\cal E}_s & = & {\bf\cal E}_{0s} + \omega_s^2 \int d^4x' {\cal G}_{s}(x;x') \nonumber \\ & & \times g(x') {\bf\cal E}_{0p}(x'){\bf\cal E}_{0s}^\dagger(x'). \end{eqnarray} We find the correlation function \begin{eqnarray} \expect{{\bf\cal E}_{s}(x_s){\bf\cal E}_{s}(x_s')} &=& \omega_s^2 \int d^4 x'' {\cal G}_s(x_s,x'') \nonumber \\ & & \times\expect{{\bf\cal E}_{0,s}(x_s'){\bf\cal E}^\dagger_{0,s}(x'')} g(x'')\nonumber \\ & & \times {\bf\cal E}_{0,p}(x'')\end{eqnarray} at which point it is clear that the only difference from the non-degenerate case of Eq. (\ref{Eq:NDPDCCorrFun})will be the replacement of idler variables with signal variables. We find \begin{eqnarray} W^{(2)} &=& \label{Eq:DegPDCPairRate} \Gamma_{\rm eff} \frac{\omega_s^2}{4\omega_p^2 }P_p {Q}_{SHG} = \frac{\Gamma_{\rm eff}}{16}P_p {Q}_{SHG}.\end{eqnarray}
\subsection{Singles rates (degenerate)}
As before, we can find the rate of detection of singles in the mode $M_S$ by equation (\ref{Eq:SinglesRateGeneral}) and \begin{eqnarray} {\cal C} &=& \int dt_s' \expect{ {\bf\cal E}_{M_S}^\dagger(t_s') {\bf\cal E}_{M_S}(t_s'')} \nonumber \\ & = & \int dt_s' d^3x_s' d^3x_s' M_{s}({\bf x}_s') M_{s}^*({\bf x}_s'') \nonumber \\ & & \times \delta(z_s'-z_0)\delta(z_s''-z_0) \expect{{\bf\cal E}_{s}^\dagger(x_s'){\bf\cal E}_{s}(x_s'') }\nonumber \\ & = &
\frac{|{\bf\cal E}_{Mp}|^2\omega_s^4}{|{\beta}_{z,s}|^2}\int d^3x' d^3x'' M_{s}({\bf x}') g({\bf x}') M_{p}^*({\bf x}') \nonumber \\ & & \times \expect{{\bf\cal E}_{0i}(x'){\bf\cal E}_{0i}^\dagger(x'') }M_{s}^*({\bf x}'')
g({\bf x}'') M_{p}({\bf x}'') \nonumber \\ & = & \frac{2 \hbar \omega_s^3 d^2}{ c^3 n_s^3 \varepsilon_0}{|{\bf\cal E}_{Mp}|^2} \int d^4 x'' \delta(z''-z_0)
\nonumber \\ & & \times \left|{\beta}_{s,z} \int d^3x G_s(x'';x) M_{s}^*({\bf x}) m({\bf x})
M_{p}({\bf x})\right|^2 .\end{eqnarray}
The singles rate is thus \begin{eqnarray} \label{Eq:DegSinglesRateSignal}W^{(1)}_s &=&
\frac{1}{4}\Gamma_{{\rm eff},s} {P_p} Q_{APG}^{(s)}.\end{eqnarray}
\subsection{Conditional efficiency (degenerate)}
The conditional efficiency is \begin{eqnarray} \label{Eq:DegPDCSignalEfficiency} \eta_s &\equiv& \frac{W^{(2)}}{{W_s^{(1)}}} =
\frac{\Gamma_{\rm eff}}{\Gamma_{{\rm eff},s}}\frac{|I_{SHG}|^2}{|I_{APG}^{(s)}|^2} \end{eqnarray} \newline
\section{Example calculations}
\newcommand{\Upsilon}{\Upsilon}
We now illustrate the preceding, general results with a few special cases. We first calculate the overlap integral for co-propagating gaussian beams. This allows us to 1) compare our results to the classical results of Boyd and Kleinman \cite{Boyd:1968}, 2) predict absolute brightness for an important geometry, type-II co-linear down-conversion in quasi-phase-matched material. Also, we compare to a recent calculation of absolute brightness for a specific geometry by Ling, et al. \cite{Ling:2008}.
We consider collinear, frequency-degenerate type-II PDC with circular gaussian beams for signal, idler and pump. We take mode shape functions \begin{equation} M_{m}({\bf x}) = \sqrt{\frac{k_m z_R}{\pi}}\frac{1}{q} e^{i k_m z} e^{i k_m \frac{r^2}{2 q}}.\end{equation} where $m \in \{s,i,p\}$, $r$ is the radial component of ${\bf x}$, and $q \equiv z-i z_R$ where $z_R$ is the Rayleigh range, assumed equal for all beams. We assume a periodically-poled material in which $\chi^{(2)}(z)$ alternates with period $2 \pi/Q$ so and we approximate $m({\bf x}) \approx \exp[i Q z]d_{\rm eff}/d $. From Equation (\ref{Eq:SingleModeSFGBrightness}) we find \begin{eqnarray} I_{SFG} &=& \sqrt{\frac{k_p k_s k_i z_R^3}{\pi^3}} \int_{-L/2}^{L/2} dz'
\frac{e^{-i \Delta k z'}}{q|q|^2 } \nonumber \\ & & \times \int 2 \pi r' dr' e^{-i \left( \frac{k_p}{q^*} - \frac{k_s+k_i}{q}\right) \frac{r'^2}{2} } \nonumber \\ & = & \frac{2 i}{k_+}\sqrt{\frac{k_p k_s k_i z_R^3}{\pi}} \nonumber \\ & & \times \int_{-L/2}^{L/2} dz' \frac{e^{-i \Delta k z'}}{(z'-iz_R)\left(R_k z'+iz_R\right)} \end{eqnarray} where $\Delta k \equiv k_p - k_s - k_i - Q$ and $R_k \equiv k_-/k_+$ and $k_\pm \equiv k_p \pm (k_s + k_i)$. In terms of the dimensionless variables $\kappa \equiv \Delta k L$, $\zeta \equiv z'/L$, $\zeta_R \equiv z_R/L$ we find \begin{eqnarray} I_{SFG} &=& \frac{2i}{k_+} \sqrt{\pi k_p k_s k_i z_R } \Upsilon
\end{eqnarray} where \begin{eqnarray} \Upsilon &\equiv & \frac{\zeta_R}{2 \pi}\int_{-1/2}^{1/2} d\zeta \frac{e^{-i \kappa \zeta}}{(\zeta-i\zeta_R)\left(R_k \zeta+i\zeta_R\right)}. \end{eqnarray}
From Equation (\ref{Eq:SingleModeSFGEff}) the upconversion efficiency is then
\begin{equation} {Q}_{SFG} = \frac{ 8 \pi \omega_p^2 }{ c^3 \varepsilon_0 n_p n_s n_i } \frac{k_p k_s k_i }{k_+^2} z_R
d_{\rm eff}^2 |\Upsilon|^2. \end{equation}
\subsection{Boyd and Klienman, 1968}
With this expression we can compare our results to those of Boyd and Kleinman \cite{Boyd:1968} for the case of second-harmonic generation. As that calculation does not include quasi-phase matching, we take $Q=0$, and then for any reasonable phase-matching we have $n_p \approx n_s$, $k_p \approx 2 k_s$ and thus $k_-=R_k=0$, $k_+ \approx 2 k_p$. We note that for $R_k = 0$, $\Upsilon$ becomes equal to the function $H$ of Boyd and Kleinman for zero absorption and walk-off angle. We find \begin{equation} P_p = \frac{4 \pi k_s \omega_s^2}{c^3 \varepsilon_0 n_s^2 n_p} z_R d^2
|\Upsilon|^2 P_s^2.\end{equation}
Boyd and Kleinman find in Equations (2.16),(2.17) and (2.20) to
(2.24) \begin{equation} P_{2} = \frac{128 \pi^2 \omega_1^2}{c^3 n_1^2 n_2} d^2 P_{1}^2 L k_1 \frac{2\pi^2 z_R}{L} |H|^2\end{equation} or \begin{equation} P_{2} =
\frac{256 \pi^4 k_1\omega_1^2}{c^3 n_1^2 n_2}z_R d^2 |H|^2 P_{1}^2.\end{equation} When converting this expression to MKS units, $d^2 \rightarrow d^2/64 \pi^3 \varepsilon_0$, and we see that the two calculations agree.
\subsection{Type-II collinear brightness}
Next we make a numerical calculation for frequency-degenerate type-II SPDC, a geometry of current interest for generation of entangled pairs, for example.
The integral $\Upsilon$ must be evaluated numerically. For a 1 cm crystal of PPKTP and a vacuum wavelength $\lambda_s = \lambda_i = 800$ nm we have $(n_s,n_i,n_p) = (1.844, 1.757, 1.964)$ and $d_{\rm eff} = 2.4$ pm/V so that $R_k = 0.04$ and the maximum of
$(z_R/L)|\Upsilon|^2\approx 0.054$ occurs at $\kappa \approx -3.0, \zeta_R \approx 0.18$. We find ${Q}_{SFG} = 2.0\times 10^{-3}$ W$^{-1}$. Used as a photon-pair source, this same crystal and geometry would yield by Equation (\ref{Eq:PDCPairRate}) \begin{equation} W^{(2)} = \Gamma_{\rm eff} P_p \frac{{Q}_{SFG}^{}}{16} \end{equation} or a pair generation efficiency of ${Q}_{SFG}/16 = 0.8 $ pairs
(s mW MHz)$^{-1}$. Note that $\Gamma_{\rm eff}$ is the filter bandwidth in angular frequency.
\subsection{Ling, Lamas-Linares, and Kurtsiefer, 2008} \label{Sec:Ling}
Recently, Ling {\em et al.} calculated the absolute emission rate into gaussian modes in the thin-crystal limit of (non-periodically-poled) nonlinear material \cite{Ling:2008}. They arrive to a down-conversion spectral brightness of \begin{equation} \label{Eq:LingRate} \frac{dR(\omega_s)}{d\omega_s} = \left(\frac{d_{\rm eff} \alpha_s \alpha_i E_p^0 \Phi(\Delta k)}{c} \right)^2 \frac{\omega_s\omega_i}{2 \pi n_s n_i} \end{equation} where $R$ is the pair collection rate and \begin{equation} \Phi(\Delta k) \equiv \int dz \int dy\, dx\, e^{i \Delta {\bf k} \cdot {\bf r} } U_p({\bf r})U_s({\bf r})U_i({\bf r}).\end{equation} Here $U_m$ describe the mode shapes of the form $U_m({\bf r}) = e^{i k_m z} e^{-(x^2 + y^2)/W_m^2}$ and
$\alpha_m = \sqrt{2/\pi W_m^2}$ are normalization constants. The field $E_p^0$ is defined such that $|E_p^0|^2 = 2\alpha_p^2 P_p/\varepsilon_0 n_p c$ where $P_p$ is the pump power, giving \begin{equation} \frac{dR(\omega_s)}{d\omega_s} = \frac{\omega_s\omega_i d^2}{\pi c^3\varepsilon_0 n_p n_s n_i} P_p \frac{}{
}|\alpha_p\alpha_s\alpha_i \Phi(\Delta k)|^2 \end{equation} Assuming the output is collected with narrow-band filters of transmission $T_s (\omega_s),T_i(\omega_i)$ for signal and idler, respectively, the integrated rate is \begin{equation} R = \frac{dR(\omega_s)}{d\omega_s} \int d\Omega T_s(\omega_p/2+\Omega) T_i(\omega_p/2-\Omega) \end{equation} where we have assumed $dR(\omega_s)/d\omega_s$ constant over the width of the filters. For comparison, using Eqs (\ref{Eq:SingleModeSFGEff}) and (\ref{Eq:PDCPairRate}), we find \begin{equation} W^{(2)} = \frac{\omega_s
\omega_i d^2 }{2c^3 \varepsilon_0 n_p n_s n_i} P_p |I_{SFG}|^2
\Gamma_{\rm eff}.\end{equation} Then with Equation (\ref{Eq:GammaArea}) and noting that in the thin crystal limit $|I_{SFG}| =
|\alpha_p\alpha_s\alpha_i \Phi(\Delta k)|$, we see that the two results are identical.
\section{Conclusions} \label{sec:Conclusions}
Using the approach of coupled wave-equations, familiar from nonlinear optics, we have calculated the absolute brightness and temporal correlations of spontaneous parametric down-conversion in the narrow-band regime. The results are obtained with a Green function method and are generally valid within the paraxial regime. We find that efficiencies of SFG and SPDC can be expressed in terms of mode overlap integrals, and are proportional for corresponding geometries. Also, we find pair time correlations in terms of signal and idler filter impulse response functions. Results for both degenerate and non-degenerate SPDC are found. Comparison to classical calculations by Boyd and Kleinman, and to a recent calculation by Ling et al. show the connection to classical nonlinear optics and ``golden rule''-style brightness calculations, while considerably generalizing the latter. We expect these results to be important both for designing SPDC sources, as the results of well-known classical calculations can be used, and for building and optimizing such sources.
\appendix{}
\section{alternate propagator}
We can use Equations (\ref{Eq:BackFromTheFuture}) and (\ref{Eq:EqualTimeCommutator}) to express the propagator as \begin{eqnarray}
\expect{{\bf\cal E}_{}(x){\bf\cal E}_{}^\dagger(x') }&=& |{\beta}_{t}|^2 \int d^4 x'' d^4 x''' \delta(t''-t_f) \nonumber \\ & & \times \delta(t'''-t_f) \expect{{\bf\cal E}_{}(x''){\bf\cal E}_{}^\dagger(x''') } \nonumber \\ & & \times {\cal G}^*(x'';x)
{\cal G}(x''';x').
\nonumber \\ & = & |A_{\gamma} {\beta}_{t}|^2 \int d^4 x'' \delta(t''-t_f) \nonumber \\ & & \times {\cal G}^*(x'';x)
{\cal G}(x'';x') \end{eqnarray} Noting that $\int d^4 x'' \delta(t''-t_f) {\cal G}^*(x'';x)
{\cal G}(x'';x') = v_g \int d^4 x'' \delta(z''-z_0) {\cal G}^*(x'';x)
{\cal G}(x'';x') $ we find \begin{eqnarray} \label{Eq:AlternatePropagator} \expect{{\bf\cal E}_{}(x){\bf\cal E}_{}^\dagger(x') }&=&
\frac{2\hbar n \omega^3 }{c^3 \varepsilon_0} \int d^4 x'' \delta(z''-z_0) \nonumber \\ & & \times {\cal G}^*(x'';x) {\cal G}(x'';x') \end{eqnarray}
\section{Lorentzian filter}
A common filter has a Lorentzian transfer function and an exponential impulse response \begin{equation} F(\tau) = \frac{\Gamma}{2} \theta(\tau) \exp[-\Gamma \tau /2 ].\end{equation} The spectral transmission is $T(\Omega) = \Gamma^2/(\Gamma^2 + 4 \Omega^2)$, i.e., unit transmission for constant ${\bf\cal E}$, a full-width at half-maximum of $\Delta \Omega_{\rm FWHM} = \Gamma$ and an area $\int d\Omega\, T(\Omega) = \pi\Gamma /2$. If we put a filter of this sort in each arm, the output has \begin{eqnarray} f(t_s-t_i) &= & \frac{\Gamma_s\Gamma_i}{4}\int dt' \theta(t_s-t') \theta(t_i-t') \nonumber \\ & & \times \exp[-\Gamma_i(t_i-t')/2]\nonumber \\ & & \times \exp[-\Gamma_s(t_s-t')/2] \end{eqnarray} or \begin{eqnarray} f(\tau) &=& \frac{\Gamma_s\Gamma_i}{2(\Gamma_s + \Gamma_i)} \left\{ \begin{array}{ll}\exp[-\Gamma_s\tau/2] & \tau > 0 \\ \exp[\Gamma_i\tau/2] & \tau < 0 \end{array} \right. \nonumber \\ \end{eqnarray} The effective bandwidth is \begin{equation} \Gamma_{\rm eff} = 4 \int d\tau
|f(\tau)|^2= \frac{\Gamma_s\Gamma_i}{\Gamma_s+\Gamma_i} \end{equation}
It is worth noting that in the limit $\Gamma_s \rightarrow \infty$ (the limit of a broad-band filter in the signal beam, or in practical terms, not having a filter there at all), filter becomes \begin{eqnarray} f(t_s-t_i) &= & \frac{\Gamma_i}{2} \left\{ \begin{array}{rl}0 & t_i < t_s \\ \exp[-\Gamma_i(t_i-t_s)/2] & t_i > t_s \end{array}\right. \nonumber \\ \end{eqnarray} That is, the idler photon will always arrive later, and with a distribution (after the signal arrival) that is precisely the transfer function of the idler-beam filter. Another interesting limit is for matched filters, $\Gamma_s = \Gamma_i = \Gamma$. Then we find \begin{eqnarray} f(t_s-t_i) &= &
\frac{\Gamma}{4}\exp[-\Gamma|t_s-t_i|/2] .\end{eqnarray}
Note that for $\Gamma_s \rightarrow \infty$, the detection rate is
$|{\cal A}|^2 \Gamma_i /{4} $, i.e., proportional to the idler filter bandwidth $\Gamma_i$. The reverse, $s \leftrightarrow i$ is also true, of course. From this we can get an idea of the conditional efficiency: The rate for filtered signal with {any} idler is proportional to \begin{equation} \Gamma_s \ge \frac{\Gamma_i \Gamma_s}{\Gamma_s + \Gamma_i}.\end{equation} For example, putting matched filters $\Gamma_s = \Gamma_i = \Gamma$ will give a rate proportional to $\Gamma_i \Gamma_s/({\Gamma_s + \Gamma_i})$ i.e., half of the rate without the idler filter. This indicates that, of the signal photons that pass the the signal filter, half of their ``twin'' idler photons do not pass the idler filter.
\end{document}
\end{document} | arXiv |
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation $X\hookrightarrow Y$.
In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X,
$f\circ g_{1}=f\circ g_{2}\implies g_{1}=g_{2}.$
Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.
In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks.
The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism.
Relation to invertibility
Left-invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and $l\circ f=\operatorname {id} _{X}$), then f is monic, as
$f\circ g_{1}=f\circ g_{2}\Rightarrow l\circ f\circ g_{1}=l\circ f\circ g_{2}\Rightarrow g_{1}=g_{2}.$
A left-invertible morphism is called a split mono or a section.
However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group homomorphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
A morphism f : X → Y is monic if and only if the induced map f∗ : Hom(Z, X) → Hom(Z, Y), defined by f∗(h) = f ∘ h for all morphisms h : Z → X, is injective for all objects Z.
Examples
Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of all groups, of all rings, and in any abelian category.
It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense. For example, in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map q : Q → Q/Z, where Q is the rationals under addition, Z the integers (also considered a group under addition), and Q/Z is the corresponding quotient group. This is not an injective map, as for example every integer is mapped to 0. Nevertheless, it is a monomorphism in this category. This follows from the implication q ∘ h = 0 ⇒ h = 0, which we will now prove. If h : G → Q, where G is some divisible group, and q ∘ h = 0, then h(x) ∈ Z, ∀ x ∈ G. Now fix some x ∈ G. Without loss of generality, we may assume that h(x) ≥ 0 (otherwise, choose −x instead). Then, letting n = h(x) + 1, since G is a divisible group, there exists some y ∈ G such that x = ny, so h(x) = n h(y). From this, and 0 ≤ h(x) < h(x) + 1 = n, it follows that
$0\leq {\frac {h(x)}{h(x)+1}}=h(y)<1$
Since h(y) ∈ Z, it follows that h(y) = 0, and thus h(x) = 0 = h(−x), ∀ x ∈ G. This says that h = 0, as desired.
To go from that implication to the fact that q is a monomorphism, assume that q ∘ f = q ∘ g for some morphisms f, g : G → Q, where G is some divisible group. Then q ∘ (f − g) = 0, where (f − g) : x ↦ f(x) − g(x). (Since (f − g)(0) = 0, and (f − g)(x + y) = (f − g)(x) + (f − g)(y), it follows that (f − g) ∈ Hom(G, Q)). From the implication just proved, q ∘ (f − g) = 0 ⇒ f − g = 0 ⇔ ∀ x ∈ G, f(x) = g(x) ⇔ f = g. Hence q is a monomorphism, as claimed.
Properties
• In a topos, every mono is an equalizer, and any map that is both monic and epic is an isomorphism.
• Every isomorphism is monic.
Related concepts
There are also useful concepts of regular monomorphism, extremal monomorphism, immediate monomorphism, strong monomorphism, and split monomorphism.
• A monomorphism is said to be regular if it is an equalizer of some pair of parallel morphisms.
• A monomorphism $\mu $ is said to be extremal[1] if in each representation $\mu =\varphi \circ \varepsilon $, where $\varepsilon $ is an epimorphism, the morphism $\varepsilon $ is automatically an isomorphism.
• A monomorphism $\mu $ is said to be immediate if in each representation $\mu =\mu '\circ \varepsilon $, where $\mu '$ is a monomorphism and $\varepsilon $ is an epimorphism, the morphism $\varepsilon $ is automatically an isomorphism.
• A monomorphism $\mu :C\to D$ is said to be strong[1][2] if for any epimorphism $\varepsilon :A\to B$ and any morphisms $\alpha :A\to C$ and $\beta :B\to D$ such that $\beta \circ \varepsilon =\mu \circ \alpha $, there exists a morphism $\delta :B\to C$ such that $\delta \circ \varepsilon =\alpha $ and $\mu \circ \delta =\beta $.
• A monomorphism $\mu $ is said to be split if there exists a morphism $\varepsilon $ such that $\varepsilon \circ \mu =1$ (in this case $\varepsilon $ is called a left-sided inverse for $\mu $).
Terminology
The companion terms monomorphism and epimorphism were originally introduced by Nicolas Bourbaki; Bourbaki uses monomorphism as shorthand for an injective function. Early category theorists believed that the correct generalization of injectivity to the context of categories was the cancellation property given above. While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms. Saunders Mac Lane attempted to make a distinction between what he called monomorphisms, which were maps in a concrete category whose underlying maps of sets were injective, and monic maps, which are monomorphisms in the categorical sense of the word. This distinction never came into general use.
Another name for monomorphism is extension, although this has other uses too.
See also
• Embedding
• Nodal decomposition
• Subobject
Notes
1. Borceux 1994.
2. Tsalenko & Shulgeifer 1974.
References
• Bergman, George (2015). An Invitation to General Algebra and Universal Constructions. Springer. ISBN 978-3-319-11478-1.
• Borceux, Francis (1994). Handbook of Categorical Algebra. Volume 1: Basic Category Theory. Cambridge University Press. ISBN 978-0521061193.
• "Monomorphism", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Van Oosten, Jaap (1995). "Basic Category Theory" (PDF). Brics Lecture Series. BRICS, Computer Science Department, University of Aarhus. ISSN 1395-2048.
• Tsalenko, M.S.; Shulgeifer, E.G. (1974). Foundations of category theory. Nauka. ISBN 5-02-014427-4.
External links
• monomorphism at the nLab
• Strong monomorphism at the nLab
| Wikipedia |
\begin{definition}[Definition:Variable/Value]
A variable $x$ may be (temporarily, conceptually) identified with a particular object.
If so, then that object is called the '''value''' of $x$.
\end{definition} | ProofWiki |
Transfers on Bloch groups and scissors congruence groups
I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups.
To fix notation and recall definitions:
From the general algebraic K-theory machinery, we get transfers on $K_3$. In particular, for a finite field extension $E/F$, we get a map $\operatorname{tr}_{E/F}:K_3(E)\to K_3(F)$ such that the composition $K_3(F)\to K_3(E)\stackrel{\operatorname{tr}_{E/F}}{\longrightarrow}K_3(F)$ is multiplication with the degree $[E:F]$.
Now by the work of Bloch, Dupont-Sah, Suslin and others, we have another description of $K_3(F)^{\operatorname{ind}}=K_3(F)/K_3^M(F)$ in terms of an exact sequence $$ 0\to \widetilde{\operatorname{Tor}}(\mu(F),\mu(F))\to K_3(F)^{\operatorname{ind}}\to B(F)\to 0. $$ In the above, the Bloch group $B(F)$ is defined as $$ B(F)=\ker\left(\mathcal{P}(F)\to \Lambda^2(F^\times):[x]\to x\wedge(1-x)\right) $$ and the group $\mathcal{P}(F)$ is the pre-Bloch group or scissors congruence group $$ \mathcal{P}(F)=\left(\bigoplus_{x\in F^\times\setminus\{1\}}\mathbb{Z}[x]\right) /\left([x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]\right), $$ the relation coming from the five-term relation satisfied by the dilogarithm. In particular, elements of the Bloch group $B(F)$ can be written down as linear combinations of symbols $[x], x\in F^\times\setminus\{1\}$ satisfying certain relations.
Now I can formulate my questions on transfers on Bloch groups and scissors congruence groups.
Does the transfer on $K_3$ induce a transfer on the Bloch group? I think that this is not the case in general. The element $[x]+[1-x]\in B(F)$ is independent of $x$, and is typically denoted by $c_F$. The element $c_{\mathbb{R}}$ has exact order $6$ in $B(\mathbb{R})$, and the element $c_{\mathbb{C}}$ is trivial in $B(\mathbb{C})$. This seems to contradict transfers for the Bloch group (it does not contradict transfers for $K_3^{\operatorname{ind}}$ because the torsion moves from $B(F)$ to $\widetilde{\operatorname{Tor}}(\mu(F),\mu(F))$). Are there more torsion elements like this, in particular with other odd orders? Are there further obstructions to the existence of transfers on Bloch groups? If $F$ contains an algebraically closed fields, it follows from work of Suslin and Levine that $B(F)$ is uniquely $\ell$-divisible for $\ell$ different from the characteristic - in particular $c_F=0$. Does the $K$-theory transfer induce a transfer on the Bloch group in this situation? What would be a good reference?
Is there an explicit description of what the transfer map on $K_3$ does on the Bloch group? I would be interested in a description that only uses the definition of the Bloch group via points on $\mathbb{P}^1(F)\setminus\{0,1,\infty\}$ given above.
More generally, are there transfers known on scissors congruence groups/pre-Bloch groups? As written above, these groups are defined in terms of points on $\mathbb{P}^1\setminus\{0,1,\infty\}$ modulo the five-term relation. A very naive approach to the definition of transfers for pre-Bloch groups in an extension $E/F$ would be to sum over $E$-points lying over $F$-points of $\mathbb{P}^1\setminus\{0,1,\infty\}$. Has anyone ever tried to work this out, or are there known obstruction why this cannot provide a transfer? Assuming it works, how would one relate such a naive definition to the definition of transfers for algebraic K-theory? In a related direction, what torsion elements besides those in $B(F)$ are known in the scissors congruence groups over fields which are not algebraically closed (in the algebraically closed case, the scissors congruence groups are uniquely divisible)?
ag.algebraic-geometry group-cohomology algebraic-k-theory
Matthias Wendt
Matthias WendtMatthias Wendt
Here is some partial information to the question which I figured out in the meantime. The main question (on transfers for scissors congruence groups) still stands...
Concerning 1: The torsion in $B(F)$ is cyclic, and is related to the roots of unity which are roots of irreducible polynomials of degree $2$ over $F$. For example, $K_3(\mathbb{F}_q)^{\operatorname{ind}}\cong\mathbb{Z}/(q^2-1)$ where $\mathbb{Z}/(q-1)$ comes from the roots of unity in $\mathbb{F}_q$, and $\mathbb{Z}/(q+1)$ comes from the Bloch group. Torsion elements in the Bloch group (such as $[x]+[x^{-1}]$ or $[x]+[1-x]$) seem to be related to automorphisms of $\mathbb{P}^1\setminus\{0,1,\infty\}$. However, I do not know of a general way to write down explicit generators for the torsion in $B(F)$. These torsion elements do obstruct the existence of transfers for $B(F)$ because the torsion element corresponding to $\zeta_n$ is killed in the extension $F(\zeta_n)$.
Concerning 2 and 3: If $E/F$ is a finite Galois extension with group $G$, then $K_3(E)^{\operatorname{ind}}$ is a $G$-module, and by the work of Levine, $K_3(F)^{\operatorname{ind}}=(K_3(E)^{\operatorname{ind}})^G$. I guess the transfers are given by summing over Galois orbits, and that this procedure should also work for the Bloch group.
In the case of a radical extension the relevant facts can be found in papers of Dupont and Sah: assume $F$ is such that $\zeta_n\in F$, let $a\in F$ and consider the Galois extension $F(\sqrt[n]{a})/F$. Then Dupont-Sah prove the following equality in the scissors congruence group: $$ \frac{[a]}{n}=\sum_{0\leq i\leq n-1}[\zeta_n^i a]. $$ In particular, the sum over elements of the Galois orbit in $F(\sqrt[n]{a})$ has a representative over $F$.
Ok, as said before: this is only partial information. I guess, what I want to know is, if the above works generally for arbitrary Galois extensions. How could one generally obtain $F$-representatives of sums over Galois orbits in $E/F$?
I have worked on this question for some time (in the $\mathbb{Q}$-coefficients case), trying to construct norm maps on $B_2(F)$ in a way, similar to Milnor $K-$theory. Eventually, I was able to construct norm maps on the middle cohomology groups of complexes $$ B_3(F) \otimes \Lambda^{n-3} F^{\times} \longrightarrow B_2(F) \otimes \Lambda^{n-2} F^{\times} \longrightarrow \Lambda^n F^{\times}, $$ which conjecturally coincide with $H^{n-1}_{M}(F,\mathbb{Q}(n)),$ for which existance of norm maps is known (see https://arxiv.org/abs/1511.00520). Unfortunately, I had to use the fact that the kernel of the map $$B_2(F)\longrightarrow \Lambda^2 F^{\times}$$ does not change under a simple transcendental extension of the field, so the construction of the norm is not completely explicit at the end of the day.
Daniil RudenkoDaniil Rudenko
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Introduction to Reynolds experiment
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written 10 months ago by Syedahina Mohi • 10 modified 10 months ago by Sanket Shingote ♦♦ 290
Reynold's Apparatus
The type of flow is determined from Reynolds number.
$R.E=\dfrac {\rho V\times d}{\mu}$ (by 0. Reynolds in 1883)
Apparatus consists of
1) A tank containing water at constant head,
2) A small tank containing some dye,
3) A glass tube having a bell-mouthed entrance at one end and regulating valve at the other end.
The water from the tank was allowed to flow through the glass tube. The velocity of flow was varied by the regulating valve. A liquid dye having same specific weight as water was introduced into the glass tube as shown in the figure.
Observations:-
1) When the velocity of flow was low, the dye filament in the glass tube was in the form of a straight line. This straight line of dye filament was parallel to glass tube, which was the case of laminar flow.
2) With the increase of velocity of flow, the dye filament was no longer a straight line but it became a wavy one as shown in the figure below. The figure shows that the floor is no longer laminar.
3) With further increase of velocity of flow, the wavy dye filament broke up and finally diffused in water. This means that the fluid particles of the dye at this higher velocity are moving in random fashion, which shows the case of turbulent flow. Thus in case of turbulent flow the mixing of the filament and water is intense and flow is irregular random and disorderly.
Critical velocity
The velocity of flow at the critical depth is known as critical velocity. It is denoted by '$V_C$'. The expression for critical velocity is obtained by,
$h_C=\left(\dfrac{q^2}g\right)^{1/3}$
Taking cube to both sides, we get
$h_C^3= \dfrac{q^2}g$
Or, $gh_c^3=q^2………………(1)$
But q= discharge per unit width
$q=\dfrac Qb$
$q=\dfrac {\text{Area}\times V}b$
$q=\dfrac {b\times h\times V}b$
$q=h\times V$
$\therefore q=h_C\times V_C$
Substituting the value of 'q' in equation (1),
$gh_C^3=(h_C\times V_C)^2$
$gh_C^3=h_C^2\times V_C^2$
Divide by $h_C^2$
$V_C=\sqrt{g\times h_C}$
Kinetic energy correction factor
It is defined as the ratio of the kinetic energy of the flow per second based on actual velocity across a section to the kinetic energy of flow per second based on average velocity across the same section. It is denoted by 'a'
$\therefore a =\dfrac{\text{K.E/sec based on actual velocity}}{\text{ K.E/sec based on average velocity}}$
Momentum correction factor
It is defined as the ratio of momentum of the flow per second based on the actual velocity to the momentum of the flow per second based on average velocity across a section.
It is denoted by '$\beta$'.
$\therefore \beta =\dfrac{\text{momentum/sec based on actual velocity}}{\text{ momentum/sec based on average velocity}}$
Numericals
Q1) Show that the momentum correction factor and energy correction correction factor for laminar flow through circular pipe are $\dfrac43$ and 2.0 respectively.
Solution:-
(i) Momentum correction factor or $\beta$
The velocity distribution through a circular pipe for laminar flow at any radius 'r' is given by,
$u=\dfrac1{4\mu}\left(-\dfrac{dP}{dx}\right)(R^2-r^2)…………..(1)$
Consider an elementary area dA in the form of a ring at the radius 'r' and width 'dr' then
$dA=2\pi rdr$
Rate of fluid flowing through the ring,
=dQ = Velocity× area of the ring element
$=u\times 2\pi rdr$
Momentum of the fluid through ring per second
=mass × velocity
$=\rho\times dQ\times u$
$=\rho\times2\pi r\times u\times u$
$=2\pi \rho u^2rdr$
Therefore the total actual momentum of the fluid per second across the section $\int_0^R2\pi \rho u^2rdr$
Substituting the value of 'u' from equation (1),
$=2\pi \rho\int_0^R\left[\dfrac1{4\mu}\left(-\dfrac{dP}{dx}\right)(R^2-r^2)\right]^2rdr$
$=2\pi \rho\left[\dfrac1{4\mu}\left(-\dfrac{dP}{dx}\right)\right]^2\int_0^R(R^2-r^2)^2rdr$
$=2\pi \rho\left[\dfrac1{16μ^2}\left(\dfrac{dP}{dx}\right)^2\right]\int_0^R(R^4+r^4-2R^2r^2)rdr$
$=\dfrac{\pi \rho}{8\mu^2}\left(\dfrac{dP}{dx}\right)^2\int_0^R(R^4r+r^5-2R^2r^3)dr$
$=\dfrac{\pi \rho}{8\mu^2}\left(\dfrac{dP}{dx}\right)^2\left[\dfrac{R^4r^2}2+\dfrac{r^6}6-\dfrac{2R^2r^4}4\right]^R_0$
$=\dfrac{\pi \rho}{8\mu^2}\left(\dfrac{dP}{dx}\right)^2\left[\dfrac{R^6}2+\dfrac{R^6}6-\dfrac{2R^6}4\right]$
$=\dfrac{\pi \rho}{8\mu^2}\left(\dfrac{dP}{dx}\right)^2\left[\dfrac{6R^6+2R^6-6R^6}{12}\right]$
$=\dfrac{\pi \rho}{8\mu^2}\left(\dfrac{dP}{dx}\right)^2\times\dfrac{R^6}{6}$
$=\dfrac{\pi \rho}{48\mu^2}\left(\dfrac{dP}{dx}\right)^2\times R^6……………(2)$
Momentum of the fluid per second based on average velocity
$=\dfrac{\text{mass of fluid}}{\text{second}}\times\text{Average Velocity}$
$=\rho A\overline{u}×\overline{u}=\rho A\overline{u}^2$
Where A = area of cross-section = $\pi R^2$
Average velocity = $\overline{u}=\dfrac {U_{max}}2$
$=\dfrac 12\times\dfrac1{4\mu}\left(-\dfrac{dP}{dx}\right)R^2$
$=\dfrac 1{8\mu}\left(-\dfrac {dP}{dx}\right)R^2$
Therefore Momentum per second based on average velocity,
$=\rho\times\pi R^2\left[\dfrac1{8\mu}\left(-\dfrac{dP}{dx}\right)R^2\right]^2$
$=\rho\times\pi R^2\times\dfrac1{64\mu^2}\left(-\dfrac{dP}{dx}\right)^2R^4$
$=\dfrac{\rho\pi\left(-\dfrac{dP}{dx}\right)^2R^6}{64\mu^2}.......................(3)$
$\beta=\dfrac{\dfrac{\pi \rho}{48\mu^2}\left(-\dfrac{dP}{dx}\right)^2\times R^6}{\dfrac{\rho\pi}{64\mu^2}\left(-\dfrac{dP}{dx}\right)^2R^6}$
$∴ \beta=\dfrac {64}{48}=\dfrac 43$
2) Energy correction factor 'a'
Kinetic energy of the fluid flowing through the elementary ring of radius 'r' and of width 'dr' per second,
$=\dfrac 12\times \text{mass}\times u^2$
$=\dfrac12\times\rho dQ\times u^2$
$=\dfrac12\times\rho\times(u\times2\pi rdr)\times u^2$
$=\dfrac12\rho\times2\pi ru^3dr$
$=\pi\rho ru^3dr$
Therefore total actual kinetic energy of flow per second
$=\int^R_0\pi \rho ru^3dr$
$=\int^R_0\pi \rho r\left[\dfrac1{4\mu}\left(-\dfrac{dP}{dx}\right)(R^2-r^2)\right]^3dr$
$=\pi \rho \left[\dfrac1{4\mu}\left(-\dfrac{dP}{dx}\right) \right]^3\int^R_0(R^2-r^2)^3rdr$
$=\pi \rho \left[\dfrac1{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \right]\int^R_0(R^6-r^6-3R^4r^2+3R^2r^4)rdr$
$=\pi \rho \left[\dfrac1{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \right]\int^R_0(rR^6-r^7-3R^4r^3+3R^2r^5)dr$
$=\pi \rho \left[\dfrac1{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \right]\left[\dfrac{R^6r^2}{2}-\dfrac{r^8}8-\dfrac{3R^4r^4}4+\dfrac{3R^2r^6}6\right]^R_0$
$=\dfrac{\pi ρ}{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \left[\dfrac{R^8}{2}-\dfrac{R^8}8-\dfrac{3R^8}4+\dfrac{3R^8}6\right]$
$=\dfrac{\pi ρ}{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \left[R_8\left(\dfrac{12-3-18+12}{24}\right)\right]$
$=\dfrac{\pi ρ}{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \left(\dfrac{R^8}{8}\right)………….(4)$
Kinetic energy of floor based on average velocity
$=\dfrac 12\times \text{mass}\times \overline{u}^{2}$
$=\dfrac12\times\rho A\overline{u}\times \overline{u}^{2}$
$=\dfrac12\times\rho\times A\times \overline{u}^{3}$
Substituting the value of $A=\pi R^2$
And $\overline{u}=\dfrac1{8\mu}\left(-\dfrac{dP}{dx}\right)R^2$
Kinetic energy of the flow per second
$=\dfrac12\times\rho\times\pi R^2\times \left[\dfrac1{8\mu}\left(-\dfrac{dP}{dx}\right)R^2\right]^{3}$
$=\dfrac12\times\rho\times\pi R^2\times \dfrac1{64\times8\mu^3}\left(-\dfrac{dP}{dx}\right)^3R^6$
$=\dfrac{\rho\timesπ}{128\times8\mu^3}\left(-\dfrac{dP}{dx}\right)^3R^8……………(5)$
$\therefore a=\dfrac {equation (4)}{equation(5)}$
$a=\dfrac {\dfrac{\pi ρ}{64\mu^3}\left(-\dfrac{dP}{dx}\right)^3 \left(\dfrac{R^8}{8}\right)}{ \dfrac{\rho\timesπ}{128\times8\mu^3}\left(-\dfrac{dP}{dx}\right)^3R^8}$
$=\dfrac {128\times8}{64\times8}$
$a=2.0$
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\begin{document}
\title{A hybrid mass transport finite element method\ for Keller--Segel type systems}
\begin{abstract} We propose a new splitting scheme for general reaction-taxis-diffusion systems in one spatial dimension capable to deal with simultaneous concentrated and diffusive regions as well as travelling waves and merging phenomena. The splitting scheme is based on a mass transport strategy for the cell density coupled with classical finite element approximations for the rest of the system. The built-in mass adaption of the scheme allows for an excellent performance even with respect to dedicated mesh-adapted AMR schemes in original variables. \end{abstract}
{\bf Keywords:} mass transport schemes, reaction-aggregation-diffusion systems, splitting schemes, tumor invasion models
\section{Introduction} The aim of the present work is to design a numerical scheme capable to deal with concentrations and diffusion phenomena typically arising in one-dimensional taxis-diffusion systems of the form \begin{equation} \label{eq:systemtype}\left\{ \begin{aligned} \partial_t \rho &= \partial_x \(D_\rho \partial_x \rho - \chi \rho \partial_x c \) + R_\rho(\rho)& \text{in }(0,\infty) \times (a,b), \\ \varepsilon \partial_t c &= D_c \partial_x ^2 c + R_c(\rho, c) & \text{in }(0,\infty) \times (a,b),\\
\partial_x \rho(\cdot, r) &= \partial_x c(\cdot, r)=0, &r\in\{a,b\},\\
\rho(0,\cdot) &= \rho_0 \geq 0, \quad c(0,\cdot) = c_0 \geq 0 & \end{aligned}\right. \end{equation} with Lipschitz continuous source terms $R_\rho, \, R_c$ that satisfy $R_\rho(0), R_c(\rho, 0)\geq 0$. Here $\rho$ denotes the cell density and $c$ the concentration of a chemo-attractant. These systems constitute adaptations of the classical cell migration model by Patlak, Keller and Segel \cite{Patlak.1953, KS.1970}. They have been widely used in the modeling of biological processes such as cell organization in tissue, immune system dynamics and cancer growth \cite{Esipov.1998, Anderson.2000, EMT_paper}. The dynamics of their solutions are quite rich; apart from traveling waves \cite{Kong.2008} the aggregation phenomenon studied in \cite{jager1992explosions,BDP} that leads to blowup in finite time is of specific interest. One has moreover observed the occurrence of high concentrations that can emerge in a smooth solution, split, and merge with each other \cite{Painter.2011}. Nonlinear diffusions or saturated responses in the chemotactic sensitivity are natural ways to include volume filling effects into the models, see \cite{HPVolume,CCVolume}. They usually avoid blow-up in a biologically meaningful way and lead to interesting phenomena and asymptotic stabilization. Finally, these models are basic building bricks for a variety of cancer invasion models in the literature \cite{Chaplain.2005, EMT_paper, Preziosi.2003, Stinner.2015, Johnston.2010} in which the coupling with extracellular matrix, enzymatic activators and other substances are taken into account. One of the common features in all of these models is the simultaneous occurrence of regions of high concentrated densities with diffuse profiles leading to numerical difficulties in choosing well-adapted meshes. The numerical approximation of all of these simultaneous phenomena is particularly challenging.
In \cite{Carillo.2008} a mass transport steepest descent scheme has been proposed to resolve a modified 1D Keller-Segel system for the log interaction kernel proposed in \cite{Calvez.2007}. The method satisfies a discrete free energy dissipation principle by design being based on the variational schemes for Fokker-Planck type equations introduced in \cite{jordan1998variational,KW} and applied to Keller-Segel type models in \cite{Carillo.2008,Blan.sys}. By considering the problem in transformed variables the method can resolve areas of high concentrations accurately without any mesh refinement. This approach has been extended to several dimensions for nonlinear aggregation-diffusion equations and with different approaches in the discretization in \cite{carrillo2009numerical,MO2014,CRW,JMO} and the references therein.
The aim of this work is to extend the mass transport approach to the general class of systems \eqref{eq:systemtype}. We will test different scenarios that feature in particular the splitting, traveling and emerging of concentrations. For the adjustment of the scheme we propose a splitting method, where we employ the technique from \cite{Carillo.2008} to the Keller-Segel part of the system (i.e. the first equation of \eqref{eq:systemtype} with $R_c = 0$). The remaining system of an ODE and a diffusion reaction equation will be decoupled and solved by a suitable finite element method. The advantage of the mass transport approach for the cell densities equations is that the mesh adapts naturally to the mass distribution, and then coarse meshes in the mass variable can still lead to good numerical approximations as we will discuss below.
In more details, we split \eqref{eq:systemtype} into two subsystems. The solution of the full system \eqref{eq:systemtype} can then be approximated by appropriately combining short time solution of the subsystems. We introduce at first the diffusion-advection system given by \begin{equation} \label{eq:split_KS} \tag{I}\left\{ \begin{aligned} \partial_t \rho &= \partial_x \(D_\rho \partial_x \rho - \chi \rho \partial_x c \) & \text{in }(0,\infty) \times (a,b), \\ \partial_t c &= 0 & \text{in }(0,\infty) \times (a,b), \\
\partial_x \rho(\cdot, r) &=0, &r\in\{a,b\},\\ \rho(0,\cdot) &= \rho^I_0 \geq 0, \quad c(0,\cdot) = c^I_0 \geq 0. \end{aligned}\right. \end{equation} This system makes the assumption of a steady chemo-attractant density $c$ and mass conservation in the cell density $\rho$. Second, we consider the reaction-diffusion system \begin{equation} \label{eq:split_ReacDiff} \tag{II}\left\{ \begin{aligned} \partial_t \rho&= R_\rho(\rho )& \text{in }(0,\infty) \times (a,b), \\ \partial_t c &= D_c \partial_x ^2 c + R_c(\rho, c)& \text{in }(0,\infty) \times (a,b),\\ \partial_x c(\cdot, r)&=0, &r\in\{a,b\}, \\ \rho(0,\cdot) &= \rho^{II}_0 \geq 0, \quad c(0,\cdot) = c^{II}_0\geq 0& \end{aligned}\right. \end{equation} that contains the remaining terms of the system. Following the mass transport algorithm \cite{Carillo.2008} we transform the system \eqref{eq:split_KS} into new variables. With this aim, we consider the pseudo inverse cumulative distribution of the cell density $\rho$, \begin{equation}\label{eq:pseudoinv}
V(t, w) = \inf \left\{ y : \int_{a_I}^y c(x,t)\, dx > w \right\}, \end{equation} which is defined by $$ 0\leq w \leq \int_a^b \rho(t,x)\, dx = m(t) \,. $$ The system \eqref{eq:split_KS} can now be rewritten, following e.g. \cite{Carrillo.2005}, as \begin{equation} \label{eq:split_KS_trans} \tag{I'}\left\{ \begin{aligned}
\partial_t V &= - D_\rho \partial_w \( \left[ \partial_w V \right]^{-1} \) + \chi \partial_x c |_{(x=V(w))}& \text{in }(0,\infty) \times (0,m),\\ \partial_t c &= 0 & \text{in }(0,\infty) \times (a,b),\\ V(\cdot, 0) &= a, \qquad V(\cdot, M) = b,\\ V(0,\cdot) &= V_I , \quad c(0,\cdot) = c_I \geq 0, & \end{aligned}\right. \end{equation} where $m$ denotes a given mass during this splitting step. The advantage of the proposed splitting is that the mass of cell densities does not change over the first step and the cell density is fixed over the second step.
The details of the full discretization of the proposed splitting scheme will be given in Section 2. In Section 3 we discuss the choice of the constraints in the time, spatial and mass steppings due to the choice of the full discrete schemes. Section 4 is devoted to study in detail the performance of this splitting scheme in many complex situations ranging from the simpler Keller-Segel type systems and their small variations to quite more biologically relevant systems in tumor invasion as discussed above. We will analyze the experimental convergence and the computational cost of this discretization with respect to previous schemes with mesh-refinement algorithms in original spatial variables. Finally we conclude in Section 5.
\section{Numerical method} \label{section:num} In what follows, we describe a numerical treatment for both systems \eqref{eq:split_KS_trans} and \eqref{eq:split_ReacDiff}. The inverse distribution $V$ is given on the time evolving \emph{mass space} $(0,m_h(t))$, whereas the chemo-attractant $c$ is given in the Eulerian coordinates in $(a,b)$. This leads to two meshes that the proposed numerical method employs.
First, we discretize the normalized mass domain $(0,1)$, on which the pseudo inverse distribution $V$ resides by the mesh \begin{equation*}
0 = w_0 < w_1 <\dots <w_M=1, \quad w_j = j h_w, \quad j = 0,\dots, M \end{equation*} with length $M \in \mathbb{N}$ and width $h_w = 1/M$ that corresponds to the width $\Delta w(t)=m_h(t) h_w$ in the time evolving mass domain $(0,m_h(t))$. We denote the point values of $V$ by $V_j (t) = V(m_h(t)\,w_j, t)$ for $j=0,\dots,M$ and introduce the linear spline in $w$ connecting the discrete values that we denote by $V_h(t,m_h(t)\,w)$. Here we have used the discrete mass of the cells \begin{equation*} m_h(t) = \int_a^b \rho_h(t,x)\, dx, \end{equation*} where $\rho_h$ is a discrete representation of the cell density to be defined later on.
A second mesh partitions the physical space $(a,b)$ for the chemo-attractant density $c$ into
\begin{equation}\label{eq:c_grid}
a = x_0 < x_1 <\dots <x_N=b, \quad x_k = a + k \Delta x, \quad k = 0,\dots, N.
\end{equation}
The chemo-attractant mesh is thus of length $N$ and width $\Delta x = (b-a)/N$. We employ a linear finite element representation for the chemo-attractant density $c$. Therefore let $\{ \phi_k, ~k=1,\dots, N-2\}$ be the basis of piecewise linear hat functions on the grid \eqref{eq:c_grid} satisfying the boundary conditions. In particular, we have \begin{equation*} \phi_k(x) = \begin{cases} (x-x_{k-1}) /\Delta x,& x_{k-1} \leq x \leq x_k, \\ (x_{k+1}- x) /\Delta x,& x_{k} \leq x \leq x_{k+1},\\ 0, &\text{otherwise} \end{cases}, \qquad k= 2, \dots, N-2. \end{equation*} in the center of the domain and \begin{align*} \phi_1(x) &= \begin{cases} 1 ,& a \leq x \leq x_1, \\ (x_2 - x) /\Delta x,& x_{1} \leq x \leq x_{2},\\ 0, &\text{otherwise}, \end{cases}\\[1ex] \phi_{N-1}(x) &= \begin{cases} (x-x_{N-2} ) /\Delta x,& x_{N-2} \leq x \leq x_{N-1}, \\ 1,& x_{N-1} \leq x \leq b,\\ 0, &\text{otherwise} \end{cases} \end{align*} near the boundary. By using the basis functions we can define the approximate chemo-attractant density as \begin{equation*} c_h(x,t) = \sum_{k=1}^N c_i(t) \phi_i(x). \end{equation*} For the construction of the splitting method we define solution operators for both systems \eqref{eq:split_KS_trans} and \eqref{eq:split_ReacDiff}.
To this end we design $T$ to be a numerical solution operator of system \eqref{eq:split_KS_trans} in the following sense: if $(V_h(\tilde t), c_h(\tilde t), m_h(\tilde t))$ is a numerical solution at $t=\tilde t$ then $ T_{\Delta t}(V_h(\tilde t), c_h(\tilde t), m_h(\tilde t)) $ is a numerical solution of system \eqref{eq:split_KS_trans} at time $t = \tilde t + \Delta t$. In the same manner, we define also a solution operator $S$ for system \eqref{eq:split_ReacDiff}.
\subsection{The solution operator $T$ for system \eqref{eq:split_KS_trans}}
For a discretization of the system $\eqref{eq:split_KS_trans}$ we need to evaluate the derivative of the chemo-attractant concentration in the state variable $V$. With this aim we consider an interpolation by cubic splines of the discrete chemo-attractant concentration. Let $(V_h(t), c_h(t), m_h(t))$ be given initial data. By $\hat c_h$ we denote the cubic spline over the data points $(x_k, c_h(t,x_k))$ for $k=1,\dots,N$ that satisfies the boundary conditions $\partial_x \hat c_h(a) = \partial_x \hat c_h(b)=0$. We use this spline for the approximation of the advection term. Concerning the time integration we split the taxis and diffusion terms and treat the stiff diffusion terms implicitly. In this way we allow for both large time steps and stability of the scheme. We apply in particular the two stage implicit-explicit midpoint scheme (see e.g. \cite{pareschi2005implicit}) that reads in our case \begin{subequations} \begin{equation}\label{eq:V_scheme_I} - 2\frac{\tilde V_j (t) - V_j(t)}{\Delta t} = \frac{D_\rho}{\tilde V_{j+1}(t) - \tilde V_{j}(t)} - \frac{D_\rho}{\tilde V_j(t) - \tilde V_{j-1}(t)} - \chi \partial_x \hat c_h(V_j(t)), \end{equation}
\begin{equation}\label{eq:V_scheme_II} - \frac{T_{\Delta t} V_j (t) - V_j(t)}{\Delta t} = \frac{D_\rho}{\tilde V_{j+1}(t) - \tilde V_{j}(t)} - \frac{D_\rho}{\tilde V_j(t) - \tilde V_{j-1}(t)} - \chi \partial_x \hat c_h(\tilde V_j(t)) \end{equation} \end{subequations} both for $j=0, \dots,M$. We have approximated the diffusion terms above by a central difference formula as in \cite{Carillo.2008}. At the boundary we impose Neumann boundary conditions, i.e. $$ \frac{1}{\tilde V_{M+1}(t) -\tilde V_{M}(t)} = \frac{1}{\tilde V_{0}(t) -\tilde V_{-1}(t)}=0\,. $$ The intermediate stage $\tilde V_j(t)$ is given by a nonlinear implicit equation \eqref{eq:V_scheme_I} and we use the Newton's method for its computation. For the computation of the taxis terms in \eqref{eq:V_scheme_I} and \eqref{eq:V_scheme_II} we evaluate the afore determined spline $\hat c_h$.
The chemo-attractant density as well as the mass of the cells are not affected by system $\eqref{eq:split_KS_trans}$, hence we define the numerical operator accordingly by $$ T_{\Delta t} c_h(t) = c_h(t), \quad T_{\Delta t} m_h(t) = m_h(t). $$
Note that if instead of linear diffusion, i.e. $D_\rho$ constant, we have a power-law nonlinear diffusion $D_\rho(\rho)=D_\rho \rho^{\gamma-1}$, $\gamma>1$, modelling cell volume size effects as in \cite{HPVolume,CCVolume}, we obtain a similar approximation \begin{subequations}
\begin{equation}\label{eq:V_scheme_IKSLnD}
- 2\frac{\tilde V_j (t) - V_j(t)}{\Delta t} = \frac{\tilde D(t)}{(\tilde V_{j+1}(t) - \tilde V_{j}(t))^{\gamma}} - \frac{\tilde D(t)}{(\tilde V_j(t) - \tilde V_{j-1}(t))^{\gamma}} - \chi \partial_x \hat c_h(V_j(t)), \end{equation}
\begin{equation}\label{eq:V_scheme_IIKSLnD}
- \frac{T_{\Delta t} V_j (t) - V_j(t)}{\Delta t} = \frac{\tilde D(t)}{(\tilde V_{j+1}(t) - \tilde V_{j}(t))^{\gamma}} - \frac{\tilde D(t)}{(\tilde V_j(t) - \tilde V_{j-1}(t))^{\gamma}}
- \chi \partial_x \hat c_h(\tilde V_j(t))
\end{equation} \end{subequations} with $\tilde D(t)=D_\rho\gamma^{-1}\Delta w(t)^{\gamma-1}$, $j=0,\dots, M$ and similar boundary conditions as above. Remember that the continuous function $T_{\Delta t} V_h(t)$ is built as the linear interpolant of the values $T_{\Delta t} V_{j}(t)$ for $j=0,\dots, M$, and thus we can define a reconstructed density $T_{\Delta t}\rho_h (t)$ by its own definition \begin{equation}\label{eq:recden} T_{\Delta t}\rho_h (t) = \left( \frac{\partial T_{\Delta t}V_h(t)}{\partial w}\right)^{-1} \end{equation} as long as the sequence $V_{j}(t)$ is strictly increasing.
\subsection{The solution operator $S$ for system \eqref{eq:split_ReacDiff}}
In the splitting method that we propose we will apply the reaction-diffusion operator $S$ starting with the data $(T_{\Delta t} V_h(t), T_{\Delta t}c_h( t), T_{\Delta t}m_h(t))$ obtained from a previous evaluation of the operator $T$. For simplicity we will describe the numerical operator $S$ for general initial data $(V_h(t), c_h( t), m_h(t))$.
System \eqref{eq:split_ReacDiff} is formulated for physical concentrations of cells. To provide adequate initial data using the given approximations $(V_h(t), c_h(t), m_h(t))$ we transform the discrete pseudo inverse distribution $V_h(t)$ on $(0,m_h(t))$ to a finite volume representation of $\rho(t, \cdot)$ on $(a,b)$. Since the approximate density $\rho_h$ satisfies \begin{equation*} \int_{V_{j-1}(t)}^{V_j(t)} \rho_h(t,x) \, dx = \Delta w(t), \end{equation*} for all $j=1,\dots,M$ by construction \eqref{eq:recden}, we can introduce the cell averages and the piecewise constant function $\rho_h$ in the following way \begin{equation*} \rho_j(t) = \frac{\Delta w(t)}{V_j(t)- V_{j-1}(t)}, \quad j = 1,\dots,M, \quad \rho_h(t,x) = \sum_{j=1}^{M} \rho_j(t)\chi_{(V_{j-1}(t), V_j(t))}(x). \end{equation*} This approximation of the cell density resides on physical space $(a,b)$. Note though that the cell averages are given on a non-uniform grid which differs from the grid for the chemo-attractant density $c$ given in \eqref{eq:c_grid}.
Now, we are in the position to write down the scheme for system \eqref{eq:split_ReacDiff}. Again we split diffusion from reaction and apply the implicit-explicit midpoint scheme and obtain \begin{subequations}\label{eq:reac_update} \begin{equation}\label{eq:cell_reac_update1} \tilde \rho_j(t) = \rho_j(t) + \frac{\Delta t}{2} R_\rho (\rho_j(t)), \quad j=1, \dots,M, \end{equation} \begin{equation}\label{eq:attr_update1} 2 \varepsilon \,\frac{\tilde c_k(t) - c_k(t) }{\Delta t}\!\int_{a}^{b} \!\!\phi_k \phi_l \,dx = -\tilde c_k(t) D_c \int_{a}^{b} \frac{\partial \phi_k}{\partial x} \frac{\partial \phi_l}{\partial x}\,dx + \int_{a}^{b} \!\!R_c(\rho_h(t), c_h(t)) \phi_l\, dx, \quad k,l = 1,\dots, N-1, \end{equation} \begin{equation} \label{eq:cell_reac_update2} S_{\Delta t}\rho_j(t) = \rho_j(t) + \frac{\Delta t}{2} R_\rho (\tilde \rho_j(t)), \quad j=1, \dots,M, \end{equation} \begin{equation}\label{eq:attr_update2} \varepsilon \,\frac{S_{\Delta t} c_k(t) - c_k(t) }{\Delta t}\!\!\int_{a}^{b} \!\!\phi_k \phi_l \,dx = -\tilde c_k(t) D_c \!\! \int_{a}^{b} \!\frac{\partial \phi_k}{\partial x} \frac{\partial \phi_l}{\partial x}\,dx + \!\!\int_{a}^{b} \!\!\!R_c(\tilde \rho_h(t), \tilde c_h(t)) \phi_l\, dx, \quad k,l = 1,\dots, N-1. \end{equation} \end{subequations} As usual, we employ precomputed integrals of the basis functions $$
\int_{a}^{b} \phi_k \phi_l \,dx \qquad \mbox{and} \qquad \int_{a}^{b} \frac{\partial \phi_k}{\partial x} \frac{\partial \phi_l}{\partial x}\,dx $$ in the computation of the linear systems \eqref{eq:attr_update1} and \eqref{eq:attr_update2}. The integrals of the form
$\int_{a}^{b} R_c(\rho_h(t), c_h(t)) \phi_l\, dx$ are dependent on $V_h(t)$. For their computation we use suitable quadratures together with an indicator function to identify the position of a particular point $x\in [a,b]$ on the grid corresponding to the cell density $\rho_h$. The reaction update in the cell density $c_h$ alters the mass of the cells over the interval $\Omega$. Thus we update $m_h(t)$ by \begin{equation*} S_{\Delta t} m_h(t) = \sum_{j=1}^{M} S_{\Delta t}\rho_j(t) (V_j(t)- V_{j-1}(t)). \end{equation*} To be able to apply the advection-diffusion operator after the reaction-diffusion update we transform $S_{\Delta t}\rho_h(t)$ to its inverse distribution representation $S_{\Delta t} V_j(t)$. Therefore, we use the formula \begin{equation}\label{eq:V_reaction_update} \int_{S_{\Delta t} V_{j-1}(t)}^{S_{\Delta t} V_j(t)} \sum_{j=1}^{M} S_{\Delta t} \rho_j(t)\, \chi_{(V_{j-1}(t), V_j(t))}(x)\, dx = S_{\Delta t} m_h(t) h_w, \quad j = 1,\dots,M. \end{equation} As long as $S_{\Delta t} V_j(t)$ is monotonically increasing in $j$, identity \eqref{eq:V_reaction_update} allows for an efficient update of the inverse distribution $V_h$.
\subsection{The splitting method} To approximate the full system \eqref{eq:systemtype} we propose the classical Strang splitting method \cite{strang} employing both numerical operators defined above. For given non-negative and sufficiently smooth initial conditions $\rho_0$ and $c_0$ of system \eqref{eq:systemtype} we deduce discrete initial data $(V_h(0), c_h(0), m_h(0))$. To compute a discrete representation $V_h(0)$ of the normalized concentration $\rho_0/m_h(0)$ we integrate as in \eqref{eq:V_reaction_update}.
Then we define the fully discrete Strang splitting scheme for system \eqref{eq:systemtype} iteratively by \begin{equation}\label{eq:full_strang} (V_h(t^{n+1}), c_h( t^{n+1}), m_h(t^{n+1})) = T_{\Delta t^n /2} S_{\Delta t^n} T_{\Delta t^n /2} (V_h(t^n), c_h( t^n), m_h(t^n)), \quad n = 0,1,2, \dots , \end{equation} where $0=t^0< t^n = \sum_{i=1}^{n} \Delta t^i$ is a discretization of the time axis. In this way we alternate between applying the diffusion-taxis and the diffusion-reaction operator. The symmetrical structure leads to the second order splitting error.
To optimize the efficiency we adapt the time increment $\Delta t$ in each time step. Since the discretization of system \eqref{eq:split_KS} is more sensitive to instabilities that are caused by large time increments $\Delta t$ than the discretization of the diffusion--reaction system, we start the method in each time step with the numerical operator $T$ in which we determine $\Delta t^n$. We will elaborate on the stability of the scheme in Section \ref{sec:monotonicity}.
The scheme \eqref{eq:full_strang} is not limited to the case of a single pair of a cell and an chemo-attractant. An extension to multiple attractants (i.e. a replacement of $\chi \rho \nabla c$ by a sum $\chi_1 \rho \nabla c_1 + \dots + \chi_n \rho \nabla c_n$ in \eqref{eq:systemtype}) is straightforward. The case of multiple cell densities coupled through the taxis terms, such as in the model discussed in \cite{EMT_paper}, can be treated as well. Note though that each cell species brings along another non-uniform mesh on the domain $(a,b)$ which requires further projections in the numerical operator $S$.
\section{Monotonicity preservation of the diffusion-taxis operator}\label{sec:monotonicity} As demonstrated in \cite{Chertock.2008} unphysical negative values that arise in the numerical solutions of the Keller-Segel type systems can cause instabilities and wrong behavior of the scheme. Hence, the so called \emph{positivity preserving} finite volumes schemes for these kind of models have been developed, e.g. in \cite{Chertock.2008}. A non-negative density $\rho$ implies a monotonously increasing pseudo inverse distribution $V$ by its definition \eqref{eq:pseudoinv}. If a method operates on inverse distributions it should in turn preserve the discrete monotonicity of $V$. This \emph{monotonicity preserving} property of such schemes was studied in the case of filtration and convolution-diffusion equations in \cite{Gosse.2006.Lagrangian, Gosse.2006.Filtration}. In more details, We call a method monotonicity preserving if from $V_j(t)-V_{j-1}(t) > 0$ for all $0<j\leq M$ follows that also $V_j(t+\Delta t)-V_{j-1}(t + \Delta t) > 0$ for all $0<j\leq M$.
In the rest of this section we focus on a simplified problem that motivates a way to adapt the time increment $\Delta t$ in such a way, that non-monotone solutions and thus possible related instabilities are avoided. We consider in particular the system \eqref{eq:split_KS_trans} for the case of a steady chemo-attractant $c \in C^1(a,b)$. For the numerical resolution we consider a forward Euler scheme of the form \begin{equation}\label{eq:forward_simple_diff} V_j (t+\Delta t) = V_j(t) + \Delta t\, \chi \partial_x c(V_j(t)) -\Delta t \left[\frac{\tilde D(t)}{( V_{j+1}(t) - V_{j}(t))^{\gamma}} - \frac{\tilde D(t)}{(V_j(t) - V_{j-1}(t))^{\gamma}} \right], \end{equation} for a discrete inverse distribution as defined in Section \ref{section:num}. This scheme can be understood as an explicit first-order version of the advection-diffusion operator introduced in the previous section. In this setting we can follow the lines of \cite{Gosse.2006.Lagrangian, Gosse.2006.Filtration} and derive a bound on $\Delta t$ that makes the scheme \eqref{eq:forward_simple_diff} monotonicity preserving:
\begin{lemma}\label{lem:cfl} The scheme \eqref{eq:forward_simple_diff} is monotonicity preserving, if for a fixed $\theta \in(0,1)$ both CFL conditions \begin{subequations} \begin{align} \Delta t ^n &< \frac{\theta}{2\, D_\rho\, \Delta w ^{\gamma-1}} \,\min_{0 \leq j < M} \frac{ (V_{j+1}(t^n) - V_j(t^n)) (V_{j}(t^n) - V_{j-1}(t^n))}{\max_{k=j-1,j} ~(V_{k+1}(t^n) - V_k(t^n))^{-(\gamma -1)} }, \label{eq:cfl_diff}\\[1ex]
\Delta t ^n &< \frac{1-\theta}{\chi} \, \min_{0 \leq j < M} \frac{ (V_{j+1}(t^n) - V_j(t^n)) }{ \left| \partial_x c(V_{j+1}(t^n)) - \partial_x c(V_{j}(t^n)) \right|}\label{eq:cfl_chemo} \end{align} \end{subequations}
are satisfied. \end{lemma} \begin{proof}
We consider a single time step in the scheme \eqref{eq:forward_simple_diff} and drop the superscript $n$. For brevity we will use the notation $\Delta V_{j+1/2} = V_{j+1}- V_j$. We assume the monotonicity of the discrete inverse distribution at the time instance $t$ and compute for an arbitrary $0\leq j<M$ the difference
\begin{align*}
\Delta V_{j+1/2}(t + \Delta t) &= \Delta V_{j+1/2}(t) + \Delta t\, \chi \( \partial_x c(V_{j+1}(t)) - \partial_x c(V_j(t))\) \nonumber \\[1ex]
&\quad - \frac{\Delta t \,D_\rho}{\Delta w} \left[\frac{(\Delta w) ^\gamma}{\gamma \, ( \Delta V_{j+3/2}(t))^{\gamma}} - \frac{(\Delta w) ^\gamma}{\gamma \, ( \Delta V_{j+1/2}(t))^{\gamma}} \right] \nonumber \\&\quad+ \frac{\Delta t \,D_\rho}{\Delta w} \left[\frac{(\Delta w) ^\gamma}{\gamma \,(\Delta V_{j+1/2}(t))^{\gamma}} - \frac{(\Delta w) ^\gamma}{\gamma \,( \Delta V_{j-1/2}(t))^{\gamma}} \right].
\end{align*}
By applying the mean value theorem to the function $f(x) = x^{\gamma}/\gamma$ we find two function evaluations of its derivative, $\kappa_{j}$ and $\kappa_{j+1}$, such that we obtain
\begin{align*}
\Delta V_{j+1/2}(t + \Delta t) &= \Delta V_{j+1/2}(t) + \Delta t\, \chi \( \partial_x c(V_{j+1}(t)) - \partial_x c(V_j(t))\) \\[1ex]
&\quad - \Delta t \,D_\rho\, \kappa_{j+1} \left[\frac{1}{\Delta V_{j+3/2}(t)} - \frac{1}{\Delta V_{j+1/2}(t)} \right] \quad+ \Delta t \,D_\rho \,\kappa_{j}\left[\frac{1}{\Delta V_{j+1/2}(t)} - \frac{1}{\Delta V_{j-1/2}(t)} \right].
\end{align*}
Note that by the non-negativity of $\Delta V_{j+1/2}$ both $\kappa_{j}$ and $\kappa_{j+1}$ are non-negative.
In the next step, we define $L_{j+1/2}= (\partial_x c(V_{j+1}(t)) - \partial_x c(V_j(t)))/(V_{j+1}(t) - V_j(t))$ and rewrite
\begin{align*}
\Delta V_{j+1/2}(t + \Delta t) &=\Delta V_{j+1/2}(t) \( 1 + \Delta t\, \chi L_{j+1/2} - \frac{\Delta t \,D_\rho\, \kappa_{j+1}}{\Delta V_{j+3/2}(t) \, \Delta V_{j+1/2}(t)} - \frac{\Delta t \,D_\rho\, \kappa_{j}}{\Delta V_{j+1/2}(t) \Delta V_{j-1/2}(t)}\) \\[1ex]
&\quad + \frac{\Delta t \,D_\rho\, \kappa_{j+1}}{\Delta V_{j+3/2}(t) \, \Delta V_{j+1/2}(t)}\, \Delta V_{j+3/2}(t)
+ \frac{\Delta t \,D_\rho\, \kappa_{j}}{\Delta V_{j+1/2}(t) \Delta V_{j-1/2}(t)} \, \Delta V_{j-1/2}(t).
\end{align*}
Finally we estimate by the monotonicity at time instance $t$
\begin{equation}\label{eq:dv_est}
\Delta V_{j+1/2}(t + \Delta t) \geq \Delta V_{j+1/2}(t) \( 1 - \Delta t\, \chi |L_{j+1/2}| \!- \!\frac{\Delta t \,D_\rho\, \kappa_{j+1}}{\Delta V_{j+3/2}(t) \, \Delta V_{j+1/2}(t)}\! -\! \frac{\Delta t \,D_\rho\, \kappa_{j}}{\Delta V_{j+1/2}(t) \Delta V_{j-1/2}(t)}\)\!.
\end{equation}
By using the conditions \eqref{eq:cfl_diff} and \eqref{eq:cfl_chemo}, the non-negativity of the right hand side in \eqref{eq:dv_est} follows. This implies the monotonicity-preserving property of the scheme \eqref{eq:forward_simple_diff}. \end{proof}
For our splitting method \eqref{eq:full_strang} we assume that we avoid time step restrictions due to the diffusion terms by our implicit treatment and take a closer look on the condition \eqref{eq:cfl_chemo} ($\theta = 0$). The point values of the inverse distribution $V_j(t)$ for $0\leq j \leq M$ coincide with the mesh cell interfaces of the non-uniform mesh corresponding to the cell densities $\rho_h(t)$. Thus the quantity $L_{j+1/2}$ in the proof of Lemma \ref{lem:cfl} can be understood as a finite difference formula for the second derivative of the chemo-attractant density $c$. In effect, the above CFL condition \eqref{eq:cfl_chemo} motivates to choose the time increment $\Delta t^n$ according to \begin{equation}\label{eq:splschemecfl}
\Delta t^n \propto \( \chi \sup_{\{ x \in I\}}|\partial_x ^2 c(x)| \)^{-1}. \end{equation}
For our numerical experiments with the more complex scheme \eqref{eq:full_strang} we have accordingly computed the time increments by \begin{equation}\label{eq:gf_cfl}
\Delta t ^n = CFL ~ \min \left\{\min_{0 \leq j < M} \frac{ (V_{j+1}(t^n) - V_j(t^n)) }{\chi \, \left| \partial_x c(V_{j+1}(t^n)) - \partial_x c(V_{j}(t^n))\right|}, ~ K \Delta w \right\} \end{equation} for constants $CFL, K>0$. The additional bound proportional to $\Delta w$ balances the temporal and the spatial errors; large values of $K$ can be used in practice. We have chosen $CFL = 0.49$ and $K=100$ in our numerical experiments. Using this condition we have not observed any non-monotone numerical solutions in our experiments and no instabilities have occurred.
\section{Numerical experiments}\label{sec:experiments}
In this section we apply our newly developed mass transport method to several models arising in biomedical applications that bring along merging, emerging, and traveling concentrations phenomena. In particular, we consider the classical Keller-Segel model both elliptic and parabolic. We study also a simple as well as a detailed cancer invasion model. The latter takes the role of the serine protease urokinase-type plasminogen activator into account. The numerical study of such systems constitutes a challenge due to the complex behavior that the solutions exhibit. Numerical experiments presented below demonstrate the robustness and reliability of our newly developed mass transport finite element method.
\subsection{A parabolic-elliptic Keller-Segel model with logistic growth} In the first test case we consider the modified KS system from \cite{Calvez.2007} with added logistic growth which reads \begin{equation}\label{eq:PKS_lgg}\left\{ \begin{aligned} \partial_t \rho &= \partial_x \( \partial_x \rho - \chi \rho \partial_x c\) + \mu \rho (1- \rho)& \text{in }(0,\infty) \times \mathbb{R}, \\
c(\cdot, x) &= - \frac{1}{\pi} \int_\mathbb{R} \log(|x-y|)\rho(\cdot,y)\,dy &\text{in }(0,\infty) \times \mathbb{R},\\ \rho(0,\cdot) &= \rho_0 \geq0. \end{aligned}\right. \end{equation} Note that the adaptation of system \eqref{eq:PKS_gf} to $2D$ with $\mu =0$ is equivalent to the simplified Keller-Segel system from \cite{jager1992explosions}, where the chemo-attractant $c$ is determined by a Poisson equation. The logistic term accounts for additional cell growth that is locally limited by resources and space. Global existence of solutions to the parabolic-parabolic model with logistic growth in 2D was shown in \cite{osaki2002exponential}. Except for the logistic source term this model has been numerically investigated by the mass transport scheme employing only inverse distributions in \cite{Carillo.2008}.
Since the chemo-attractant density $c$ is given by a convolution term, we do not need to use a finite element approximation. Instead we proceed as in \cite{Carillo.2008} and expand the diffusion taxis operator by \begin{subequations}\label{eq:V_scheme_IKS}
\begin{equation}\label{eq:V_scheme_IKSL}
- 2\frac{\tilde V_j (t) - V_j(t)}{\Delta t} = \frac{1}{\tilde V_{j+1}(t) - \tilde V_{j}(t)} - \frac{1}{\tilde V_j(t) - \tilde V_{j-1}(t)} +\frac{\chi\, \Delta w}{ \pi} \lim\limits_{\varepsilon \rightarrow 0} \sum_{i: |\tilde V_j(t) - \tilde V_i(t)|\geq \varepsilon} \frac{1}{\tilde V_j(t)- \tilde V_i(t)}, \end{equation}
\begin{equation}\label{eq:V_scheme_IIKSL}
- \frac{T_{\Delta t} V_j (t) - V_j(t)}{\Delta t} = \frac{1}{\tilde V_{j+1}(t) - \tilde V_{j}(t)} - \frac{1}{\tilde V_j(t) - \tilde V_{j-1}(t)}
+\frac{\chi\, \Delta w}{ \pi} \lim\limits_{\varepsilon \rightarrow 0} \sum_{i: |\tilde V_j(t) - \tilde V_i(t)|\geq \varepsilon} \frac{1}{\tilde V_j(t)- \tilde V_i(t)}.
\end{equation} \end{subequations} During the computation of \eqref{eq:V_scheme_IKSL} we control the convergence of the Newton method by comparing subsequent iterates. If the iteration fails to converge, we abort the computation assuming blowup of the numerical solution. The time increments in these experiments have been adapted such, that the Jacobian of \eqref{eq:V_scheme_IKSL} that occurs in the Newton iteration is strictly diagonally dominant. The second operator $S$ in this setting accounts only for the logistic growth term. For the numerical simulations we use a grid with only $M=50$ points.
We consider a numerical experiment with the parameters $D_\rho = 1,~ \chi =2.5 \pi$ and the initial datum given by \begin{equation}\label{eq:V_singlePeak_ini} V_0(w) = \frac{w- 0.5}{\sqrt[4]{(w + 0.01)\, (1.01 - w)}}. \end{equation} This experiment has been studied in the case $\mu =0$ in \cite{Carillo.2008}, where blowup in final time around $t=0.33$ has been obtained numerically. We confirm the same phenomenon using the splitting method, see Figure \ref{fig:exp431}. The blowup was indicated by the method during the computation.
When conducting the experiment with altered $\mu = 0.2$, no blowup occurs, as can be seen in Figure \ref{fig:exp431prolif}. The aggregation stops and reverses since the logistic term attracts the cell concentration to a lower density. The total mass of the cells decreases after the aggregation stops and increases again after around $t=1.5$. No blowup could be observed even for later times, instead the numerical solution seems to converge to a stationary state. The CFL condition given by \eqref{eq:splschemecfl} has caused an increase of the time increment over the computation time.
\begin{figure}
\caption{Numerical results (cell concentration and inverse cumulative function) for the parabolic elliptic Keller-Segel model, experiment 4.3.1 in \cite{Carillo.2008}. The cell concentration blows up. The numerical cell concentration has attained a maximum of approximately $176$. }
\label{fig:exp431}
\end{figure}
\begin{figure}
\caption{Numerical results (cell concentration, inverse cumulative function, and mass) for the parabolic elliptic Keller-Segel model with added logistic growth \eqref{eq:PKS_lgg}. The additional reaction term has prevented blowup.}
\label{fig:exp431prolif}
\end{figure}
\subsection{Nonlinear diffusion and chemotaxis models} Our method can also resolve models that include generalized diffusion and migration terms as we will demonstrate in this section. To this end we consider at first the model \begin{equation}\label{eq:PKS_varm}\left\{ \begin{aligned} \partial_t \rho &= \partial_x \( \rho^{\gamma-1} \partial_x \rho - \chi \rho \partial_x c\)& \text{in }(0,\infty) \times \mathbb{R}, \\
c(\cdot, x) &= - \frac{1}{\pi} \int_\mathbb{R} \log(|x-y|)\rho(\cdot,y)\,dy &\text{in }(0,\infty) \times \mathbb{R},\\ \rho(0,\cdot) &= \rho_0 \geq0 \end{aligned}\right. \end{equation} with exponent $\gamma>1$. In the case $\chi=0$ the first equation in \eqref{eq:PKS_varm} is known as the \emph{porous media equation} modeling the gas flow through a porous interface. We refer to \cite{Vazquez.2007} for an introduction to the subject.
Similar as in \eqref{eq:V_scheme_IKS}, the scheme to resolve \eqref{eq:PKS_varm} corresponds to \eqref{eq:V_scheme_IKSLnD}-\eqref{eq:V_scheme_IIKSLnD} where the chemo-attractant gradient is computed as \begin{equation}\label{chemoconv}
- \partial_x \hat c_h(V_j(t)) = \frac{\Delta_w(t)}{\pi} \lim\limits_{\varepsilon \rightarrow 0} \sum_{i: |V_j(t) - V_i(t)|\geq \varepsilon} \frac{1}{ V_j(t)- V_i(t)}. \end{equation}
\begin{figure}
\caption{Numerical results (cell concentration and inverse cumulative function) for the nonlinear diffusion model \eqref{eq:PKS_varm}, initial condition \eqref{eq:V_singlePeak_ini}, $\chi=2.5\pi$ and $\gamma=2$ (top row), $\gamma=1.5$ (bottom row). For both chosen values of $m$ the numerical solution converges to a steady state. We have used $M=500$ points.}
\label{fig:nonlinear_diff}
\end{figure}
We have tested our scheme using again the initial condition \eqref{eq:V_singlePeak_ini} and the chemo-sensitivity $\chi=2.5\pi$. Figure \ref{fig:nonlinear_diff} exhibits the results from the numerical simulation for the exponents $\gamma =2$ and $\gamma=1.5$. In both cases the nonlinear diffusion prevents the blowup that would occur for $\gamma =1$ and the numerical solution converges to a stationary state.
Another model that we consider here has been proposed in \cite{HPVolume}. In this work the authors endowed the Keller-Segel model with a volume filling mechanism. For this purpose they reconsidered the derivation of the model from a random walk and added a function $q(\rho)$ describing the probability that a cell finds sufficient space to jump to a particular position. We adopt here the probability function $q(\rho) = 1- \rho^\gamma$ that models the volume filling together with enhanced diffusion for $\gamma >1$ and reduced diffusion for $\gamma<1$ \cite{HPVolume}. Independent of the choice of $\gamma>1$, the model does not allow for cell migration to a position where the maximal density $\rho=1$ has already been reached. The corresponding model for the cell density includes nonlinear diffusion and advection terms and reads \begin{equation}\label{eq:vol_filling}
\partial_t \rho = \partial_x \( D_\rho (1 + (\gamma -1)\rho^\gamma) \partial_x \rho - \chi (1-\rho^\gamma) \rho \partial_x c\)\qquad \text{in }(0,\infty) \times \mathbb{R}. \end{equation}
For the numerical experiments with the volume filling model \eqref{eq:vol_filling} we have adapted the update steps \eqref{eq:V_scheme_IKSLnD} and \eqref{eq:V_scheme_IIKSLnD} in the diffusion-advection operator by \begin{subequations}\label{eq:vol_filling_scheme}
\begin{align*}
- 2\frac{\tilde V_j (t) - V_j(t)}{\Delta t} &= \frac{1}{\tilde V_{j+1}(t) - \tilde V_{j}(t)} - \frac{1}{(\tilde V_j(t) - \tilde V_{j-1}(t))}
+\frac{(\gamma -1)\tilde D(t)}{(\tilde V_{j+1}(t) - \tilde V_{j}(t))^{\gamma}} - \frac{(\gamma -1)\tilde D(t)}{(\tilde V_j(t) - \tilde V_{j-1}(t))^{\gamma}} \nonumber \\[1ex]
& \quad - \chi \left[1 - \(\frac{2 \, \Delta w}{V_{j+1}(t) - V_{j-1}(t)}\)^\gamma\right] \partial_x \hat c_h( V_j(t)),
\end{align*}
\begin{align*}
- \frac{T_{\Delta t} V_j (t) - V_j(t)}{\Delta t} &= \frac{1}{\tilde V_{j+1}(t) - \tilde V_{j}(t)} - \frac{1}{(\tilde V_j(t) - \tilde V_{j-1}(t))}
+\frac{(\gamma -1)\tilde D(t)}{(\tilde V_{j+1}(t) - \tilde V_{j}(t))^{\gamma}} - \frac{(\gamma -1)\tilde D(t)}{(\tilde V_j(t) - \tilde V_{j-1}(t))^{\gamma}} \nonumber \\[1ex]
& \quad - \chi \left[1 - \(\frac{2 \, \Delta w}{V_{j+1}(t) - V_{j-1}(t)}\)^\gamma\right] \partial_x \hat c_h( \tilde V_j(t)),
\end{align*} \end{subequations} where we set $\frac{2 \, \Delta w}{V_{j+1}(t) - V_{j-1}(t)}=0$ for $j=1$ and $j=M$ to account for the boundary conditions. \begin{figure}
\caption{Numerical results (cell concentration, inverse cumulative function) for the model \eqref{eq:vol_filling} with chemo-attractant given by a convolution as in \eqref{eq:PKS_lgg}. Results are shown for $\gamma=2$ (top row) and $\gamma=0.5$ (bottom row). We have used $M=50$ points in the numerical computation.}
\label{fig:vol_filling}
\end{figure}
In Figure \ref{fig:vol_filling} we present simulation results with the parabolic-elliptic model \eqref{eq:PKS_varm} where we have replaced the original evolution equation of the cell density by the volume filling approach \eqref{eq:vol_filling}. Again we have used the initial condition \eqref{eq:V_singlePeak_ini} and the chemo-sensitivity parameter $\chi=2.5\pi$. We exhibit the numerical results for $\gamma=2$ and $\gamma=0.5$. The computed cell densities do not exceed a density of one in both cases and no blowup occurs. Instead the cells evolve quickly to a bounded spatial profile from which they slowly diffuse afterwards. The parameter choice $\gamma = 2$ leads to a larger maximal cell density throughout the computation when compared to the case $\gamma=0.5$.
\subsection{The parabolic-parabolic Keller-Segel model} In this section we apply our scheme to the well known parabolic-parabolic Keller-Segel system which reads \begin{equation}\label{eq:PKS_gf}\left\{ \begin{aligned} \partial_t \rho &= \partial_x \( D_\rho \partial_x \rho - \chi \rho \partial_x c\),&\text{in }(0,\infty) \times (a,b), \\ \partial_t c &= D_c \partial_x ^2 c + \rho - c, &\text{in }(0,\infty) \times (a,b),\\
\partial_x \rho(\cdot, r) &= \partial_x c(\cdot, r)=0, &r\in\{a,b\},\\ \rho(0,\cdot) &= \rho_0 \geq0,\quad c(0,\cdot)= c_0 \geq 0.& \end{aligned}\right. \end{equation} As opposed to \eqref{eq:PKS_lgg} this system features an additional parabolic equation to be treated by the splitting method. In order to exhibit the phenomena that the scheme can resolve, we consider two test cases with distinct initial chemo-attractant densities that control the cell movement. In both tests we adopt the initial datum \eqref{eq:V_singlePeak_ini} for the inverse distribution $V$.
\subsubsection*{Peak movement} For our first numerical experiment with the system \eqref{eq:PKS_gf} we use the parameters $D_\rho =0.1, ~D_c = 0.01,~\chi = 2.5,~\alpha = 0.5,~\beta = 1$ and the domain $\Omega=(a,b)$ with boundaries chosen $a=V_0(0)\approx-1.58,~b=V_0(1)\approx1.58$. As initial chemo-attractant concentration we take the logistic function \[ c_0(x) = \frac{1}{1+ \mathbf e^{-5\, x}}, \quad x \in \Omega.\] For the simulation we employ meshes with $M=45$ and $N=230$ points and the CFL condition \eqref{eq:gf_cfl}.
Figure \ref{fig:movement} presents the cell dynamics, showing their movement to the right side of the domain. As the cells produce the chemical with density $c$, a negative gradient is created that leads to an aggregation of the cells which counteracts the movement. We point out that both the migration and the growth are well resolved by the splitting scheme.
\begin{figure}
\caption{Numerical results (cell concentration, inverse cumulative function, and chemo-attractant density in space and time) for the parabolic-parabolic KS model \eqref{eq:PKS_gf} in the \enquote{peak movement} experiment. The movement and aggregation is accurately resolved using $M=45$ grid points.}
\label{fig:movement}
\end{figure}
\subsubsection*{Peak splitting} In the next test we use the parameters $D_\rho =D_c = 0.1,~\chi = 5,~\alpha = \beta = 1$ and the computational domain $(a,b)$ with boundaries chosen as in the \enquote{peak movement} experiment. The initial chemo-attractant density though is replaced by the function \[c_0(x) = 1 - \mathbf e^{-20\, x^2}, \quad x \in (a,b).\] Figure \ref{fig:splitting} shows the computational results for the discretization parameters $M=90$ on the mass space mesh and $N=450$ on the Finite Element mesh. The cells move out of the center of the domain on which the most part of the attracting chemical is already consumed. The symmetrical movement to both sides leads to a splitting of the initial concentration into two peaks. The discretization grid for the cells on the density level concentrates its grid points on the locations of both peaks and adapts to the solution over time.
\begin{figure}
\caption{Numerical results (cell concentration, inverse cumulative function, and chemo-attractant density in space and time) for the parabolic-parabolic KS model \eqref{eq:PKS_gf} in the \enquote{peak splitting} experiment. At the final time we show the approximated cell averages of the density $\rho$ at their respective position on the grid (top). The grid for the cell density adapts to the two splitting peaks.}
\label{fig:splitting}
\end{figure}
In the setting of the present experiment we study the convergence of the introduced splitting scheme experimentally. For a fixed instance in time and given $M$, let $V^h_i,~i=1,\dots,M-1$ denote a numerical solution corresponding to the mesh discretization parameter $M$. Then we define the approximate $L^1$ finite difference error by \begin{equation}\label{eq:fd_error}
E^V_M = \frac{1}{M} \sum_{j= 1}^{M-1} \left| V^{h}_j - V^{h/2}_{2j} \right|, \end{equation} where we have used a numerical solution on a finer mesh with $2M$ points, $V^{h/2}_j,~j=1,\dots,2M-1$, as the reference solution. The \emph{experimental order of convergence} (EOC) for the discretization error in $V_h$ can now be defined by \begin{equation}\label{eq:eoc_v} EOC^V(M) = log_2(E^V_M) - \log_2(E^V_{M/2}) \end{equation} for any even integer $M$. Similarly, we define the EOC for the cell densities on their non-uniform mesh. To this end let $\rho^h_i,~i=1,\dots,M$ denote the finite volume representation corresponding to $V^h_i,~i=1,\dots,M-1$ and let $E^\rho(M)$ denote the discrete $L^1$ error using as reference $\rho^\text{ref}_i,~i=1,\dots,2M$ the finite volume representation of $V^{h/2}_j,~j=1,\dots,2M-1$. This $L^1$ error is computed by projecting the reference solution to the coarser non-uniform grid corresponding to the cell densities $\rho^h_i,~i=1,\dots,M$. Then we define for even integers $M$ analog to \eqref{eq:eoc_v} \[EOC^\rho(M) = log_2(E^\rho_M) - \log_2(E^\rho_{M/2}).\]
\begin{table}
\centering
\begin{tabular}{r | r r |r r}
\Tstrut \Bstrut $M$/$M^\text{ref}$& error $E^V_M$ & $EOC^V$ & error $E^\rho_M$ & $EOC^\rho$ \\ \hline
20 / 40 & 7.231e-04 & & 1.360e-02 & \Tstrut\\
40 / 80 & 7.427e-05 & 3.283 & 1.719e-03 & 2.984 \\
80 / 160 & 1.416e-05 & 2.391 & 2.768e-04 & 2.635 \\
160 / 320 & 4.698e-06 & 1.592 & 7.085e-05 & 1.966 \\
320 / 640 & 1.615e-06 & 1.541 & 2.591e-05 & 1.451 \\
\end{tabular}
\caption{Mesh convergence in the peak splitting experiment up to $T = 0.01$ with respect to the discretization parameter $M$. We have adapted the number of points on the Finite Element mesh by $N= M$. The EOCs approach two in the inverse distributions and in the corresponding densities. Yet, for large $M$ the EOC drops which is probably due to limitations by the Finite Element mesh.} \label{tbl:EOC_peak_splitting} \end{table}
In Table \ref{tbl:EOC_peak_splitting} we exhibit the errors and the EOCs computed at the final time $T=0.01$ and with constant time increment $\Delta t = 10^{-4}$ when doubling the mesh resolution on the mass space mesh iteratively. We have coupled the resolution of the Finite Element mesh to the number of points for the inverse distribution by using $N= M$. We can clearly see that the method converges as the mesh size is refined. The EOCs indicate a convergence order of two in both, the inverse distributions and the densities. However, we see that the EOC decreases as the grids becomes very fine. We suppose that this is caused by the finite element mesh that is only uniformly but not locally refined: as $M$ increases the number of mesh cells on the non-uniform finite volume mesh for the cell densities aggregates around the positions of the peaks. Throughout the computation the finite element solution $c_h$ must in turn be interpolated in many points in a small physical area which leads to a loss of accuracy as the number $\max_i|\{j: x_i\leq V_j\leq x_{i+1}\}|$ increases. Nevertheless, Table \ref{tbl:EOC_peak_splitting} demonstrates that the method has provided accurate numerical results using only a few mesh points.
\subsection{A cancer invasion system}\label{sec:cancer1} In this test case we address a model of cancer invasion of the extracellular matrix (ECM), the first step in cancer metastasis. The macroscopic modeling of this process commonly uses an Keller-Segel approach that models the densities of the cancer cells, the concentration of the extracellular fibers on which cancer cells adhere and move, and the density of an enzyme of the matrix metallopreteinases (MMPs) family that is produced by the cancer cells and is responsible for the degradation of the ECM.
There is a wide variety of cancer invasion models in the literature, see e.g. \cite{Chaplain.2005, EMT_paper, Preziosi.2003, Stinner.2015, Johnston.2010}. In order to test our scheme we employ a simple test case based on the pioneering model \cite{Anderson.2000} augmented with a proliferation term in the cancer cell density equation which reads
\begin{equation}\label{eq:splitting_invasion}\left\{ \begin{aligned} \partial_t \rho &= \partial_x \( D_\rho \partial_x \rho - \chi \rho \partial_x v\)+ \mu \rho(1-\rho)&\text{in }(0,\infty) \times (a,b), \\ \partial_t v &= - \delta v m &\text{in }(0,\infty) \times (a,b), \\ \partial_t m &= D_m \partial_x ^2 m + \alpha \rho - \beta m &\text{in }(0,\infty) \times (a,b)\\
\partial_x \rho(\cdot, r) &= \partial_x m(\cdot, r)=0, &r\in\{a,b\},\\ \rho(0,\cdot) &= \rho_0 \geq0,\quad v(0,\cdot)=v_0, \quad m(0,\cdot)= m_0. & \end{aligned}\right. \end{equation}
In this model the cancer cells with the density $\rho$ move using their motility apparatus with a preferred direction towards higher concentrations of the ECM with concentration denoted by $v$. This is the haptotaxis phenomenon. Being a network in a static equilibrium the ECM does not translocate. The MMPs however, whose density we denote by $m$, diffuse freely in the extracellular environment. Additionally, the cancer cells proliferate towards a preferred density $\rho=1$ and they produce MMPs with a constant rate. The MMPs attach to the ECM which they dissolve upon contact.
\begin{figure}
\caption{Numerical results (cell concentration, inverse cumulative function, tissue and MMP density) for the cancer invasion model \eqref{eq:splitting_invasion} using only $N=M=45$ mesh points. At the final time we show the approximated cell averages of the tumor density $\rho$ at their respective position on the grid (top left). A high concentration of tumor cells emerges and invades the tissue. The grid for the tumor density omits the part of the tissue that is not yet invaded.}
\label{fig:gradflow_cancer}
\end{figure}
For a numerical experiment with this model we consider the computational domain $(0,1)$ with initial conditions \begin{equation}
\rho_0(x) = \mathbf e^{-x^2 / \varepsilon}, \quad
v_0(x) = 1- 0.5\, \mathbf e^{-x^2 / \varepsilon},\quad
m_0(x) = 0.5 \, \mathbf e^{-x^2 / \varepsilon}, \end{equation} where we use $\varepsilon = 10^{-2}$. Moreover we employ the parameter values $D_c = 2 \cdot 10^{-4}, ~ \chi = 5 \cdot 10^{-3},~\mu = 0.2,~D_a= 10^{-3}, ~\delta = 10, ~\alpha = 0.1,$ and $\beta = 0$. We apply the splitting scheme \eqref{eq:full_strang} using meshes of $M=N=45$ points and the CFL condition \eqref{eq:splschemecfl}.
In our method we discretize both the ECM density $v$ and the MMP concentration $m$ on the same finite element basis. The corresponding approximations are updated in the reaction-diffusion operator of the splitting method. The interpolations are only needed with respect to the ECM density $v$. We resolve the migration of the cancer cells in transformed variables with the advection-diffusion operator and the cell proliferation in original variables with the reaction-diffusion operator.
The considered numerical experiment simulates the propagation of cancer cells into the ECM on the right side of the computational domain. To account for the corresponding temporal expansion of the support of the cancer cell density $c$ we have adapted the treatment of the right boundary. In more details, we have neglected the discrete cancer cell density entry adjacent to the right boundary in the proliferation update \eqref{eq:reac_update}, i.e. $S_{\Delta t}\rho_M(t) = \rho_M(t) $. Though we have not excluded the corresponding boundary entry in the cumulative function, $V_{M-1}$, from the diffusion and haptotaxis updates of the scheme.
We present the according numerical results in Figure \ref{fig:gradflow_cancer}. Apart from the propagation of the cells into the tissue, we observe a build up of cancer cells at the leading front of the tumor. Degradation of the tissue and MMP production are also visible. Throughout the computation the not invaded part of the tissue is resolved by a single grid cell in the cancer cell density.
\begin{table}
\centering
\begin{tabular}{r | r r }
\Tstrut \Bstrut $\Delta t$/$\Delta t^\text{ref}$& error $E^V_{\Delta t}$ & $EOC^t$ \\ \hline
\Tstrut 0.1/ 0.05 & 2.244e-04 & \\
0.05/ 0.025 & 4.728e-05 & 2.247 \\
0.025 / 0.0125 & 1.107e-05 & 2.094 \\
0.0125 / 0.00625 & 2.766e-06 & 2.001 \\
\end{tabular}\hspace{2em}
\begin{tabular}{r | r r}
\Tstrut \Bstrut $M$/$M^\text{ref}$& error $E^V_M$ & $EOC^V$ \\ \hline
\Tstrut 10 / 20 & 7.867e-03 & \\
20 / 40 & 1.919e-03 & 2.035 \\
40 / 80 & 4.475e-04 & 2.100\\
80 / 160 & 1.514e-04 & 1.563 \\
\end{tabular}
\caption{Experimental convergence in time (left) and in space (right) in the numerical experiment with system \eqref{eq:splitting_invasion} at $T = 1$. In all computations we have set $N= M$. The EOCs suggest a convergence of second order in time and space.} \label{tbl:EOC_invasion_gf} \end{table} In this experiment we have also studied the convergence of the scheme experimentally. Along with the errors in space, we have also computed the errors in time by the formula \begin{equation}\label{eq:fd_error_time}
E^V_{\Delta t} = \frac{1}{M} \sum_{j= 1}^{M-1} \left| V^{h,\,\Delta t}_j - V^{h,\,2\Delta t}_{j} \right|, \end{equation} where $V^{h,\,\Delta t}_i,~i=1,\dots,M-1$ denotes a numerical solution computed on $M$ mesh points with constant time increment $\Delta t$. For the computation of the temporal errors we have considered a fine spatial resolution with $M=N=600$ mesh cells. The corresponding EOC is given by $EOC^t=log_2(E^V_{\Delta t}) - \log_2(E^V_{2\Delta t})$. The spatial errors and EOCs are computed according to \eqref{eq:fd_error} and \eqref{eq:eoc_v} with constant time increment $\Delta t = 2 \times 10^{-4}$ and coupled $N=M$. Both, temporal and spatial errors have been computed at the final time $T=1$.
In Table \ref{tbl:EOC_invasion_gf} we present the computed errors and EOCs in the invasion experiment. We see that the method converges as either the mesh size or the time increment is refined. The EOCs in time and space range around two which confirms our expected second order. As in the \enquote{peak splitting} experiment, the EOC decreases slowly as the mesh is refined to very high resolutions. We point out that previous numerical tests which did not employ our proposed boundary treatment have yield only a spatial EOC of one.
\subsection{The uPA model}\label{sec:uPA} In the last series of experiments we apply our scheme to a detailed tumor invasion system derived in \cite{Chaplain.2005}. This model focuses on the enzymatic \emph{urokinase plasminogen activator} (uPA) system which is known to play an essential role in the context of cancer progression and metastasis. The uPA is an extracellular serine protease which is responsible for the activation of the protease plasmin. This activation occurs mainly if uPA is bound to its uPA receptors (uPAR) on the cancer cell membrane. The receptor bound uPA enhances the affinity of uPAR to the ECM constituent vitronectin \cite{Wei.1994} and integrins. Thus, the uPA/uPAR-complex regulates indirectly also the vitronectin-integrin interactions. Both proteases plasmin and uPA catalyze the degradation of vitronectin and other ECM components. Another actor in the system is the plasminogen activator inhibitor type 1 (PAI-1) which is produced by the tumor cells and limits the activation of plasmin to prevent tissue damage and to maintain homeostasis.
The considered model complements the system \eqref{eq:splitting_invasion} by chemotactic movement of the cells due to uPA and PAI-1, remodeling of the ECM modeled by a logistic term and the dynamics of the uPA system modeled in terms of mass-action kinetics. We refer to \cite{Chaplain.2005} for more details. The full model reads
\begin{equation}\label{eq:uPA} \left\{ \begin{aligned} \partial_t \rho &= \partial_x \( D_\rho \partial_x \rho\right. \left. - \chi_u \rho\partial_x u - \chi_p \rho \partial_x p - \chi_v \rho\partial_x v\) +\mu_1\rho(1-\rho) &\text{in }(0,\infty) \times (a,b),\\ \partial_t v & = - \delta vm + \phi_{21}up - \phi_{22}vp + \mu_2v(1-v) &\text{in }(0,\infty) \times (a,b),\\ \partial_t u &= D_u \partial_x ^2 u -\phi_{31}up - \phi_{33}\rho u + \alpha_3 \rho &\text{in }(0,\infty) \times (a,b),\\ \partial_t p &= D_p \partial_x ^2 p - \phi_{41}up - \phi_{42}vp + \alpha_{4}m &\text{in }(0,\infty) \times (a,b),\\ \partial_t m &= D_m \partial_x ^2 m + \phi_{52} vp + \phi_{53}\rho u - \alpha_{5}m &\text{in }(0,\infty) \times (a,b),\\
\partial_x \rho(\cdot, r) &= \partial_x u(\cdot, r)= \partial_x p(\cdot, r)=\partial_x m(\cdot, r)=0, &r\in\{a,b\},\\ \rho(0,\cdot) &= \rho_0,\quad v(0,\cdot)=v_0 , u(0,\cdot)= u_0\quad p(0,\cdot)= p_0, \quad m(0,\cdot)= m_0, & \end{aligned} \right. \end{equation} where the cancer cell concentration is represented by $\rho$, the ECM by the density of its constituent vitronectin $v$, and uPA, PAI-1, and plasmin densities are denoted by $u$, $p$, and $m$. We assume non-negative initial data.
We consider a numerical experiment that we have studied in \cite{Urokinase_paper} by a Finite Volume method. It employs the parameter values from \cite{Andasari.2011} given by \begin{equation*} \begin{array}{lll} D_c = 3.5 \times 10^{-4}, & \chi_u = 3.05\times 10^{-2}, & \mu_1 = 0.25, \\ D_u = 2.5\times 10^{-3}, & \chi_p=3.75\times 10^{-2}, & \mu_2=0.15, \\ D_p=3.5\times 10^{-3}, & \chi_v = 2.85\times 10^{-2}, & \delta=8.15, \\ D_m=4.91\times 10^{-3}, & \phi_{21}=0.75, & \phi_{22}= 0.55,\\ \phi_{31}=0.75, & \phi_{33}=0.3, & \phi_{41}=0.75, \\ \phi_{42}=0.55, & \phi_{52}=0.11, & \phi_{53}=0.75, \\ \alpha_3 = 0.215, & \alpha_4 = 0.5, & \alpha_5=0.5, \end{array} \end{equation*} and the computational domain $I=(0,10)$ with the initial date \begin{align*}
c_0(x) &= \mathbf e^{-x^2/\varepsilon},
&& v_0(x)= 1- \frac 1 2 \mathbf e^{-x^2/\varepsilon},
&&u_0(x) = \frac 1 2\mathbf e^{-x^2/\varepsilon},\\
p_0(x) &= \frac {1}{20} \mathbf e^{-x^2/\varepsilon}, &&m_0(x) = 0, &&\varepsilon= 5 \times 10^{-3}. \end{align*}
As done to treat the model \eqref{eq:splitting_invasion} we use a single finite element basis to discretize the concentrations of the ECM, the uPA, the PAI-1, and the plasmin. The cubic spline in the advection-diffusion operator interpolates the linear combination $\chi_v v+ \chi_u u + \chi_p p$. Similar as in the models \eqref{eq:PKS_gf} and \eqref{eq:splitting_invasion} the scheme approximates the cell proliferation in Eulerian coordinates but diffusion and advection of the cancer cells in transformed variables. We have used the same boundary treatment as in Section \ref{sec:cancer1}.
\begin{figure}
\caption{Numerical results (cancer cell concentration, inverse cumulative function, ECM, uPA, PAI-1 and plasmin density in space and time) in an numerical experiment with the model \eqref{eq:uPA} computed by the new scheme. The dynamics, particularly the steep peaks in the cancer cell density, are well resolved by the scheme. We have used $M=N=400$ grid points on both meshes in the numerical simulation.}
\label{fig:gradflow_chaplol}
\end{figure}
In Figure \ref{fig:gradflow_chaplol} we present the simulation results obtained by our scheme with mesh parameters $M=N=400$. The method is capable to approximate accurately the dynamics that we have obtained in \cite{Urokinase_paper} including the emergence and movement of multiple steep peeks. The present simulation clearly demonstrates the robustness of the newly developed scheme to simulate complex taxis-diffusion systems arising in cell biology.
To investigate the dynamics of such a cancer invasion system in the case that the cell migration is restricted by the occupied extracellular space we have endowed the model \eqref{eq:uPA} with the volume filling approach \eqref{eq:vol_filling}. In more details we have replaced the evolution equation for the tumor cell density in \eqref{eq:uPA} by \begin{equation}\label{eq:uPA_vol_filling}
\partial_t \rho = \partial_x \( D_\rho (1 + (\gamma -1)\rho^\gamma)\partial_x \rho - (1-\rho^\gamma)(\chi_u \rho\partial_x u + \chi_p \rho \partial_x p + \chi_v \rho\partial_x v)\) +\mu_1\rho(1-\rho) \end{equation} and resolved the same numerical experiment as above. To this end the scheme has been adapted in a similar way as in \eqref{eq:vol_filling_scheme}. \begin{figure}
\caption{Numerical results (cancer cell concentration, inverse cumulative function, ECM, uPA, PAI-1 and plasmin density in space and time) in the model \eqref{eq:uPA} with volume filling by \eqref{eq:uPA_vol_filling} and exponent $\gamma=2$. We have used $M=N=400$ grid points on both meshes in the numerical simulation.}
\label{fig:chaplol_vf2}
\end{figure} \begin{figure}
\caption{Numerical results (cancer cell concentration, inverse cumulative function, ECM, uPA, PAI-1 and plasmin density in space and time) in the model \eqref{eq:uPA} with volume filling by \eqref{eq:uPA_vol_filling} and exponent $\gamma=0.5$. We have used $M=N=400$ grid points on both meshes in the numerical simulation.}
\label{fig:chaplol_vf0_5}
\end{figure}
In Figures \ref{fig:chaplol_vf2} and \ref{fig:chaplol_vf0_5} we show the simulation results for the exponents chosen $\gamma=2$ and $\gamma=0.5$, where we have used $M=N=400$ mesh points in the computation. Contrary to the simulations without volume filling, the cancer cells do not exhibit the rich dynamics, i.e. the formation of multiple clusters. Instead a single concentration of tumor cells invades the ECM and leaves a homogeneous distribution of tumor cells of maximal density $\rho=1$ behind. Reducing the diffusivity of the cells by decreasing the exponent $\gamma$ results in a slower invasion of the tissue and to a lower concentration at the invading front of tumor cells. This can be seen when comparing Figure \ref{fig:chaplol_vf2} ($\gamma=2$) and Figure \ref{fig:chaplol_vf0_5} ($\gamma=0.5$).
To study how the new method compares in efficiency to more conventional numerical methods we consider again the above experiment without volume filling. For the comparison we consider the Finite Volume/Finite Difference from \cite{Urokinase_paper} for both uniform and adaptive meshes. In particular we have chosen a second order method with implicit-explicit Strang operator splitting. For the adaptive mesh refinement (AMR) method we have chosen the gradient monitor function to determine the mesh-cells to be either refined or coarsened\footnote{In more details we have used the refinement and coarsening threshold values $\theta_\text{ref} = 10,~\theta_\text{coars} = 2.5$, a single refinement and coarsening operation per time step $n_\text{ref}=n_\text{coars}=1$ and a maximal refinement level of $l_\text{max}=2$, cf \cite{Urokinase_paper}.}. For brevity, we will refer to the adaptive method as AMR and to the uniform method as FVFD. The new mass-transport/finite element method will be denoted by MTFE.
\begin{figure}
\caption{Relation between the CPU time and the error (left) and between the (average) number of cells and the error (right) for the FVFD, AMR, and MTFE scheme in log-log scale in a numerical experiment with the uPA model \eqref{eq:uPA}. The new MTFE method seems to be most efficient in terms of error per CPU time, its relation between the error and the average number of cells is similar as in the FVFD scheme.}
\label{fig:mtfe_vs_amr}
\end{figure}
For our comparism we consider the set $S= \{ 40, 80, 160, 320, 640, 1280\}$ and run the MTFE method for $M \in S$, the FVFD method for $N=6k$ for any $k \in S$, and the AMR method for $N_0 \in S$ with $N_0$ denoting the number of cells on the lowest level. We couple the two meshes in the MTFE scheme by setting $N= M$. We do not consider finer resolutions due to restrictions by the uniform reference solution in the error computations of solutions obtained by the MTFE scheme. For comparison reasons we let $N$ denote the average number of cells in the AMR method. In addition, all three methods employ the same Courant number $CFT=0.49$ and all numerical solutions are computed on the domain $\Omega = (0,5)$.
We compute the numerical solutions of the considered experiment at the time instance $t=23$ that features two steep peaks in the cancer cell concentration. In this process we measure the CPU time that is needed for the corresponding simulations and compute the error of the approximation at the final time. For the error computation we have used a reference solution that employs a uniform mesh with cell size $h=1.25 \times 10^{-5}$ in the relevant part of the domain\footnote{We have computed a uniform solution in $(0,2)$ with our uniform method using $N=160\,000$ mesh cells.}. The discrete $L^1$ error is then computed with respect to the densities using a suitable projection of the reference solution. Note that the following test results are dependent on our (non-reference) implementation of the numerical methods.
We show the results of our comparison in Figure \ref{fig:mtfe_vs_amr}. Here we present the relation between the error and the computation time and the relation between the error and the average number of cells for all three methods. We see that for all tested methods the error decreases as either the cell number or the CPU time increases. Figure \ref{fig:mtfe_vs_amr} (left) exhibits an advantage of the new MTFE method over the other schemes in efficiency for most of the conducted simulations. This can be seen as the MTFE method achieves in most cases lower errors than the FVFD or the AMR scheme using the same CPU time. As the runtime increases the MTFE method approaches the efficiency of the AMR method with the new method being at a slight advantage over the mesh refinement method. Clearly, the AMR and the MTFE scheme both outperform the FVFD method for sufficiently large CPU times.
Figure \ref{fig:mtfe_vs_amr} (right) shows that the AMR method achieves the lowest errors when compared with simulations by the FVFD and MTFE scheme employing the same average number of cells. The error of the MTFE scheme has a similar dependence on the number of cells as the error of the FVFD scheme. We conjecture thus that the better efficiency of the MTFE scheme in terms of CPU time seen in Figure \ref{fig:mtfe_vs_amr} (left) is probably caused by the CFL condition in the MTFE scheme allowing for larger time steps compared to the FVFD method.
\section{Conclusion} In this paper we have proposed a new splitting scheme for one-dimensional reaction-taxis-diffusion systems related to the Keller-Segel system. The solutions of these systems are well known for having concentrated and diffusive regions simultaneously. In addition, traveling waves and merging phenomena typically occur.
Our splitting has separated a part of the model which is mass conservative in the cell density from the rest of the system. The latter has been approximated by a classical linear finite element method, whereas the approximation of the conservative part has been based on the mass transport strategy. More precisely, we have first transformed the cell density to the corresponding pseudo-inverse cumulative distribution. Then we have discretized the transformed system by the finite difference method and used a cubic spline to account for the chemo-attractant whose evolution is described in the rest subsystem. The splitting method is described in Section~\ref{section:num}. In Lemma~\ref{lem:cfl} we have studied the stability of the explicit mass transport method for the conservative part in which we allowed for general nonlinear diffusion. The obtained result has been used to derive a time-step restriction for our scheme.
In Section \ref{sec:experiments} we have presented a series of numerical experiments demonstrating the robustness and reliability of the scheme. In particular, we have used the new method to resolve the Keller-Segel model in the parabolic-elliptic and in the parabolic-parabolic form numerically. We have applied our scheme also to augmentations of these systems by reaction terms, nonlinear diffusion and a volume filling approach. The method has resolved the movement, splitting and aggregation phenomena accurately. We have verified the mesh convergence of the scheme in both time and space in an application to a simple tumor invasion system in Section \ref{sec:cancer1}. The obtained experimental order of convergence has ranged around two spatially and temporally. Moreover, we have applied the scheme to the uPA-tumor invasion model from \cite{Chaplain.2005} in Section \ref{sec:uPA}. The proposed hybrid mass transport finite element scheme has been capable to resolve its complex dynamics featuring multiple peaks in the cancer cell concentration without using a fine spatial discretization. By the help of our new method we could also study a combination of the uPA model with the volume filling approach from \cite{HPVolume}. In addition, we have compared the efficiency of the hybrid mass transport finite element method with a finite volume scheme with adaptive mesh refinement from \cite{Urokinase_paper}. The hybrid mass transport finite element method has not only outperformed the uniform finite volume scheme but it has also delivered slightly better results than the finite volume scheme equipped with adaptive mesh refinement.
\section*{Acknowledgments} JAC was partially supported by the Royal Society via a Wolfson Research Merit Award and by EPSRC grant number EP/P031587/1. NK was supported by the Max-Planck Graduate Center of the University Mainz. The research of ML was partially supported by the German Science Foundation (DFG) under the grant TRR 146 \enquote{Multiscale simulation methods for soft matter systems}.
\end{document} | arXiv |
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ROBOMECH Journal
Research Article | Open | Published: 02 May 2019
Flying watch: an attachable strength enhancement device for long-reach robotic arms
Siyi Pan ORCID: orcid.org/0000-0002-3839-60931 &
Gen Endo1
ROBOMECH Journalvolume 6, Article number: 5 (2019) | Download Citation
A long-reach robotic arm is useful for applications such as nuclear plant decommissioning, inspection, and firefighting. However, for such arms, a small external reaction force can result in large loads on proximal arm actuators because of long moment arms. This problem was previously solved by specialized arm designs that compensate external reaction forces. However, these arm designs are hard to be applied to other arms or customized to different missions. To overcome these difficulties, in this paper, we propose a modular thrust generating concept inspired by wristwatch designs, called flying watch, which can be attached to a robotic arm with mission-dependent attachment styles (attachment positions and orientations) and cooperate with actuators to enhance arm strength. We first introduce flying watch concept, design, and dynamics. Then we propose two levels of watch-actuator cooperation in quasistatic situations by introducing a problem called Watch Actuator Cooperation for Arm Enhancement (WACAE) and providing an example solution. The first level of cooperation is only watches adapt their thrusts to minimize actuator loads, which is generally applicable to varieties of arms. The second level cooperation is that not only do the watches adapt their thrusts but also the actuators cooperatively position the watches to optimal positions and orientations to counteract external reaction forces, which is suitable for redundant arms and can counteract external reaction forces more effectively. Finally, we present simulations to verify that flying watches can significantly reduce actuator loads using both levels of watch-actuator cooperation (the first level by 36.9% and the second level by 43.7%).
A long-reach robotic arm is usually designed for applications such as nuclear power plant decommissioning, [1, 2] inspection, [3,4,5], and firefighting [6]. Some arms, such as [2,3,4], are redundant since long arms for those applications usually need to operate in environments filled with obstacles and the redundant Degree of Freedom (DOF) can negotiate the obstacles.
One major problem for designing and operating a long-reach arm is that a small external reaction force exerted on the arm may result in a large load on proximal actuators that can exceed their actuation ability because of long moment arms. Here the external reaction forces are reaction forces from the external environment that include gravity.
To solve this problem, previous studies focus on designing specialized arms with certain force compensation mechanisms. These arms can be classified into two classes, passive force compensating arms and active force compensating arms, based on whether an energy source is required for force compensation. Regarding passive force compensating arms, in [3], a 20-m-long arm with helium balloon body was proposed to carry small loads like camera. The torque of gravity was passively counteracted by the buoyancy of helium. In [4], a 9.5-m-long modular arm with spring based weight compensation mechanisms was built for inspection of an ITER related nuclear fusion device. Additionally, some gravity compensating arm designs based on springs [7, 8] and weight [9] may also be used for designing passive force compensating arms. Regarding active force compensating arms, in [1, 2, 10], specialized coupled tendon-driven arms that can compensate gravity were proposed. In [5, 6], arms that apply water jets to counteract gravity were proposed. In [11, 12], arms with passive joints driven by thrusts to counteract external reaction forces were proposed.
However, specialized arm designs have two major problems. (1) They are difficult to be customized to different missions. (2) Their designs are difficult to be applied to other long arms. For example, the arm designed in [11] can counteract gravity using thrusts, which is useful for carrying sensors and picking up objects. However, if we want the arm to push aside objects on the ground, the external reaction force is horizontal and the arm may not be able to provide sufficient end effector forces. Therefore, the arm in [11] cannot be customized to different missions. For pushing objects aside, the arm design in [12] is suitable since it can counteract external reaction forces in different directions using thrusts. However, adapting the arm design in [12] to the existing arm in [11] requires significant mechanism modification on the arm in [11].
Therefore, how to more easily give a long arm higher strength to counteract larger external reaction forces and better versatility to accomplish more missions is an interesting and practical question. In order to answer this question, in this paper, we propose a modular concept called flying watch as shown in Fig. 1, which can be attached to a robotic arm using mission-dependent attachment styles and enhance arm strength by cooperating with arm actuators. The structure of this paper is as follows. We will first propose the flying watch concept in "Flying watch concept" section. Then we will present the design of flying watch in "Flying watch design" section. Next we will introduce the dynamics of flying watch in "Flying watch dynamics" section. After that, we will propose two levels of watch-actuator cooperation for arm enhancement in quasistatic situations in "Flying watch based arm enhancement" section. Then, we will use simulations to verify the effectiveness of flying watches in quasistatic situations in "Simulation" section. Finally, this paper will be concluded in "Conclusion" section.
A flying watch attached to one link of a robotic arm. A wristwatch is included in the figure to demonstrate the similarity between flying watch and wristwatch and the size of the flying watch
The contributions of this paper are as follows. (1) Proposing a modular attachable concept, called flying watch, that can enhance arm strength by attaching to the arm and adapt to different missions by changing its attachment styles. (2) Proposing two levels of arm enhancement watch-actuator cooperation in quasistatic situations by introducing a problem called Watch Actuator Cooperation for Arm Enhancement (WACAE) and providing an example solution. (3) Verifying through simulations that flying watch can effectively reduce the maximum arm actuator loads through watch-actuator cooperation.
Flying watch concept
People with different wrist dimensions and demands (such as time keeping and health monitoring, etc.) wear different wristwatches. In order to adapt to users' diverse wrist dimensions and demands, wristwatches come with different sizes and functions and watch bands are adjustable and usually replaceable.
Similarly, robotic arms have different designs and are demanded to achieve different missions. Inspired by the wristwatch designs and thrust actuated arm designs [5, 6, 11, 12], we imagine there could be a watch-like thrust generating module that also come with different sizes, actuation abilities, and adjustable and replaceable watch bands so that such modules can be attached to different robotic arms with mission-dependent attachment styles and cooperate with arm actuators to enhance arm strength. We call this concept flying watch.
Flying watch concept is further illustrated in Fig. 2. Multiple flying watches are attached to a long-reach robotic arm. All joints of the long arm are actuated by motors and the flying watches generate thrusts in cooperation with the existing arm motors to enhance arm strength (or in other words, reduce the maximum motor loads). The position and orientation of watch thrusts depend on attachment styles of the watches. By changing the attachment style, multiple watches can enhance the same arm in different ways to adapt to different missions. For example, when the long arm in Fig. 2 is required to pick up a heavy object as shown in Fig. 2a, four flying watches are attached with their thrust generating axes on vertical plane in order to counteract gravity. When the long arm is required to push heavy debris on the ground in different directions as shown in Fig. 2b, two watches are attached with thrust generating axes on the vertical plane and the other two watches are attached with thrust generating axes on the horizontal plane. Such attachment style helps counteracting arm gravity and horizontal reaction forces from debris.
a Flying watches help a long arm pick up an object. b Flying watches help a long arm push debris on the ground in different directions. Rotation axes of red flying watches are on horizontal planes. Rotation axes of green flying watches are on vertical planes
Our concept is closely related to researches about specialized robotic arm designs actuated by propeller thrusts [11, 12] or water jets [5, 6]. However, our concept is different from these previous researches in two aspects. Firstly, these previous researches focused on specialized arms which cannot easily adapt to different missions or be applied to other arms. Instead of focusing on arm designs, our study is targeted on an attachable module that can enhance robotic arms and can adapt to different missions by changing its attachment styles. The concept has a watch-like design in order to be attached to robotic arms and a compact sandwich-like propeller arrangement (two propellers wrapping an attached link). Such concept and design have never been proposed by previous research. Secondly, the purpose of the arm thrusters in those previous research is to drive passive joints instead of reducing actuator loads. On the contrary, the purpose of flying watch is to reduce actuator loads by cooperating with existing arm actuators. Such cooperation between thrusters and arm actuators is never explored in previous research. We will explore such cooperation by proposing a problem call WACAE and providing an example solution.
Additionally, our concept is also relevant to researches about aerial transformation and manipulation such as [13, 14] because propeller thrusts are applied in combination with actuators. Our concept is different from those researches in that our flying watch is designed only for manipulators with bases on the ground.
Flying watch design
Design requirements
As an initial physical experiment in the future, we plan to test flying watches on an existing planar arm in our lab, called Planar Inspection Arm (PIA). This arm is chosen for our future experiment because it is originally designed for inspecting only horizontal plane using end effector sensors and the arm motors are generally not strong enough to support the arm weight when the workspace is not horizontal. In order to demonstrate flying watches can enhance the strength of an existing arm, we will attach two flying watches to PIA to enhance its strength so that it can work on a 30° slope plane with regard to horizon plane. PIA and the planned attachment style are shown in Fig. 3 and the specification of PIA is shown in Table 1. The flying watches are attached on the middle of links.\(l_{1}\), \(l_{2}\), and \(l_{3}\) are corresponding link length. \(m_{1}\), \(m_{2}\), and \(m_{3}\) are corresponding link mass. \(T_{1}\) and \(T_{2}\) are the thrusts of flying watch 1 and flying watch 2. In this paper, we will design a flying watch prototype that can accomplish this mission.
PIA and the attachment style of the flying watches for future physical experiments
Table 1 PIA specification
Therefore, based on flying watch concept mentioned in "Flying watch concept" section and for future physical verifications, we can summarize the following design requirements. (1) The flying watch should have adjustable and removable watch band and be able to be attached to a robotic arm. (2) The flying watch should be able to generate enough thrusts for future physical verifications on PIA. (3) The flying watch as a module should be lightweht m. (4) The flying watch should be low-cost and easy to build so that it is easily available for the public.
Propulsion mechanism design
Previous works of thrust driven arm designs [11, 12] installed propellers on extruded bar structures, which can be naturally derived from drones. In particular, in [12], four propellers are installed on a cross shape extruded structure with different orientations as an actuation unit, which can generate thrusts in different directions. However, these propeller arrangements using extruded structures are not compact and increase the chance of collisions. For compactness and simplicity, we used a sandwich-like propeller arrangement (two propellers wrapping an attached link) for flying watch and each flying watch only generate thrusts along one axis. The two propellers rotate in opposite directions to cancel reaction torques.
The counter-rotating propellers can be driven by one motor with certain transmission mechanisms. However, since the distance between the propellers needs to adapt to the attached arm diameter and designing transmission mechanisms that can adapt to such distance change is difficult, we decided to use two motors to independently drive the two propellers. Applying independent motors also gives a flying watch robustness to motor failure and allows the rotation speed of the propellers to be controlled independently, which may negotiate aerodynamic interference between the two propellers.
Based on the mentioned design, we selected two Tarot 4114 motors with Tarot 1555 carbon fiber propellers for a flying watch. The propeller has a diameter of 381 mm which can be fit on PIA links. Based on the required current of Tarot 4114 motors, Two FLYFUN-40A-V5 motor drivers are selected to independently drive the motors. The motor driver is lightweight (46 g) and small (\(48 \times 28 \times 14\) mm) and should be attached close to a flying watch to avoid power loss on cables. Regarding power supply, instead of adopting built-in batteries like a wristwatch, we choose to connect flying watch with external power source at the arm base using cables because batteries suitable for flying watches are too heavy. Regarding control signals, they can be transmitted wirelessly using Bluetooth or Zigbee. However, for simplicity, we used wires for our first prototype.
Attachment mechanism design
In a typical wristwatch design, a length adjustable shaft is used as a joint between a watch case and a watch band. The length adjustable shaft can be removed in order to replace watch band. The watch band length can be adjusted by watch band closures such as buckles and loop and hook fasteners. Similar to the wristwatch design and for 3D printability, we used fixed bar structures for watch case joints and loops of double-sided loop and hook tape as watch bands as shown in Fig. 4. The watch case has two fixed bars for mounting watch bands, three holes for mounting a motor and a dent to let motor cables go through. Also the watch case is glued with a rubber pad on the back to increase friction. The watch case dimensions are derived so that it can be attached to PIA links. The loop and hook tape (TRUSCO) used for watch band has high strength and durability and negligible tensile elongation. The watch band is easily replaceable and can be adjusted by changing its overlapping length. As a result, the watch can be attached to PIA or other similar arms.
a The design of a flying watch. b A prototype flying watch
The flying watch design is lightweight, easy to build, and low-cost. The prototype in Fig. 4 has a mass of 450 g The total additional mass resulting from a flying watch is 542 g (one watch + 2 motor drivers). The total mass of a link and a joint of PIA is 1152 g. Therefore, the flying watch is lightweight compared with the mass of PIA. The watch case can be 3D printed and all other parts are easy to find in market. The assembly process of a flying watch takes about 10 min and the total cost of a flying watch is about 210 US dollars.
Thrust and watch band strength test
In order to ensure that flying watch can generate enough thrust for future physical verifications on PIA and the watch band have enough strength, we need test the property of thrust generation and the watch band.
We first compute how much thrust is necessary for the physical experiment on PIA. Assuming there is no friction and the arm mass is equally distributed and the arm is fully supported by flying watches, from Fig. 3, we can obtain the following quasistatic equations for PIA when it is horizontally stretched (the required thrust is the largest in this configuration).
$$\left( {m_{1} + m_{2} + m_{3} } \right)g^{\prime} \times \frac{1}{2}\left( {l_{1} + l_{2} + l_{3} } \right) = T_{1} \times \left( {l_{1} + \frac{1}{2}l_{2} } \right) + T_{2} \times \left( {l_{1} + l_{2} + \frac{1}{2}l_{3} } \right)$$
$$m_{3} g^{\prime} \times \frac{1}{2}l_{3} = T_{2} \times \frac{1}{2}l_{3}$$
In (1) and (2), \(g^{\prime} = gsin\left( {\frac{\pi }{6}} \right)\) is the component of gravity acceleration along the 30° slope. We can calculate from (1) and (2) that the minimum thrust of a flying watch should be 9.72 N.
After obtaining the minimum required thrust, we tested the thrust of the propeller. The propeller thrust is controlled by the duty ratio of PWM signal given to the motor driver. Also the voltage given to the motor driver can influence the thrust. We used a device as shown in Fig. 5 to test the relation between propeller thrust/torque and PWM duty ratio of the motor driver under different voltages given to the motor driver. We found when working at the maximum input voltage of the motor driver (25.2 V), the motor easily overheats and loses speed due to heat protection. We tested several voltages and finally selected 14.8 V for flying watches. With this voltage, the propeller can generate a maximum thrust of 9.7 N with a reaction torque of 0.21 Nm. The reaction torque of one propeller is not negligible compare to the maximum torque of the joints, which justifies our design of canceling reaction torque using two counter-rotating propellers. With two propellers, a watch can approximately double the maximum thrust and generate 19.4 N thrust, which is enough for supporting PIA on a 30° slope.
A device for testing the thrusts and reaction torques of a flying watch propeller
Now we test the adhesive strength of the loop and hook watch band. Since two propellers wrap an arm link, the thrust from one propeller is always supported by arm link. Therefore, the watch band only need to endure the thrust of the other propeller. Also the watch band needs to generate enough pressure so that the friction between the watch and arm link prevents the watch sliding. In our test, we fixed one side of a watch band (a loop of double-sided loop and hook tape) and use a force gauge to pull the other side of the band until the overlapping part separates. We found that 10 mm overlapping length can endure 56.5 N shearing force, which is enough for enduring the thrust. In practice, we used 46 mm overlapping length for flying watch band and found it is appropriate to ensure enough friction to avoid sliding. For bigger flying watch with higher thrust, it is also possible to use lashing straps to build watch bands.
We successfully attached two flying watches to PIA for demonstration as shown in Fig. 6. Both flying watches can generate thrusts properly.
PIA attached with two prototype flying watches
Flying watch dynamics
In this section, we derive the Equation of Motion (EOM) of a fully actuated arm attached with \(N_{f}\) flying watches. Such EOM helps understanding how flying watch can cooperate with arm actuators to enhance arm strength.
We start from the Lagrange's Equation of Motion
$$\frac{d}{dt}\left( {\frac{\partial E}{{\partial \dot{\varvec{q}}_{k} }}} \right) - \frac{\partial E}{{\partial \varvec{q}_{k} }} = \varvec{Q}_{k}$$
\(E\) is the total kinetic energy of the arm. \(t\) is time. \(\varvec{ q}\) is the generalized coordinate describing arm configuration and its k th component is \(\varvec{q}_{k}\). \(\varvec{Q}\) is the generalized force and its \(k\) th component is \(\varvec{Q}_{k}\).
The virtual forces \(\varvec{Q}\) in (3) need to be substituted with actual forces or torques in order to make (3) more useful. The relation of virtual forces and actual forces can be find through the definition of virtual work as follows.
$$- \nabla U\left( q \right)^{\text{T}} \delta \varvec{q} +\varvec{\tau}^{{\mathbf{T}}} \delta \varvec{q} + \varvec{F}_{e}^{\varvec{T}} \delta \varvec{p} + \sum\nolimits_{k} {\varvec{T}_{k}^{T} \delta \varvec{r}_{k} = \varvec{Q}^{T} \delta \varvec{q}}$$
\(U\) is the total potential energy of the arm. \(\varvec{\tau}\) is the actuator load vector, whose real dimensions depend on the actuator mechanism (e.g. Nm for revolute actuators and N for prismatic actuators). \(\varvec{F}_{e}\) is the external reaction force and torque on the end effector. \(\varvec{p}\) is the position and orientation of the end effector. \(\varvec{T}_{k}\) is the thrust generated by the \(k\)th flying watch. \(\varvec{r}_{k}\) is the position vector of the \(k\)th flying watch.
To simplify (4), we need to substitute \(\delta \varvec{p}\) and \(\delta \varvec{r}_{k}\) with changes of generalized coordinates \(\updelta\varvec{q}\). This can be done using the following kinematics relations.
$$\delta \varvec{p} = \varvec{J}\delta \varvec{q}$$
$$\delta \varvec{r}_{k} = \varvec{J}_{k}^{f} \delta \varvec{q}$$
J and \(\varvec{J}_{k}^{f}\) are the jacobian matrices of the end effector and the \(k\)th flying watch. Substituting \(\delta \varvec{p}\) and \(\delta \varvec{r}_{k}\) in (4) using (5) and (6), (4) can be simplified as
$$- \nabla U\left( \varvec{q} \right) +\varvec{\tau}+ \varvec{J}^{T} \varvec{F}_{e} + \sum\nolimits_{k} {\varvec{J}_{k}^{fT} \varvec{T}_{k} = \varvec{Q}} \text{.}$$
The kinetic energy in (3) can be written as
$$E = \frac{1}{2}\dot{\varvec{q}}^{\varvec{T}} \varvec{M\dot{q}} .$$
\(\varvec{M}\) is the manipulator inertia tensor. Substituting \(E\) and \(\varvec{Q}\) in (3) using (8) and (7), (3) can be written as
$$\varvec{M}\left( \varvec{q} \right)\varvec{\ddot{q}} + \varvec{h}\left( {\varvec{q},\dot{\varvec{q}}} \right) - \varvec{J}^{\varvec{T}} \left( \varvec{q} \right)\varvec{F}_{e} - \sum\nolimits_{k} {\varvec{J}_{k}^{fT} \left( \varvec{q} \right)\varvec{T}_{k} + \varvec{g}\left( \varvec{q} \right) =\varvec{\tau}} \text{.}$$
In (9), to simplify the expression, we made the following definitions. The \(k\)th component of the velocity term \(\varvec{h}\) is
$$\varvec{h}_{k} \left( {\varvec{q},\dot{\varvec{q}}} \right) = \sum\nolimits_{ij} {\left( {\frac{{\partial \varvec{M}_{kj} }}{{\partial \varvec{q}_{i} }} - \frac{1}{2}\frac{{\partial \varvec{M}_{ij} }}{{\partial \varvec{q}_{k} }}} \right)\dot{\varvec{q}}_{j} \dot{\varvec{q}}_{i} } .$$
And the gravity term is
$$\varvec{g}\left( \varvec{q} \right) = \nabla U\left( \varvec{q} \right).$$
Since a flying watch is rigidly attached to the arm, the thrust from the \(k\)th flying watch can be written as
$$\varvec{T}_{k} = s_{k} \varvec{a}_{k} \left( \varvec{q} \right).$$
\(s_{k}\) and \(\varvec{a}_{k} \varvec{ }\) are the scale and the unit direction vector of the thrust (minus sign of \(s_{k}\) indicates the thrust direction is opposite to \(\varvec{a}_{k}\)). We can substitute \(\varvec{T}_{k}\) in (9) using (12) and obtain
$$\varvec{M}\left( \varvec{q} \right)\varvec{\ddot{q}} + \varvec{h}\left( {\varvec{q},\dot{\varvec{q}}} \right) - \varvec{J}^{\varvec{T}} \left( \varvec{q} \right)\varvec{F}_{e} - \sum\nolimits_{k} {s_{k} \varvec{J}_{k}^{fT} \left( \varvec{q} \right)\varvec{a}_{k} \left( \varvec{q} \right) + \varvec{g}\left( \varvec{q} \right) =\varvec{\tau}} .$$
To further simplify (13), we can define an attachment style matrix
$$\varvec{R}\left( \varvec{q} \right) = \left[ {\varvec{J}_{1}^{fT} \left( \varvec{q} \right)\varvec{a}_{1} \left( \varvec{q} \right),\varvec{ } \ldots ,\varvec{ J}_{k}^{fT} \left( \varvec{q} \right)\varvec{a}_{k} \left( \varvec{q} \right), \ldots ,\varvec{J}_{N}^{fT} \left( \varvec{q} \right)\varvec{a}_{N} \left( \varvec{q} \right)} \right].$$
And (13) is simplified as
$$\varvec{M}\left( \varvec{q} \right)\varvec{\ddot{q}} + \varvec{h}\left( {\varvec{q},\dot{\varvec{q}}} \right) - \varvec{J}^{T} \left( \varvec{q} \right)\varvec{F}_{e} - \varvec{R}\left( \varvec{q} \right)\varvec{s} + \varvec{g}\left( \varvec{q} \right) =\varvec{\tau}\text{.}$$
In (13), \(\varvec{s} = \left[ {s_{1} , \ldots , s_{k} } \right]^{T}\) is the collection of thrust magnitudes of all \(N\) flying watches. Equation (15) is EOM of a fully actuated arm with \(N_{f}\) flying arm attached.
Equation (15) is different from typical EOM of robotic arm in that it has an additional term \(- \varvec{R}\left( \varvec{q} \right)\varvec{s}\) that represents the effect of a group of flying watches to actuator loads. \(\varvec{R}\), the attachment style matrix, is a function of generalized coordinates \(\varvec{q}\) parameterized by the attachment style parameters (the positions and orientations of flying watches referring to local frames of arm links). It incorporates all information about the position and orientation of flying watches. \(\varvec{s}\), on the other hand, incorporates all information about thrust magnitudes.
Now we have understood the dynamics of flying watch in general cases. In the following sections, we will explore watch-actuator cooperation in quasistatic situations. In such situations, (15) can be simplified as
$$- \varvec{J}^{T} \left( \varvec{q} \right)\varvec{F}_{e} - \varvec{R}\left( \varvec{q} \right)\varvec{s} + \varvec{g}\left( \varvec{q} \right) =\varvec{\tau}.$$
Flying watch based arm enhancement
Two levels cooperation
In previous sections, we introduced the concept, design, and dynamics of flying watches. In this section, based on flying watch dynamics, we will discuss two levels of cooperation between watch and arm actuators for arm enhancement in quasistatic situations.
In order to enhance arm strength, the loads of the actuators need to be reduced using the effects of flying watches. As understood in "Flying watch dynamic" section, the effect of flying watch on actuator loads depends on two factors. The position and orientation of flying watches (as represented by \(\varvec{R}\)) and the thrust magnitudes (as represented by \(\varvec{s}\)). From these two factors, we can imagine two levels of cooperation between watch and arm actuators. The first level is only flying watches adapt their thrust magnitudes to minimize the actuator loads. In such case only the magnitudes of flying watch thrusts are optimized. The second and higher level is not only do watches adapt their thrusts but also actuators corporately position the watches to the optimal positions and orientations to minimize the actuator loads. In such cases, both watch thrust magnitudes and watch positions and orientations are optimized. This higher level of cooperation would more effectively reduce the actuator loads. However, it cannot be applied when the arm is not redundant or when the operator wants more control on the arm configuration. We formulate the mentioned two levels cooperation problem as follows and call such problem Watch Actuator Cooperation for Arm Enhancement (WACAE).
WACAE
$$\mathop {\hbox{min} }\limits_{{\varvec{s}, \varvec{q}}} C\left( {\varvec{s},\varvec{q}} \right) \quad s.t. \varvec{s}_{l} \le \varvec{s} \le \varvec{s}_{h} ,\varvec{q}_{l} < \varvec{q} < \varvec{q}_{h} , \varvec{p}_{d} = f\left( \varvec{q} \right)$$
\(\varvec{s}_{l}\) and \(\varvec{s}_{h}\) are the lower and upper bounds of thrust magnitudes \(\varvec{s}\). \(\varvec{q}_{l}\) and \(\varvec{q}_{h}\) are the lower and upper bounds of generalized coordinates \(\varvec{q}\) resulting from mechanism constraints. \(\varvec{p}_{d}\) is the desired end effector position and orientation. \(f\) is a function representing forward kinematics. For the first level cooperation, generalized coordinates \(\varvec{q}\) can be regard as a constant. \(C\) is the maximum normalized actuator load as follows. The \(i\)th component of \(\varvec{\tau}^{\varvec{*}}\) is defined as \(\varvec{\tau}_{\varvec{i}}^{\varvec{*}} =\varvec{\tau}_{\varvec{i}} /\varvec{\tau}_{\varvec{i}}^{\varvec{m}}\). \(\varvec{\tau}_{\varvec{i}}\) is the load that the \(i\)th joint withstands and \(\varvec{\tau}_{\varvec{i}}^{\varvec{m}}\) is the maximum load that the \(i\)th joint can withstand. The normalized actuator load for a joint represents how much potential for withstanding load that joint has used up. The joint with the maximum normalized actuator load is the bottleneck when the arm is counteracting external reaction forces. Therefore, we define the cost function as the maximum normalized actuator load.
$$C = max\left( {\left| {\varvec{\tau}^{\varvec{*}} } \right|} \right)$$
In words, WACAE is about optimizing thrust magnitudes and generalized coordinates constrained by forward kinematics, watch thrust capacity, and arm mechanism in order to minimize the maximum normalized actuator load. The WACAE solution makes the watches cooperate with arm actuators to reduce actuator loads. This problem is not explored by previous researches [5, 6, 11, 12] which only use thrusts to drive passive joints. However, WACAE is important since its solution can be used to enhance the strength of a long robotic arm. We already know some long arms need this kind of enhancement. For example, our lab developed a 10-m long arm with a payload capacity of 10 kg called super dragon [2]. We hope to improve its payload so that it can carry more sensors and we think that is achievable by attaching flying watches.
An example solution
WACAE is a nonconvex and nonlinear optimization problem. For such problem, a local optimal solution can theoretically be found by several existing optimization solvers, such as SNOPT [15] and LOQO [16], as well as arm configuration optimization methods EEIK [17] and ODLS [18]. Since proposing a new optimization solver or comparing the performance of existing optimization methods is not the interest of this paper, for simplicity, we used a modification of our previous methods EEIK [17] and ODLS [18] to solve WACAE. In the following, we present this modification as an example solution of WACAE in order to show the solvability of WACAE and for future comparison and improvement. The major difference between the example WACAE solution and EEIK and ODLS is that the example solution includes additional procedures to optimize thrusts and handle upper and lower bounds of thrusts and generalized coordinates.
In order to make this paper self-contained, we will first briefly explain Lockable Inverse Kinematics (LIK) and Automatic Optimizable Dimension Searching (AODS) proposed in [17, 18], which are components of EEIK and ODLS. After that we will present the example solution of WACAE.
Firstly, regarding LIK, it is about solving inverse kinematics while fixing certain generalized coordinates to given values. LIK is previously solved by Lockable Damped Least Squares method (LDLS) in [17, 18]. However, there is no bound on generalized coordinates. Those bound are necessary for solving WACAE in this paper. Therefore, we write LIK as the following optimization problem and use interior point method to solve it. For simplicity, we express the process of solving LIK as \(\varvec{q} = {\text{LIK}}\left( {\varvec{\mu},\varvec{ L},\varvec{ p}_{d} ,\varvec{V}_{l} ,\varvec{V}_{h} } \right)\).
$$\mathop {\hbox{min} }\limits_{\varvec{V}} \left| {\varvec{p}_{d} - f\left( {\varvec{V};\varvec{L}} \right)} \right|_{2} \quad s.t. \varvec{V}_{l} < V < \varvec{V}_{h}$$
\(\varvec{\mu}\) is the index of the locked generalized coordinates. \(\varvec{V}\) is the unlocked generalized coordinates and \(\varvec{L}\) is the locked generalized coordinates. \(\varvec{q}\) is a vector including all generalized coordinates. \(f\left( {\varvec{V};\varvec{L}} \right)\) is the forward kinematics, which is a function of \(\varvec{V}\) parameterized by \(\varvec{L}\). \(\varvec{V}_{l}\) and \(\varvec{V}_{h}\) are vectors respectively representing the lower and upper bounds of \(\varvec{V}\). The optimization problem tries to minimize the Euclidean distance between the desired and current end effector positions and orientations given the bounds on unlocked generalized coordinates \(\varvec{V}\).
We will also need Inverse Kinematics (IK) with generalized coordinate bounds to compute arm configuration when arm configuration does not need optimization. Similarly, we can write IK with bounds as the following optimization problem and use interior point method to solve it. \(f\left( \varvec{q} \right)\) is still the forward kinematics, which we express it as a function of \(\varvec{q}\).
$$\mathop {\hbox{min} }\limits_{\varvec{q}} \left| {\varvec{p}_{d} - f\left( \varvec{q} \right)} \right|_{2} \quad s.t. \varvec{q}_{l} < q < \varvec{q}_{h}$$
Secondly, regarding AODS, it is a method to automatically search for optimizable dimensions of the generalized coordinates. More specifically, given the desired end effect position and orientation \(\varvec{p}_{d}\) and the arm configuration \(\varvec{q}\), AODS searches for \(N_{r}\) dimensions of generalized coordinates indexed by \(\varvec{\xi}\) that can be optimized to minimize a certain cost function C.
The pseudocode of AODS is shown in Algorithm 1. Initially, in step 1, AODS uses LIK to add a small value \(\in\) to each dimension of \(\varvec{q}\) and record how much the cost function changes compared to the added small value. \(i^{c}\) means the complement index of \(i\), which is the index of all generalized coordinates other than \(i\). For a certain dimension, if the change is not zero, this dimension can influence the cost function and is optimizable. Then in step 2, \(N_{r}\) optimizable dimensions of \(\varvec{q}\) indexed by \(\varvec{\xi}\) is selected.
After reviewing LIK and AODS, we move forward to introduce the example WACAE solution. The major difficulty of solving WACAE is that the generalized coordinates \(\varvec{q}\) are constrained by nonlinear and nonconvex forward kinematics and each dimension of \(\varvec{q}\) is not independent. We firstly used LIK to eliminate the forward kinematics and simplify WACAE to the following problem.
Simplified WACAE
$$\mathop {\hbox{min} }\limits_{{\varvec{s}, \varvec{q}_{r} }} C\left( {\varvec{s},{\text{LIK}}\left( {\varvec{\xi},\varvec{ q}_{r} ,\varvec{ p}_{d} , \varvec{q}_{l} \left( {\varvec{\xi}^{c} } \right), \varvec{q}_{h} \left( {\varvec{\xi}^{c} } \right)} \right)} \right) \quad s.t. \varvec{s}_{l} \le \varvec{s} \le \varvec{s}_{h} ,\varvec{q}_{l} \left(\varvec{\xi}\right) \le \varvec{q}_{r} \le \varvec{q}_{h} \left(\varvec{\xi}\right)$$
In the simplified WACAE, \(\varvec{\xi}\) is the index of the optimizable redundant dimensions \(\varvec{q}_{r}\), which can be obtained using AODS. After simplification, the optimization variables are changed from \(\varvec{q}\) to \(\varvec{q}_{r}\), each dimension of which is independent. The simplified WACAE only has box shape constraints on \(\varvec{s}\) and \(\varvec{q}_{r}\) and can be solved with gradient descent. Since \(\varvec{s}\) and \(\varvec{q}_{r}\) have different physical meanings and could be high dimensional, in order to avoid ill-conditioned hessian matrix and reduce the dimension of each optimization step, we update \(\varvec{s}\) and \(\varvec{q}_{r}\) iteratively using gradient descent. The pseudocode of the example solution is shown in Algorithm 2.
In this pseudocode, the example solution takes desired end effector position and orientation (\(\varvec{p}_{d}\)), constraints (\(\varvec{s}_{l} ,\varvec{s}_{h} ,\varvec{q}_{l} , \varvec{q}_{h}\)), and index of optimizable generalized coordinates (\(\varvec{\xi}\)) as input and outputs the optimal thrust scales (\(\varvec{s}^{ *}\)) and generalized coordinates \(\varvec{q}^{\varvec{*}}\). In step 1, \(\varvec{s}\) and \(\varvec{q}\) are initialize to satisfy constraints. \(\varvec{q}\) can also be initiated by IK instead of LIK. In step 2–3, \(\varvec{s}\) and \(\varvec{q}\) are updated iteratively. BLS\(\left( {\varvec{x}, \varvec{d}, \gamma \left( \varvec{x} \right)} \right)\) in these steps means to search an appropriate step size of \(\varvec{x}\) along \(\varvec{d}\) for minimizing \(\gamma\) using backtracking line search [19]. Two kinds of restoring process are added to gradient descent in order to satisfy the box constraints on \(\varvec{s}\) and \(\varvec{q}\). (1) In updating processes, if some dimension of \(\varvec{s}\) or \(\varvec{q}_{r}\) goes beyond constraints, that dimension is restored to its original value. (2) After updating \(\varvec{s}\) or \(\varvec{q}\), if the cost function increases, then \(\varvec{s}\) or \(\varvec{q}\) goes back to the original values. With no loss of generality, we explain these restoring processes using step 2. The first kind of restoring process is equivalent to constructing an updating direction \(\varvec{\psi}\) by setting some dimensions of the opposite gradient \(-\varvec{\delta}_{\varvec{s}}\) to zero so that \(\varvec{s}\) cannot go beyond constraints in these dimensions. If we use a small enough step size \(t > 0\) so that the cost function can be linearly approximated, the cost function after updating is \(C\left( {\varvec{s} + t\varvec{\psi},\varvec{q}} \right) \approx C\left( {\varvec{s},\varvec{q}} \right) +\varvec{\delta}_{\varvec{s}}^{\text{T}} \left( {\varvec{s} + t\varvec{\psi}- \varvec{s}} \right)\, = \, C\left( {\varvec{s},\varvec{q}} \right) + t\varvec{\psi}^{\varvec{T}}\varvec{\delta}_{\varvec{s}}\). Since \(\varvec{\psi}^{\varvec{T}}\varvec{\delta}_{\varvec{s}} < 0\), \(C\left( {\varvec{s} + t\varvec{\psi},\varvec{q}} \right) < C\left( {\varvec{s},\varvec{q}} \right)\). Therefore if the updating step size \(t > 0\) is small enough for linear approximation of the cost function, the first kind of restoring process can reduce the cost function and ensure the box constraints. The second kind of restoring process check whether the updated cost function really decreases in case that the step size is not small enough. In step 4, the iteration stops when some criteria are met. Some simple criteria include setting the maximum number of iterations and the minimum acceptable decrease of the cost function. For the first level watch-actuator cooperation, step 3 can be skipped.
In this section, we simulate a redundant arm attached with flying watches and check whether the flying watches can reduce the maximum normalized actuator load through watch-actuator cooperation. In the simulations, we applied varieties of external reaction forces to the end effector under different end effector positions and evaluate the effect of flying watches on the maximum normalized actuator loads. In the following, we will first introduce the simulation setup. Then we will detail the simulation process. Finally, we will present and discuss the simulation results.
Regarding the simulation setup, we used Matlab to simulate a 9-DoF arm with 8 flying watches attached as shown in Fig. 7. The base frame is represented by purple arrows and the end effector frame is represented by red arrows. The rotation axes of flying watches are represented by arrows on cylinders. Green cylinders represent flying watches with rotation axes on vertical planes. Yellow cylinders represent flying watches with rotation axes on horizontal planes. The distance between two flying watches on the same link is 0.5 m. Since this distance is larger than the diameter of the propeller (0.381 m), the two flying watches on the same link do not collide with each other. The distance between a flying watch and the closest joint is 0.3 m. Under this distance, we can calculate the safe joint rotation range that prohibits the flying watches on both sides of a joint colliding is from − 101.2° to 101.2°. The arm is made of CFRP and has an inner diameter of 96 mm and thickness of 2 mm. The simulation arm design is derived from an existing long-reach arm in our lab [2]. In our future experiments, all joints on PIA are identical. In our simulations, we also assume that all joints of the simulation arm are identical. We set the maximum thrust of flying watch to 18 N based on our thrust test in "Thrust and watch band strength test" section.
The setup of the simulation robotic arm and flying watches
Regarding the simulation process, in a simulation case, we positioned the end effector at a certain position and applied a certain force on the end effector. Then we compare the maximum actuator loads with and without flying watch attached. More specifically about the end effector positions, since the first joint of the simulation arm rotates in yaw direction, for simplicity, the end effector is positioned on XOZ plane of the base frame. In different cases, the X coordinate of the end effector ranges from 2 to 8 m with a step size of 1 m and the Z coordinate ranges from − 7.7 to 8.3 m with a step size of 2 m. Both coordinates referred to the base frame. The end effector frame always coincides with the base frame. More specifically about the force applied on the end effector, in different cases, the force ranges from 0 to 125 N with a step size of 25 N and has directions along positive or negative direction of X, Y, or Z of the base frame. When flying watches are attached, since joint 6–8 have flying watches on both sides, we ensure that these joints range from − 90° to 90°. Since this range is narrower than the safe joint rotation range (from − 101.2° to 101.2°), the flying watches on both sides of a joint do not collide. For joint 1 (base joint), the range is from negative infinity to positive infinity since it is not related to flying watch collision. For other joints, the ranges are from − 165° to 165° so that adjacent links do not overlap. When no flying watch is attached, joint 6–8 range from − 165° to 165°. Other joint ranges do not change.
For a certain case, in order to compare the maximum normalized actuator loads with and without flying watches attached, we define the reduction rate of the maximum normalized actuator load (the cost function \(C\)), \(\zeta\), as follows. \(C_{NO\_FW}\) is the maximum normalized actuator load without flying watch attached. \(C_{FW}\) is the maximum normalized actuator load under the first or second level watch-actuator cooperation. Both \(C_{NO\_FW}\) and \(C_{FW}\) are summed over all cases concerned.
Reduction rate \(\zeta\) reflects how much flying watches have reduced the maximum normalized actuator loads.
$$\zeta = \frac{{\sum {C_{NO\_FW} } - \sum {C_{FW} } }}{{\sum {C_{NO\_FW} } }}$$
Some implementation details are as follows. Regarding the stopping criteria of the example WACAE solution, we use a very high maximum number of iterations (1000) and a very small minimum acceptable decrease of the cost function (\(10^{ - 7}\)) in order to approximate the real local minimal very precisely. When implementing the LIK and IK, the maximum end effector error is 10 mm. We found it is possible (probability = 14.3%) that LIK did not reduce the end effector error under 10 mm in step 1 of the example WACAE solution. We did not include these cases when computing the reduction rate of WACAE and computing the probability that WACAE is helpful, since no optimization really happened. When implementing AODS, for simplicity, we only compute \({\text{LIK}}\left( {i, \varvec{q}\left( i \right) + \in , \varvec{p}_{\varvec{d}} ,\varvec{q}_{\varvec{l}} \left( {i^{c} } \right), \varvec{q}_{\varvec{h}} \left( {i^{c} } \right)} \right)\) in Step 1 instead of \({{\left( {C\left( {s,{\text{LIK}}\left( {i,q(i) + \in ,p_{d} ,q_{l} (i^{c} ),q_{h} (i^{c} )} \right)} \right) - C\left( {s,{\text{LIK}}\left( {i,q(i),p_{d} ,q_{l} (i^{c} ),q_{h} (i^{c} )} \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {C\left( {s,{\text{LIK}}\left( {i,q(i) + \in ,p_{d} ,q_{l} (i^{c} ),q_{h} (i^{c} )} \right)} \right) - C\left( {s,{\text{LIK}}\left( {i,q(i),p_{d} ,q_{l} (i^{c} ),q_{h} (i^{c} )} \right)} \right)} \right)} \in }} \right. \kern-0pt} {\in}}\). If \({\text{LIK}}\left( {i, \varvec{q}\left( i \right) + \in , p_{d} ,q_{l} \left( {i^{c} } \right), q_{h} \left( {i^{c} } \right)} \right)\) exists, then the \(i\)th generalized coordinate can change without changing the end effector configuration. Therefore we regard the \(i\)th generalized coordinate as optimizable. If no flying watch is attached or the attached flying watches are working under the first level watch-actuator cooperation, IK is used to compute arm configurations. Since all joints of the simulation arm are identical, we do not need to divide the torque of a joint by the maximum torque the joint can withstand when computing the cost function in the simulations.
Until now, we have detailed the simulation setup and process, in the following, we will present and discuss the simulation results.
Firstly, we visualize some cases in our simulations in Figs. 8, 9, 10, 11, 12 and 13. The base frame is shown in Figs. 8, 9, 10, 11, 12 and 13. The positive directions of the X, Y, and Z axis of the base frame are respectively shown using magenta, black, and green line segments with one end at the origin of the base frame. The absolute values of joint torques of these cases are shown in Table 2. The positions of end effector of Figs. 8, 9, 10, 11, 12 and 13 are all [4,0,2.3]. The base joints are represented by the red dots and other joints are represented by black dots. The cyan lines on the joints represent the torques of the joints. The end effectors are represented by black dots. The external reaction forces are represented by green lines on the end effector. The flying watch thrusts are represented by red lines. When the first level watch-actuator cooperation is used, the arm is represented by magenta lines. When the second level watch-actuator cooperation is used, the arm is represented by blue lines.
The simulation arm and flying watch thrusts under the first level watch-actuator cooperation when the external reaction force on the end effector is [125,0,0] N (reduction rate = 24.1%). a Front view. b Top view. c 3-D view
The simulation arm and flying watch thrusts under the second level watch-actuator cooperation when the external reaction force on the end effector is [125,0,0] N (reduction rate = 35.6%). a Front view. b Top view. c 3-D view
The simulation arm and flying watch thrusts under the first level watch-actuator cooperation when the external reaction force on the end effector is [0,125,0] N (reduction rate = − 20.6%). a Front view. b Top view. c 3-D view
The simulation arm and flying watch thrusts under the second level watch-actuator cooperation when the external reaction force on the end effector is [0,125,0] N (reduction rate = 31.7%). a Front view. b Top view. c 3-D view
The simulation arm and flying watch thrusts under the first level watch-actuator cooperation when the external reaction force on the end effector is [0,0,− 125] N (reduction rate = − 1.0%). a Front view. b Top view. c 3-D view
The simulation arm and flying watch thrusts under the second level watch-actuator cooperation when the external reaction force on the end effector is [0,0,− 125] N (reduction rate = 20.8%). a Front view. b Top view. c 3-D view
Table 2 Joint torques of example cases
We can see by comparing Figs. 8 and 9 that the second level watch-actuator cooperation can achieve a higher reduction rate with relatively small watch thrusts. By comparing Figs. 10 and 11, when the second level watch-actuator cooperation is applied, we can observe that more horizontal thrusts are positioned closer to the end effector to obtain larger moment arms to counteract the horizontal external reaction force. From Fig. 10 and Table 2, we can also notice that the first level watch-actuator cooperation achieves a negative reduction rate in Fig. 10, which means the situation of operating watches performs worse than the situation in which no watch is attached. This is because when flying watches are attached, joint 6–8 have narrower rotation ranges to avoid flying watch collisions and the arm configuration with less actuator loads may not be available due to the narrower joint rotation ranges. However, we found in most cases operating flying watches helps to reduce the maximum actuator load. The probability that the first level watch-actuator cooperation helps to reduce the maximum actuator load is 95.41% and the probability that the second level watch-actuator cooperation helps to reduce the maximum actuator load is 98.99%. By comparing Figs. 12 and 13, we can also see when the second level watch-actuator cooperation is applied, higher reduction rate can be achieved with smaller thrusts.
Secondly, we summarize the overall simulation results in Table 3. From Table 3, we can see both levels of watch-actuator cooperation can significantly reduce the maximum actuator load by more than 36%. The second level cooperation is statistically more effective than the first level cooperation. However, for a single case, we found sometimes the second level cooperation may result in lower reduction rate than the first level cooperation. The probability that the second level cooperation performs better than the first level cooperation is 73.9%. We believe such phenomenon happens because the example WACAE solution only find local minimal instead of global minimum.
Table 3 Overall performance of flying watches
In order to further understand the composition of the reduction rate of the second level cooperation, we removed the mass of flying watches on arm links and skipped step 2 in the example WACAE solution, which means the example solution will only optimize arm configuration. The result is also shown in Table 3. We found only arm configuration optimization will result in a much lower reduction rate (8.7%) than the second level watch-actuator cooperation (43.7%). Therefore, we can understand both optimal flying watch thrusts and optimal watch positions and orientations play a role in enhancing the arm strength.
Thirdly, we tested the computation speed of the WACAE example solution. The computer we used has Intel i7 CPU (Frequency = 2.00 GHz) and 8 GB RAM. For simplicity, we changed the step size of end effector positions in X direction from 1 to 2 m. In each axis, the external reaction force of − 100 N, − 50 N, 0 N, 50 N, and 100 N are sampled for testing computation speed. The speed results are shown in Table 4.
Table 4 Speed of example WACAE solution
From Table 4, we can see the example WACAE solution is only suitable for offline thrust planning. However, since we used a very high maximum iteration number and very small minimum acceptable decrease of the cost function for the example WACAE solution in order to approximate the real minimal very precisely, the computation speed can be increased by reducing the maximum iteration number and increasing the minimum acceptable decrease of the cost function. From Table 4, we can also observe significant variation of computation time. We believe this is because the number of flying watches (8 flying watches) and the arm degree of freedom (9 DOF) are high. Even though the example solution optimizes thrusts and arm configuration iteratively, in each step (step 2 or step 3 in Algorithm 2), the search spaces still have diverse and complicated high-dimension geometries.
In this paper, we presented a modular watch-like concept, flying watch, which can be attached to robotic arms with mission-dependent attachment styles and generate thrusts in cooperation with arm actuators to enhance arm strength. Our concept is different from previous research in that it can be attached to an arm in a mission-dependent way and cooperate with actuators to enhance arm strength. We first introduced the concept, design, and dynamics of flying watch. Then two levels of watch-actuator cooperation in quasistatic situations is proposed by introducing a problem, WACAE, and providing an example solution. After that, the simulation results confirmed that flying watches can effectively enhance a robotic arm through watch-actuator cooperation.
We believe flying watch provides an important option to enhance long robotic arms. In the future, physical verification of flying watches on PIA will be done. Additionally, we will physically examine the aerodynamic interference between the two propellers on a flying watch.
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SP carried out the main part of this research and drafted the manuscript. GE contributed to the concept, design, and simulation of flying watch and revised the manuscript. All authors read and approved the final manuscript.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Present study includes the result of Advanced Research and Education Program for Nuclear Decommissioning (ARED) entrusted to Tokyo Institute of Technology by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) (Number:14532863). This work was also partially based on the results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO) (Number: 15657497).
ARED supported the construction of PIA. NEDO supported implementation of flying watch.
Department of Mechanical Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8550, Japan
Siyi Pan
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Correspondence to Siyi Pan.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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George Springer (mathematician)
George Springer (September 3, 1924 – February 18, 2019) was an American mathematician and computer scientist. He was professor emeritus of computer science at Indiana University Bloomington.[1]
George Springer
Born(1924-09-03)September 3, 1924
Cleveland, Ohio, U.S.
DiedFebruary 18, 2019(2019-02-18) (aged 94)
Newton, Massachusetts, U.S.
NationalityAmerican
Education
• Case Western Reserve University
• Brown University
• Harvard University
Scientific career
FieldsMathematics, computer science
Institutions
• Massachusetts Institute of Technology (1949–51)
• Northwestern University (1951–54)
• University of Münster (1954–55)
• University of Kansas (1955–61)
• University of Würzburg (1961–62)
• Imperial College London (1971–72)
• Indiana University Bloomington (1964–retirement)
Doctoral advisorLars Ahlfors
Springer is perhaps best known as the coauthor with Daniel P. Friedman of the widely used textbook Scheme and the Art of Computer Programming. Scheme is one of the two main dialects of LISP. Three of the pioneering books for Scheme are The Scheme Programming Language (1982) by R. Kent Dybvig, Structure and Interpretation of Computer Programs (1985) by Harold Abelson and Gerald Jay Sussman with Julie Sussman, and Scheme and the Art of Computer Programming (1989) by Springer and Friedman.
Career
Springer earned his bachelor's degree in 1945 from Case Western Reserve University (then named "Case Institute of Technology") and his master's degree in 1946 from Brown University. He earned his PhD in 1949 from Harvard University with thesis The Coefficient Problem for Univalent Mappings of the Exterior of the Unit Circle under Lars Ahlfors.[2]
From 1949 to 1951 Springer was a C.L.E. Moore Instructor at Massachusetts Institute of Technology. From 1951 to 1954 he was an assistant professor at Northwestern University. In the academic year 1954/1955 as a Fulbright Lecturer and visiting professor at the University of Münster he worked with Heinrich Behnke. In the autumn of 1955 Springer became an associate professor and subsequently a professor at the University of Kansas. In the academic year 1961/1962 he was a Fulbright Lecturer and visiting professor at the University of Würzburg. From 1964 he was a professor of mathematics and from 1987 also a professor of computer science at Indiana University Bloomington. In the academic year 1971/1972 he was a visiting professor at Imperial College in London.
Springer began his career working in function theory (of one and several complex variables) and wrote a textbook on Riemann surfaces. In the 1980s he turned more toward computer science, working on programming languages.
Personal life and death
Springer was born in Cleveland, Ohio, in 1924, to a family of Jewish immigrants from Poland.[3][4] He met his wife Annemarie (née Keiner) while at Harvard University. They were married from 1950 until her death in 2011, and had three children. Springer died on February 18, 2019, aged 94.[5]
Works
• Springer, G. (1951). "The coefficient problem for schlicht mappings of the exterior of the unit circle". Trans. Amer. Math. Soc. 70 (3): 421–450. doi:10.1090/s0002-9947-1951-0041935-5. MR 0041935.
• Springer, George (1963). "Fredholm eigenvalues and quasiconformal mapping". Bull. Amer. Math. Soc. 69 (6): 810–811. doi:10.1090/S0002-9904-1963-11043-5. MR 0161975.
• from Springer's lectures with notes prepared by Günter Scheja, Arnold Oberschelp, and Hans Rüdiger Wiehle: Einführung in die Topologie, Münster, Aschendorff 1955
• Introduction to Riemann Surfaces, Addison-Wesley 1957;[6] 2nd edition, Chelsea 1981; 3rd edition, American Mathematical Society, 2001
• with Daniel P. Friedman: Scheme and the Art of Programming, MIT Press 1989, 9th printing 1997
References
1. George Springer's Hyplan
2. George Springer at the Mathematics Genealogy Project
3. Ferguson, Niall (2015). Kissinger: 1923-1968: The Idealist. Penguin Books. p. 201. ISBN 9780698195691.
4. Biographical information from American Men and Women of Science, Thomson Gale 2005
5. George Springer 94 Hoosier Times
6. Jenkins, James A. (1958). "Review: Introduction to Riemann surfaces, by George Springer" (PDF). Bull. Amer. Math. Soc. 64 (2): 382–385. doi:10.1090/s0002-9904-1958-10245-1.
External links
• Homepage, Indiana University Bloomington
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Netherlands
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• SNAC
• IdRef
| Wikipedia |
\begin{definition}[Definition:Basis Expansion/Recurrence]
Let $b \in \N: b \ge 2$.
Let $x$ be a real number.
Let the basis expansion of $x$ in base $b$ be:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$
Let there be a finite sequence of $p$ digits of $x$:
:$\tuple {d_{r + 1} d_{r + 1} \ldots d_{r + p} }$
such that for all $k \in \Z_{\ge 0}$ and for all $j \in \set {1, 2, \ldots, p}$:
:$d_{r + j + k p} = d_{r + j}$
where $p$ is the smallest $p$ to have this property.
That is, let $x$ be of the form:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$
That is, $\tuple {d_{r + 1} d_{r + 2} \ldots d_{r + p} }$ repeats from then on, or '''recurs'''.
Then $x$ is said to '''recur'''.
\end{definition} | ProofWiki |
Fabrication of Z-scheme Ag3PO4/TiO2 Heterostructures for Enhancing Visible Photocatalytic Activity
Wenhui Liu1,
Dengdeng Liu1,
Kun Wang1,
Xiaodan Yang1,
Shuangqi Hu1 &
Lishuang Hu1
Nanoscale Research Letters volume 14, Article number: 203 (2019) Cite this article
In this paper, a synthetical study of the composite Ag3PO4/TiO2 photocatalyst, synthesized by simple two-step method, is carried out. Supplementary characterization tools such as X-ray diffraction, scanning electron microscopy, transmission electron microscopy, high-resolution transmission electron microscopy, energy dispersive X-ray spectroscopy, X-ray photoelectron spectroscopy, and UV-vis diffuse reflectance spectroscopy were adopted in this research. The outcomes showed that highly crystalline and good morphology can be observed. In the experiment of photocatalytic performance, TiO2400/Ag3PO4 shows the best photocatalytic activity, and the photocatalytic degradation rate reached almost 100% after illuminating for 25 min. The reaction rate constant of TiO2400/Ag3PO4 is the largest, which is 0.02286 min−1, twice that of Ag3PO4 and 6.6 times that of the minimum value of TiO2400. The degradation effect of TiO2400/Ag3PO4 shows good stability after recycling the photocatalyst four times. Trapping experiments for the active catalytic species reveals that the main factors are holes (h+) and superoxide anions (O·− 2), while hydroxyl radical (·OH) plays partially degradation. On this basis, a Z-scheme reaction mechanism of Ag3PO4/TiO2 heterogeneous structure is put forward, and its degradation mechanism is expounded.
Semiconductor photocatalysts have attracted increasing interest due to extenstive use in organic pollutant degradation and solar cells [1,2,3,4,5,6]. As the representative of semiconductor-based photocatalysts, TiO2 has been extensively investigated because of its excellent physical-chemical properties [7, 8]. However, the pure TiO2 photocatalyst has certain disadvantages in practical applications such as its wide band gap (3.2 eV for anatase and 3.0 eV for rutile), which leads to poor visible response.
A silver-based compound such as Ag2O, AgX (X = Cl, Br, I), Ag3PO4, Ag2CrO4, have been recently used for photocatalytic applications [9,10,11,12]. Among others, silver orthophosphate (Ag3PO4) has already attracted attention from many researchers because Ag3PO4 has a band gap of 2.45 eV and strong absorption at less than 520 nm. The quantum yield of Ag3PO4 is over 90%. It is a good visible-light photocatalyst. However, due to the formation of Ag0 on the surface of the catalyst (4Ag3PO4 + 6H2O + 12h+ + 12e− → 12Ag0 + 4H3PO4 + 3O2) during the photocatalytic reaction, the reuse of Ag3PO4 is a major problem. Therefore, it is a common practice to reduce photocatalytic corrosion of Ag3PO4 and ensure good catalytic activity of Ag3PO4. Based on literature precedence, it is known that compounding can effectively improve the photocatalytic performance of both semiconductor materials. After compounding, the separation effect of photogenerated electrons and holes is strengthened, contributing to enhance the photocatalytic activity of composite materials. Numerous researchers have investigated heterojunctions such as Bi2O3-Bi2WO6, TiO2/Bi2WO6, ZnO/CdSe, and Ag3PO4/TiO2 [2, 13,14,15]. Compared with single-phase photocatalysts, heterojunction photocatalysts can expand the light response range by coupling matched electronic structure materials. And because of the synergistic effect between components, charge can be transferred through many ways to further improve heterojunction photocatalytic activity.
Based on the above analysis, Ag3PO4-based semiconductor composites with synergistic enhancement effect were designed to improve carrier recombination defects and Ag3PO4-based semiconductor composites catalytic performance. In this paper, nano-sized TiO2 was prepared by solvothermal method, and then the nanoparticles of TiO2400 were deposited on the surface of Ag3PO4 at room temperature to obtain TiO2/Ag3PO4 composites. The photocatalytic activity of TiO2/Ag3PO4 composite was tested using RhB dye (rhodamine B).
Hydrothermal Preparation of Nano-sized TiO2
0.4 g P123 was added to a mixed solution containing 7.6 mL absolute ethanol and 0.5 mL deionized water and stirred until P123 was completely dissolved. The clarified solution was labeled as A solution. Then a mixed solution containing 2.5 mL butyl titanate (TBOT) and 1.4 mL concentrated hydrochloric acid (12 mol/L) was prepared and labeled as B solution. The solution B was added to solution A by drop. After stirring for 30 min, 32 mL ethylene glycol (EG) was added to the solution and stirred for 30 min. Then, the solution was placed in oven, at 140 °C, high temperature, and high pressure for 24 h. Natural cooling, centrifugal washing, separation, collection of sediments, and drying at 80 °C oven for 8 h. The white precipitation was calcined in muffle furnace at different temperatures (300 °C, 400 °C, 500 °C) and marked as standby of TiO2300, TiO2400, and TiO2500, respectively.
Preparation of TiO2/Ag3PO4 Photocatalyst
The 0.1 g TiO2 powder was added to the 30-mL silver nitrate solution containing 0.612 g AgNO3 and then treated by ultrasound for 30 min to make TiO2 dispersed uniformly. We added 30-mL solution containing 0.43 g Na2HPO4.12H2O and stirred for 120 min at ambient temperature. By centrifugation, cleaning with deionized water and anhydrous ethanol, the precipitates were separated, collected, and dried at 60 °C. The products were named as TiO2300/Ag3PO4, TiO2400/Ag3PO4, and TiO2500/Ag3PO4, respectively. Ag3PO4 was prepared without adding TiO2 under the same conditions as the above process.
The X-ray diffraction (XRD) patterns of the resulted samples were performed on a D/MaxRB X-ray diffractometer (Japan), which has a 35 kV Cu-Ka with a scanning rate of 0.02° s−1, ranging from 10 to 80°. Scanning electron microscopy (SEM), JEOL, JSM-6510, and JSM-2100 transmission electron microscopy (TEM) assembly with energy dispersive X-ray spectroscopy (EDX) were used to study its morphology at 10-kV acceleration voltage. X-ray photoelectron spectroscopy (XPS) information were collected by using an ESCALAB 250 electron spectrometer under 300-W Cu Kα radiation. The basic pressure was about 3 × 10−9 mbar, Combine to refer to the C1s line at amorphous carbon 284.6 eV.
Photocatalytic Activity Measure
The photocatalytic performance of TiO2/Ag3PO4 catalysts was tested by using the photodegradation of RhB in aqueous solution as the research object. Fifty milligrams of the photocatalyst was mixed with 50 mL of RhB aqueous solution (10 mg L−1) and stirred in darkness for a certain time before illumination to ensure adsorption balance. In the reaction process, cooling water is used to keep the system temperature constant at room temperature. A 1000-W Xenon lamp provides illumination to simulate visible light. LAMBDA35 UV/Vis spectrophotometer was used to characterize the concentration (C) change of RhB solution at λ = 553 nm. The decolorization rate is indicated as a function of time vs Ct/C0. Where C0 is the concentration before illumination, and Ct is the concentration after illumination. Used catalysts were recollected to detect the cycle stability of the catalysts. The experiment was repeated four times.
XRD analysis is used to determine the phase structure and crystalline type of catalyst. The XRD spectra of the prepared catalysts were shown in Fig. 1, including TiO2400, Ag3PO4, TiO2/Ag3PO4, TiO2300/Ag3PO4, TiO2400/Ag3PO4, and TiO2500/Ag3PO4. It can be obtained from the figure that the crystal structure of TiO2400 is anatase (JCPDS No. 71-1166). In the XRD spectra of Ag3PO4, the diffraction peaks located at 20.9°, 29.7°, 33.3°, 36.6°, 47.9°, 52.7°, 55.1°, 57.4°, 61.7°, and 72.0° belong to the characteristic peaks of (110), (200), (210), (211), (310), (222), (320), (321), (400), and (421) planes of Ag3PO4 (JCPDS No. 70-0702), respectively. The synthesized composite photocatalysts showed characteristic peaks consistent with TiO2 and Ag3PO4, and the characteristic peaks of TiO2 were 25.3° at the composite TiO2, TiO2300/Ag3PO4, TiO2400/Ag3PO4, TiO2500/Ag3PO4, which was consistent with the calcination temperature of TiO2 rise, the crystallinity of TiO2 becomes higher.
The XRD patterns of the as-prepared samples
Figure 2 shows the SEM, TEM, and EDX diagrams of the catalysts of TiO2400, Ag3PO4, and TiO2400/Ag3PO4. Figure 2a is the spherical nanostructure TiO2400 prepared by solvothermal method with a diameter ranging from 100 to 300 nm. Figure 2b is the Ag3PO4 crystal with a regular hexahedral structure. Its particle size ranges from 0.1 to 1.5 μm and has a fairly smooth surface. Figure 2c is the SEM image of the composite TiO2400/Ag3PO4. It can be seen that the nanoparticles of TiO2400 are deposited on the surface of Ag3PO4. The morphology of TiO2400/Ag3PO4 was further explored with TEM and the TEM diagram of TiO2400/Ag3PO4 is displayed in Fig. 2d. It can be observed that 200-nm nano-sized TiO2 particles adhere to the surface of Ag3PO4. Figure 2e is the HRTEM of TiO2400/Ag3PO4. It can be founded that TiO2 particles are closely bound to Ag3PO4, and the lattice spacing of TiO2400 and Ag3PO4 are 0.3516 and 0.245 nm, respectively, corresponding to (101) and (211) surfaces of TiO2 and Ag3PO4. Figure 2f is the EDX diagram of TiO2400/Ag3PO4. It can be seen that the sample consists of four elements: Ti, O, Ag, and P. The obvious diffraction peak of copper element is produced by the EDX excitation source, Cu Ka. EDX confirmed the corresponding chemical elements of TiO2400/Ag3PO4. In conclusion, it can be clearly judged that TiO2 is loaded on the surface of Ag3PO4 crystals in granular form and has a good hexahedron morphology.
SEM images of prepared photocatalysts: a TiO2400, b Ag3PO4, c TiO2400/Ag3PO4, d TEM image of TiO2400/Ag3PO4, e HRTEM image of TiO2400/Ag3PO4, and f corresponding EDX pattern of TiO2400/Ag3PO4
The product X-ray photoelectron spectroscopy (XPS) is investigated in Fig. 3. Figure 3a is the survey XPS spectrum of the product. Ti, O, Ag, P, and C five elements can be observed in the graph, of which C is the base, implying that composite coexisted with TiO2 and Ag3PO4. Figure 3b is the high-resolution spectrum of Ag 3d. The two main peaks centered at binding energy 366.26 eV and 372.29 eV, assigning to Ag 3d5/2 and Ag 3d3/2, respectively. It shows that Ag is mainly Ag+ in the photocatalyst of TiO2400/Ag3PO4 [16]. Figure 3c shows the XPS peak of P 2p, which corresponds to P5+ in the PO43+ structure at 131.62 eV. Two peaks located at 457.43 eV and 464.58 eV can be attributed to Ti 2p3/2 and Ti 2p1/2 in the XPS spectrum of Ti 2p orbital (Fig. 3d). Figure 3e is the XPS of O 1s. The whole peak can be divided into three characteristic peaks, 528.9 eV, 530.2 eV, and 532.1 eV. The peaks at 528.9 eV and 530.2 eV are ascribed to oxygen in Ag3PO4 and TiO2 lattices, respectively. The peaks at 532.1 eV indicate hydroxyls or the oxygen adsorbed on the surface of TiO2/Ag3PO4. The results of XPS analysis further prove that Ag3PO4 and TiO2 have been compounded.
XPS spectrum of TiO2400/Ag3PO4: a survery scan, b Ag 3d, c P 2p, d Ti 2p, and e O1s
The UV-Vis diffuse reflectance absorption spectra of the catalysts of TiO2400, Ag3PO4, and TiO2400/Ag3PO4 are exhibited in Fig. 4a. It can be seen from the figure that the optical absorption cutoff wavelengths of TiO2400 and Ag3PO4 are 400 and 500 nm, respectively. When Ag3PO4 is loaded on TiO2400, the light absorption range of the composite obviously broadens to 500–700 nm, indicating that there is interaction between Ag3PO4 and TiO2400 in the composite system of TiO2400/Ag3PO4, and the mechanism needs further study. Bandwidth of Ag3PO4, TiO2400, and TiO2400/Ag3PO4 catalysts is computed with the Kubelka-Munk formula [17]:
$$ A\mathrm{hv}=c{\left(\mathrm{hv}-\mathrm{Eg}\right)}^n $$
TiO2400, Ag3PO4, and TiO2400/Ag3PO4 catalysts: a UV-Vis DRS, b plots of (αhv)1/2 versus energy (hv)
where A, hv, c, and Eg are the absorption coefficient, incident photon energy, absorption constant, and band gap energy, respectively. The value of n for direct semiconductor is 1/2, and that for indirect semiconductor is 2. Anatase TiO2 and Ag3PO4 are indirect semiconductors, so n takes 2.
The plots depicting (αhv)1/2 versus incident photon energy (hv) from Fig. 4b indicates the band gap energy diagrams (Eg) of Ag3PO4, TiO2400, and TiO2400/Ag3PO4 catalysts are 2.45 eV, 3.1 eV, and 2.75 eV, respectively. This further proves that TiO2400/Ag3PO4 is a good visible-light photocatalyst with suitable band gap width and visible light capture ability.
Photocatalytic degradation of RhB by TiO2400, Ag3PO4, TiO2300/Ag3PO4, TiO2400/Ag3PO4, and TiO2500/Ag3PO4 was investigated in Fig. 5a. The results showed that pure TiO2400 had the worst photocatalytic effect, and the photocatalytic degradation rate was only 30% within 25 min. The photocatalytic degradation efficiency of pure Ag3PO4 was 69% after 25 min of irradiation. The photocatalytic degradation rate of TiO2300/Ag3PO4 reached 40% after 25 min. The photocatalytic degradation rate of TiO2500/Ag3PO4 was 80% after 25 min of irradiation. The best photocatalytic activity was TiO2400/Ag3PO4, and 100% of RhB was decomposed after 25 min of illumination.
a Effects of different catalysts on photocatalytic degradation of RhB under visible light. b First order kinetic fitting plots of photocatalytic degradation of RhB with different catalysts. c Cycling runs of TiO2400/Ag3PO4. d Trapping experiments of active species
Figure 5b studied the kinetics model of photocatalytic degradation of RhB. From the figure, the photodegradation of RhB was followed pseudo-first-order kinetics and the reaction rate constant (k) was calculated with the slope of fitting curves. The reaction rate constant (k) values of each sample were shown in Table 1. The reaction rate constants of TiO2400, Ag3PO4, TiO2300/Ag3PO4, TiO2400/Ag3PO4, and TiO2500/Ag3PO4 were 0.00345 min−1, 0.01148 min−1, 0.00525 min−1, 0.02286 min−1, and 0.01513 min−1, respectively. The sample TiO2400/Ag3PO4 has the largest reaction rate constant, which is 0.02286 min−1, twice that of Ag3PO4 and 6.6 times that of the minimum value of TiO2400. This indicates that the combination of Ag3PO4 and TiO2 can greatly contribute to the improvement of Ag3PO4 photocatalytic activity.
Table 1 Photo degradation rate constants and linear regression coefficients of different catalysts from equation − ln(C/C0) = kt.
Figure 5c is the stability test result of four times of degradation of RhB solution by recycling of TiO2400/Ag3PO4. The degradation effect of TiO2400/Ag3PO4 shows good stability in four times of recycling, and in the fourth cycle experiment, the degradation effect of TiO2400/Ag3PO4 was slightly higher than that of the third cycle. This may be due to the formation of composite material between Ag3PO4 and TiO2 to accelerate photogenerated electron-hole pair transfer and in situ formation of a small amount of Ag in Ag3PO4 during photocatalysis to inhibit further photo-corrosion.
The results of TiO2/Ag3PO4 capture factors are shown in Fig. 5d. After the addition of trapping agent IPA, the degradation activity decreased partially. When BQ and TEOA were added, the degradation degree of RhB decreased significantly, even close to 0. Therefore, we can infer that the main factors are holes (h+) and superoxide anions (O·− 2), while hydroxyl radical (·OH) plays partially degradation.
A possible Z-scheme photocatalytic degradation mechanism was proposed in Scheme 1 to expatiate the photocatalytic degradation of RhB by TiO2/Ag3PO4 based on free radical capture and photodegradation experiments. The band gap of Ag3PO4 is 2.45 eV, and its ECB and EVB potential are ca.0.45 eV and 2.9 eV (vs. NHE) [18], respectively. As shown in Scheme 1, under visible light irradiation, Ag3PO4 is stimulated by photons with energy greater than its band gap to produce photogenerated electron-hole pairs. The holes left in the valence band of Ag3PO4 migrated to the valence band of TiO2 and then directly participated in the RhB oxidation and decomposition process, which adsorbed on the surface of TiO2. At the same time, during the migration of photogenerated holes, the H2O and OH− adsorbed on the composite surface can also be oxidized to form ·OH, and the highly oxidizing ·OH can further oxidize and degrade pollutants. This is mainly due to the energy of holes in the valence band of Ag3PO4 which is 2.9 eV, higher than the reaction potential energy of OH−/OH (E(OH−/OH) = 1.99 eV (vs. NHE)). However, the conduction potential of Ag3PO4 is 0.45 eV, the energy of photogenerated electrons is 0.45 eV, and the activation energy of single electron oxygen is E(O2/O·− 2) = 0.13 eV (vs. NHE). The photogenerated electrons on Ag3PO4 conduction band cannot be captured by dissolved oxygen. With the accumulation of photogenerated electrons on Ag3PO4 conductive band, a small amount of Ag nanoparticles has been formed due to the photocatalytic corrosion of Ag3PO4 photocatalyst. The formed Ag nanoparticles can also be stimulated by light energy to form photogenerated electron-hole pairs. Then the electrons migrated to the conduction band of TiO2, while the holes left on the Ag nanoparticles can be compounded with the photogenerated electrons generated on the conduction band of Ag3PO4, thus preventing the further corrosion of Ag3PO4 photocatalyst. Due to the forbidden band of TiO2 is 3.1 eV, it cannot be excited under visible light and the ECB and EVB are ca. − 0.24 eV and 2.86 eV (vs. NHE), respectively. Electrons injected into TiO2 conduction band can degrade pollutants through trapping the oxygen adsorbed onto the TiO2 surface. This is mainly due to the ECB = − 0.24 eV (vs. NHE) which is more negative than E(O2/O·- 2) = 0.13 eV (vs. NHE). The results are in accordance with the trapping experiments. The main factors are holes (h+) and superoxide anions (O·- 2), while hydroxyl radical (·OH) plays partially degradation.
Schematic illustration of the photocatalytic mechanism of TiO2/Ag3PO4
Basing on the above discussion, the degradation reaction of TiO2/Ag3PO4 is expressed by the chemical equation as follows:
Generation of photoelectron hole pairs:
$$ {\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4+\mathrm{hv}\to {\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4\left({\mathrm{e}}^{-}\right)+{\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4\left({\mathrm{h}}^{+}\right) $$
$$ {\mathrm{Ag}}^{+}+{\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4\left({\mathrm{e}}^{-}\right)\to \mathrm{Ag}+{\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4 $$
$$ \mathrm{Ag}+\mathrm{hv}\to \mathrm{Ag}\left({\mathrm{e}}^{-}\right)+\mathrm{Ag}\left({\mathrm{h}}^{+}\right) $$
Migration and transformation of photogenerated hole electron pairs:
$$ {\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4\left({\mathrm{h}}^{+}\right)+\mathrm{Ti}{\mathrm{O}}_2\to \mathrm{Ti}{\mathrm{O}}_2\left({\mathrm{h}}^{+}\right)+{\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4 $$
$$ {\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4\left({\mathrm{e}}^{-}\right)+\mathrm{Ag}\left({\mathrm{h}}^{+}\right)\to \mathrm{Ag}+{\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4 $$
$$ \mathrm{Ag}\left({\mathrm{e}}^{-}\right)+\mathrm{Ti}{\mathrm{O}}_2\to \mathrm{Ti}{\mathrm{O}}_2\left({\mathrm{e}}^{-}\right)+\mathrm{Ag} $$
$$ \mathrm{Ti}{\mathrm{O}}_2\left({\mathrm{e}}^{-}\right)+{\mathrm{O}}_2\to {\mathrm{O}}_2^{\cdotp -}+\mathrm{Ti}{\mathrm{O}}_2 $$
$$ {\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4\left({\mathrm{h}}^{+}\right)+0{\mathrm{H}}^{-}\to \mathrm{OH}\cdotp +{\mathrm{Ag}}_3\mathrm{P}{\mathrm{O}}_4 $$
Degradation of pollutants:
$$ \mathrm{Ti}{\mathrm{O}}_2\left({\mathrm{h}}^{+}\right)+\mathrm{RhB}\to \mathrm{Degradation}\ \mathrm{product}+{\mathrm{CO}}_2+{\mathrm{H}}_2\ \mathrm{O} $$
$$ {\mathrm{O}}_2^{\cdotp -}+\mathrm{RhB}\to \mathrm{Degradation}\ \mathrm{product}+{\mathrm{CO}}_2+{\mathrm{H}}_2\ \mathrm{O} $$
$$ \mathrm{OH}\cdotp +\mathrm{RhB}\to \mathrm{Degradation}\ \mathrm{product}+{\mathrm{CO}}_2+{\mathrm{H}}_2\ \mathrm{O}+{\mathrm{Cl}}^{-} $$
In summary, a comprehensive investigation of the composite Ag3PO4/TiO2 photocatalyst, prepared by a simple two-step method is presented. Complementary characterization tools such as X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HR-TEM), energy dispersive X-ray spectroscopy (EDX), X-ray photoelectron spectroscopy (XPS), and UV-vis diffuse reflectance spectroscopy (DRS) were utilized in this study. The results showed that the composite Ag3PO4/TiO2 photocatalyst is highly crystalline and has good morphology. For Ag3PO4/TiO2 degradation of RhB, TiO2400/Ag3PO4 shows the highest photocatalytic activity. After 25 min of reaction, the photocatalytic degradation rate reached almost 100%. The reaction rate constant of TiO2400/Ag3PO4 is 0.02286 min−1, which is twice that of Ag3PO4 and 6.6 times that of the minimum value of TiO2400. The TiO2400/Ag3PO4 also exhibits good stability after recycling four times. The main active catalytic species are holes (h+) and superoxide anions (O·− 2), while hydroxyl radical (·OH) plays partially degradation from trapping experiments. In addition, a Z-scheme reaction mechanism of Ag3PO4/TiO2 heterogeneous structure is proposed to explain the RhB degradation mechanism. The accumulation of photogenerated electrons on Ag3PO4 conductive band causes photoetching of Ag3PO4 photocatalyst to form a small amount of Ag nanoparticles, consequently, accelerating photogenerated electron transfer in the Ag3PO4 conduction band, thus preventing further Ag3PO4 photocatalyst corrosion.
The authors declare that materials and date are promptly available to readers without undue qualifications in material transfer agreements. All data generated in this study are included in this article.
BQ:
p-benzoquinone
DRS:
UV-vis diffuse reflectance spectroscopy
EDX:
Energy dispersive X-ray spectrometer
HR-TEM:
High-resolution transmission electron microscopy
IPA:
RhB:
TEM:
TEOA:
XPS:
X-ray photoelectron spectroscopy
XRD:
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This work is greatly indebted to professors Shuangqi Hu for his meticulous instruction and Lishuang Hu who helped us with great encouragement.
No funding support.
Environmental and Safety Engineering Institute, North University of China, Taiyuan, 030051, Shanxi, People's Republic of China
Wenhui Liu
, Dengdeng Liu
, Kun Wang
, Xiaodan Yang
, Shuangqi Hu
& Lishuang Hu
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This work presented here was performed in collaboration of all the authors. All authors read and approved the final manuscript.
Correspondence to Shuangqi Hu or Lishuang Hu.
Liu, W., Liu, D., Wang, K. et al. Fabrication of Z-scheme Ag3PO4/TiO2 Heterostructures for Enhancing Visible Photocatalytic Activity. Nanoscale Res Lett 14, 203 (2019) doi:10.1186/s11671-019-3041-8
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June 2013, 2(2): 403-422. doi: 10.3934/eect.2013.2.403
Nonlinear instability of solutions in parabolic and hyperbolic diffusion
Stephen Pankavich 1, and Petronela Radu 2,
Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, United States
Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588
Received November 2012 Revised December 2012 Published March 2013
We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including evolution equations with supercritical and exponential nonlinearities.
Keywords: steady states., variable coefficients, instability, Evolution equations, sign-changing damping.
Mathematics Subject Classification: Primary: 35B35, 35B05, 35B30; Secondary: 35L70, 35K5.
Citation: Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403
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Stephen Pankavich Petronela Radu | CommonCrawl |
The Annals of Mathematical Statistics
Ann. Math. Statist.
Volume 37, Number 2 (1966), 425-434.
On the Sample Size and Simplification of a Class of Sequential Probability Ratio Tests
Adnan F. Ifram
More by Adnan F. Ifram
Full-text: Open access
PDF File (915 KB)
Article info and citation
$T$ is the sequential probability ratio test (SPRT) based on the sequence $\{X_n\}$ whose family of distributions $\{P_\theta, \theta \varepsilon \Theta\}$ satisfies certain sufficiency, monotone likelihood ratio, and consistency assumptions. Sufficiency reduces the criterion of $T$ to $q\theta_2^n/q\theta_1^n$, where $q_{\theta n}$ is the density of $X_n, \theta_1$ and $\theta_2$ are the values of $\theta$ specified by the hypothesis and alternative respectively, and $\theta_1 < \theta_2$. It is assumed that $q_{\theta n}(x)$ has the asymptotic form: $q_{\theta n}(x) \sim f_{\theta n}(x) \equiv K(n)C(\theta, x)e^{nh(\theta, x)}$ as $n \rightarrow \infty$. The Simplified SPRT $T^\ast$ is proposed where $T^\ast$ uses the criterion $e^{nh(\theta_2,\cdot)}/e^{nh(\theta_1,\cdot)}$. The following conditions are relevant: Condition C states that $h(\theta,x)$ has a unique maximum at $x = \theta$ and $h(\theta,\theta)$ is free of $\theta$; Condition D(i) requires $\Delta_{\theta n}(x) \equiv q_{\theta n}(x)/f_{\theta n}(x)$ to be bounded for all $\theta, x$, and $n$; and Condition D(ii) states that for each $\theta,\Delta_{\theta n}(x) \rightarrow 1$ uniformly in $x$ for $x$ in a neighborhood of $x = \theta$. Let $N$ and $N^\ast$ be the sample sizes of $T$ and $T^\ast$ respectively, and let $\theta_0$ be the solution of $h(\theta_2, x) = h(\theta_1, x)$. It is shown in Section 3 that Conditions C and D imply: $P_\theta(N^\ast > n) < \gamma\delta^n/n^{\frac{1}{2}}$, where $0 < \delta < 1, \gamma < \infty$ and $\theta \neq \theta_0$. The same is true for $N$. Thus, the moment generating functions of $N$ and $N^\ast$ exist, and inequalities for the expected values of $N$ and $N^\ast$ are readily obtained with respect to any $P_\theta, \theta \neq \theta_0$. The following monotonicity properties of $E_\theta N$ and $E_\theta N^\ast$ are established under an additional condition in Section 4: the expected values increase for $\theta$ less than a certain interval containing $\theta_0$ and decrease for $\theta$ greater than this interval. Several examples are discussed in Section 5, and the conditions are checked in Section 6.
Ann. Math. Statist., Volume 37, Number 2 (1966), 425-434.
First available in Project Euclid: 27 April 2007
Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699524
Digital Object Identifier
doi:10.1214/aoms/1177699524
Mathematical Reviews number (MathSciNet)
MR187359
Zentralblatt MATH identifier
links.jstor.org
Ifram, Adnan F. On the Sample Size and Simplification of a Class of Sequential Probability Ratio Tests. Ann. Math. Statist. 37 (1966), no. 2, 425--434. doi:10.1214/aoms/1177699524. https://projecteuclid.org/euclid.aoms/1177699524
On the Asymptotic Behavior of Densities with Applications to Sequential Analysis
Ifram, Adnan F., The Annals of Mathematical Statistics, 1965
Asymptotically Optimum Properties of Certain Sequential Tests
Wong, Seok Pin, The Annals of Mathematical Statistics, 1968
Sequential Life Tests in the Exponential Case
Epstein, Benjamin and Sobel, Milton, The Annals of Mathematical Statistics, 1955
Contributions to the Theory of Sequential Analysis, II, III
Girshick, M. A., The Annals of Mathematical Statistics, 1946
Estimation of the Larger Translation Parameter
Blumenthal, Saul and Cohen, Arthur, The Annals of Mathematical Statistics, 1968
Limiting Distributions for Some Random Walks Arising in Learning Models
Norman, M. Frank, The Annals of Mathematical Statistics, 1966
Estimation of Two Ordered Translation Parameters
A Necessary and Sufficient Condition that for Regular Multiple Decision Problems of Type I Every Unbiased Procedure Has Minimax Risk
Stefansky, Wilhelmine, The Annals of Mathematical Statistics, 1970
Asymptotically Optimal Tests in Markov Processes
Johnson, Richard A. and Roussas, George G., The Annals of Mathematical Statistics, 1970
An Exponential Subfamily which Admits UMPU Tests Based on a Single Test Statistic
Bar-Lev, Shaul K. and Reiser, Benjamin, The Annals of Statistics, 1982
euclid.aoms/1177699524 | CommonCrawl |
How old is the ship of Theseus?
The ship of Theseus consists of n parts. The first part needs to be replaced every day, the second part every second day, the third part every three days and so on.
Theseus is concerned that replacing all parts on the same day may cause the ship to lose its identity, and become something different than the ship of Theseus. But he has a strategy to avoid this situation: whenever all parts are scheduled for replacement on the same day, before starting to replace the old parts, he adds a new part to his ship, which only needs to be replaced in $n + 1$ days.
The ship originally consisted of only one part and today Theseus has announced that he's added the 100th part to it.
mathematics calculation-puzzle
GOTO 0GOTO 0
$\begingroup$ It's not really related to the puzzle itself, but I don't think Theseus has actually solved his problem. How does adding a new part (that wasn't part of the original ship) help preserve its identity? $\endgroup$ – KSmarts Mar 26 '15 at 16:17
$\begingroup$ @KSmarts "Now Theseus has two problems" $\endgroup$ – Michael Mar 26 '15 at 19:02
$\begingroup$ @KSmarts: If you accept Theseus's implicit premise (i.e., "the ship retains its identity if there is always at least one part intact"), then it should retain its identity. You add the new piece, and that piece is now part of the ship; that piece will remain intact for the remainder of the day while the others are replaced, so at no point does the ship lose its identity. $\endgroup$ – wchargin Mar 27 '15 at 4:06
First parts
Starting from the first day, Theseus has a single-part ship: this means that this part can only work for a day before being replaced, so Theseus should replace the first part on the second day. Given that replacing the first and only part will let the ship "lose its identity", then before replacing the first part, Theseus adds a second part: so the second part of his ship is added on the second day, before removing the first one.
On the second day, the first part has got to be changed again, but the second part has still got one day. On the third day the parts will be changed together, meaning that Theseus will have to add a third part.
Adding the nth part
When will Theseus have to change all the $n$ parts together again? It is easy to understand that Theseus will have to change all the parts when the number of days becomes equal to $lcm(1, 2, 3, ..., n-1, n) + 1$, which is the least common multiple of the duration of each part plus one, because the first day the only existing part doesn't have to be changed.
We can write a table representing the day when Theseus has to change all the $n$ parts together based on $n$:
n day
4 13
7 421
9 2521
10 2521
... ...
60 9690712164777231700912801
61 591133442051411133755680801
We can clearly see that this number becomes dramatically large when dealing with these amounts of parts.
When Theseus adds the $100$th part, this means that he already changed all the parts together $99$ times (adding a new part every time), and that it has come the day to change all the parts together again. So the age $A$ of the ship can be obtained solving the following equation:
$A = 1 + \sum \limits_{i=1}^{99} lcm(1, 2, ..., i) \simeq 2.149 \times 10^{41}$
The ship is $214959977860203405582463952869483994880562$ days old.
Which is around $588.93$ billions of billions of billions of billions of years, WOW!
I created a small Python 2.7 script to calculate the number of days, here it is:
from fractions import gcd
def lcm(a, b):
return (a/(gcd(a, b))*b)
age = 1
for i in xrange(1, 100):
age += reduce(lcm, range(1, i+1))
print "Theseus' ship is", age, "days old!"
Marco BonelliMarco Bonelli
$\begingroup$ This is correct. Excellent. $\endgroup$ – GOTO 0 Mar 26 '15 at 15:10
$\begingroup$ Just curious: How many years is that? $\endgroup$ – Mark Mar 26 '15 at 15:56
$\begingroup$ @Mark Around $588.93$ millions of millions of millions of millions of millions of millions of years. $\endgroup$ – Marco Bonelli Mar 26 '15 at 16:08
$\begingroup$ @Mark $5.885*10^{38}$ years, for those of us who prefer scientific notation to "millions of ...". $\endgroup$ – Tim S. Mar 26 '15 at 18:21
$\begingroup$ Given that the age of our universe is "only" $1.38×10^{10}$ years, and Theseus had to wait for quite a time while the life on Earth gets complicated enough to produce Theseuses (which did not happen until ~55000 years ago, which we can round to 0, as if the Earth started producing Odysseys and Donalds Trumps just 0 seconds ago. If you compare that to the $6×10^{38}$, well, you probably do not care about the previous number anyway. It's going to be $10^{28}$ the current age of the Universe, and no one knows if elem. partcls will be ripped apart by dark energy then. No conclusions yet. $\endgroup$ – kkm Nov 22 '19 at 4:11
Very old indeed - Theseus is effectively immortal, and his ship is almost certainly space-faring, as the number of days is rather more than the current age of the universe, and probably more than the expected life of most suns.
The actual answer is more than $32\times 27\times 35\times 97\#$, where $97\#$ (97 primorial) is the product of all primes up to $97$.
This number is the least common multiple of all the numbers up to $100$. However there are some extra multiples in there too, so the actual age is larger than this, although it is less than $100!$ (100 factorial).
By way of concrete example, when Theseus added the 22nd part, the ship was approximately 2 million years old.
JoffanJoffan
$\begingroup$ This is roughly what I was just writing in my answer, so I'll upvote you and leave mine unfinished. :) $\endgroup$ – Ian MacDonald Mar 26 '15 at 14:51
The ship is
days old.
If the nth part is added today, the (n+1)th part will be added after lcm(1…n) days. Thus the 100th part gets added on day number $\sum_{i=1}^{99} \mathrm{lcm}(1,\ldots i)$.
Raziman T VRaziman T V
$\begingroup$ You didn't consider the first day $\endgroup$ – Marco Bonelli Mar 26 '15 at 15:09
$\begingroup$ @Marco - the age on the first day would be zero days old, though. $\endgroup$ – Joffan Mar 26 '15 at 17:35
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Riemann–Siegel formula
In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by Siegel (1932) in unpublished manuscripts of Bernhard Riemann dating from the 1850s. Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably. When used along the critical line, it is often useful to use it in a form where it becomes a formula for the Z function.
If M and N are non-negative integers, then the zeta function is equal to
$\zeta (s)=\sum _{n=1}^{N}{\frac {1}{n^{s}}}+\gamma (1-s)\sum _{n=1}^{M}{\frac {1}{n^{1-s}}}+R(s)$
where
$\gamma (s)=\pi ^{{\tfrac {1}{2}}-s}{\frac {\Gamma \left({\tfrac {s}{2}}\right)}{\Gamma \left({\tfrac {1}{2}}(1-s)\right)}}$
is the factor appearing in the functional equation ζ(s) = γ(1 − s) ζ(1 − s), and
$R(s)={\frac {-\Gamma (1-s)}{2\pi i}}\int {\frac {(-x)^{s-1}e^{-Nx}}{e^{x}-1}}dx$
is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most 2πM. The approximate functional equation gives an estimate for the size of the error term. Siegel (1932) and Edwards (1974) derive the Riemann–Siegel formula from this by applying the method of steepest descent to this integral to give an asymptotic expansion for the error term R(s) as a series of negative powers of Im(s). In applications s is usually on the critical line, and the positive integers M and N are chosen to be about (2πIm(s))1/2. Gabcke (1979) found good bounds for the error of the Riemann–Siegel formula.
Riemann's integral formula
Riemann showed that
$\int _{0\searrow 1}{\frac {e^{-i\pi u^{2}+2\pi ipu}}{e^{\pi iu}-e^{-\pi iu}}}\,du={\frac {e^{i\pi p^{2}}-e^{i\pi p}}{e^{i\pi p}-e^{-i\pi p}}}$
where the contour of integration is a line of slope −1 passing between 0 and 1 (Edwards 1974, 7.9).
He used this to give the following integral formula for the zeta function:
$\pi ^{-{\tfrac {s}{2}}}\Gamma \left({\tfrac {s}{2}}\right)\zeta (s)=\pi ^{-{\tfrac {s}{2}}}\Gamma \left({\tfrac {s}{2}}\right)\int _{0\swarrow 1}{\frac {x^{-s}e^{\pi ix^{2}}}{e^{\pi ix}-e^{-\pi ix}}}\,dx+\pi ^{-{\frac {1-s}{2}}}\Gamma \left({\tfrac {1-s}{2}}\right)\int _{0\searrow 1}{\frac {x^{s-1}e^{-\pi ix^{2}}}{e^{\pi ix}-e^{-\pi ix}}}\,dx$
References
• Berry, Michael V. (1995), "The Riemann–Siegel expansion for the zeta function: high orders and remainders", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 450 (1939): 439–462, doi:10.1098/rspa.1995.0093, ISSN 0962-8444, MR 1349513, Zbl 0842.11030
• Edwards, H.M. (1974), Riemann's zeta function, Pure and Applied Mathematics, vol. 58, New York-London: Academic Press, ISBN 0-12-232750-0, Zbl 0315.10035
• Gabcke, Wolfgang (1979), Neue Herleitung und Explizite Restabschätzung der Riemann-Siegel-Formel (in German), Georg-August-Universität Göttingen, hdl:11858/00-1735-0000-0022-6013-8, Zbl 0499.10040
• Patterson, S.J. (1988), An introduction to the theory of the Riemann zeta-function, Cambridge Studies in Advanced Mathematics, vol. 14, Cambridge: Cambridge University Press, ISBN 0-521-33535-3, Zbl 0641.10029
• Siegel, C. L. (1932), "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Quellen Studien zur Geschichte der Math. Astron. Und Phys. Abt. B: Studien 2: 45–80, JFM 58.1037.07, Zbl 0004.10501 Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966.
External links
• Gourdon, X., Numerical evaluation of the Riemann Zeta-function
• Weisstein, Eric W. "Riemann–Siegel Formula". MathWorld.
Bernhard Riemann
• Cauchy–Riemann equations
• Generalized Riemann hypothesis
• Grand Riemann hypothesis
• Grothendieck–Hirzebruch–Riemann–Roch theorem
• Hirzebruch–Riemann–Roch theorem
• Local zeta function
• Measurable Riemann mapping theorem
• Riemann (crater)
• Riemann Xi function
• Riemann curvature tensor
• Riemann hypothesis
• Riemann integral
• Riemann invariant
• Riemann mapping theorem
• Riemann form
• Riemann problem
• Riemann series theorem
• Riemann solver
• Riemann sphere
• Riemann sum
• Riemann surface
• Riemann zeta function
• Riemann's differential equation
• Riemann's minimal surface
• Riemannian circle
• Riemannian connection on a surface
• Riemannian geometry
• Riemann–Hilbert correspondence
• Riemann–Hilbert problems
• Riemann–Lebesgue lemma
• Riemann–Liouville integral
• Riemann–Roch theorem
• Riemann–Roch theorem for smooth manifolds
• Riemann–Siegel formula
• Riemann–Siegel theta function
• Riemann–Silberstein vector
• Riemann–Stieltjes integral
• Riemann–von Mangoldt formula
• Category
| Wikipedia |
Why does the electron wavefunction not collapse within atoms at room temperature in gas, liquids or solids due to decoherence?
Decoherence theory predicts that any quantum particle coupled to any "large" environment should undergo decoherence and its wavefunction should collapse. This explains why measurement leads to wavepacket reduction.
However, in solids, liquids or gases, electrons within atoms don't reduce and stay as wavefunctions (orbits) somehow protected from the environment of the atoms.
This is surprising since the atoms are at room temperature, with a lot of things to interact with such as neighbouring atoms, light, thermal excitations, etc. So any idea why electrons seem 'protected' from a wavepacket reduction in atoms?
quantum-mechanics wavefunction atomic-physics wavefunction-collapse decoherence
A.J BeahvA.J Beahv
Welcome to SE -- good question! Decoherence does not mean that there won't be a wavefunction anymore, it just means that if the electron becomes coupled to the surrounding environment, its state will be described by a probabilistic mixture of orbital wavefunctions rather than a (coherent) superposition thereof. The electron in an atom doesn't have some "non-quantum" state(s) it can collapse into -- collapse just means that it will end up in one of the orbital states.
As a simplified example, consider spin states of an electron (simpler than orbitals because there are only two of them). Let $|0\rangle$ and $|1\rangle$ be some (orthonormal) basis states for this system. Then if the electron is initially in the state $$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),$$ a (coherent) superposition of the two basis states, after it has interacted with some noisy environment for a while we would expect its state to evolve towards a probabilistic mixture of the states $|0\rangle$ and $|1\rangle$ (assuming we are still representing in this basis), with probabilities 0.5 each unless there is some other factor to bias them. But the electron's spin cannot magically enter some other state that is not a linear combination of these; similarly, the orbital state remains an orbital state even when it decoheres.
edited May 9 at 1:17
WillWill
$\begingroup$ "assuming we are still interested in this basis" How does what we are "interested" in affect the behavior of an electron? By "interested in", do you mean "subjecting the wavefunction to an operator that has these basis states as eigenstates"? $\endgroup$ – Acccumulation May 8 at 18:18
$\begingroup$ @A.JBeahv again, good questions. First of all, you cannot measure the position of particle exactly; the best you can do is measure whether it is inside some small region of space or not, and you can make that region arbitrarily small, but not a single point. If you do that and localize the electron within a very small region, then its momentum uncertainty will be be large, and so after the measurement it might not even be in a bound state any more! (Full disclosure: I'm not sure how you would actually do such a measurement.) See more in next comment. $\endgroup$ – Will May 9 at 1:15
$\begingroup$ @A.JBeahv But if it is still in a bound state, such a state can always be written as a superposition of orbital states, since these are a complete basis for the bound states. Again, if you localize the electron (via measurement) in some small region, there will be very high energy orbitals involved in such a superposition, such that there may be significant probability of dislodging the electron entirely, but the bound part of the wavefunction will still be a superposition of orbital states. This, though, brings us to Rococo's answer, which points out that the actual energies required for $\endgroup$ – Will May 9 at 1:22
$\begingroup$ @A.JBeahv localization are much higher than those typically found in every-day-scale noisy environments, so you probably wouldn't get such extreme localization due to environmental interactions in, say, a laboratory on Earth (unless you are creating high-energy noise on purpose!) The bottom line is: the type of effective "measurement" due to coupling with the environment that you describe is actually quite improbable (vanishingly improbable as the volume in which you are localizing the electron goes to 0.) Hope this helps! $\endgroup$ – Will May 9 at 1:27
$\begingroup$ Thanks again. That really helped. Just to be sure I got it right : if I compare it to the usual double slit exp with single electrons detection on the screen, here in the atom, there is nothing close to that process because not enough energy and coupling with a measurement device. And this mainly comes from the Heisenberg principle which keeps the electron delocalized over a typical angstrom-size area around the nucleus (by balancing momentum and position within the atom) with typical eV energies. Would you agree ? $\endgroup$ – A.J Beahv May 9 at 8:27
I agree with Will's answer, but since there are multiple ways of looking at this here is another one: For an electron that is initially in its ground state to become spatially localized necessarily requires that some energy be added. For a hydrogen atom, the needed energy is at least 10 eV (to get to the second shell), and increasingly more than this to make an increasingly localized wavepacket. This requires high-energy photons, and there normally (at temperatures we find on Earth) aren't that many of those around, nor are there enough low-energy photons for multiple-photon transitions to be likely.
In a high-temperature environment in which there are lots of x-ray and gamma ray photons to drive these transitions, you probably would no longer have neutral hydrogen but instead a plasma. The electrons in this plasma might indeed be localized on a smaller scale than the hydrogen orbitals, depending on parameters such as the density.
This theme of needing higher energies to resolve smaller locations might sound familiar- it is just another manifestation of why we need huge accelerators like the LHC to directly probe the physics on very small length scales within a nucleon.
RococoRococo
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Home Journals EJEE MPPT for Photovoltaic System Using Adaptive Fuzzy Backstepping Sliding Mode Control
MPPT for Photovoltaic System Using Adaptive Fuzzy Backstepping Sliding Mode Control
Attoui Hadjira* | Behih Khalissa | Bouchama Ziyad | Zerroug Nadjat
QUERE Laboratory, Faculty of Technology, Ferhat Abbas University of Setif, Setif 19000, Algeria
LSI Laboratory, Faculty of Technology, Ferhat Abbas University of Setif, Setif 19000, Algeria
Department of Sciences and Technology, Bachir El Ibrahimi University, Bordj Bou Arreridj 34000, Algeria
[email protected]
This paper presents an intelligent monitoring control strategy for a maximum power point tracking (MPPT) in photovoltaic (PV) system applications. The design of the proposed nonlinear adaptive control law (AFBSMC) is formulated based on adaptive fuzzy systems, backstepping approach and sliding mode technique to maximize the power output of a PV system under various sets of conditions and parameters variation. Unlike many conventional controllers, the main contribution of the present paper provides a soften control law which useful to handle parameters variations due to the different operating conditions occurring on the PV system and makes the controller easy to implement. This aim is achieved using fuzzy systems in an adaptive scheme to approximate the switching control function of the global control law while backstepping sliding mode control compensates uncertainties and external disturbances. The analytical stability proof of the closed-loop system is corroborated via Lyapunov synthesis while numerical simulations of different operating conditions of a PV system is conducted to validate the effectiveness of the proposed approach.
backstepping sliding mode control, adaptive fuzzy control, Lyapunov stability, MPPT, photovoltaic system, DC-DC converter
Renewable energy has drawn much attention in recent years due to the high demand for green energy resources. The importance of solar panels (solar energy system) is greater nowadays as renewable sources since they exhibit many merits such as producing clean electrical energy, little maintenance and unlike other sources of renewable energy it has no geographical restrictions.
Solar PV systems have complex configurations that consist of components with nonlinear behaviors mainly due to the weather varying conditions such as temperature, irradiance and among others. These operating conditions are changed significantly with time which requires an adaptive control scheme to maintain adequate maximum power production in a practical operating environment.
Therefore, ensuring maximum electrical power extraction from photovoltaic systems, regardless of load changes and environmental conditions, is the primary objective control strategy, known as the Maximum Power Point Tracking Problem (MPPT) [1].
Many methods have been developed to determine the maximum power point (MPP) under all conditions [2-8]. There are numerous approaches, some of which are based on the well-known perturb and observe (P&O) concept [2, 3], others on the sliding mode control method [4, 5], artificial neural networks or fuzzy based algorithms [6, 7], and synergetic control [1, 7, 8].
Maximum power voltage (MPV)-based techniques using a two-loop MPPT control system are proposed in Refs. [9-11]. The first loop determines the PV array's MPV reference, while the second regulates the PV array's voltage to the reference voltage. The MPV reference search and PV voltage tracking are repeated until the maximum power is obtained. A hybrid technique consisting of two loops is presented in Ref. [12] to track MPP more effectively. MPP is estimated using an incremental conductance approach in the first loop. To regulate the system to the searched reference MPP, a second loop terminal sliding mode controller is constructed. Whereas Dahech et al. [13] proposed backstepping sliding mode control (BSMC) for the second loop. However, the main drawback of the SMC is the chattering. We propose in this paper to approximate discontinuous control using an adaptive fuzzy system based on the universal approximation theorem to tackle the problem of chattering mentioned in Ref. [13]. To ensure the system's global stability, the parameter of the fuzzy system is modified using an adaptation law based on the Lyapunov synthesis.
The goal of this work is to develop an adaptive fuzzy Backstepping sliding mode controller (AFBSMC) for MPPT, in order to overcome the problem of chattering. The whole system is modeled and simulated in Matlab/Simulink. Simulation results prove and confirm the effectiveness of the proposed approach in the elimination of the chattering.
The performance of this approach proves a high efficiency compared to the BSMC. The remainder of this paper is organized as follows. In Section 2, the MPPT system modeling is presented. The design of AFBSMC is exposed in Section 3. Section 4 uses numerical simulations to demonstrate the controller's usefulness in tracking MPP. The paper comes to a close with a conclusion.
2. MPPT System Modeling
Consider a boost type converter connected to a PV module with a resistive load as illustrated in Figure 1.
Figure 1. MPPT system schematic
2.1 PV model
PV array is a p-n junction semiconductor, which converts light into electricity. When the incoming solar energy exceeds the band-gap energy of the module, photons are absorbed by materials to generate electricity. The equivalent-circuit model of PV is shown in Figure 2.
It consists of a light-generated source, diode (D), series and parallel resistances [14].
Figure 2. Equivalent model of solar cell
where Iph indicates photocurrent, which depends on the level of light intensity, Ipv (Photovoltaic panel current) is output current, Vpv is the PV module output voltage, Rp is the equivalent shunt resistance, and Rs is the intrinsic series resistance.
In this work the PV module used is the KC200GH-2P. The parameters of this module are exposed in Table 1.
Figures 3 and 4 show the PV characteristic under different irradiance levels, and under different temperatures respectively.
As illustrated in the figures, the open-circuit voltage is dominated by temperature, and solar irradiance has preeminent influence on short- circuit current (Isc). We can conclude that high temperature and low solar irradiance will reduce the power conversion capability.
Table 1. Parameters of the PV module KC200GH-2P
Maximum power Pmpp
200 [W]
Short circuit current Iscr
8.21 [A]
Open circuit voltage Voc
32.9 [V]
Voltage at maximum power point Vmpp
Current at maximum power point Impp
P-N junction characteristic factor A
Figure 3. PV characteristic under different irradiances levels (temperature =25°C)
Figure 4. PV characteristic under diffe
2.2 Boost converter model
The converter is used to regulate the PV module output voltage Vpv in order to extract as much power as possible from the PV module. Referring to Ref. [12], the dynamics of the boost converter is given by Eq. (1):
$\frac{d V_{p v}}{d t}=\frac{1}{C_{1}}\left(I_{p v}-I_{L}\right)$
$\frac{d I_{L}}{d t}=\frac{1}{L} V_{p v}-\frac{R_{C}(1-d)}{L\left(1+\frac{R_{C}}{R}\right)} I_{L}$
$+\frac{1-d}{L}\left(\frac{R_{C}}{R_{C}+R}-1\right) V_{C 2}-\frac{V_{D}(1-d)}{L}$
$\frac{d V_{C 2}}{d t}=\frac{1-d}{C_{2}\left(1+\frac{R_{C}}{R}\right)} I_{L}-\frac{1}{C_{2}\left(R_{C}+R\right)} V_{C 2}$ (1)
where, the three state variables Vpv, IL and VC2 are respectively the output voltage of the PV module, the inductor current and the voltage of the capacitor C2. VD is the forward voltage of the power diode; d is the duty ratio of the PWM control input signal; R is the load resistance.
By taking $x(t)=\left[V_{p v}(t) I_{L}(t) V_{C 2}(t)\right]^{T}$, the Eq. (1) can be written in the following form [12].
$\left\{\begin{array}{c}\frac{d V_{p v}}{d t}=\frac{1}{C_{1}}\left(I_{p v}-I_{L}\right) \\ \frac{d I_{L}}{d t}=f_{1}(x)+g_{1}(x) d(t) \\ \frac{d V_{C 2}}{d t}=f_{2}(x)+g_{2}(x) d(t)\end{array}\right.$ (2)
where, $x=\left[\begin{array}{lll}x_{1} & x_{2} & x_{3}\end{array}\right]^{T}$.
Eqns. (3)-(6) show the expression of the functions $f_{1}, f_{2}, g_{1}$ and $g_{2}$
$f_{1}(x)=\frac{x_{1}}{L}-\frac{R_{C}}{L\left(1+\frac{R_{C}}{R}\right)} x_{2}+\frac{1}{L}\left(\frac{R_{C}}{R_{C}+R}-1\right) x_{3}-\frac{V_{D}}{L}$ (3)
$g_{1}(x)=-\frac{R_{C}}{L\left(1+\frac{R_{C}}{R}\right)} x_{2}-\frac{1}{L}\left(\frac{R_{C}}{R_{C}+R}-1\right) x_{3}+\frac{V_{D}}{L}$ (4)
$f_{2}(x)=\frac{1}{C_{2}\left(1+\frac{R_{C}}{R}\right)} x_{2}-\frac{1}{C_{2}\left(R_{C}+R\right)} x_{3}$ (5)
$g_{2}(x)=-\frac{1}{C_{2}\left(1+\frac{R_{C}}{R}\right)} x_{2}$ (6)
3. Design of Adaptive Fuzzy Backstepping Sliding Mode MPPT Controller
To achieve MPP under the changing atmosphere, the Figure 5 illustrates the overall control structure. Here, Ipv and Vpv are measured from PV array and sent to the MPP searching algorithm, which generates the reference maximum power voltage $V_{\text {ref }}$. Then, the reference voltage $V_{\text {ref }}$ is given to the maximum power voltage based AFBSM controller for the maximum power tracking.
Figure 5. Proposed system
3.1 MPP searching algorithm
To achieve the maximum power operation, we use an incremental conductance method to search the MPP voltage $V_{\text {ref }}$.
The power slope $d P_{p v} / d V_{p v}$ can be expressed as:
$\frac{d P_{p v}}{d V_{p v}}=I_{p v}+V_{p v} \frac{d I_{p v}}{d V_{p v}}$
When the power slope $\frac{d P_{p v}}{d V_{p v}}=0$, i.e., $\frac{d i_{p v}}{d V_{p v}}=-\frac{I_{p v}}{V_{p v}}$, the PV system operates at the maximum power generation.
Therefore, the update law for $V_{\text {ref }}$ is given by the following rules [12]:
$\left\{\begin{array}{l}V_{r e f}=V_{r e f}(k-1)+\Delta V, \text { for } \frac{d i_{p v}}{d V_{p v}}>-\frac{I_{p v}}{V_{p v}} \\ V_{r e f}=V_{r e f}(k-1)-\Delta V, \text { for } \frac{d i_{p v}}{d V_{p v}}<-\frac{I_{p v}}{V_{p v}}\end{array}\right.$
3.2 Backstepping sliding mode controller
To extract the maximum power from a PV panel a backstepping sliding mode controller is designed. Where the objective of this is to let the panel PV voltage $V_{p v}$ track the reference maximum power voltage $V_{\text {ref }}$ by acting on the duty cycle $d(t)$ of the boost converter switch.
The recursive nature of the propose control design is similar to the standard Backstepping methodology. However, the proposed control design uses Backstepping to design controllers with a zero-order sliding surface at the last step [13]. The design proceeds as follows:
For the first step we consider zero-order sliding surface represented by following equation:
$e_{1}=x_{1}-x_{d}$ (7)
where: $x_{d}=V_{\text {ref }}$.
Considering an auxiliary tracking error variable:
$e_{2}=\dot{e}_{1}+\alpha_{1}$ (8)
Let the first Lyapunov function candidate:
$V_{1}=\left(\frac{1}{2}\right) e_{1}^{2}$ (9)
The time derivation of Eq. (5) is given by the Eq. (10):
$\dot{V}_{1}=e_{1} \dot{e}_{1}=e_{1}\left(e_{2}-\alpha_{1}\right)=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}$ (10)
The stabilization of $\mathrm{e}_{1}$ can be obtained by introducing a new virtual control $\alpha_{1}$ Eq. (11), such that:
$\alpha_{1}=\lambda_{1} e_{1}, \lambda_{1}>0$ (11)
where $\lambda_{1}$ a positive feedback gain, such that $\alpha_{1}$ has been chosen in order to eliminate the non linearity and getting $\dot{V}_{1}(s)<0$. The term $e_{1} e_{2}$ of $\dot{V}_{1}$ will be eliminated in the next step, so the first sub system is stabilized.
For the second step we consider the following sliding surface:
$s=\lambda_{2} e_{1}-e_{2}$ (12)
The augmented Lyapunov function is given by Eq. (13):
$V_{2}=V_{1}+\left(\frac{1}{2}\right) s^{2}$ (13)
Eqns. (14)-(15) show the time derivative of $V_{2}$:
$\dot{V}_{2}=\dot{V}_{1}+s . \dot{s}$ (14)
$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+s .\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)-\dot{e}_{2}\right]$ (15)
The Eqns. (16)-(18) represent the time derivative of $e_{2}$:
$\dot{e}_{2}=\ddot{e}_{1}+\dot{\alpha}_{1}$ (16)
$\dot{e}_{2}=\ddot{V}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1}$ (17)
$\dot{e}_{2}=-\frac{1}{C_{1}}\left[f_{1}(x)+g_{1}(x) d(t)\right]+\frac{1}{C_{1}} \dot{I}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1}$ (18)
One can obtain the Eq. (19):
$\dot{s}=\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)+\frac{1}{C_{1}}\left[f_{1}(x)+g_{1}(x) d(t)\right]\right.$
$\left.-\frac{1}{C_{1}} I_{p v}+\ddot{x}_{1 d}-\dot{\alpha}_{1}\right]$ (19)
We impose the following dynamic to the sliding surface:
$\dot{s}=-h \cdot(s+\beta \operatorname{sgn}(s))$ (20)
$\operatorname{sign}(.)$ is the usual sign function., where $h>0$ and $\beta>0$ .
Substituting Eq. (18) into Eq. (15) we obtain:
$\begin{aligned} \dot{V}_{2}=-\lambda_{1} \mathrm{e}_{1}^{2}+\mathrm{e}_{1} \mathrm{e}_{2} & \\+& \mathrm{s}\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)\right.\\ &-\left(-\frac{1}{C_{1}}\left[f_{1}(x)+g_{1}(x) d(t)\right]\right.\\ &\left.\left.+\frac{1}{C_{1}} I_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1}\right)\right] \end{aligned}$ (21)
The control law is defined as:
$d(t)=d_{e q}(t)+\frac{C_{1}}{g_{1}(x)} d_{s w}(t)$ (22)
$d_{s w}(t)=-h(s+\beta \operatorname{sgn}(s))$ (23)
where, $d_{s w}(t)$ in the Eq. (23) is the switching control.
So we have:
$d(t)=\frac{1}{g_{1}(x)}\left[-f_{1}(x)-C_{1} \lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)+I_{p v}\right.$$-C_{1} \ddot{x}_{1 d}+C_{1} \dot{\alpha}_{1}-C_{1} h(\mathrm{~s}$$+\beta s g n(s)]$ (24)
The Eq. (15) is developed to:
$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}-s[h(\mathrm{~s}+\beta \operatorname{sgn}(s)]$ (25)
Introducing the norm in the Eq. (25), we get the Eqns. (26) and (27):
$\dot{V}_{2} \leq-\lambda_{1} e_{1}^{2}+e_{1} e_{2}-h s^{2}-\beta|s|$ (26)
$\dot{V}_{2} \leq-e^{T} p e-\beta|s|$ (27)
where, $e=\left[\begin{array}{ll}e_{1} & e_{2}\end{array}\right]^{T}$and P is a symmetric matrix defined as:
$P=\left[\begin{array}{cc}\lambda_{1}+h \lambda_{2}^{2} & -h \lambda_{2}-\frac{1}{2} \\ -h \lambda_{2}-\frac{1}{2} & h\end{array}\right]$
This proves the decreasing of Lyapunov function, which ensures that the closed-loop system is stable and robust.
This kind of sliding mode is certainly robust and stabilized but has two major drawbacks. The first lies in the presence of the sign function, where the control signal causes the phenomenon of chattering. The second disadvantage lies in the difficulty of the calculation of the constant $\beta, h$. To overcome these drawbacks several solutions have been presented in literature [15-17].
In order to resolve this problem, we propose to modify the following in the previous control law by using a fuzzy adaptive system. By letting the sliding area has input to approximate the term $d_{s w}(t)$. The fuzzy kind of lathing allows elimination of the phenomenon of chattering perfectly. At the same time as the adaptive appearance is designed to approximate the constant.
3.3 AFBSM controller of PV system
In this work we use the following If-Then rules [18, 19] to construct the fuzzy logic system:
$R_{i}:$ If $x_{1}$ is $F_{1}^{i}$ and ... and $x_{n}$ is $F_{n}^{i}$ then $y$ is $B^{i}, i=1,2, \ldots, n$
The fuzzy logic system with the singleton fuzzifier, product inference and center average defuzzifier is written as follows:
$y(x)=\frac{\sum_{i=1}^{n} \theta_{i} \prod_{j=1}^{n} \mu_{F_{j}^{i}\left(x_{j}\right)}}{\sum_{i=1}^{n}\left[\prod_{j=1}^{n} \mu_{F_{j}^{i}}\left(x_{j}\right)\right]}$ (28)
$x=\left[x_{1}, \ldots, x_{n}\right]^{T} \in R^{n}, \mu_{F_{j}^{i}}\left(x_{j}\right)$ is the membership of $F_{j}^{i}$
$\theta_{i}=\max _{y \in R} \mu_{B^{i}}(y)$, let:
$\xi_{i}(x)=\frac{\prod_{j=1}^{n} \mu_{F_{j}^{i}}\left(x_{j}\right)}{\sum_{i=1}^{n}\left[\prod_{j=1}^{n} \mu_{F_{j}^{i}}\left(x_{j}\right)\right]}$ (29)
$\xi(x)=\left[\xi_{1}(x), \xi_{2}(x), \ldots, \xi_{n}(x)\right]^{T}$ and $\theta=\left[\theta_{1}, \theta_{2}, \ldots, \theta_{n}\right]^{T}$.
Then the fuzzy logic system can be rewritten by the Eq. (30):
$y(x)=\theta^{T} \xi(x)$ (30)
The following Lemma, points out that the above fuzzy logic systems are capable to uniformly approximating any continuous nonlinear function, over a compact set Ωx.
Lemma: [18-19].
For any given continuous function f(x) on a compact set $\Omega_{x} \subset R^{n}$; there exists a fuzzy logic system y(x) In the form (30), such that for any given positive constant ε.$\sup _{x \in \Omega_{x}}|f(x)-y(x)| \leq \varepsilon$.
Then we propose to use a fuzzy system in the form (30) which can approximate the discontinuous control $d_{s w}(x)$, this latter is modeled by fuzzy system $\hat{h}(s)$. Then we have the following Eqns. (31) and (32):
$d_{s w}(x)=\hat{z}(s)+\Delta z(x)$ (31)
$\hat{z}(s)=\hat{\theta}_{z}^{T} \xi_{z}(s)$ (32)
Such that: w= Δz(x), is the approximation error.
So the control law is:
$d(t)=\frac{1}{g_{1}(x)}\left[-f_{1}(x)-C_{1} \lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)+\dot{I}_{p v}\right.$
$\left.-C_{1} \ddot{x}_{1 d}+C_{1} \dot{\alpha}_{1}-C_{1} \hat{z}(s)\right]$ (33)
$V_{2}=V_{1}+\left(\frac{1}{2}\right) s^{2}+\frac{1}{2 \eta_{z}} \tilde{\theta}_{z}^{T} \tilde{\theta}_{z}$ (34)
The derivative of this latter introducing Eq. (31), is given by Eqns. (35) and (36):
$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+s \cdot\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)-(-\frac{1}{C 1}\left[f_{1}(x)+g_{1}(x)\left(d_{e q}(t)\right)\right]+\tilde{\theta}_{z} \xi(s) \underbrace{-\hat{z}(s)}+w+\frac{1}{C 1} \dot{I}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1})\right]-\frac{1}{\eta_{h}} \tilde{\theta}_{z}^{\tau} \hat{\theta}_{z}$ (35)
$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+s \cdot\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)-(-\frac{1}{\mathrm{C} 1}\left[f_{1}(x)+g_{1}(x)\left(d_{e q}(t)\right)\right] \underbrace{-\hat{z}(s)}+w+\frac{1}{C 1} \dot{I}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1})\right]-\frac{1}{\eta_{h}} \tilde{\theta}_{z}^{T}\left[\dot{\hat{\theta}}_{z}-\eta_{z} s \cdot \xi(s)\right]$ (36)
Choosing the adaptive law as follow:
$\dot{\hat{\theta}}_{z}=\eta_{z} s \xi(s)$ (37)
The optimal value of $\hat{z}(s)$ is such that:
$\left|\hat{z}^{*}(s)\right| \geq|w|$ (38)
$\dot{V}_{2} \leq=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+|s|\left[-\left|\hat{z}^{*}(s)\right|+|w|\right]$ (39)
$\dot{V}_{2} \leq-e^{T} Q e-\varphi|s| \rightarrow \dot{V}_{2} \leq 0$ (40)
$\varphi=\left|\hat{z}^{*}(s)\right|-|w|$.
$e=\left[\begin{array}{ll}e_{1} & e_{2}\end{array}\right]^{T}$ and Q is a symmetric matrix with the following form:
$Q=\left[\begin{array}{cc}\lambda_{1}^{2} & -\frac{1}{2} \\ -\frac{1}{2} & 0\end{array}\right]$
To validate the proposed approach, we used the PV module KC200GH-2P, a boost converter and a resistive load.
We consider that the parameters of the boost are as follows:
L = 1.21 mH, RL = 0.15 Ω, RC = 39.6 Ω, C1 = 1000 µF, C2 = 1000 µF, R = 25 Ω, and VD = 0.82 V.
In this section we present the simulation results when applying the BSMC MPPT controller [16] and the Adaptive Fuzzy Backstepping Sliding Mode control law under different atmospheric conditions and load variation using Matlab/Simulink.
In order to construct the fuzzy system for the signal, we divide the discourse universe (the surface) in to three sets; "Positive", "Zero" and "Negative" which are associated with the following membership functions Eqns. (41)-(43):
$\mu_{\text {negative }}(s)=1 /(1+8 \cdot \exp (s-0.1))$ (41)
$\mu_{\text {zero }}(s)=1 /\left(-\exp (s / 0.5)^{2}\right)$ (42)
$\mu_{\text {positive }}(s)=1 /(1-8 . \exp (s-0.1))$ (43)
To deduce the signal h ̂(S) we used the Three fuzzy rules:
$\begin{array}{ccccc}R^{1}: \text { if } & \boldsymbol{s} & \text { is } & \text { Negative } & \text { then } & \hat{\mathrm{z}}(s)=-C \\ R^{2}: \text { if } & s & \text { is } & \text { Zero } & \text { then } & \hat{z}(s)=0 \\ R^{3}: \text { if } & s & \text { is } & \text { Positive } & \text { then } & \hat{z}(s)=C\end{array}$
4.1 Simulation results with standard operating conditions
Simulation results at Standard Test Condition: S=1000 W/m² and T = 25°C for both types of control, BSMC MPPT and the proposed AFBSMC are illustrate in Figures 6-9.
Figure 6. Evolution of $P_{P v}$
Figure 7. Evolution of duty cycle $d(t)$
Figure 8. Evolution of $V_{P v}$ and $V_{r e f}$
Figure 9. Evolution of sliding surface
For all the simulation results above, the fuzzy adaptive backstepping sliding mode control approach is able to maintain the output at optimum point and provides a good performance but also rejects the chattering drawback appeared in backstepping sliding mode control.
4.2 Simulation results under irradiation variations
For verifying the effect of changing irradiation conditions, as shown in Figure 10. The temperature is considered constant with a value of 25°C.
As shown in Figures 11-13 below, when the irradiance level changes, the proposed controller can track quickly the maximum power point (the response time of the AFBSMC is 0.02s).
Figure 10. Irradiation's variation
Figure 11. Evolution of $V_{P v}$ and $V_{r e f}$
Figure 12. Evolution of $P_{P v}$
Figure 13. Evolution of duty cycle d(t)
Figure 14. $V_{P v}-P_{P v}$ characteristics under solar irradiation variations
During the simulation the trace of the operating point is staying close to the MPP as shown in Figure 14.
4.3 Simulation results under temperature variations
For verifying the effect of changing temperature conditions, as shown Figure 15. The solar irradiation is considered constant with a value of 1000 W/m².
It is clear from Figures 16-18 below that the proposed AFBSMC provides a good performance and proves a high efficiency compared to the BSMC.
Figure 19 illustrates that the proposed controller follows the trajectory of the MPP perfectly.
Figure 15. Temperature's variation
Figure 19. $V_{P v}-P_{P v}$ characteristics under temperature variations
4.4 Simulation results under load variations
To show the robustness of the proposed AFBSMC, considering load change from 15 Ω to 80 Ω and from 80 Ω to 45 Ω under the Standard Test Condition (S=1000 W/m² and T = 25°C), the corresponding results are shown in Figures 20-22. It can be easily concluded that the proposed controller achieves strong robustness and has satisfactory response under these types of disturbance.
We constate that the AFBSMC not only performs its principal motion to drive the dynamics of the system to operate in the desired performance but also rejects the chattering drawback appeared in BSMC.
Figure 20. Load variation
Figure 21. $P_{P v}$ and $V_{o}$ under load variation
Figure 22. Evolution of duty cycle $d(t)$
This paper develops an adaptive MPPT control algorithm to extract and maximize the power created from PV system. Hence, the design of the proposed controller is based on fuzzy logic systems to generate a continuous time varying duty cycle for a DC converter which is inserted between the PV system and the load. Backstepping sliding mode methodology was derived based on the Lyapunov theory to render the controller more robust and to provide global system stability. The effectiveness and the robustness of the proposed approach have been verified using Simulink software. The obtained results show that the proposed control scheme is indeed appropriate for experimental applications.
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[2] Gradella, V.M., Rafael, G.J., Ruppert, F.E. (2009). Analysis and simulation of the P&O MPPT algorithm using a linearized PV array model. 2009 35th Annual Conference of IEEE Industrial Electronics, pp. 231-236. https://doi.org/10.1109/IECON.2009.5414780
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[4] El Idrissi, R., Abbou, A., Mokhlis, M. (2020). Backstepping integral sliding mode control method for maximum power point tracking for optimization of PV system operation based on high-gain observer. International Journal of Intelligent Engineering and Systems, 13(5): 133-144. https://doi.org/10.22266/ijies2020.1031.13
[5] Mohammadinodoushan, M., Abbassi, R., Jerbi, H., Ahmed, F. W., Rezvani, A. (2021). A new MPPT design using variable step size perturb and observe method for PV system under partially shaded conditions by modified shuffled frog leaping algorithm-SMC controller. Sustainable Energy Technologies and Assessments, 45: 101056. https://doi.org/10.1016/j.seta.2021.101056
[6] Loukil, K., Abbes, H., Abid, H., Abid, M., Toumi, A. (2020). Design and implementation of reconfigurable MPPT fuzzy controller for photovoltaic systems. Ain Shams Engineering Journal, 11(2): 319-328. https://doi.org/10.1016/j.asej.2019.10.002
[7] Akoro, E., Tevi, G.J., Faye, M.E., Doumbia, M.L., Maiga, A.S. (2020). Artificial neural network photovoltaic generator maximum power point tracking method using synergetic control algorithm. International Journal on Emerging Technologies, 11(2): 590-594.
[8] Nguimfack-Ndongmo, J., Kenné, G., Kuate-Fochie, R., Njomo, A.F.T., Nfah, E.M. (2021). Adaptive neuro-synergetic control technique for SEPIC converter in PV systems. International Journal of Dynamics and Control, 1-14. https://doi.org/10.1007/s40435-021-00808-1
[9] Kim, I.S. (2007). Robust maximum power point tracker using sliding mode controller for the three-phase grid-connected photovoltaic system. Solar energy, 81(3): 405-414. https://doi.org/10.1016/j.solener.2006.04.005
[10] Koutroulis, E., Kalaitzakis, K., Voulgaris, N.C. (2001). Development of a microcontroller-based, photovoltaic maximum power point tracking control system. IEEE Transactions on Power Electronics, 16(1): 46-54. https://doi.org/10.1109/63.903988
[11] Veerachary, M., Senjyu, T., Uezato, K. (2003). Neural-network-based maximum-power-point tracking of coupled-inductor interleaved-boost-converter-supplied PV system using fuzzy controller. IEEE Transactions on Industrial Electronics, 50(4): 749-758. https://doi.org/10.1109/TIE.2003.81476
[12] Chiu, C.S., Ouyang, Y.L., Ku, C.Y. (2012). Terminal sliding mode control for maximum power point tracking of photovoltaic power generation systems. Solar Energy, 86(10): 2986-2995. https://doi.org/10.1016/j.solener.2012.07.008
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A Liouville theorem of degenerate elliptic equation and its application
Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations
October 2013, 33(10): 4531-4547. doi: 10.3934/dcds.2013.33.4531
Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems
Jaume Giné 1, , Maite Grau 1, and Jaume Llibre 2,
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
Received November 2012 Revised January 2013 Published April 2013
In this paper we find necessary and sufficient conditions in order that a planar quasi--homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi--homogeneous polynomial differential system can be transformed into a differential system of the form $\dot{u} \, = \, u f(v)$, $\dot{v} \, = \, g(v)$ with $f(v)$ and $g(v)$ polynomials, and vice versa.
Keywords: polynomial first integral, rational first integral., Quasi--homogeneous polynomial differential equations, integrability problem.
Mathematics Subject Classification: Primary: 34C05, 34A34, 34C2.
Citation: Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531
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Jaume Giné Maite Grau Jaume Llibre | CommonCrawl |
Ernesto Laura
Ernesto Laura (23 March 1879 – 29 December 1949) was an Italian mathematician born in Porto Maurizio.[1][2]
Ernesto Laura
Born(1879-03-23)23 March 1879
Porto Maurizio
Died29 December 1949(1949-12-29) (aged 60)
Padua
NationalityItalian
Alma materUniversity of Turin
Scientific career
FieldsMathematics
Biography
He graduated in mathematics in 1901 at the University of Turin, where he was a student of Morera and of Somigliana.
He taught rational mechanics at the Universities of Messina, Pavia and Padua.
Laura has dealt with uncommon elasticity problems, namely related to indefinitely extended elastic means and, especially in the last part of his University career, the mechanics of flexible and inextensible surfaces.
Notes
1. An Italian short biography of Ernesto Laura in Edizione Nazionale Mathematica Italiana online.
2. Tonolo, Angelo (1950), "Necrologio di Ernesto Laura" (PDF), Bollettino dell'Unione Matematica Italiana, 3 (in Italian), 5 (3–4): 398–400
External links
• An Italian short biography of Ernesto Laura in Edizione Nazionale Mathematica Italiana online.
Authority control
International
• VIAF
National
• Italy
Academics
• zbMATH
| Wikipedia |
Estimation of sodium and chloride storage in critically ill patients: a balance study
Lara Hessels1Email authorView ORCID ID profile,
Annemieke Oude Lansink-Hartgring1,
Miriam Zeillemaker-Hoekstra1, 2 and
Maarten W. Nijsten1
Annals of Intensive Care20188:97
© The Author(s) 2018
Received: 2 April 2018
Accepted: 3 October 2018
Nonosmotic sodium storage has been reported in animals, healthy individuals and patients with hypertension, hyperaldosteronism and end-stage kidney disease. Sodium storage has not been studied in ICU patients, who frequently receive large amounts of sodium chloride-containing fluids. The objective of our study was to estimate sodium that cannot be accounted for by balance studies in critically ill patients. Chloride was also studied. We used multiple scenarios and assumptions for estimating sodium and chloride balances.
We retrospectively analyzed patients admitted to the ICU after cardiothoracic surgery with complete fluid, sodium and chloride balance data for the first 4 days of ICU treatment. Balances were obtained from meticulously recorded data on intake and output. Missing extracellular osmotically active sodium (MES) was calculated by subtracting the expected change in plasma sodium from the observed change in plasma sodium derived from balance data. The same method was used to calculate missing chloride (MEC). To address considerable uncertainties on the estimated extracellular volume (ECV) and perspiration rate, various scenarios were used in which the size of the ECV and perspiration were varied.
A total of 38 patients with 152 consecutive ICU days were analyzed. In our default scenario, we could not account for 296 ± 35 mmol of MES in the first four ICU days. The range of observed MES in the five scenarios varied from 111 ± 27 to 566 ± 41 mmol (P < 0.001). A cumulative value of 243 ± 46 mmol was calculated for MEC in the default scenario. The range of cumulative MEC was between 62 ± 27 and 471 ± 56 mmol (P = 0.001 and P = 0.003). MES minus MEC varied from 1 ± 51 to 123 ± 33 mmol in the five scenarios.
Our study suggests considerable disappearance of osmotically active sodium in critically ill patients and is the first to also suggest rather similar disappearance of chloride from the extracellular space. Various scenarios for insensible water loss and estimated size for the ECV resulted in considerable MES and MEC, although these estimates showed a large variation. The mechanisms and the tissue compartments responsible for this phenomenon require further investigation.
Intracellular volume
Extracellular volume
When long-term balance studies in humans demonstrated that sodium could accumulate without weight gain or hypernatremia, this challenged the generally accepted model on sodium homeostasis [1]. This model states that changes in sodium homeostasis can primarily be explained by a two-compartment model with an intracellular (ICV) and extracellular volume (ECV), where key ions are completely dissolved—i.e., osmotically active. An extra compartment that stores sodium nonosmotically without causing an expansion of the ECV has been proposed by Titze et al. [2]. In both animal and human studies, they found that sodium is stored nonosmotically in the skin [2, 3]. Nonosmotic sodium storage is presumably facilitated by large strongly negatively charged polymers such as glycosaminoglycans [4, 5]. The accumulation of chloride in the skin has been suggested in animal models [6, 7], but has not been as extensively studied as sodium storage.
Patients admitted to the intensive care unit (ICU) typically receive large amounts of sodium and chloride during their ICU treatment [8]. Both hypernatremia and hyperchloremia are a frequent complication in critically ill patients and are associated with adverse outcome [9–12]. The infusion of high amounts of chloride is also recognized as cause of hyperchloremic acidosis [11, 12]. Improved understanding of sodium chloride homeostasis in this patient group is therefore of utmost importance. To our knowledge, no studies have tried to measure missing sodium as evidence of stored sodium in ICU patients. Likewise, a potentially similar phenomenon for chloride has not been studied yet.
The objective of our study was therefore to estimate sodium and chloride that might 'disappear' in balance studies in ICU patients. Since random and systematic errors as well as different assumptions on the size of the ECV and perspiration strongly affect the calculated sodium or chloride deficit, five scenarios were tested in which the assumed sizes of the ECV or perspiration were varied.
This observational retrospective balance study involved all patients of ≥ 18 years admitted to a tertiary cardiothoracic ICU from October 2010 until December 2014 with a minimal ICU length of stay of 4 days.
Data that were collected and analyzed included basic demographics, reason of admission, acute physiology and chronic health evaluation (APACHE-IV) score for disease severity, acute kidney injury according to the KDIGO AKI criteria in the first 7 days and in-hospital mortality [13]. Fluid, sodium and chloride balances were derived from meticulously recorded input records (including enteral and parenteral feeding and administered fluids, including creep fluids such as solvent solutions) and output records, including daily 24-h urine collections. Our ICU did not have a full electronic patient database management system during the study period. Therefore, all data were derived from nursing and medical charts. All electrolyte concentrations, determined in blood or 24-h urine, were collected.
Estimation of missing extracellular osmotically active sodium (MES) and chloride (MEC)
The most important components to determine electrolyte balances are detailed records of fluids, administered to or lost by the patient, including 24-h urine analyses. The detailed calculations used for determining water, sodium and chloride balances, including the estimation for (in)sensible perspiration, have been described earlier and are specified in detail in Additional file 1: Tables S1–S3 [8]. Insensible perspiration was calculated as:
Insensible perspiration = 10 mL/kg/day + 2.5 mL/kg/day per degree centigrade above 37 °C (max body weight in equation 100 kg) (× 0.6 if intubated)(× 0.5 on admission day) [14].
For insensible perspiration, core temperature measured via the bladder catheter was used.
To estimate MES for each ICU patient, we compared the observed changes in estimated extracellular ΔNaobs with the expected change (ΔNaexp).
Of every ICU calender day, last measured plasma sodium was compared with the last measured plasma sodium of the previous day. For electrolyte measurements, the direct ion-selective method was used. For the patients studied, we defined the ECV at 40% in our default model, since surgical patients receive a considerable fluid load perioperatively [15]. We corrected for different sizes of the ECV at the beginning of the day versus the end of day, due to infused fluids.
Only ECVfirst that was calculated for admission day used measured body weight:
$${\text{ECV}}_{\text{first}} \, = \,0.4\, \times \,{\text{body}}\,{\text{weight}}\,\,\left( {\text{kg}} \right).$$
The extracellular volume at the end of the day (i.e., 23:59) was defined as:
$${\text{ECV}}_{23:59} \, = \,{\text{ECV}}_{\text{previous}} \, + \,{\text{fluid}}\,{\text{balance}}\,\,\left( {\text{L}} \right),$$
where ECVprevious is the ECV from 24 h earlier, or in the case it concerns the end of the first ICU day it relates to ECVfirst.
The extracellular volume at the beginning of the next day (i.e., 00:00) was defined as:
$${\text{ECV}}_{00:00} = {\text{ECV}}_{{{\text{last}}\,{\text{of}}\,{\text{the}}\,{\text{previous}}\,{\text{day}}}} \,\,\left( {\text{L}} \right).$$
The expected change in total amount of sodium in the ECV over a calender day was defined as:
$$\Delta {\text{Na}}_{ \exp }^{ + } \, = \,\left[ {{\text{Na}}^{ + } } \right]_{\text{last}} \, \times \,{\text{ECV}}_{\text{last}} \, - \,\left[ {{\text{Na}}^{ + } } \right]_{\text{previous}} \, \times \,{\text{ECV}}_{\text{previous}} \,\,\left( {\text{mmol}} \right).$$
The observed change in total extracellular sodium on the basis of administrated and excreted sodium was thereafter defined as:
$$\Delta {\text{Na}}_{\text{obs}}^{ + } \, = \,{\text{sodium}}\,{\text{balance}}\, = \,{\text{Na}}_{\text{in}} \,{-}\,{\text{Na}}_{\text{out}} \,\,\left( {\text{mmol}} \right).$$
The missing extracellular osmotically active sodium that apparently 'disappeared' from the ECV was defined as:
$${\text{MES}}\, = \,\Delta {\text{Na}}_{\text{obs}} \,{-}\,\Delta {\text{Na}}_{ \exp } \,\,\left( {\text{mmol}} \right).$$
For chloride, the same method as described above was used to calculate MEC, where instead of sodium, chloride should be read.
As a sensitivity analysis to test the robustness of our results, we tested 2 × 2 additional more extreme scenarios with respect to our assumptions on the ECV and perspiration. Where the default model assumed an ECV of 40% of the body weight and an insensible perspiration of 10 mL/kg/day, we tested both an extracellular compartment of 20% of body weight [16] and 60% of body weight [17]. In order to encapsulate the wide uncertainty in estimating actual perspiration, we also tested both lower and upper published extremes in perspiration rate of 5 mL/kg/day plus 2.5 mL/kg/day per degree centigrade above 37 °C versus and a perspiration rate of 20 mL/kg/day plus 2.5 mL/kg/day per degree centigrade above 37 °C.
To assess the differences in ECV between males and females, we performed a sub-analysis. In this analysis, an ECV of 40% of the body weight was assumed for males and an ECV of 30% was assumed for females.
Means are given ± SE, medians with interquartile range, unless otherwise indicated. MES and MEC were compared with a Student's t test. A two-sided P < 0.05 was considered significant. Cumulative calculations took account of increases in cumulative errors with the Pythagorean theory of error propagation. Balance calculations and statistical analysis were performed with SPSS 23.0 (IBM, Chicago, IL).
A total of 38 patients with 152 consecutive ICU days were included. Their baseline characteristics are given in Table 1.
Patients characteristics
n = 38
Sex, male
Reason of admission
LOS ICU (days)
7.4 (4.8–13.7)
Patients on diuretics
APACHE-IV
Data are depicted as mean (SD), n (%) or median (interquartile range) as appropriate
APACHE Acute Physiology and Chronic Health Evaluation
The included patients received large amounts of fluids (13.6 ± 0.6 L), sodium (1441 ± 75 mmol) and chloride (1377 ± 76 mmol) in the 4-day period, resulting in a cumulative fluid balance of 3.9 ± 0.6 L, a sodium balance of 822 ± 76 mmol and a chloride balance of 556 ± 82 mmol. Both the mean plasma sodium and chloride concentrations did not significantly change during the first four ICU days (Table 2).
Cumulative data on fluid and electrolyte administration
Fluid (L)
3.6 ± 0.4
11.0 ± 0.5
Sodium (mmol)
460 ± 52
1235 ± 72
Chloridea (mmol)
Plasma sodium (mmol/L)
137.1 ± 0.5
Plasma chloridea (mmol/L)
Data are depicted as mean ± SE
* Difference between day 1 and day 4
aChloride data were available for 27, 27, 24 and 28 patients, respectively, on day 1 to day 4
Missing extracellular osmotically active sodium and chloride
Based on our calculations, for sodium a MES of 74 ± 15 mmol per day was observed. This resulted in a cumulative MES of 296 ± 35 mmol during four ICU days (Fig. 1). For chloride, a MEC of 61 ± 23 mmol per day was seen with a cumulative MEC of 243 ± 46 mmol over the first four ICU days (Fig. 1).
Time course of estimated cumulative MES and MEC for the first four ICU days. Values are depicted as mean ± SE. The first values reflect levels at ICU admission, when storage was assumed defined as zero. The values at the subsequent time points reflect levels at the end (i.e., midnight) of each ICU day. As can be seen under normal and stable circulating electrolyte levels (Table 2), a significant amount of sodium (MES) and chloride (MEC) 'disappears' from the balances over the first four ICU days
We also calculated the difference between MES and MEC. The cumulative difference was 56 ± 40 mmol over the first four ICU days.
In the four scenarios in addition to the default scenario, we changed the assumed ECV and assumed perspiration and assessed their impact on MES and MEC (Figs. 2 and 3).
Scenarios for both estimated cumulative MES and MEC. Values are depicted as mean ± 95% CI. The 95% CI is represented by the dotted lines. The first values reflect levels at ICU admission, when storage was assumed to be zero. In all scenarios, there were considerable MES and MEC after 4 days of ICU admission. a With stable sodium levels, MES is mostly influenced by altering the insensible perspiration. b MEC showed a similar pattern as MES, but was slightly more affected by the changes in the extracellular compartment than MES
Estimated cumulative MES and MEC according to different scenarios. Values are depicted as means. The calculated MES (blue), MEC (red) and their difference (light gray) on ICU day 4 according to the scenarios with different assumptions on perspiration and the size of the extracellular volume
When perspiration was increased to 20 mL/kg/day, MES doubled for both the 20% and 60% ECV scenario (551 ± 35 mmol and 566 ± 41 mmol, respectively, both P < 0.001, Fig. 2a). After decreasing perspiration to 5 mL/kg/day, in both the ECV of 20% and 60% scenario, the amount of sodium we could not account for decreased more than 2.5 times the initial calculated MES (111 ± 27 mmol and 126 ± 31 mmol, respectively, both P < 0.001).
When these four scenarios were repeated for the calculations of MEC, similar results were obtained. When perspiration was increased to 20 mL/kg/day, for both the 20% and 60% ECV scenario we observed a similar increase in MEC from 243 ± 46 mmol to 414 ± 39 mmol (P = 0.006) to and 471 ± 56 mmol (P = 0.003, Fig. 2b). After decreasing perspiration to 5 mL/kg/day, MEC decreased. A MEC of 62 ± 27 mmol was observed when ECV was set at 20% in the low perspiration scenario (P = 0.001). However, when ECV was defined as 60%, MEC did not significantly change (243 ± 46 mmol vs. 119 ± 55 mmol, respectively, P = 0.09).
For all five depicted scenarios, the difference between MES and MEC was also calculated (Fig. 3) to identify potential structural differences between sodium and chloride disappearance. These differences varied from 1 ± 51 to 123 ± 33 mmol, indicating that MES and MEC were of the same order of magnitude.
The sub-analysis to assess sexual differences in ECV is found in Additional file 1: Fig. S1.
This is the first balance study that aimed to estimate missing extracellular sodium (MES) in ICU patients and missing extracellular chloride (MEC) in any patient group. Although we found considerable variations in estimated MES and MEC according to the various scenarios, the results suggest a considerable MES and a somewhat lower MEC (Fig. 3).
To calculate MES and MEC, we used one default and four more extreme scenarios, which we believe cover the scope of published sizes of the ECV and rates of perspiration. An ECV of 20% of body weight is a conservative choice in patients arriving at the ICU after major surgery [16], while 60% is an extreme estimate [17]. Regarding perspiration, defining the extremes was more difficult, but nearly all sources assume a perspiration ≥ 400 ml/day for both the skin and the respiratory tract without fever [14]. Our estimate of 5 mL/kg/day probably is thus the lower limit, while 20 mL/kg/day is a large estimate. We believe that the true value of both MES and MEC should be somewhere in between the four more extreme scenarios as depicted in Fig. 3. MES and MEC were mainly influenced by perspiration and MEC also somewhat by the ECV. This underscores that both the size of the ECV and insensible perspiration are important determinants in the estimated size of MES and MEC. In sex-specific models (Additional file 1: Fig. S1), males showed slightly higher MES and MEC compared with females.
More sodium and chloride disappeared from the balances during the first two days of ICU admission than in the subsequent days (Fig. 1). Resuscitation fluids, often high in sodium and chloride content, are frequently administered during surgery and in the early postoperative period. Whereas the recommended limits for dietary sodium intake are 2.3 g/day [18], our patients received an average of 8.3 g (i.e., 360 mmol) sodium per day, with positive sodium balances but stable sodium concentrations. This resulted in a MES of 296 mmol after four ICU days. When this MES is expressed in terms of NaCl 0.9% infusion, 1.9 L of this fluids sodium went missing in our patients.
Nonosmotic sodium storage has been studied in several non-critically ill patient groups. In healthy individuals, it has recently been observed that half of an acute intravenous hypertonic saline load of 201 mmol appears to be briefly stored nonosmotically [19], possibly in interaction with the endothelial glycocalyx. Sodium storage has been reported to increase with advancing age, to be greater in men and patients with hypertension, hyperaldosteronism, end-stage kidney disease and infection [20–22]. Tissue sodium levels are variable and may be altered by dialysis and diuretic treatment [23, 24]. However, the precise clinical significance of nonosmotic sodium storage has not been defined yet. The existence of nonosmotic sodium storage has not been examined in critically ill patients. Nonosmotic sodium storage could also be relevant in ICU-acquired hypernatremia (IAH) [25] and could explain the relatively long duration of IAH once it develops, although sodium balances were not performed in this study. It is believed that the electrical binding capacity of various tissues for sodium is altered during inflammation [26], which may interact with the development of IAH in critically ill patients. Irrespective of a potential relation between IAH and sodium storage, a strategy in which infusion fluids with lower sodium chloride content are used to reduce IAH is probably desirable [27]. We reported earlier [10] that changes in bulk intravenous fluid constitution paralleled changes in the incidence of ICU-acquired hypernatremia. Recently, it was elegantly shown that maintenance fluid therapy constitutes a higher sodium, chloride and water burden than acute resuscitation fluid administration [28].
With regard to chloride, which also disappeared in our balance calculations, both sodium and chloride storage may affect changes in blood pressure [6, 29].
As we cannot explain MES and MEC by the conventional two-compartment model where sodium is extracellular and potassium intracellular, a specific storage compartment may be the buffer of these sodium and chloride loads. The key alternative to nonosmotic storage is loss of sodium and chloride to the ICV. This effect has been demonstrated by healthy persons who sustained muscular injury [30]. Critical illness is often accompanied with critical illness myopathy, and loss of sodium and chloride to the ICV might then also be conceivable [31]. As we reported in an earlier study [8], our patients displayed a negative potassium balance of 101 mmol, which is another argument for possible intracellular uptake of sodium in exchange for potassium release. Moreover, we also observed a negative electrolyte-free water (EFW) balance in these patients. Together, this suggests that no ICV expansion occurred [8]. Therefore, we assumed that all fluids administered (including EFW) remained in the ECV. However, if part of the EFW would enter the ICV, this would result in lower increases and thus even higher MES and MEC estimates.
The presence of nonosmotic storage could be verified through direct tissue analysis or via specialized MRI [2, 3, 20]. Sodium changes in the tissues of ICU patients resulting from MES could be imaged via 23Na MRI. To our knowledge, 35Cl MRI has not yet been used to study MEC, but it is a promising and intriguing technique to identify the anatomical spaces where salt is stored [32, 33]. Importantly, this technique should be able to differentiate between the two main explanations for missing sodium and chloride: nonosmotic storage or intracellular uptake.
Our study has a number of limitations. Due to its retrospective design, we could not control for many variations in standard care. We had to make several assumptions, as, for example, for the insensible perspiration or the size of the ECV. However, we believe that the extreme scenarios on perspiration and ECV in our sensitivity analyses covered all realistic scenarios. We did not account for fecal losses, as we could not retrieve this information. Since we observed early postoperative patients, fecal production was absent or very low and, moreover, loss of sodium and chloride through the gut is usually very limited [34]. We did not measure weight changes as this is not routine procedure at our unit. Daily weight measurements could be added in the future to further validate our results.
On the relatively short term, differences in body weight measured in kg as measured in ICU patients will be less accurate than fluid balances measured in mL. Therefore, we only used initial recorded weight to estimate the ECV. Fluid balances in critically ill patients often have a poor correlation with changes in body weight [35, 36]. Especially cumulative fluid balance is prone to errors, as measuring errors get cumulated [36], which we accounted for in our error estimates. It must be noted that body weight measurement also has multiple possible errors, which could be the explanation of the lack of association between fluid balance and differences in body weight [36]. However, we believe that due to the short time this study covers and the meticulous recalculation of the fluid balance, including gastric retention, drain fluids and insensible perspiration, we have minimized errors as far as realistically possible.
Insensible perspiration remains very challenging to measure. As MES and MEC were most influenced by insensible perspiration, the lack of direct measurement of perspiration is an important limitation of our study. We tried, however, to maximize the chance to include the true value as much as possible with our five different scenarios. Direct measurement of (in)sensible perspiration would make estimated of MES and MEC more accurate (Fig. 3). Unfortunately, we are not aware of reliable tools to measure (in)sensible perspiration.
In this first observational balance study, we selected our patients based on complete balance data, which could have induced selection bias. The Androque-Madias [37] and Nguyen-Kurtz [38] formulas are frequently used when estimating the plasma sodium level after a saline infusion in dysnatremic ICU patients [17, 19]. However, we choose not to use these formulas in our study, as they do not account for excretion of sodium or chloride or they use empirically derived constants which were not suitable for using in our model. However, predictions on the size of the ECV from both formulas fall within the four scenarios.
In conclusion, our detailed sodium and chloride balances in ICU patients after cardiothoracic surgery suggest a loss of osmotically active sodium and chloride from the ECV. The estimates depend considerably on the scenarios used. Whether these ions are nonosmotically stored or transferred to the intracellular space needs further study.
APACHE-IV:
Acute Physiology and Chronic Health Evaluation—IV
ECV:
ICV:
MEC:
missing extracellular osmotically active chloride
missing extracellular osmotically active sodium
[Na+]first :
plasma sodium concentration on day of measurement
[Na+]last :
last plasma sodium concentration previous day
ΔNaobs :
observed change in estimated extracellular sodium
ΔNaexp :
expected change in estimated extracellular sodium
LH collected, analyzed and interpreted the data and was the main writer of the manuscript. AOL and MZH made important intellectual contributions to the manuscript. MWN conceived and oversaw the collection, analysis and interpretation of the data and made important intellectual contributions to the manuscript. All authors read and approved the final manuscript.
We would like to thank Flip Baardman, Rients de Boer and André Fitze for meticulously recording patient data.
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
The study was approved by the medical ethics committee (IRB) of our institution (Medisch Ethische Toetsingscommissie, METc 2015.089). As a retrospective study of routinely collected and anonymized data, informed consent was not required by our IRB.
No funding was used in conducting this research.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
13613_2018_442_MOESM1_ESM.docx Additional file 1. Detailed information on constants and calculations and sex-specific model for MES and MEC.
Department of Critical Care, University of Groningen, University Medical Center Groningen, Hanzeplein 1, 9700 RB Groningen, The Netherlands
Department of Anesthesiology, University of Groningen, University Medical Center Groningen, Groningen, The Netherlands
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\begin{document}
\title{Adaptive Approximation for Multivariate Linear Problems\\ with Inputs Lying in a Cone}
\author{Yuhan Ding, Fred J. Hickernell, Peter Kritzer, Simon Mak}
\maketitle
\begin{abstract} \noindent We study adaptive approximation algorithms for general multivariate linear
problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform
essentially no better than non-adaptive algorithms for convex input sets, the
situation may be different for non-convex sets. A typical example
considered here is function approximation based on series expansions. Given
an error tolerance, we use series coefficients of the input to construct an
approximate solution such that the error does not exceed this tolerance. We study the situation where we can bound the norm of the input based on a pilot sample, and the situation where we keep track of the decay rate of the series coefficients of the input. Moreover,
we consider situations where it makes sense to infer coordinate and smoothness importance. Besides performing an error analysis, we also study the information cost of our algorithms and the computational complexity
of our problems, and we identify conditions under which we can avoid a curse of
dimensionality. \end{abstract}
\section{Introduction}
In many situations, adaptive algorithms can be rigorously shown to perform \emph{essentially no better} than non-adaptive algorithms. Yet, in practice adaptive algorithms are appreciated because they relieve the user from stipulating the computational effort required to achieve the desired accuracy. The key to resolving this seeming contradiction is to construct a theory based on assumptions that favor adaptive algorithms. We do that here.
Adaptive algorithms infer the necessary computational effort based on the function data sampled. Adaptive algorithms may perform better than non-adaptive algorithms if the set of input functions is \emph{non-convex}. We construct adaptive algorithms for general multivariate linear problems where the input functions lie in non-convex cones. Our algorithms use a finite number of series coefficients of the input function to construct an approximate solution that satisfies an absolute error tolerance. We show our algorithms to be essentially optimal. We derive conditions under which the problem is tractable, i.e., the information cost of constructing the approximate solution does not increase exponentially with the dimension of the input function domain. In the remainder of this section we define the problem and essential notation. But first, we present a helpful example.
\subsection{An Illustrative Example}\label{DHKM:secexamp}
Consider the case of approximating functions defined over $[-1,1]^d$, using a Chebyshev polynomial basis. The input function is denoted $f$, and the solution is $\SOL(f) = f$. In this case,
\begin{align*}
f & = \sum_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d} \widehat{f}({\boldsymbol{k}}) u_{\boldsymbol{k}} =: \SOL(f),
\qquad {\boldsymbol{k}} = (k_1, \ldots, k_d) \in {\mathbb{N}}_0^d,\\
u_{\boldsymbol{k}} & := \prod_{\ell =1}^d \Tilde{u}_{k_\ell} ,
\qquad \Tilde{u}_{k}(x) := \cos( k \cos^{-1}(x)) \quad \forall k \in {\mathbb{N}}_0.
\end{align*} Approximating $f$ well by a finite sum requires knowing which terms in the infinite series for $f$ are more important. Let ${\mathcal{F}}$ denote a Hilbert space of input functions where the norm of ${\mathcal{F}}$ is a ${\boldsymbol{\lambda}}$-weighted norm of the series coefficients: \begin{equation*}
\norm[{\mathcal{F}}]{f} := \norm[2]{\left(\frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}}\right)_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}}, \qquad \text{where } {\boldsymbol{\lambda}} = \bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}, \quad \lambda_{\boldsymbol{k}} := \prod_{\substack{\ell =1\\ k_\ell > 0}}^d \frac{w_\ell}{k^r_\ell}, \quad r > 0. \end{equation*} The $w_\ell$ are non-negative \emph{coordinate weights}, which embody the assumption that $f$ may depend more strongly on coordinates with larger $w_\ell$ than those with smaller $w_\ell$. The definition of the ${\mathcal{F}}$-norm implies that an input function must have series coefficients that decay quickly enough as the degree of the polynomial increases. Larger $r$ implies smoother input functions.
The ordering of the weights, \begin{equation} \label{DHKM:lambda_order}
\lambda_{{\boldsymbol{k}}_1} \ge \lambda_{{\boldsymbol{k}}_2} \ge \cdots >0, \end{equation} implies an ordering of the wavenumbers, ${\boldsymbol{k}}$. It is natural to approximate the solution using the first $n$ series coefficients as follows: \begin{equation*}
\APP(f,n) := \sum_{i=1}^n \widehat{f}({\boldsymbol{k}}_i) u_{{\boldsymbol{k}}_i} \qquad \forall f \in {\mathcal{F}}, \ n \in {\mathbb{N}}. \end{equation*}
Here, we assume that it is possible to sample the series coefficients of the input function. This is a less restrictive assumption than being able to sample any linear functional, but it is more restrictive than only being able to sample function values. An important future problem is to extend the theory in this chapter to the case where the only function data available are function values.
The error of this approximation in terms of the norm on the output space, ${\mathcal{G}}$, can be expressed as \begin{gather*}
\norm[{\mathcal{G}}]{\SOL(f) - \APP(f,n)} = \norm[2]{\bigl(\widehat{f}({\boldsymbol{k}}_i) \bigr)_{i=n+1}^{\infty}}, \\
\text{where}\qquad
\norm[{\mathcal{G}}]{\sum_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d} \widehat{g}({\boldsymbol{k}}) u_{{\boldsymbol{k}}}} : = \norm[2]{\bigl(\widehat{g}({\boldsymbol{k}}) \bigr)_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}}. \end{gather*} If one has a fixed data budget, $n$, then $\APP(f,n)$ is the best answer.
However, our goal is an algorithm, $\ALG(f,\varepsilon)$ that satisfies the error criterion \begin{equation} \label{DHKM:err_crit}
\norm[{\mathcal{G}}]{\SOL(f) - \ALG(f,\varepsilon)} \le \varepsilon \qquad \forall \varepsilon > 0, \ f \in {\mathcal{C}}, \end{equation} where $\varepsilon$ is the error tolerance, and ${\mathcal{C}} \subset {\mathcal{F}}$ is the set of input functions for which $\ALG$ is successful. This algorithm contains a rule for choosing $n$---depending on $f$ and $\varepsilon$---so that $\ALG(f,\varepsilon) = \APP(f,n)$. The objectives of this chapter are to \begin{itemize}
\item construct such a rule,
\item choose a set ${\mathcal{C}}$ of input functions for which the rule is valid,
\item characterize the information cost of $\ALG$,
\item determine whether $\ALG$ has optimal information cost, and
\item understand the dependence of this cost on the number of input variables, $d$, as well as the error tolerance, $\varepsilon$. \end{itemize} We return to this example in Section \ref{DHKM:revisexamp} to discuss the answers to some of these questions. We perform some numerical experiments for this example in Section \ref{DHKM:numexamp_sec}.
\subsection{General Linear Problem}
Now, we define our problem more generally. A solution operator maps the input function to an output, $\SOL:{\mathcal{F}} \to {\mathcal{G}}$. As in the illustrative example above, the Banach spaces of inputs and outputs are defined by series expansions: \begin{gather*}
{\mathcal{F}} := \left \{f = \sum_{{\boldsymbol{k}} \in \mathbb{K}} \widehat{f}({\boldsymbol{k}}) u_{{\boldsymbol{k}}} : \norm[{\mathcal{F}}]{f} : = \norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in \mathbb{K}}} < \infty \right\}, \quad 1 \le \rho \le \infty, \\
{\mathcal{G}} := \left \{g = \sum_{{\boldsymbol{k}} \in \mathbb{K}} \widehat{g}({\boldsymbol{k}}) v_{{\boldsymbol{k}}} : \norm[{\mathcal{G}}]{g} : = \norm[\tau]{\bigl( \widehat{g}({\boldsymbol{k}}) \bigr)_{{\boldsymbol{k}} \in \mathbb{K}}} < \infty \right\}, \quad 1 \le \tau \le \rho. \end{gather*} Here, $\{u_{{\boldsymbol{k}}}\}_{{\boldsymbol{k}} \in \mathbb{K}}$ is a basis for the input Banach space ${\mathcal{F}}$, $\{v_{{\boldsymbol{k}}}\}_{{\boldsymbol{k}} \in \mathbb{K}}$ is a basis for the output Banach space ${\mathcal{G}}$, $\mathbb{K}$ is a countable index set, and ${\boldsymbol{\lambda}} = (\lambda_{\boldsymbol{k}})_{{\boldsymbol{k}} \in \mathbb{K}}$ is the sequence of weights. These bases are defined to match the solution operator: \begin{equation} \label{DHKM:basis_relate}
\SOL(u_{{\boldsymbol{k}}}) = v_{{\boldsymbol{k}}} \qquad \forall {\boldsymbol{k}} \in \mathbb{K}. \end{equation} The $\lambda_{{\boldsymbol{k}}}$ represent the importance of the series coefficients of the input function. The larger $\lambda_{{\boldsymbol{k}}}$ is, the more important $\widehat{f}({\boldsymbol{k}})$ is.
Although this problem formulation is quite general in some aspects, condition \eqref{DHKM:basis_relate} is somewhat restrictive. In principle, the choice of basis can be made via the singular value decomposition, but in practice, if the norms of ${\mathcal{F}}$ and ${\mathcal{G}}$ are specified without reference to their respective bases, it may be difficult to identify bases satisfying \eqref{DHKM:basis_relate}.
To facilitate our derivations below, we establish the following lemma via H\"older's inequality:
\begin{lemma} \label{DHKM:Key_Lem} Let ${\mathcal{K}}$ be some proper or improper subset of the index set ${\mathbb{K}}$. Moreover, let $\rho'$ be defined by the relation \begin{equation*}
\frac 1\rho + \frac 1 {\rho'} = \frac 1 \tau, \qquad \text{i.e., } \rho' := \frac{\rho \tau}{\rho - \tau}, \end{equation*} so $\tau \le \rho' \le \infty$. Let $\Lambda := \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}}$ be the norm of a subset of the weights. Then the following are true for $f = \sum_{{\boldsymbol{k}} \in {\mathcal{K}}} \widehat{f}({\boldsymbol{k}}) u_{{\boldsymbol{k}}}$: \begin{equation} \label{DHKM:SOL_ineq}
\norm[{\mathcal{G}}]{\SOL(f)} = \norm[\tau]{\bigl(\widehat{f}({\boldsymbol{k}}) \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}} \le \norm[{\mathcal{F}}]{f} \, \Lambda,
\end{equation}
\begin{multline}
\label{DHKM:SOL_tight_ineq}
\bigabs{\widehat{f}({\boldsymbol{k}})} = \begin{cases}
\displaystyle
\frac{R \lambda_{{\boldsymbol{k}}}^{\rho'/\rho + 1}}{\Lambda^{\rho'/\rho}}, & \forall {\boldsymbol{k}} \in {\mathcal{K}}, \quad \text{if}\ \rho'<\infty, \\
R \Lambda \delta_{{\boldsymbol{k}},{\boldsymbol{k}}^*}, & \forall {\boldsymbol{k}} \in {\mathcal{K}}, \ {\boldsymbol{k}}^* \in {\mathcal{K}} \text{ satisfies } \lambda_{{\boldsymbol{k}}^*} = \Lambda, \quad\text{if}\ \rho' = \infty,
\end{cases}
\\
\implies \ \ \norm[{\mathcal{F}}]{f} = R \ \mbox{and} \ \norm[{\mathcal{G}}]{\SOL(f)} = R \Lambda.
\end{multline} Equality \eqref{DHKM:SOL_tight_ineq} illustrates how inequality \eqref{DHKM:SOL_ineq} may be made tight. \end{lemma} \begin{proof} We give the proof for $\rho' < \infty$. The proof for $\rho' = \infty$ follows similarly. The proof of inequality \eqref{DHKM:SOL_ineq} proceeds by applying H\"older's inequality: \begin{align}
\label{DHKM:SOL_A}
\norm[{\mathcal{G}}]{\SOL(f)}
& = \biggnorm[{\mathcal{G}}]{\sum_{{\boldsymbol{k}} \in {\mathcal{K}}} \widehat{f}({\boldsymbol{k}}) v_{{\boldsymbol{k}}}} = \norm[\tau]{\bigl( \widehat{f}({\boldsymbol{k}}) \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}} = \left [\sum_{{\boldsymbol{k}} \in {\mathcal{K}}} \left\lvert\frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right\rvert^{\tau} \lambda_{{\boldsymbol{k}}}^{\tau} \right]^{1/\tau} \\
\nonumber
& \le \biggnorm[\rho]{\biggl( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \biggr)_{{\boldsymbol{k}} \in {\mathcal{K}}}} \, \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}} = \norm[{\mathcal{F}}]{f} \, \Lambda \qquad \text{since }\frac 1\rho + \frac 1 {\rho'} = \frac 1 \tau. \end{align} Substituting the formula for $\bigabs{\widehat{f}({\boldsymbol{k}})}$ in \eqref{DHKM:SOL_tight_ineq} into equation \eqref{DHKM:SOL_A} and applying the relationship between $\rho$, $\rho'$, and $\tau$ yields \begin{equation*}
\norm[{\mathcal{G}}]{\SOL(f)}
= \frac{R \bignorm[\tau]{\bigl( \lambda_{{\boldsymbol{k}}}^{\rho'/\rho + 1} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}}} {\Lambda^{\rho'/\rho}}
= \frac{R \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}}^{\rho'/\rho + 1}}
{\Lambda^{\rho'/\rho}} = R \Lambda. \end{equation*} Moreover, \begin{equation*}
\norm[{\mathcal{F}}]{f}
= \norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathcal{K}}}}
= \frac{R \bignorm[\rho]{\bigl( \lambda_{{\boldsymbol{k}}}^{\rho'/\rho} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}}}{\Lambda^{\rho'/\rho}}
= \frac{R \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}}}^{\rho'/\rho}}
{\Lambda^{\rho'/\rho}} = R. \end{equation*} This completes the proof. \end{proof} \
Taking ${\mathcal{K}} = {\mathbb{K}}$ in the lemma above, the norm of the solution operator can be expressed in terms of the norm of ${\boldsymbol{\lambda}}$ as follows: \begin{equation} \label{DHKM:SOLNorm}
\norm[{\mathcal{F}} \to {\mathcal{G}}]{\SOL} = \sup_{\norm[{\mathcal{F}}]{f} \le 1} \norm[{\mathcal{G}}]{\SOL(f)} = \bignorm[\rho']{{\boldsymbol{\lambda}}}. \end{equation} We assume throughout this chapter that the weights are chosen to keep this norm is finite, namely, \begin{equation} \label{DHKM:SOLNormFinite}
\bignorm[\rho']{{\boldsymbol{\lambda}}} < \infty. \end{equation}
As in Section \ref{DHKM:secexamp}, here in the general case the $\lambda_{{\boldsymbol{k}}}$ are assumed to have a known order as was specified in \eqref{DHKM:lambda_order}. We also assume that all $\lambda_{{\boldsymbol{k}}}$ are positive to avoid the trivial case where $\SOL(f)$ can be expressed exactly as a finite sum for all $f \in {\mathcal{F}}$.
\subsection{An Approximation and an Algorithm}
The optimal approximation based on $n$ series coefficients of the input function is defined in terms of the series coefficients of the input function corresponding to the largest $\lambda_{{\boldsymbol{k}}}$ as follows: \begin{equation} \label{DHKM:APP_def}
\APP : {\mathcal{F}} \times {\mathbb{N}}_0 \to {\mathcal{G}}, \quad \APP(f,0) = 0, \ \ \APP(f,n) := \sum_{i=1}^n \widehat{f}({\boldsymbol{k}}_i) v_{{\boldsymbol{k}}_i} \ \forall n \in {\mathbb{N}}. \end{equation} By the argument leading to \eqref{DHKM:SOL_A} it follows that \begin{equation} \label{DHKM:APP_Err_Coef}
\norm[{\mathcal{G}}]{\SOL(f) - \APP(f,n)} = \norm[\tau]{\bigl(\widehat{f}({\boldsymbol{k}}_i)\bigr)_{i=n+1}^\infty}. \end{equation} An upper bound on the approximation error follows from Lemma \ref{DHKM:Key_Lem}: \begin{equation} \label{DHKM:Refined_APP_err}
\norm[{\mathcal{G}}]{\SOL(f) - \APP(f,n) } \le \norm[\rho]{\left(\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}\right)_{i=n+1}^\infty}
\bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}}. \end{equation} This leads to the following theorem.
\begin{theorem} \label{DHKM:APP_optimality_thm} Let ${\mathcal{B}}_{R} : = \{ f \in {\mathcal{F}} : \norm[{\mathcal{F}}]{f} \le R \}$ denote the ball of radius $R$ in the space of input functions. The error of the approximation defined in \eqref{DHKM:APP_def} is bounded tightly above as \begin{equation} \label{DHKM:APP_errorBd}
\sup_{f \in {\mathcal{B}}_R} \norm[{\mathcal{G}}]{\SOL(f) - \APP(f,n)} \le R \, \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}}. \end{equation} Moreover, the worst case error over ${\mathcal{B}}_R$ of $\APP'(\cdot,n)$, for any approximation based on $n$ series coefficients of the input function, can be no smaller. \end{theorem}
\begin{proof} The proof of \eqref{DHKM:APP_errorBd} follows immediately from \eqref{DHKM:Refined_APP_err} and Lemma \ref{DHKM:Key_Lem}. The optimality of $\APP$ follows by bounding the error of an arbitrary approximation, $\APP'$, applied to functions that mimic the zero function.
Let $\APP'(0,n)$ depend on the series coefficients indexed by ${\mathcal{J}} = \{{\boldsymbol{k}}'_1, \ldots, {\boldsymbol{k}}'_n\}$. Use Lemma \ref{DHKM:Key_Lem} with ${\mathcal{K}} = {\mathbb{K}} \setminus {\mathcal{J}}$ to choose $f$ to mimic the zero function, have norm $R$, and have as large a solution as possible, i.e., \begin{gather} \nonumber
\widehat{f}({\boldsymbol{k}}'_1) = \cdots = \widehat{f}({\boldsymbol{k}}'_n) = 0, \qquad \norm[{\mathcal{F}}]{f} = R, \\
\norm[{\mathcal{G}}]{\SOL(f)} = R \norm[\rho']{\left( \lambda_{{\boldsymbol{k}}} \right)_{{\boldsymbol{k}} \notin {\mathcal{J}}}} \qquad \text{by \eqref{DHKM:SOL_tight_ineq}}. \label{DHKM:FoolFun} \end{gather} Then $\APP'(\pm f,n) = \APP'(0,n)$ because $f$ mimics the zero function, and \begin{align*} \MoveEqLeft{\sup_{f \in {\mathcal{B}}_R} \norm[{\mathcal{G}}]{\SOL(f) - \APP(f,n)}} \\ & \ge \max_{\pm} \norm[{\mathcal{G}}]{\SOL(\pm f) - \APP'(\pm f,n)} = \max_{\pm} \norm[{\mathcal{G}}]{\SOL(\pm f) - \APP'(0,n)} \\ & \ge \frac 12 \left [ \norm[{\mathcal{G}}]{\SOL(f) - \APP'(0,n)} + \norm[{\mathcal{G}}]{- \SOL(f) - \APP'(0,n)}\right] \\ & \ge \norm[{\mathcal{G}}]{\SOL(f)}
= R \norm[\rho']{\left( \lambda_{{\boldsymbol{k}}} \right)_{{\boldsymbol{k}} \notin {\mathcal{J}}}} \qquad \text{by \eqref{DHKM:FoolFun}}. \end{align*} The ordering of the $\lambda_{{\boldsymbol{k}}}$ implies that $\norm[\rho']{\left( \lambda_{{\boldsymbol{k}}} \right)_{{\boldsymbol{k}} \notin {\mathcal{J}}}}$ for arbitrary ${\mathcal{J}}$ can be no smaller than the case ${\mathcal{J}} = \{{\boldsymbol{k}}_1, \ldots, {\boldsymbol{k}}_n\}$. This completes the proof. \end{proof} \
While approximation $\APP$ is a key piece of the puzzle, our ultimate goal is an algorithm, $\ALG : {\mathcal{C}} \times [0,\infty)$, satisfying the absolute error criterion \eqref{DHKM:err_crit}. The non-adaptive Algorithm \ref{DHKM:BallAlg} satisfies this error criterion for ${\mathcal{C}} = {\mathcal{B}}_R$.
\begin{algorithm}[H] \caption{Non-Adaptive $\ALG$ for a Ball of Input Functions \label{DHKM:BallAlg}}
\begin{algorithmic}
\PARAM the Banach spaces ${\mathcal{F}}$ and ${\mathcal{G}}$, including the weights ${\boldsymbol{\lambda}}$; the ball radius, $R$; $\APP$ satisfying \eqref{DHKM:APP_errorBd}
\INPUT a black-box function, $f$; an absolute error tolerance, $\varepsilon>0$
\Ensure Error criterion \eqref{DHKM:err_crit} for ${\mathcal{C}} = {\mathcal{B}}_{R}$
\State Choose $n^* = \min \left \{n \in {\mathbb{N}}_0 : \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}} \le \varepsilon /R \right \}$
\RETURN $\ALG(f,\varepsilon) = \APP(f,n^*)$ \end{algorithmic} \end{algorithm}
After defining the information cost of an algorithm and the problem complexity in the next subsection, we demonstrate that this non-adaptive algorithm is optimal when the set of inputs is chosen to be ${\mathcal{C}} = {\mathcal{B}}_R$. However, typically one cannot bound the norm of the input function a priori, so Algorithm \ref{DHKM:BallAlg} is impractical.
The key difficulty is that error bound \eqref{DHKM:APP_errorBd} depends on the norm of the input function. In contrast, we will construct error bounds for $\APP(f,n)$ that only depend on function data. These will lead to \emph{adaptive} algorithms $\ALG$ satisfying error criterion \eqref{DHKM:err_crit}. For such algorithms, the set of allowable input functions, ${\mathcal{C}}$, will be a \emph{cone}, not a ball.
Note that algorithms satisfying error criterion \eqref{DHKM:err_crit} cannot exist for ${\mathcal{C}} = {\mathcal{F}}$. Any algorithm must require a finite sample size, even if it is huge. Then, there must exist some $f \in {\mathcal{F}}$ that looks exactly like the zero function to the algorithm but for which $\norm[{\mathcal{G}}]{\SOL(f)}$ is arbitrarily large. Thus, algorithms satisfying the error criterion exist only for some strict subset of ${\mathcal{F}}$. Choosing that subset well is both an art and a science.
\subsection{Information Cost and Problem Complexity}
The information cost of $\ALG(f,\varepsilon)$ is denoted $\COST(\ALG,f,\varepsilon)$ and defined as the number of function data---in our situation, series coefficients---required by $\ALG(f,\varepsilon)$. For adaptive algorithms this cost varies with the input function $f$. We also define the information cost of the algorithm in general, recognizing that it will tend to depend on $\norm[{\mathcal{F}}]{f}$: \begin{equation*}
\COST(\ALG, {\mathcal{C}}, \varepsilon,R) : = \max_{f \in {\mathcal{C}} \cap {\mathcal{B}}_{R}} \COST(\ALG,f,\varepsilon). \end{equation*} Note that while the cost depends on $\norm[{\mathcal{F}}]{f}$, $\ALG(f,\varepsilon)$ has no knowledge of $f$ beyond the fact that it lies in ${\mathcal{C}}$. It is common for $\COST(\ALG, {\mathcal{C}}, \varepsilon,R)$ to be ${\mathcal{O}}(\varepsilon^{-p})$, or perhaps asymptotically $c\log(1 + \varepsilon^{-1})$.
Let ${\mathcal{A}}({\mathcal{C}})$ denote the set of all possible algorithms that may be constructed using series coefficients and that \emph{satisfy error criterion \eqref{DHKM:err_crit}}. We define the \emph{computational complexity} of a problem as the information cost of the best algorithm: \begin{equation*}
\COMP({\mathcal{A}}({\mathcal{C}}), \varepsilon,R) := \min_{\ALG \in {\mathcal{A}}({\mathcal{C}})} \COST(\ALG, {\mathcal{C}}, \varepsilon,R) . \end{equation*} These definitions follow the information-based complexity literature \cite{TraWer98, TraWasWoz88}. We define an algorithm to be \emph{essentially optimal} if there exist some fixed positive $\omega$, $\varepsilon_{\max}$, and $R_{\min}$ for which \begin{multline} \label{DHKM:EssentialOpt}
\COST(\ALG, {\mathcal{C}}, \varepsilon,R) \le \COMP({\mathcal{A}}({\mathcal{C}}), \omega \varepsilon,R) \\ \forall \, \varepsilon \in (0, \varepsilon_{\max}], \ R \in [R_{\min}, \infty). \end{multline} If the complexity of the problem is ${\mathcal{O}}(\varepsilon^{-p})$, the cost of an essentially optimal algorithm is also ${\mathcal{O}}(\varepsilon^{-p})$. If the complexity of the problem is asymptotically $c \log(1 + \varepsilon^{-1})$, then the cost of an essentially optimal algorithm is also asymptotically $c \log(1 + \varepsilon^{-1})$. We will show that our adaptive algorithms presented in Sections \ref{DHKM:pilot_sec} and \ref{DHKM:tracking_sec} are essentially optimal.
\begin{theorem}\label{DHKM:thm_cost_non_adapt} The non-adaptive Algorithm \ref{DHKM:BallAlg} has an information cost for the set of input functions ${\mathcal{C}} = {\mathcal{B}}_R$ that is given by \[ \COST(\ALG, {\mathcal{B}}_R, \varepsilon,R') = \min \left \{n \in {\mathbb{N}}_0 : \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}} \le \varepsilon /R \right \}. \] This algorithm is essentially optimal for the set of input functions ${\mathcal{B}}_R$, namely, \begin{multline*} \COST(\ALG, {\mathcal{B}}_R, \varepsilon,R') \le \COMP({\mathcal{A}}({\mathcal{B}}_R),\omega \varepsilon ,R') \\ \forall \, \varepsilon \in (0, \varepsilon_{\max}], \ R \in [R_{\min}, \infty), \end{multline*} where $\varepsilon_{\max}$ and $R_{\min}$ are arbitrary and fixed, and $\omega = R_{\min}/R$. \end{theorem}
\begin{proof} Fix positive $\varepsilon_{\max}$, $R_{\min}$, $R$, and $\omega$ as defined above. For $0 < \varepsilon \le \varepsilon_{\max}$ and $R_{\min} \le R' \le R$, the information cost of non-adaptive Algorithm \ref{DHKM:BallAlg} follows from its definition. Let \[ n^*(\varepsilon,R) : = \COST(\ALG, {\mathcal{B}}_R, \varepsilon,R'). \] Construct an input function $f \in {\mathcal{B}}_{R'}$ as in the proof of Theorem \ref{DHKM:APP_optimality_thm} with ${\mathcal{J}} = \{{\boldsymbol{k}}_1, \ldots, {\boldsymbol{k}}_{n^*(\omega \varepsilon,R')} \}$. By the argument in the proof of Theorem \ref{DHKM:APP_optimality_thm}, any algorithm in ${\mathcal{A}}({\mathcal{B}}_{R'})$ that can approximate $\SOL(f)$ with an error no greater than $\omega \varepsilon$ must use at least $n^*(\omega \varepsilon,R')$ series coefficients. Thus, \begin{align*} \COST(\ALG, {\mathcal{B}}_R, \varepsilon,R') & = n^*(\varepsilon,R) = n^*(\varepsilon R'/R,R') \\ & \le n^*(\omega \varepsilon, R') \qquad \text{since } R'/R \ge \omega \\ & \le \COMP({\mathcal{A}}({\mathcal{B}}_{R'}),\omega \varepsilon, R') \le \COMP({\mathcal{A}}({\mathcal{B}}_{R}),\omega \varepsilon, R'). \end{align*} Thus, Algorithm \ref{DHKM:BallAlg} is essentially optimal. \end{proof} \
For Algorithm \ref{DHKM:BallAlg}, the information cost, $\COST(\ALG,{\mathcal{B}}_R,\varepsilon, R)$, depends on the decay rate of the tail norm of the $\lambda_{{\boldsymbol{k}}_i}$. This decay may be algebraic or exponential and also determines the problem complexity, $\COMP({\mathcal{A}}({\mathcal{B}}_R),\varepsilon, R)$, as a function of the error tolerance, $\varepsilon$.
This theorem illustrates how an essentially optimal algorithm for solving a problem for a ball of input functions, ${\mathcal{C}} = {\mathcal{B}}_R$, can be non-adaptive. However, as alluded to above, we claim that it is impractical to know a priori which ball your input function lies in. On the other hand, in the situations described below where ${\mathcal{C}}$ is a cone, we will show that ${\mathcal{A}}({\mathcal{C}})$ actually contains only adaptive algorithms via the lemma below. The proof of this lemma follows directly from the definition of non-adaptivity.
\begin{lemma} \label{DHKM:NoNonAdpatLem} For a given set of input functions, ${\mathcal{C}}$, if ${\mathcal{A}}({\mathcal{C}})$ contains any non-adaptive algorithms, then for every $\varepsilon > 0$, \begin{equation*}
\COMP({\mathcal{A}}({\mathcal{C}}),\varepsilon) : = \sup_{R > 0} \COMP({\mathcal{A}}({\mathcal{C}}),\varepsilon, R) < \infty. \end{equation*} \end{lemma}
\subsection{Tractability}\label{DHKM:secTractability}
Besides understanding the dependence of $\COMP({\mathcal{A}}({\mathcal{C}}), \varepsilon,R)$ on $\varepsilon$, we also want to understand how $\COMP({\mathcal{A}}({\mathcal{C}}), \varepsilon,R)$ depends on the dimension of the domain of the input function. Suppose that $f: \Omega^d \to {\mathbb{R}}$, for some $\Omega \subseteq {\mathbb{R}}$, and let ${\mathcal{F}}_d$ denote the dependence of the input space on the dimension $d$. The set of functions for which our algorithms succeed, ${\mathcal{C}}_d$, depends on the dimension, too. Also, $\SOL$, $\APP$, $\COST$, and $\COMP$ depend implicitly on dimension, and this dependence is sometimes indicated explicitly by the subscript $d$.
Different dependencies of $\COMP({\mathcal{A}}({\mathcal{C}}_d), \varepsilon,R)$ on the dimension $d$ and the error tolerance $\varepsilon$ are formalized as different notions of tractability. Since the complexity is defined in terms of the best available algorithm, tractability is a property that is inherent to the problem, not to a particular algorithm. We define the following notions of tractability (for further information on tractability we refer to the trilogy \cite{NovWoz08a}, \cite{NovWoz10a}, \cite{NovWoz12a}). Note that in contrast to these references we explicitly include the dependence on $R$ in our definitions. This dependence is natural for cones ${\mathcal{C}}$ and might be different if ${\mathcal{C}}$ is not a cone.
\begin{itemize} \item We say that the adaptive approximation problem is strongly polynomially tractable if and only if there are non-negative $C$, $p$, $\varepsilon_{\max}$, and $R_{\min}$ such that $$ \COMP({\mathcal{A}}({\mathcal{C}}_d), \varepsilon,R)\le C\,R^p\,\varepsilon^{-p} \qquad \forall d\in{\mathbb{N}},\ \varepsilon \in (0, \varepsilon_{\max}], \ R \in [R_{\min}, \infty). $$ The infimum of $p$ satisfying the bound above is denoted by $p^*$ and is called the exponent of strong polynomial tractability. \newline \qquad
\item We say that the problem is polynomially tractable if and only if there are non-negative $C,p$, $q$, $\varepsilon_{\max}$, and $R_{\min}$ such that $$ \COMP({\mathcal{A}}({\mathcal{C}}_d), \varepsilon,R)\le C\,d^{\,q}\,R^p\, \varepsilon^{-p}\qquad \forall d\in{\mathbb{N}},\ \varepsilon \in (0, \varepsilon_{\max}], \ R \in [R_{\min}, \infty). $$ \vskip 0.5pc
\item We say that the problem is weakly tractable iff $$ \lim_{d+R\,\varepsilon^{-1}\to\infty}\ \frac{\log\, \COMP({\mathcal{A}}({\mathcal{C}}_d), \varepsilon,R)} {d+R\,\varepsilon^{-1}}\,=\,0. $$ \end{itemize}
Necessary and sufficient conditions on these tractability notions will be studied for different types of algorithms in Sections \ref{DHKM:SecPilotTract} and \ref{DHKM:SecDecayTract}.
We remark that, for the sake of brevity, we focus here on tractability notions that are summarized as algebraic tractability in the recent literature (see, e.g., \cite{KriWoz19}). Theoretically, one could also study exponential tractability, where one would essentially replace $\varepsilon^{-1}$ by $\log (1 + \varepsilon^{-1})$ in the previous tractability notions. A more detailed study of tractability will be done in a future paper.
\subsection{The Illustrative Example Revisited}\label{DHKM:revisexamp}
The example in Section \ref{DHKM:secexamp} chooses $\rho = \tau = 2$ and $\rho' = \infty$. Thus, we obtain by Theorem \ref{DHKM:thm_cost_non_adapt}: \begin{align*}
\COMP({\mathcal{A}}({\mathcal{B}}_{R}),\varepsilon,R) &= \COST(\ALG,{\mathcal{B}}_{R},\varepsilon,R)
=\min\{ n \in {\mathbb{N}}_0 : \lambda_{{\boldsymbol{k}}_{n+1}} \le \varepsilon/R \}. \end{align*} Using the non-increasing ordering of the $\lambda_{{\boldsymbol{k}}_i}$, we employ a standard technique for bounding the $n+1^\text{st}$ largest $\lambda_{{\boldsymbol{k}}}$ in terms of the sum of the $p^\text{th}$ power of all the $\lambda_{{\boldsymbol{k}}}$. For $0 < 1/r < p$, \begin{align*} (n+1)\lambda_{{\boldsymbol{k}}_{n+1}}^{p} &\le \sum_{i=1}^{n+1}\lambda_{{\boldsymbol{k}}_{i}}^{p} \le \sum_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}\lambda_{{\boldsymbol{k}}}^{p} = \prod_{\ell =1}^d \left[1 + w_\ell^{p}\sum_{k = 1}^\infty \frac {1}{k^{pr}} \right ] \\ & = \prod_{\ell =1}^d \left[1 + w_\ell^{p}\zeta(pr)\right ] = \exp\left(\sum_{\ell = 1}^d \log\bigl(1 + w_\ell^{p}\zeta(pr) \bigr) \right) \\ & \le \exp\left(\zeta(pr) \sum_{\ell = 1}^{\infty} w_\ell^{p} \right) \qquad \text{since }\log(1+x) \le x \ \forall x\ge 0. \end{align*} Hence, substituting the above upper bound on $\lambda_{{\boldsymbol{k}}_{n+1}}$ into the formula for the complexity of the problem, we obtain an upper bound on the complexity: \begin{align*}
\MoveEqLeft{\COMP({\mathcal{A}}({\mathcal{B}}_{R}),\varepsilon,R)} \\
&\le\min\left\{ n \in {\mathbb{N}}_0 : \frac{1}{n+1}\exp\left(\zeta(pr) \sum_{\ell = 1}^\infty w_\ell^{p} \right) \le \left( \frac{\varepsilon}{R} \right)^{p} \right\} \\
&= \left \lceil \left( \frac{R}{\varepsilon} \right)^{p} \exp\left(\zeta(pr) \sum_{\ell = 1}^{\infty} w_\ell^{p} \right) \right \rceil - 1. \end{align*}
If $p^\dagger$ is the infimum of the $p$ for which $\sum_{\ell = 1}^{\infty} w_\ell^{p}$ is finite, and $p^\dagger$ is finite, then we obtain strong polynomial tractability and an exponent of strong tractability that is $p^* = \max(1/r,p^\dagger)$. On the other hand, if the coordinate weights are all unity, $w_1 = w_2 = \cdots = 1$, then there are $2^d$ different $\lambda_{{\boldsymbol{k}}}$ with a value of $1$, and so $\COMP({\mathcal{A}}({\mathcal{B}}_{R}),\varepsilon,R) \ge 2^d$, and the problem is not tractable.
\subsection{What Comes Next} In the following section we define a cone of input functions, ${\mathcal{C}}$, in \eqref{DHKM:pilot_cone} whose norms can be bounded above in terms of the series coefficients obtained from a pilot sample. Adaptive Algorithm \ref{DHKM:PilotConeAlg} is shown to be optimal for this ${\mathcal{C}}$. We also identify necessary and sufficient conditions for tractability.
Section \ref{DHKM:tracking_sec} considers the situation where function data is relatively inexpensive, and we track the decay rate of the series coefficients. Adaptive Algorithm \ref{DHKM:TrackConeAlg} is shown to be optimal in this situation.
Section \ref{DHKM:smoothimportance_sec} considers the case where the most suitable weights ${\boldsymbol{\lambda}}$ are not known a priori and are instead inferred from function data. Adaptive Algorithm \ref{DHKM:InfPilotConeAlg} combines this inference step with Algorithm \ref{DHKM:PilotConeAlg} to construct an approximation that satisfies the error criterion.
\section{Bounding the Norm of the Input Function Based on a Pilot Sample} \label{DHKM:pilot_sec}
\subsection{The Cone and the Optimal Algorithm}
The premise of an adaptive algorithm is that the finite information we observe about the input function tells us something about what is not observed. Let $n_1$ denote the number of pilot observations, based on the set of wavenumbers \begin{equation} \label{DHKM:KOnedef}
{\mathcal{K}}_1 := \{{\boldsymbol{k}}_1, \ldots, {\boldsymbol{k}}_{n_1} \}, \end{equation} where the ${\boldsymbol{k}}_i$ are defined by the ordering of the $\lambda_{{\boldsymbol{k}}}$ in \eqref{DHKM:lambda_order}. Let $A$ be some constant inflation factor greater than one. The cone of functions whose norm can be bounded well in terms of a pilot sample, $\{\widehat{f}({\boldsymbol{k}}_1), \ldots, \widehat{f}({\boldsymbol{k}}_{n_1})\}$, is given by \begin{equation} \label{DHKM:pilot_cone}
{\mathcal{C}} = \left \{ f \in {\mathcal{F}} : \norm[{\mathcal{F}}]{f} \le A \norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}} \right\}. \end{equation}
Referring to error bound \eqref{DHKM:Refined_APP_err}, we see that the error of $\APP(f,n)$ depends on the series coefficients not sampled. The definition of ${\mathcal{C}}$ allows us to bound these as follows: \begin{align*}
\norm[\rho]{\left(\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}\right)_{i=n+1}^\infty} & = \left[ \norm[{\mathcal{F}}]{f}^\rho - \norm[\rho]{\left(\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}\right)_{i=1}^n}^\rho
\right]^{1/\rho} \qquad \forall f \in {\mathcal{F}} \\
& \le \left[ A^\rho \norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}}^\rho - \norm[\rho]{\left(\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}\right)_{i=1}^n}^\rho
\right]^{1/\rho} \quad \forall f \in {\mathcal{C}}. \end{align*} This inequality together with error bound \eqref{DHKM:Refined_APP_err} implies the data-based error bound \begin{subequations} \label{DHKM:pilot_errbd} \begin{equation} \norm[{\mathcal{G}}]{\SOL(f) - \APP(f,n)} \le \ERR\bigl(\dataN,n\bigr) \qquad \forall f \in {\mathcal{C}}, \end{equation} where \begin{multline} \ERR\bigl(\dataN,n\bigr) \\
: =
\left[ A^\rho \norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}}^\rho - \norm[\rho]{\left(\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}\right)_{i=1}^n}^\rho \right]^{1/\rho}
\, \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}} ,
\\ n \ge n_1. \end{multline} \end{subequations} This error bound decays as $\bignorm[\rho]{\bigl( f({\boldsymbol{k}}_i)/\lambda_{{\boldsymbol{k}}_i} \bigr)_{i=1}^n}$ increases and as the tail norm of the $\lambda_{{\boldsymbol{k}}_i}$ decreases. This data-driven error bound underlies Algorithm \ref{DHKM:PilotConeAlg}, which is successful for ${\mathcal{C}}$ defined in \eqref{DHKM:pilot_cone}:
\begin{algorithm}
\caption{$\ALG$ Based on a Pilot Sample\label{DHKM:PilotConeAlg}}
\begin{algorithmic}
\PARAM the Banach spaces ${\mathcal{F}}$ and ${\mathcal{G}}$, including the weights ${\boldsymbol{\lambda}}$; an initial sample size, $n_1 \in {\mathbb{N}}$; an inflation factor, $A > 1$; $\APP$ satisfying \eqref{DHKM:Refined_APP_err}
\INPUT a black-box function, $f$; an absolute error tolerance,
$\varepsilon>0$
\Ensure Error criterion \eqref{DHKM:err_crit} for the cone defined in \eqref{DHKM:pilot_cone}
\State Let $n \leftarrow n_1 -1$ \Repeat
\State Let $n \leftarrow n + 1$
\State Compute $\ERR\bigl(\dataN,n\bigr)$ as defined in \eqref{DHKM:pilot_errbd}
\Until $\ERR\bigl(\dataN,n\bigr) \le \varepsilon$
\RETURN $\ALG(f,\varepsilon) = \APP(f,n)$
\end{algorithmic} \end{algorithm}
\begin{theorem} \label{DHKM:PilotCostThm} Algorithm \ref{DHKM:PilotConeAlg} yields an answer satisfying absolute error criterion \eqref{DHKM:err_crit}, i.e., $\ALG \in {\mathcal{A}}({\mathcal{C}})$ for ${\mathcal{C}}$ defined in \eqref{DHKM:pilot_cone}. The information cost is \begin{multline} \label{DHKM:PilotConeAlg_cost}
\COST(\ALG,{\mathcal{C}},\varepsilon,R) \\
= \min \left \{n \ge n_1 : \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}} \,
\le \varepsilon/[(A^\rho -1)^{1/\rho}R] \right \}. \end{multline} There exist positive $\varepsilon_{\max}$ and $R_{\min}$ for which the computational complexity has the lower bound \begin{multline} \label{DHKM:PilotConeAlg_comp}
\COMP({\mathcal{A}}({\mathcal{C}}),\varepsilon,R) \ge \min \left \{n \ge n_1 : \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}} \,
\le 2\varepsilon/[(1 - 1/A) R] \right \} \\
\forall \varepsilon \in (0, \varepsilon_{\max}], \ R \in [R_{\min}, \infty). \end{multline} Algorithm \ref{DHKM:PilotConeAlg} is essentially optimal. Moreover, ${\mathcal{A}}({\mathcal{C}})$ contains only adaptive algorithms. \end{theorem}
\begin{proof} The upper bound on the computational cost of this algorithm is obtained by noting that \begin{align*}
\MoveEqLeft[1]{\COST(\ALG,{\mathcal{C}},\varepsilon,R)} \\
& = \max_{f \in {\mathcal{C}} \cap {\mathcal{B}}_{R}} \min \left \{n \ge n_1 : \ERR\bigl(\dataN,n\bigr) \le \varepsilon \right \} \\
& \le \max_{f \in {\mathcal{C}} \cap {\mathcal{B}}_{R}} \min \left \{n \ge n_1 :
(A^\rho -1)^{1/\rho} \norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathcal{K}}_1 }} \,
\bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}}
\le \varepsilon \right \} \\
& \le \min \left \{n \ge n_1 :
(A^\rho -1)^{1/\rho} R \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}}
\le \varepsilon \right \}, \end{align*} since $\bignorm[\rho]{\bigl( \widehat{f}({\boldsymbol{k}})/\lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}} \le \bignorm[\rho]{\bigl( \widehat{f}({\boldsymbol{k}}_i)/\lambda_{{\boldsymbol{k}}_i} \bigr)_{i=1}^{n}} \le \norm[{\mathcal{F}}]{f} \le R$ for all $f \in {\mathcal{B}}_R$, $n \ge n_1$. Moreover, this inequality is tight for some $f \in {\mathcal{C}} \cap {\mathcal{B}}_{R}$, namely, those certain $f$ for which $\widehat{f}({\boldsymbol{k}}_i) = 0$ for $i > n_1$. This completes the proof of \eqref{DHKM:PilotConeAlg_cost}.
To prove the lower complexity bound, choose $\varepsilon_{\max}$ and $R_{\min}$ such that \[ \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n_1+1}^{\infty}} \,
> 2\varepsilon_{\max}/[(1 - 1/A) R_{\min}].
\] Let $\ALG'$ be any algorithm that satisfies the error criterion, \eqref{DHKM:err_crit}, for this choice of ${\mathcal{C}}$ in \eqref{DHKM:pilot_cone}. Fix $R \in [R_{\min},\infty)$ and $\varepsilon \in (0,\varepsilon_{\max}]$ arbitrarily. Two fooling functions will be constructed of the form $f_\pm = f_1 \pm f_2$.
The input function $f_1$ is defined via its series coefficients as in Lemma \ref{DHKM:Key_Lem}, having nonzero coefficients only for ${\boldsymbol{k}} \in {\mathcal{K}}_1$: \begin{equation*}
\bigabs{\widehat{f}_1({\boldsymbol{k}})} = \begin{cases} \displaystyle \frac{R (1+1/A) \lambda_{{\boldsymbol{k}}}^{\rho'/\rho + 1}}{2\bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}}^{\rho'/\rho}}, & {\boldsymbol{k}} \in {\mathcal{K}}_1, \\
0, & {\boldsymbol{k}} \notin {\mathcal{K}}_1,
\end{cases}
\qquad \norm[{\mathcal{F}}]{f_1} = \frac{R(1 + 1/A)}{2}. \end{equation*} Suppose that $\ALG'(f_1,\varepsilon)$ samples the series coefficients $\widehat{f}_1({\boldsymbol{k}})$ for ${\boldsymbol{k}} \in {\mathcal{J}}$, and let $n$ denote the cardinality of ${\mathcal{J}}$.
Now, construct the input function $f_2$, having zero coefficients for ${\boldsymbol{k}} \in {\mathcal{J}}$ and also as in Lemma \ref{DHKM:Key_Lem}: \begin{gather} \nonumber
\bigabs{\widehat{f}_2({\boldsymbol{k}})} = \begin{cases} \displaystyle \frac{R (1-1/A) \lambda_{{\boldsymbol{k}}}^{\rho'/\rho + 1}}{2\bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \notin {\mathcal{J}}}}^{\rho'/\rho}}, & {\boldsymbol{k}} \notin {\mathcal{J}}, \\
0, & {\boldsymbol{k}} \in {\mathcal{J}},
\end{cases}
\qquad \norm[{\mathcal{F}}]{f_2} = \frac{R(1 - 1/A)}{2}, \\
\label{DHKM:SOLf2bd}
\norm[{\mathcal{G}}]{\SOL(f_2)} = \frac{R(1 - 1/A)}{2} \, \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \notin {\mathcal{J}}}}. \end{gather} Let $f_{\pm} = f_1 \pm f_2$. By the definitions above, it follows that \begin{align} \nonumber
\norm[{\mathcal{F}}]{f_{\pm}} &= \norm[{\mathcal{F}}]{ f_1 \pm f_2 } \le \norm[{\mathcal{F}}]{ f_1} + \norm[{\mathcal{F}}]{ f_2 } = R, \\
\nonumber
\norm[\rho]{\left( \frac{\widehat{f}_\pm({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}} \right)_{i=1}^{n_1}}
& = \norm[\rho]{\left( \frac{\widehat{f}_1({\boldsymbol{k}}_i) \pm \widehat{f}_2({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}} \right)_{i=1}^{n_1}} \\
\nonumber
& \ge \norm[\rho]{\left( \frac{\widehat{f}_1({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}} \right)_{i=1}^{n_1}} - \norm[\rho]{\left( \frac{\widehat{f}_2({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}} \right)_{i=1}^{n_1}} \\
\nonumber
& \ge \norm[{\mathcal{F}}]{ f_1} - \norm[{\mathcal{F}}]{ f_2 } = \frac{R}{A} \ge \frac{\norm[{\mathcal{F}}]{f_{\pm}}}{A}. \end{align} Therefore, $f_\pm \in {\mathcal{C}} \cap {\mathcal{B}}_R$. Moreover, since the series coefficients for $f_\pm$ are the same for ${\boldsymbol{k}} \in {\mathcal{J}}$, it follows that $\ALG'(f_+,\varepsilon) = \ALG'(f_-,\varepsilon)$. Thus, $\SOL(f_{+})$ must be quite similar to $\SOL(f_{-})$.
Using an argument like that in the proof of Theorem \ref{DHKM:APP_optimality_thm}, it follows that \begin{align*} \varepsilon & \ge \max_{\pm} \norm[{\mathcal{G}}]{\SOL(f_{\pm}) - \ALG'(f_{\pm},\varepsilon)} = \max_{\pm} \norm[{\mathcal{G}}]{\SOL(f_{\pm}) - \ALG'(f_{+},\varepsilon)} \\ & \ge \frac 12 \left [ \norm[{\mathcal{G}}]{\SOL(f_{+}) - \ALG'(f_{+},\varepsilon)} + \norm[{\mathcal{G}}]{\SOL(f_{-}) - \ALG'(f_{+},\varepsilon)} \right] \\ & \ge \frac 12 \norm[{\mathcal{G}}]{\SOL(f_+ - f_-)} = \norm[{\mathcal{G}}]{\SOL(f_2)} =\frac{R(1 - 1/A)}{2} \, \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \notin {\mathcal{J}}}} \qquad \text{by \eqref{DHKM:SOLf2bd}} \\ & \ge \frac{R(1 - 1/A)}{2} \, \bignorm[\rho']{\bigl( \lambda_{{\boldsymbol{k}}_i} \bigr)_{i = n+1}^\infty}, \end{align*} by the ordering of the ${\boldsymbol{k}}$ in \eqref{DHKM:lambda_order}. By the choice of $R_{\min}$ and $\varepsilon_{\max}$ above, it follows that $n > n_1$. This inequality then implies lower complexity bound \eqref{DHKM:PilotConeAlg_comp}. Because $\lim_{R \to \infty} \COMP({\mathcal{A}}({\mathcal{C}}), \varepsilon,R) = \infty$ it follows from Lemma \ref{DHKM:NoNonAdpatLem} that ${\mathcal{A}}({\mathcal{C}})$ contains only adaptive algorithms.
The essential optimality of Algorithm \ref{DHKM:PilotConeAlg} follows by observing that \[ \COST(\ALG,{\mathcal{C}},\varepsilon,R) \le \COMP({\mathcal{A}}({\mathcal{C}}),\omega \varepsilon,R) \qquad \text{for } \omega = \frac{1-1/A}{2(A^\rho -1)^{1/\rho}}. \] This satisfies definition \eqref{DHKM:EssentialOpt}. \end{proof} \
The above derivation assumes that $A > 1$. If $A =1$, then our cone consists of functions whose series coefficients vanish for wavenumbers outside ${\mathcal{K}}_1$. The exact solution can be constructed using only the pilot sample. Our algorithm is then non-adaptive, but succeeds for input functions in the cone ${\mathcal{C}}$, which is an unbounded set.
We may not be able to guarantee that a particular $f$ of interest lies in our cone, ${\mathcal{C}}$, but we may derive necessary conditions for $f$ to lie in ${\mathcal{C}}$. The following proposition follows from the definition of ${\mathcal{C}}$ in \eqref{DHKM:pilot_cone} and the fact that the term on the left below underestimates $\norm[{\mathcal{F}}]{f}$.
\begin{proposition} If $f \in {\mathcal{C}}$, then \begin{equation} \label{DHKM:PilotConeNecessary}
\norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}} \right)_{{\boldsymbol{k}}_i =1}^{n}} \le A
\norm[\rho]{\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}} \qquad \forall n \in {\mathbb{N}}. \end{equation} \end{proposition}
If condition \eqref{DHKM:PilotConeNecessary} is violated in practice, then $f \notin {\mathcal{C}}$, and Algorithm \ref{DHKM:PilotConeAlg} may output an incorrect answer. The remedy is to make ${\mathcal{C}}$ more inclusive by increasing the inflation factor, $A$, and/or the pilot sample size, $n_1$.
\subsection{Tractability}\label{DHKM:SecPilotTract}
In this section, we write ${\mathcal{C}}_d$ instead of ${\mathcal{C}}$, to stress the dependence on $d$, and for the same reason we write $\lambda_{d,{\boldsymbol{k}}_i}$ instead of $\lambda_{{\boldsymbol{k}}_i}$. Recall that we assume that $\lambda_{d,{\boldsymbol{k}}_1}\ge \lambda_{d,{\boldsymbol{k}}_2}\ge \cdots >0$. Let \[ n(\delta,d) :=\min \left \{n\ge 0: \bignorm[\rho']{\bigl( \lambda_{d,{\boldsymbol{k}}_i} \bigr)_{i = n+1}^{\infty}} \,
\le \delta \right \} \qquad \forall \delta > 0. \] From Equations \eqref{DHKM:PilotConeAlg_cost} and \eqref{DHKM:PilotConeAlg_comp}, we obtain that \begin{multline*}
\COMP({\mathcal{A}}({\mathcal{C}}_d),\omega_{\textup{lo}} \varepsilon,R) \le n(\varepsilon/R,d) \le
\COMP({\mathcal{A}}({\mathcal{C}}_d),\omega_{\textup{hi}} \varepsilon,R) \\
\forall \varepsilon \in (0, \varepsilon_{\max}], \ R \in [R_{\min}, \infty), \end{multline*} where the positive constants $\omega_{\textup{lo}}$ and $\omega_{\textup{hi}}$ depend on $A$, but not depend on $d$, $\varepsilon$, or $R$. From the equation above, it is clear that tractability depends on the behavior of $n(\varepsilon/R,d)$ as $R/\varepsilon$ and $d$ tend to infinity. We would like to study under which conditions we obtain the various tractability notions defined in Section \ref{DHKM:secTractability}.
To this end, we distinguish two cases, depending on whether $\rho'$ is infinite or not. This distinction is useful because it allows us to relate the computational complexity of the algorithms considered in this chapter to the computational complexity of linear problems on certain function spaces considered in the classical literature on information-based complexity, as for example \cite{NovWoz08a}. The case $\rho'=\infty$ corresponds to the worst-case setting, where one studies the worst performance of an algorithm over the unit ball of a space. The results in Theorem \ref{DHKM:thmtract1} below are indeed very similar to the results for the worst-case setting over balls of suitable function spaces. The case $\rho<\infty$ corresponds to the so-called average-case setting, where one considers the average performance over a function space equipped with a suitable measure. For both of these settings there exist tractability results that we will make use of here.
\paragraph*{CASE 1: $\rho'=\infty$:}
If $\rho'=\infty$, we have, due to the monotonicity of the $\lambda_{d,{\boldsymbol{k}}_i}$, \[ n(\varepsilon/R,d)=\min \left \{n\ge 0\colon \lambda_{d,{\boldsymbol{k}}_{n+1}} \,
\le \varepsilon/R \right \}. \] We then have the following theorem.
\begin{theorem} \label{DHKM:thmtract1} Using the same notation as above, the following statements hold for the case $\rho'=\infty$.
\begin{itemize}
\item[1.]
We have strong polynomial tractability if and only if there exist $\eta>0$ and $i_0\in{\mathbb{N}}$ such that
\begin{equation}\label{DHKM:eq:condspt}
\sup_{d\in{\mathbb{N}}} \sum_{i=i_0}^\infty \lambda_{d,{\boldsymbol{k}}_i}^\eta < \infty.
\end{equation}
Furthermore, the exponent of strong polynomial tractability is then equal to the infimum of those $\eta>0$ for which \eqref{DHKM:eq:condspt} holds.
\item[2.]
We have polynomial tractability if and only if there exist $\eta_1, \eta_2 \ge 0$ and $\eta_3, K>0$ such that
\[
\sup_{d\in{\mathbb{N}}} d^{-\eta_1}\, \sum_{i=\lceil K d^{\eta_2} \rceil}^\infty \lambda_{d,{\boldsymbol{k}}_i}^{\eta_3} < \infty.
\]
\item[3.]
We have weak tractability if and only if
\begin{equation}\label{DHKM:eq:condwt}
\sup_{d\in{\mathbb{N}}} \, \exp(-cd) \sum_{i=1}^\infty \exp\left(-c\left(\frac{1}{\lambda_{d,{\boldsymbol{k}}_i}}\right)\right) <\infty\quad \mbox{for all}\quad c>0.
\end{equation} \end{itemize} \end{theorem}
\begin{proof} Letting $\widetilde{\varepsilon}:=\sqrt{\varepsilon/R}$, we see that $n(\varepsilon/R,d)=\min \left \{n\ge 0\colon \lambda_{d,{\boldsymbol{k}}_{n+1}} \,\le \widetilde{\varepsilon}^{\,2} \right \}$. The latter expression is well studied in the context of tractability of linear problems in the worst-case setting defined on unit balls of certain spaces, and if and only if conditions on the $\lambda_{d,{\boldsymbol{k}}_i}$ for various tractability notions are known. These conditions can be found in \cite[Chapter 5]{NovWoz08a} for (strong) polynomial tractability and \cite{WerWoz17} for weak tractability.
Since, in this chapter, we consider $\min \left \{n\ge 0\colon \lambda_{d,{\boldsymbol{k}}_{n+1}} \,\le \varepsilon/R \right \}$, and in \cite{NovWoz08a} and \cite{WerWoz17} $\varepsilon /R$ is replaced by the square of the error tolerance, there are slight differences between the results here and those in the aforementioned references; to be more precise, the exponent of strong polynomial tractability is $\eta$ here, whereas it is $2\eta$ in \cite{NovWoz08a}, and $1/\lambda_{d,{\boldsymbol{k}}_i}$ in \eqref{DHKM:eq:condwt} corresponds to $1/\sqrt{\lambda_{d,{\boldsymbol{k}}_i}}$ in \cite{WerWoz17}. \end{proof}
\paragraph*{CASE 2: $\rho'<\infty$:}
In this case, letting $\widetilde{\varepsilon}:=(\varepsilon/R)^{\rho'/2}$ and $\widetilde{\lambda}_{d,i}:=\lambda_{d,{\boldsymbol{k}}_i}^{\rho'}$, we have \begin{align}
\nonumber
n(\varepsilon/R,d) &=\min \left \{n\ge 0\colon \sum_{i=n+1}^\infty \lambda_{d,{\boldsymbol{k}}_i}^{\rho'}\,
\le (\varepsilon/R)^{\rho'} \right \} \\
\label{DHKM:eqaveragetract}
& =\min \left \{n\ge 0\colon \sum_{i=n+1}^\infty \widetilde{\lambda}_{d,i}\,
\le \widetilde{\varepsilon}^{\,2} \right \}. \end{align} However, the latter expression corresponds exactly to the average-case tractability (with respect to the parameters $\widetilde{\lambda}_{d,i}$ and $\widetilde{\varepsilon}$) defined on certain spaces as studied in, e.g., \cite{NovWoz08a}. This leads us to the following theorem. \begin{theorem} \label{DHKM:thmtract2} Using the same notation as above, the following statements hold for the case $\rho'<\infty$.
\begin{itemize}
\item[1.]
We have strong polynomial tractability if and only if there exist $\eta\in (0,1)$ and $i_0\in{\mathbb{N}}$ such that
\begin{equation}\label{DHKM:eq:condspt1}
\sup_{d\in{\mathbb{N}}} \sum_{i=i_0}^\infty \lambda_{d,{\boldsymbol{k}}_i}^{\rho'\,\eta} < \infty.
\end{equation}
Furthermore, the exponent of strong polynomial tractability is then
\[
\inf\left\{\rho'\eta/(1-\eta)\colon \mbox{$\eta$ satisfies \eqref{DHKM:eq:condspt1}}\right\}.
\]
\item[2.]
We have polynomial tractability if and only if there exist $\eta_1, \eta_2 \ge 0$ and $\eta_3\in (0,1), K>0$ such that
\[
\sup_{d\in{\mathbb{N}}} d^{-\eta_1}\, \sum_{i=\lceil K d^{\eta_2} \rceil}^\infty \lambda_{d,{\boldsymbol{k}}_i}^{\rho'\,\eta_3} < \infty.
\]
\item[3.] Let $t_{d,i}:=\sum_{k=i}^\infty \lambda_{d,{\boldsymbol{k}}_i}$.
We have weak tractability if and only if
\[
\lim_{i\to\infty} t_{d,i}\, (\log i)^2=0\quad\mbox{for all $d$},
\]
and there exists a function $f:[0,1/2)\to \{1,2,3,\ldots\}$ such that \[
\sup_{\beta\in (0,1/2]}\, \beta^{-2} \,
\sup_{d\ge f(\beta)}\,\, \sup_{i\ge \lceil \exp (d\sqrt{b}) \rceil +1}\, \, \lim_{i\to\infty} t_{d,i}\, (\log i)^2
< \infty. \]
\end{itemize} \end{theorem} \begin{proof}
The proof of the theorem is similar to that of Theorem \ref{DHKM:thmtract1}, using \eqref{DHKM:eqaveragetract}. \end{proof} \begin{remark}
Results for further tractability notions, such as quasi-polynomial tractability or $(s,t)$-weak
tractability, can be shown using similar arguments as above and results from \cite{KriWoz19}, \cite{NovWoz10a}, \cite{WerWoz17}, and the papers cited therein. \end{remark}
To be more concrete, we consider the situation where the $\lambda_{{\boldsymbol{k}}}$ are specified in terms of positive \emph{coordinate weights}, $w_1, \ldots, w_d$, and positive \emph{smoothness weights}, $s_1, s_2, \ldots$: \begin{equation}
\label{DHKM:prodwts} \lambda_{d,{\boldsymbol{k}}} := \prod_{\substack{\ell =1\\ k_\ell > 0}}^d w_\ell s_{k_\ell}, \qquad {\boldsymbol{k}} \in {\mathbb{N}}_0^d, \ d \in {\mathbb{N}}. \end{equation} This is a generalization of the example in Section \ref{DHKM:secexamp}, where $s_{k} = k^{-r}$. This form of the $\lambda_{d,{\boldsymbol{k}}}$ is considered in greater detail in Section \ref{DHKM:smoothimportance_sec}. The same argument as in Section \ref{DHKM:revisexamp} implies that the sum of the $\lambda_{d,{\boldsymbol{k}}_i}^\eta$ is bounded above as \[ \sum_{i=1}^{\infty}\lambda_{d, {\boldsymbol{k}}_{i}}^{\eta} \le \exp\left(\sum_{k=1}^\infty s_k^\eta \sum_{\ell = 1}^{d} w_\ell^{\eta} \right)\le \exp\left(\sum_{k =1 }^{\infty} s_k^\eta \sum_{\ell = 1}^{\infty} w_\ell^{\eta} \right). \] Moreover, it also follows that for any fixed positive integer $i_0$, the sum of the $\lambda_{d,{\boldsymbol{k}}_i}^\eta$ is bounded below as \begin{align*}
\sup_{d \in {\mathbb{N}}} \, \sum_{i=i_0}^{\infty}\lambda_{d, {\boldsymbol{k}}_{i}}^{\eta} & \ge w_1^\eta \sum_{k=i_0}^{\infty}s_{\kappa_i}^{\eta} \qquad \parbox{6cm}{considering only ${\boldsymbol{k}}_i$ of the form $(k, 0, 0, \ldots, 0)$, \\
and ordering ${\boldsymbol{s}}$ so that $s_{\kappa_1} \ge s_{\kappa_2} \ge \cdots$,} \\
\sup_{d \in {\mathbb{N}}} \, \sum_{i=i_0}^{\infty}\lambda_{d, {\boldsymbol{k}}_{i}}^{\eta} & \ge s_{1}^{\eta} \sum_{i=i_0}^{\infty} w_{\ell_i}^\eta
\qquad
\\
& \qquad \parbox{8cm}{considering only ${\boldsymbol{k}}_i$ of the form $(0, \ldots, 0,1,0 , \ldots, 0)$,\\
where the non-zero component is at the $\ell$-th position with \\
$i_0\le \ell \le d$, and ordering ${\boldsymbol{w}}$ so that $w_{\ell_1} \ge w_{\ell_2} \ge \cdots$.} \end{align*} Thus, we have necessary and sufficient conditions for strong tractability. \begin{corollary} \label{DHKM:sptexample_cor} For the $\lambda_{d,{\boldsymbol{k}}}$ of the form \eqref{DHKM:prodwts} we have strong polynomial tractability if and only if there exists $\eta>0$ such that
\begin{equation*}
\sum_{k=1}^\infty s_{k}^\eta < \infty \text{ and }
\sum_{\ell = 1}^{\infty} w_\ell^{\eta} < \infty. \end{equation*} \end{corollary}
\begin{remark} Note that in the setting of this example, the term $\sum_{i=1}^{\infty}\lambda_{d, {\boldsymbol{k}}_{i}}^{\eta}$ will usually depend exponentially on $d$ unless the coordinate weights decay to zero fast enough with increasing $\ell$. Hence, we can only hope for tractability under the presence of decaying $w_\ell$. For further details on weighted approximation problems and tractability, we refer to \cite{NovWoz08a}. \end{remark}
\section{Tracking the Decay Rate of the Series Coefficients of the Input Function} \label{DHKM:tracking_sec}
From error bound \eqref{DHKM:APP_Err_Coef} it follows that the faster the $\widehat{f}({\boldsymbol{k}}_i)$ decay, the faster $\APP(f,n)$ converges to the solution. Unfortunately, adaptive Algorithm \ref{DHKM:PilotConeAlg} does not adapt to the decay rate of the $\widehat{f}({\boldsymbol{k}}_i)$ as $i \to \infty$. It simply bounds $\norm[{\mathcal{F}}]{f}$ based on a pilot sample. The algorithm presented in this section tracks the rate of decay of the $\widehat{f}({\boldsymbol{k}}_i)$ and terminates sooner if the $\widehat{f}({\boldsymbol{k}}_i)$ decay more quickly. Similar algorithms for quasi-Monte Carlo integration are developed in \cite{HicJim16a}, \cite{JimHic16a}, and \cite{HicEtal17a}.
There is an implicit assumption in this section that function data are cheap and we can afford a large sample size. A large sample size is required to do meaningful tracking of the decay of the series coefficients. The previous section and the next section are more suited to the case when function data are expensive and the final sample size must be modest.
Let $(n_j)_{j\ge 0}$ be a strictly increasing sequence of non-negative integers. This sequence may increase geometrically or algebraically. Define the sets of wavenumbers analogously to \eqref{DHKM:KOnedef}, \begin{equation*}
n_{-1}=0, \qquad {\mathcal{K}}_j := \{{\boldsymbol{k}}_{n_{j-1}+1}, \ldots, {\boldsymbol{k}}_{n_j}\} \quad \text{for } j \in {\mathbb{N}}_0. \end{equation*} If $n_0 = 0$, then ${\mathcal{K}}_0$ is empty. For any $f \in {\mathcal{F}}$, define the norms of subsets of series coefficients: \begin{equation} \label{DHKM:SigmaDef} \sigma_j (f):=\norm[\rho]{\biggl(\frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \biggr)_{{\boldsymbol{k}} \in {\mathcal{K}}_j}} \qquad \text{for } j \in {\mathbb{N}}. \end{equation} Thus, $\norm[{\mathcal{F}}]{f} = \norm[\rho]{\bigl(\sigma_j(f) \bigr)_{j \in {\mathbb{N}}_0}}$.
For this section, we define the cone of input functions by \begin{equation} \label{DHKM:TrackConeDef}
{\mathcal{C}} : =\left\{f\in{\mathcal{F}} \colon \sigma_{j+r} (f)\le ab^r \sigma_j (f)\ \forall j,r\in{\mathbb{N}}\right\}. \end{equation} Here, $a$ and $b$ are positive reals with $b< 1 < a$. The constant $a$ is an inflation factor, and the constant $b$ defines the general rate of decay of the $\sigma_j(f)$ for $f \in {\mathcal{C}}$. Because $ab^r$ may be greater than one, we do not require the series coefficients of the solution, $\SOL(f)$, to decay monotonically. However, we expect their partial sums to decay steadily. The series coefficients for wavenumbers ${\boldsymbol{k}} \in {\mathcal{K}}_0$ do not affect the definition of ${\mathcal{C}}$ and may behave erratically. Lemma \ref{DHKM:Key_Lem} implies that \begin{equation} \label{DHKM:LambdaDef}
\norm[\tau]{\bigl(\widehat{f}({\boldsymbol{k}}) \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}_j}} \le \sigma_j(f) \Lambda_j, \qquad \text{where } \Lambda_j : = \norm[\rho']{\bigl(\lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}_j}}. \end{equation} From \eqref{DHKM:SOLNorm} and \eqref{DHKM:SOLNormFinite} it follows that the norm of the solution operator is \begin{equation} \label{DHKM:NormLambdaFinite}
\norm[{\mathcal{F}} \to {\mathcal{G}}]{\SOL} = \bignorm[\rho']{\bigl(\Lambda_j \bigr)_{j \in {\mathbb{N}}_0}} < \infty \end{equation}
If $f$ belongs to the ${\mathcal{C}}$ defined in \eqref{DHKM:TrackConeDef} and $n_0 = 0$, then \begin{align*}
\norm[{\mathcal{F}}]{f} & = \norm[\rho]{\Biggl(\,\norm[\rho]{\biggl(\frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \biggr)_{{\boldsymbol{k}} \in {\mathcal{K}}_j}}\,\Biggr)_{j \in {\mathbb{N}}}} = \bignorm[\rho]{\bigl(\sigma_j(f) \bigr)_{j \in {\mathbb{N}}}} \\
& \le \bignorm[\rho]{\bigl(\sigma_1(f), ab \sigma_1(f), ab^2 \sigma_1(f), \ldots \bigr)} \\
& = \left(1 + \frac{a^\rho b^\rho}{1 - b^\rho} \right)^{1/\rho} \norm[\rho]{\biggl(\frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \biggr)_{{\boldsymbol{k}} \in {\mathcal{K}}_1}}. \end{align*} Comparing this inequality to the definition of ${\mathcal{C}}$ in the previous section, it can be seen that ${\mathcal{C}}$ defined in \eqref{DHKM:TrackConeDef} is a subset of ${\mathcal{C}}$ defined in \eqref{DHKM:pilot_cone} if we choose $A=\left(1 + \frac{a^\rho b^\rho}{1 - b^\rho} \right)^{1/\rho}$ in \eqref{DHKM:pilot_cone}.
From the expression for the error in \eqref{DHKM:APP_Err_Coef} and the definition of the cone in \eqref{DHKM:TrackConeDef}, we can now derive a data-driven error bound for all $f \in {\mathcal{C}}$ and $j \in {\mathbb{N}}$: \begin{align} \nonumber \MoveEqLeft{\norm[{\mathcal{G}}]{\SOL(f)-\APP(f,n_j)}} \\ \nonumber &= \norm[\tau]{\left(\widehat{f}({\boldsymbol{k}}_i) \right)_{i = n_j+1}^\infty} = \norm[\tau]{ \left(\norm[\tau]{\bigl(\widehat{f}({\boldsymbol{k}}) \bigr)_{{\boldsymbol{k}} \in {\mathcal{K}}_l}} \right)_{l=j+1}^\infty} \\ \nonumber & \le \norm[\tau]{ \bigl(\sigma_l(f) \Lambda_l \bigr)_{l=j+1}^\infty} \qquad \text{by \eqref{DHKM:LambdaDef}} \\ \nonumber & = \norm[\tau]{ \bigl(\sigma_{j+r}(f) \Lambda_{j+r} \bigr)_{r=1}^\infty} \\ & \le a \sigma_j(f) \norm[\tau]{ \bigl(b^r\Lambda_{j+r} \bigr)_{r=1}^\infty} =:\ERR\bigl(\dataNj,n_j\bigr)
\qquad \text{by \eqref{DHKM:TrackConeDef}.}
\label{DHKM:algoineq} \end{align}
This upper bound depends only on the function data and the parameters defining ${\mathcal{C}}$. The error vanishes as $j \to \infty$ because $\sigma_j(f) \le ab^{j-1} \sigma_1(f) \to 0$ and $\Lambda_j \to 0$. Moreover, the error bound for $\APP(f,n_j)$ depends on $\sigma_j(f)$, whose rate of decay need not be postulated in advance.
These assumptions accommodate both the cases where the approximation converges algebraically and exponentially. To illustrate the algebraic case, suppose that $\widehat{f}({\boldsymbol{k}}_i)/\lambda_{{\boldsymbol{k}}_i} = {\mathcal{O}}(i^{-r_\Delta})$ for some positive $r_\Delta > 1/\rho$. For this algebraic case one would normally define ${\mathcal{C}}$ in terms of an exponentially increasing sequence, $(n_j)_{j\ge 0}$, e.g., $n_j = n_0 2^j$, which implies that \begin{align*}
\sigma_j(f) &= \left[ \sum_{i=n_0 2^{j-1} + 1}^{n_0 2^j} \biggabs{\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}}^\rho \right]^{1/\rho}
= {\mathcal{O}} \left( \left [ \sum_{i=n_0 2^{j-1} + 1}^{n_0 2^j} i^{-\rho r_\Delta} \right]^{1/\rho} \right) \\
& = {\mathcal{O}} \left( 2^{-j(r_\Delta-1/\rho)} \right). \end{align*} Reasonable functions would satisfy \begin{equation*}
C_{\textup{lo}} 2^{-j(r_\Delta-1/\rho)} \le \sigma_j(f) \le C_{\textup{up}} 2^{-j(r_\Delta-1/\rho)} \end{equation*} for some constants $C_{\textup{lo}}$ and $C_{\textup{up}}$. Choosing $a \ge C_{\textup{up}}/C_{\textup{lo}} $ and $b \ge 2^{-(r_\Delta-1/\rho)}$ causes the cone ${\mathcal{C}}$ to include such functions. Note that only the ratio of $C_{\textup{up}}$ to $C_{\textup{lo}}$ need be assumed to determine $a$, and choosing $b$ larger than necessary does not affect the order of the decay of the error bound.
To illustrate the exponential case, suppose that $\widehat{f}({\boldsymbol{k}}_i)/\lambda_{{\boldsymbol{k}}_i} = {\mathcal{O}}(\textup{e}^{-r_\Delta i})$. For this exponential case one would normally define ${\mathcal{C}}$ in terms of an arithmetic sequence, $(n_j)_{j\ge 0}$, e.g., $n_j = n_0 + j s$, where $s$ is a positive integer. This implies that \begin{align*}
\sigma_j(f) &= \left[ \sum_{i=n_0 + j s -s + 1}^{n_0 + j s} \biggabs{\frac{\widehat{f}({\boldsymbol{k}}_i)}{\lambda_{{\boldsymbol{k}}_i}}}^\rho \right]^{1/\rho}
= {\mathcal{O}} \left( \left [ \sum_{i=n_0 + j s -s + 1}^{n_0 + j s} \textup{e}^{- \rho r_\Delta i} \right]^{1/\rho} \right) \\
& = {\mathcal{O}} \left( \textup{e}^{-j r_{\Delta} s} \right). \end{align*} Analogous to the algebraic case, reasonable functions would satisfy
$C_{\textup{lo}} \textup{e}^{-j r_{\Delta} s} \le \sigma_j(f) \le C_{\textup{up}} \textup{e}^{-j r_{\Delta} s}$ for some constants $C_{\textup{lo}}$ and $C_{\textup{up}}$. Choosing $a \ge C_{\textup{up}}/C_{\textup{lo}} $ and $b \ge \textup{e}^{- r_{\Delta} s}$ causes the cone ${\mathcal{C}}$ to include such functions. Again, only the ratio of $C_{\textup{up}}$ to $C_{\textup{lo}}$ need be assumed to determine $a$, and choosing $b$ larger than necessary does not affect the order of the decay of the error bound.
\subsection{The Adaptive Algorithm and Its Computational Cost} \label{DHKM:SecAdapAlgTrackDecay}
The data-driven error bound in \eqref{DHKM:algoineq} forms the basis for an adaptive Algorithm \ref{DHKM:TrackConeAlg}, which solves our problem for input functions in the cone ${\mathcal{C}}$ defined in \eqref{DHKM:TrackConeDef}. The following theorem establishes its viability and computational cost. In deriving upper bounds on the computational cost and lower bounds on the complexity, we may sacrifice tightness for simplicity.
\begin{algorithm}
\caption{Adaptive ALG for a Cone of Input Functions Tracking the Series Coefficient Decay Rate \label{DHKM:TrackConeAlg}}
\begin{algorithmic}
\PARAM the Banach spaces ${\mathcal{F}}$ and ${\mathcal{G}}$, including the weights ${\boldsymbol{\lambda}}$; a strictly increasing sequence of non-negative integers, $(n_j)_{j\ge 0}$; an inflation factor, $a$; the general decay rate, $b$; $\APP$ satisfying \eqref{DHKM:APP_Err_Coef}
\INPUT a black-box function, $f$; an absolute error tolerance,
$\varepsilon>0$
\Ensure Error criterion \eqref{DHKM:err_crit} for the cone defined in \eqref{DHKM:TrackConeDef}
\State Let $j \leftarrow 0$ \Repeat
\State Let $j \leftarrow j + 1$
\State Compute $\ERR\bigl(\dataNj,n_j\bigr)$ as defined in \eqref{DHKM:algoineq}
\Until $\ERR\bigl(\dataNj,n_j\bigr) \le \varepsilon$
\RETURN $\ALG(f,\varepsilon) = \APP(f,n_{j})$ \end{algorithmic} \end{algorithm}
\begin{theorem}\label{DHKM:TractConeCompCost} Algorithm \ref{DHKM:TrackConeAlg} yields an answer satisfying absolute error criterion \eqref{DHKM:err_crit}, i.e., $\ALG \in {\mathcal{A}}({\mathcal{C}})$ for ${\mathcal{C}}$ defined in \eqref{DHKM:TrackConeDef}. The information cost is $\COST(\ALG,f,\varepsilon)=n_{j^*}$, where $j^*$ is defined implicitly as \begin{equation} \label{DHKM:EqTractConejstar} j^* = \min\left \{ j \in {\mathbb{N}} : \ERR\bigl(\dataNj,n_j\bigr) \le \varepsilon \right\}. \end{equation} Moreover, $\COST(\ALG,{\mathcal{C}},\varepsilon,R) \le n_{j^\dagger}$, where $j^\dagger$ is defined as follows: \begin{equation} \label{DHKM:TractConejdagger} j^\dagger = \min \left \{j \in {\mathbb{N}} : \norm[\tau]{ \bigl(b^{j+r}\Lambda_{j+r} \bigr)_{r=1}^\infty} \le \frac{b\varepsilon}{Ra^2} \left( \frac{1 - b^{j\rho}}{1 - b^\rho} \right)^{1/\rho} \right\}. \end{equation} \end{theorem}
\begin{proof} The value of $j^*$ in \eqref{DHKM:EqTractConejstar} follows directly from the error criterion. The success of the algorithm follows from the error bound in \eqref{DHKM:algoineq}.
For the remainder of the proof consider $R$ and $\varepsilon$ to be fixed. For any $f \in {\mathcal{C}} \cap {\mathcal{B}}_R$ and for any $j^\dagger$ defined as in \eqref{DHKM:TractConejdagger}, it follows that \begin{align*} R &\ge \norm[{\mathcal{F}}]{f} = \norm[\rho]{\left(\frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \right)_{{\boldsymbol{k}} \in {\mathbb{K}}}}
\ge \norm[\rho]{\left(\sigma_j(f)\right)_{j=1}^{j^\dagger}}
\qquad \text{by \eqref{DHKM:SigmaDef} } \\ & \ge \norm[\rho]{\left(a^{-1}b^{1-j^\dagger}\sigma_{j^\dagger}(f), \ldots, a^{-1} b^{-1}\sigma_{j^\dagger}(f), \sigma_{j^\dagger}(f) \right) } \quad \text{by \eqref{DHKM:TrackConeDef}}\\ & \ge \frac {\sigma_{j^\dagger}(f)} a \norm[\rho]{\bigl(b^{1-j^\dagger}, \ldots, b^{-1}, 1 \bigr) } = \frac {b\sigma_{j^\dagger}(f)} a \left( \frac{b^{-j^\dagger\rho} -1}{1 - b^\rho} \right)^{1/\rho}\\ & \ge \sigma_{j^\dagger}(f)\,\frac{Ra}{\varepsilon} \norm[\tau]{ \bigl(b^r\Lambda_{j^\dagger+r} \bigr)_{r=1}^\infty} \qquad \text{by the definition of } j^\dagger \text{ in \eqref{DHKM:TractConejdagger}} \\ & = \frac{R}{\varepsilon} \ERR\bigl(\dataNjd,n_{j^\dagger}\bigr)\, . \end{align*} From this last inequality, it follows that $j^\dagger \ge j^*$. \end{proof} \
Although Algorithm \ref{DHKM:TrackConeAlg} tracks the decay rate of the $\widehat{f}({\boldsymbol{k}}_i)$, the information cost bound and complexity bound in the theorem above do not reflect different decay rates of the $\widehat{f}({\boldsymbol{k}}_i)$. That is a subject for future investigation.
\subsection{Essential Optimality of the Algorithm}
To establish the essential optimality of Algorithm \ref{DHKM:TrackConeAlg} requires some additional, reasonable assumptions on the sequences $(n_j)_{j \in {\mathbb{N}}_0}$ and $\bigl(\sigma_j(f) \bigr)_{j \in {\mathbb{N}}_0}$. Recall from \eqref{DHKM:NormLambdaFinite} that $\bigl(\Lambda_j \bigr)_{j \in {\mathbb{N}}_0}$ has a finite $\rho'$ norm. We require that the $\Lambda_j$ must decay steadily with $j$: \begin{equation} \label{DHKM:LambdaDecayCond}
\alpha^{-1} \beta^r \Lambda_j \le \Lambda_{j+r} \le \alpha \gamma^r \Lambda_j \quad \forall j,r \in {\mathbb{N}}_0, \qquad \text{for some } \beta, \gamma < 1 \le \alpha. \end{equation} We also assume that the ratio of the largest to smallest $\lambda_{{\boldsymbol{k}}}$ in a group is bounded above: \begin{equation} \label{DHKM:MinMaxCond}
\sup_{j \in {\mathbb{N}}} \frac{\lambda_{{\boldsymbol{k}}_{n_{j-1}+1}}}{\lambda_{{\boldsymbol{k}}_{n_{j}}}} \le S_1 < \infty. \end{equation} For the illustrative choices of $(n_j)_{j \in {\mathbb{N}}_0}$ and $\bigl({\boldsymbol{k}}_i \bigr)_{i \in {\mathbb{N}}}$ preceding Section \ref{DHKM:SecAdapAlgTrackDecay} this assumption holds. Let $\card(\cdot)$ denote the cardinality of a set. We assume that if ${\mathcal{J}}$ is an arbitrary set of wavenumbers with $\card({\mathcal{J}}) \le n_j$, then there exists some $l \le n_{j+1}$ for which ${\mathcal{K}}_l \setminus {\mathcal{J}}$ retains some significant fraction of the original ${\mathcal{K}}_l$ elements: \begin{equation} \label{DHKM:PropCond}
\inf_{j \in {\mathbb{N}}} \ \min_{{\mathcal{J}} \subset {\mathbb{K}} \, : \, \card({\mathcal{J}}) \le n_j} \ \max_{0 \le l \le j+1} \frac{\card({\mathcal{K}}_l \setminus {\mathcal{J}})}{\card({\mathcal{K}}_l)} \ge S_2 > 0. \end{equation} Again, for the illustrative choices of $(n_j)_{j \in {\mathbb{N}}_0}$ and $\bigl({\boldsymbol{k}}_i \bigr)_{i \in {\mathbb{N}}}$ preceding Section \ref{DHKM:SecAdapAlgTrackDecay} this assumption holds.
The following theorem establishes a lower bound on the complexity of our problem for input functions in ${\mathcal{C}}$. The theorem after that shows that the cost of our algorithm as given in Theorem \ref{DHKM:TractConeCompCost} is essentially no worse than this lower bound.
\begin{theorem} \label{DHKM:TractConeLowBdComp} A lower bound on the complexity of the linear problem is \begin{align*}
&\COMP({\mathcal{A}}({\mathcal{C}}),\varepsilon,R) > n_{j^\ddagger}, \intertext{where}
j^\ddagger & = \max \left \{ j \in {\mathbb{N}} : b^{j+1} \Lambda_{j+1} >
\frac{2a\alpha \varepsilon}{R(a-1)(1 - b^\rho)^{1/\rho}} \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right]^{1/\rho} \right \}. \end{align*} \end{theorem}
\begin{proof}
As in the proof of Theorem \ref{DHKM:PilotCostThm} we consider fixed and arbitrary R and $\varepsilon$. We proceed by carefully constructing the test input functions, $f_1$ and $f_{\pm} = f_1 \pm f_2$, lying in ${\mathcal{C}} \cap {\mathcal{B}}_{R}$, which yield the same approximate solution but different true solutions. This leads to a lower bound on $\COMP({\mathcal{A}}({\mathcal{C}}),\varepsilon,R)$. The proof is provided for $\rho' < \infty$. The proof for $\rho' = \infty$ is similar.
The first test function $f_1 \in {\mathcal{C}}$ is defined in terms of its series coefficients---inspired by Lemma \ref{DHKM:Key_Lem}---as \begin{align} \nonumber f_1 &= f_{10} + f_{11} + \cdots, \qquad \widehat{f}_{1j}({\boldsymbol{k}}) := \begin{cases} \displaystyle \frac{c_1 b^{j} \lambda_{{\boldsymbol{k}}}^{\rho'/\rho+1}}{\Lambda_j^{\rho'/\rho}}, & {\boldsymbol{k}} \in {\mathcal{K}}_j, \\ 0, & {\boldsymbol{k}} \notin {\mathcal{K}}_j, \end{cases} \\ \nonumber c_1 &:= \frac{R(a+1)(1 - b^\rho)^{1/\rho}}{2a}. \end{align} It can be verified that the test function lies both in ${\mathcal{B}}_{R}$ and in ${\mathcal{C}}$: \begin{align} \nonumber \sigma_j(f_1) & = \norm[\rho]{\biggl(\frac{\widehat{f}_{1j}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \biggr)_{{\boldsymbol{k}} \in {\mathcal{K}}_l}} = c_1 b^j, \qquad j \in {\mathbb{N}}_0,\\ \nonumber \norm[{\mathcal{F}}]{f_1} &= \norm[\rho]{\bigl( \sigma_j(f) \bigr)_{j \in {\mathbb{N}}_0} } = \frac{c_1}{(1 - b^\rho)^{1/\rho}} = \frac{R(a+1)}{2a} \le R, \\ \nonumber \sigma_{j+r}(f_1) &= b^{r} \sigma_j(f_1) \le a b^r \sigma_j(f_1), \qquad j,r \in {\mathbb{N}}_0. \end{align}
Now let $\ALG'$ be an arbitrary algorithm in ${\mathcal{A}}({\mathcal{C}})$, and suppose that $\ALG'(f_1,\varepsilon)$ samples $f_1({\boldsymbol{k}})$ for ${\boldsymbol{k}} \in {\mathcal{J}}$. Let $\widetilde{\calK}_j = {\mathcal{K}}_j \setminus {\mathcal{J}}$ for all non-negative integers $j$. Construct the function $f_2$, having zero coefficients for ${\boldsymbol{k}} \in {\mathcal{J}}$, but otherwise looking like $f_1$: \begin{align} \nonumber f_2 &= f_{20} + f_{21} + \cdots, \qquad \widehat{f}_{2j}({\boldsymbol{k}}) := \begin{cases} \displaystyle \frac{c_2 b^{j} \lambda_{{\boldsymbol{k}}}^{\rho'/\rho+1}}{\widetilde{\Lambda}_j^{\rho'/\rho}}, & {\boldsymbol{k}} \in \widetilde{\calK}_j, \\ 0, & \text{otherwise}, \end{cases} \\ \nonumber c_2 &:= \frac{R(a-1)(1 - b^\rho)^{1/\rho}}{2a}, \qquad \widetilde{\Lambda}_j := \norm[\rho']{\bigl(\lambda_{\boldsymbol{k}} \bigr)_{{\boldsymbol{k}} \in \widetilde{\calK}_j}} \le \Lambda_j, \\ \nonumber \sigma_j(f_2) & = \norm[\rho]{\biggl(\frac{\widehat{f}_{2j}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}} \biggr)_{{\boldsymbol{k}} \in \widetilde{\calK}_j}} = \begin{cases} c_2 b^j, & \widetilde{\calK}_j \ne \emptyset, \\ 0, & \widetilde{\calK}_j = \emptyset, \end{cases} \qquad j \in {\mathbb{N}}_0, \\ \nonumber \norm[{\mathcal{F}}]{f_2} &= \norm[\rho]{\bigl( \sigma_j(f_{2}) \bigr)_{j \in {\mathbb{N}}_0} } \le \frac{c_2}{(1 - b^\rho)^{1/\rho}} = \frac{R(a - 1)}{2a}\le R, \\ \nonumber \norm[{\mathcal{G}}]{\SOL(f_{2j})} &= \sigma_j(f_{2j}) \widetilde{\Lambda}_j = c_2 b^{j} \widetilde{\Lambda}_j, \qquad j \in {\mathbb{N}}_0, \\ \norm[{\mathcal{G}}]{\SOL(f_2)} & = \norm[\tau]{\bigl(c_2 b^{j} \widetilde{\Lambda}_j \bigr)_{j \in {\mathbb{N}}_0}} = \frac{R(a-1)(1 - b^\rho)^{1/\rho}}{2a} \norm[\tau]{\bigl(b^{j} \widetilde{\Lambda}_j \bigr)_{j \in {\mathbb{N}}_0}}. \label{DHKM:SOLftwo} \end{align}
Furthermore, define $f_{\pm} = f_1 \pm f_2$. It can be verified that $f_{\pm}$ also lie both in ${\mathcal{B}}_{R}$ and in ${\mathcal{C}}$: \begin{align} \nonumber \norm[{\mathcal{F}}]{f_\pm} &\le \norm[{\mathcal{F}}]{f_1} + \norm[{\mathcal{F}}]{f_2} \le \frac{c_1 + c_2}{(1 - b^\rho)^{1/\rho}} = R, \\ \nonumber \sigma_j(f_\pm) & \ge \sigma_j(f_1) - \sigma_j(f_2) \ge \left(c_1 - c_2 \right) b^{j} = \frac{R(1 - b^\rho)^{1/\rho}b^{j}}{a}, \qquad j \in {\mathbb{N}}_0, \\ \nonumber \sigma_{j+r}(f_\pm) & \le \sigma_{j+r}(f_1) + \sigma_{j+r}(f_2) \le (c_1+c_2) b^{j+r} \\ \nonumber & = R(1 - b^\rho)^{1/\rho}b^{j+r} \le a b^r \sigma_j(f_\pm), \qquad j \in {\mathbb{N}}_0. \end{align}
Since $\widehat{f}_2({\boldsymbol{k}}) = 0$ for ${\boldsymbol{k}} \in {\mathcal{J}}$, it follows that $\ALG'(f_\pm,\varepsilon) = \ALG'(f_1,\varepsilon)$. But, even though the two test functions $f_\pm$ lead to the same approximate solution, they have different true solutions. In particular, \begin{align} \nonumber \varepsilon &\ge \max \bigl\{\norm[{\mathcal{G}}]{\SOL(f_+) - \ALG'(f_+,\varepsilon)}, \norm[{\mathcal{G}}]{\SOL(f_-) - \ALG'(f_-,\varepsilon)} \bigr\} \\ \nonumber &\ge \frac 12 \bigl[\norm[{\mathcal{G}}]{\SOL(f_+) - \ALG(f_1,\varepsilon)} + \norm[{\mathcal{G}}]{\SOL(f_-) - \ALG'(f_1,\varepsilon)} \bigr] \\ \nonumber &\qquad \qquad \text{since } \ALG'(f_\pm,\varepsilon) = \ALG'(f_1,\varepsilon) \\ \nonumber &\ge \frac 12 \norm[{\mathcal{G}}]{\SOL(f_+) - \SOL(f_-)} \quad \text{by the triangle inequality}\\ \nonumber &\ge \frac 12 \norm[{\mathcal{G}}]{\SOL(f_+ - f_-)} \quad \text{since $\SOL$ is linear}\\ &= \norm[{\mathcal{G}}]{\SOL(f_2)} = \frac{R(a-1)(1 - b^\rho)^{1/\rho}}{2a} \norm[\tau]{\bigl(b^{j} \widetilde{\Lambda}_j \bigr)_{j \in {\mathbb{N}}_0}} \qquad \text{by \eqref{DHKM:SOLftwo}.} \label{DHKM:eps_LBA} \end{align}
Suppose that $\card({\mathcal{J}}) = \COST(\ALG',f_{\pm},\varepsilon) \le n_{j^{\star}}$. Then by condition \eqref{DHKM:PropCond}, there exists an $l^\star \le j^\star+1$ where $\card(\widetilde{\calK}_{l^\star}) \ge S_2 \card({\mathcal{K}}_{l^\star})$. This implies a lower bound on $\widetilde{\Lambda}_{l^\star}$. Let $m = n_{l^\star} - n_{l^\star-1} = \card({\mathcal{K}}_{l^\star})$. Then, $m_{\textup{in}} = \lceil S_2 m \rceil \ge S_2 m$ is a lower bound on $\card(\widetilde{\calK}_{l^\star})$, and $m_{\textup{out}} = m - m_{\textup{in}} \le (1 - S_2) m$ is an upper bound on $\card({\mathcal{K}}_{l^\star} \setminus \widetilde{\calK}_{l^\star})$. Moreover,
\begin{align*}
\Lambda_{l^\star}^\rho & = \sum_{i \in {\mathcal{K}}_{l^\star}} \lambda_{{\boldsymbol{k}}_i}^{\rho} = \widetilde{\Lambda}_{l^\star}^\rho + \sum_{i \in {\mathcal{K}}_{l^\star} \setminus \widetilde{\calK}_{l^\star}} \lambda_{{\boldsymbol{k}}_i}^{\rho}
\\
&\le \widetilde{\Lambda}_{l^\star}^\rho + m_{\textup{out}} \lambda_{{\boldsymbol{k}}_{n_{l^\star -1}+1}}^\rho \qquad \text{by the ordering of the } \lambda_{{\boldsymbol{k}}_i}\\
&\le \widetilde{\Lambda}_{l^\star}^\rho + m_{\textup{out}} S_1^\rho \lambda_{{\boldsymbol{k}}_{n_{l^\star}}}^\rho \qquad \text{by \eqref{DHKM:MinMaxCond}}\\
& \le \widetilde{\Lambda}_{l^\star}^\rho + \frac{m_{\textup{out}}}{m_{\textup{in}}} S_1^\rho \widetilde{\Lambda}_{l^\star}^\rho \qquad \text{by the definition of } \widetilde{\Lambda}_{l^\star}\\
& \le \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right] \widetilde{\Lambda}_{l^\star}^\rho \qquad \text{by the bounds on } m_{\textup{in}} \text{ and } m_{\textup{out}}
\\
& \le \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right] b^{-\rho l^\star} \norm[\tau]{\bigl(b^{j} \widetilde{\Lambda}_j \bigr)_{j \in {\mathbb{N}}_0}}^\rho\, . \end{align*}
Returning to \eqref{DHKM:eps_LBA}, the above inequality implies that \begin{equation*}
\varepsilon \ge \frac{R(a-1)(1 - b^\rho)^{1/\rho}}{2a} \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right]^{-1/\rho} b^{l^\star} \Lambda_{l^\star}. \end{equation*} Since $l^\star \le j^{\star}+1$ it follows that $ b^{l^\star} \ge b^{j^\star+1}$ and from condition \eqref{DHKM:LambdaDecayCond} it follows that $\Lambda_{l^\star} \ge \Lambda_{j^\star+1}/\alpha$. Thus, \begin{equation*}
\varepsilon \ge \frac{R(a-1)(1 - b^\rho)^{1/\rho}}{2a\alpha} \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right]^{-1/\rho} b^{j^\star+1} \Lambda_{j^\star+1}. \end{equation*} If any algorithm satisfies the error tolerance $\varepsilon$ for all input functions in ${\mathcal{C}} \cap {\mathcal{B}}_R$ and has information cost no greater than $n_{j^\star}$, then $j^\star$ must satisfy the above inequality. By contrast, if the above inequality is violated for any $j^\star$, then the information cost of the successful algorithm must be greater than $n_{j^\star}$. This completes the proof. \end{proof}
\begin{theorem} \label{DHKM:TrackConeAlgOptThm} Adaptive Algorithm \ref{DHKM:TrackConeAlg} is essentially optimal for the cone of input functions defined in \eqref{DHKM:TrackConeDef}. \end{theorem} \begin{proof} Let $j^\dagger(\varepsilon)$ be defined as in \eqref{DHKM:TractConejdagger}, with the $\varepsilon$ dependence made explicit. Choose $\varepsilon_{\max}$ and $R_{\min}$ in \eqref{DHKM:EssentialOpt} such that $j^\dagger(\varepsilon) \ge 2$. This definition implies that
\begin{align*}
b^{j^\dagger(\varepsilon)} \Lambda_{j^\dagger(\varepsilon) }
& = \frac{[1 - (\gamma b)^\tau]^{1/\tau}}{\alpha}
\norm[\tau]{ \bigl(b^{j^\dagger(\varepsilon)-1+r} \alpha \gamma^{r-1} \Lambda_{j^\dagger(\varepsilon)} \bigr)_{r=1}^\infty}
\\
& \ge \frac{[1 - (\gamma b)^\tau]^{1/\tau}}{\alpha} \norm[\tau]{ \bigl(b^{j^\dagger(\varepsilon)-1+r}\Lambda_{j^\dagger(\varepsilon)-1+r} \bigr)_{r=1}^\infty} \qquad \text{by \eqref{DHKM:LambdaDecayCond}}
\\
&
> \frac{b [1 - (\gamma b)^\tau]^{1/\tau} \varepsilon}{Ra^2 \alpha} \left( \frac{1 - b^{\rho(j^\dagger(\varepsilon)-1)}}{1 - b^\rho} \right)^{1/\rho} \qquad \text{by \eqref{DHKM:TractConejdagger}}
\\
&
\ge \frac{b [1 - (\gamma b)^\tau]^{1/\tau} \varepsilon}{Ra^2 \alpha} \qquad \text{since } j^\dagger(\varepsilon) \ge 2
\\
& = \frac{\alpha^2}{(b\beta)^2} \times \frac{2a \alpha \omega \varepsilon}{R(a-1)(1 - b^\rho)^{1/\rho}} \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right]^{1/\rho}
\\
\intertext{where }
\omega & = \frac{(a-1)b^3 \beta^2 (1 - b^\rho)^{1/\rho} [1 - (\gamma b)^\tau]^{1/\tau}}{2a^3 \alpha^4 } \left[1 + \left(\frac 1 {S_2} -1 \right) S_1^\rho \right]^{-1/\rho}. \end{align*}
Making the $\varepsilon$ dependence explicit in the definition of $j^\ddagger(\varepsilon)$ in Theorem \ref{DHKM:TractConeLowBdComp} it follows from the above inequality that \begin{equation*}
b^{j^\dagger(\varepsilon)} \Lambda_{j^\dagger(\varepsilon) }
> \frac{\alpha^2}{(b\beta)^2} b^{j^\ddagger(\omega\varepsilon)+2} \Lambda_{j^\ddagger(\omega \varepsilon) + 2 }
\ge \alpha b^{j^\ddagger(\omega\varepsilon)} \Lambda_{j^\ddagger(\omega \varepsilon)} \qquad \text{by \eqref{DHKM:LambdaDecayCond}}. \end{equation*}
If $j^{\dagger}(\varepsilon) \ge j^{\ddagger}(\omega \varepsilon)$, then \eqref{DHKM:LambdaDecayCond} implies that \[
b^{j^\dagger(\varepsilon)} \Lambda_{j^\dagger(\varepsilon) } \le \alpha (\gamma b)^{j^{\dagger}(\varepsilon) - j^{\ddagger}(\omega \varepsilon)} b^{j^{\ddagger}(\omega \varepsilon) } \Lambda_{j^\ddagger(\omega \varepsilon) } \le \alpha b^{j^{\ddagger}(\omega \varepsilon) } \Lambda_{j^\ddagger(\omega \varepsilon) }. \] But, this contradicts the above inequality. Thus, $j^\dagger(\varepsilon) < j^\ddagger(\omega \varepsilon)$, and so \[ \COST(\ALG,{\mathcal{C}},\varepsilon,R) \le n_{j^\dagger(\varepsilon)} < n_{j^\ddagger(\omega \varepsilon)} < \COMP({\mathcal{A}}({\mathcal{C}}),\omega \varepsilon,R). \] Thus, Algorithm \ref{DHKM:TrackConeAlg} is essentially optimal. \end{proof}
\subsection{Tractability}\label{DHKM:SecDecayTract}
We again would like to study tractability. As it turns out, by using the relation between the cones defined in \eqref{DHKM:pilot_cone} and \eqref{DHKM:TrackConeDef}, respectively, we easily obtain sufficient conditions for tractability.
\begin{theorem} \label{DHKM:thmtract3} The respective conditions presented in Theorem \ref{DHKM:thmtract1} for the case where $\rho'=\infty$ and in Theorem \ref{DHKM:thmtract2} for the case where $\rho'<\infty$ are sufficient for strong polynomial, polynomial, and weak tractability of the approximation problem defined on cones as in \eqref{DHKM:TrackConeDef}. \end{theorem} \begin{proof} As pointed out above, ${\mathcal{C}}$ defined in \eqref{DHKM:TrackConeDef} is a subset of ${\mathcal{C}}$ defined in \eqref{DHKM:pilot_cone}, by choosing $A=\left(1 + \frac{a^\rho b^\rho}{1 - b^\rho} \right)^{1/\rho}$ in \eqref{DHKM:pilot_cone}. This means that the approximation problem on ${\mathcal{C}}$ defined in \eqref{DHKM:TrackConeDef} is essentially (i.e., up to constants depending on $A,a,b$ and $\rho$) no harder than the same problem on ${\mathcal{C}}$ defined in \eqref{DHKM:pilot_cone}. This, however, implies that all sufficient conditions in Theorem \ref{DHKM:thmtract1} are also sufficient in the case considered in Theorem \ref{DHKM:thmtract3}. \end{proof}
Theorem \ref{DHKM:thmtract3} yields sufficient conditions for the tractability notions considered here. A general result for necessary conditions seems to be more difficult to obtain and is left open for future research.
\section{Inferring Coordinate and Smoothness Importance} \label{DHKM:smoothimportance_sec}
In Sections \ref{DHKM:pilot_sec} and \ref{DHKM:tracking_sec}, the weights ${\boldsymbol{\lambda}} = (\lambda_{{\boldsymbol{k}}})_{{\boldsymbol{k}} \in {\mathbb{K}}}$, which appear in the definition of the cone of inputs, ${\mathcal{C}}$, are taken as given and fixed. One may assume the form suggested in \eqref{DHKM:prodwts}, which defines ${\boldsymbol{\lambda}}$ in terms of coordinate weights and smoothness weights. However, practically speaking it may be difficult to know a priori the values of these weights. This section explores a situation where the initial data collected for the input function data can be used to learn ${\boldsymbol{\lambda}}$, inferring which input variables in $f$ may be more important and the smoothness of the function.
The motivation for this section is situations where the relative importance of the $d$ input variables of the function is not known from physical considerations. We also envision situations where the cost of function data is large, e.g., the result of an expensive computer simulation. Thus, we are not concerned with the cost of the algorithm beyond the information cost, which we hope to limit to ${\mathcal{O}}(d)$.
\subsection{Product, Order and Smoothness Dependent (POSD) Weights} The $u_{{\boldsymbol{k}}}$ and the $\lambda_{{\boldsymbol{k}}}$ in this section are defined as \begin{equation} u_{\boldsymbol{k}} = \prod_{\ell = 1}^d \Tilde{u}_{k_\ell}, \quad
\lambda_{{\boldsymbol{k}}} = \Gamma_{\|{\boldsymbol{k}}\|_0} \prod_{\substack{\ell=1\\ k_\ell>0}}^d w_\ell s_{k_\ell}, \quad \Gamma_0 = s_1 = 1, \quad {\boldsymbol{k}} \in \mathbb{N}_0^d, \label{DHKM:posdeq} \end{equation} where ${\boldsymbol{w}} = (w_\ell)_{l=1}^d$ is the vector of coordinate weights, ${\boldsymbol{s}} = (s_k)_{k=1}^\infty$ is the vector of smoothness weights, $\boldsymbol{\Gamma} = (\Gamma_m)_{m=1}^d$ is the vector of \emph{order weights}, and $\norm[0]{{\boldsymbol{k}}}$ denotes the number of nonzero elements of ${\boldsymbol{k}}$. The intuition behind these weights is as follows: \begin{itemize}
\item Coordinate weights quantify the importance for the $d$ input variables in $f$.
\item Smoothness weights quantify the importance of the $\Tilde{u}_k$. E.g., if the $\Tilde{u}_k$ are polynomials of degree $k$ as in Section \ref{DHKM:secexamp}, then the faster the $s_k$ decay, the smoother $f$ is.
\item Order weights quantify the importance of effects with different orders; ${\boldsymbol{k}}$ having one nonzero element corresponds to a first-order or main effect, ${\boldsymbol{k}}$ having two nonzero elements corresponds to a second-order (interaction) effect. (e.g., first-order, second-order). \end{itemize} \
This parametrization is motivated by several guiding principles from the experimental design literature \cite{WuHam2009}, which are briefly described below. In statistical parlance, the terms $\widehat{f}({\boldsymbol{k}}) u_{{\boldsymbol{k}}}$ are effects. \begin{itemize} \item \emph{Effect sparsity} assumes that only a small number of inputs in $f$ are important. In \eqref{DHKM:posdeq}, this sparsity means that only a small number of product weights ${\boldsymbol{w}}$ are large. This principle arises in the sufficient condition for strong tractability in Corollary \ref{DHKM:sptexample_cor}. \item \emph{Effect heredity} assumes that lower-order effects are more important than higher-order effects. E.g., $\lambda_{(1,0, 0,\ldots, 0)}$ should be larger than $\lambda_{(1, 1, 0, \ldots, 0)}$. In \eqref{DHKM:posdeq}, this heredity can be enforced by assuming that the order weights $\Gamma_{m}$ decrease with $m$. \item \emph{Effect hierarchy} assumes that an effect is active \emph{only} when all its component effects are active. For example, $\lambda_{(1, 1, 0, \ldots, 0)} > 0$ only when $\lambda_{(1, 0, 0, \ldots, 0)}$ and $\lambda_{(0, 1, 0, \ldots, 0)}$ are both nonzero. This hierarchy is implicitly enforced by the product structure of the weights in \eqref{DHKM:posdeq}. \item \emph{Effect smoothness} assumes that lower-degree effects are more important than higher-degree effects. For example, when the $(\Tilde{u}_k)_{k \in \mathbb{N}_0}$ are polynomials, this means that linear effects are more important than quadratic effects, which are in turn more significant than cubic effects, and so on. Effect smoothness can be imposed by assuming ${\boldsymbol{s}}$ to be a decreasing sequence. \end{itemize} \
The $\lambda_{{\boldsymbol{k}}}$ defined in \eqref{DHKM:posdeq} are called product, order and smoothness dependent (POSD) weights. From a quasi-Monte Carlo (QMC) perspective, the POSD weights in \eqref{DHKM:posdeq} generalize upon the product-and-order dependent (POD) weights in \cite{KuoEtal12a}, which were introduced for analyzing QMC methods in partial differential equations with random coefficients. The latter POD weights can be recovered by ignoring the smoothness weights.
Our POSD weights differ from the smoothness-driven product-and-order dependent (SPOD) weights in \cite{Dea2014}, which were recently used to analyze higher-order QMC methods for stochastic partial differential equations. These SPOD weights take the form: \begin{equation*}
\gamma_{{\mathfrak{u}}} = \sum_{{\boldsymbol{k}} \in \{1, \ldots, \alpha\}^{|{\mathfrak{u}}|}} \|{\boldsymbol{k}}\|_1 ! \prod_{\ell \in {\mathfrak{u}}} \left( 2^{\delta(k_\ell,\alpha)} w_\ell^{k_\ell} \right), \quad \|{\boldsymbol{k}}\|_1 = \sum_{l=1}^d k_\ell, \quad {\mathfrak{u}} \subseteq \{1, \ldots, d\},
\end{equation*} where $\delta (k_\ell,\alpha)$ is 1 if $k_\ell=\alpha$ and 0 otherwise. Intuitively, the SPOD weights quantify the importance of each \textit{subspace} (indexed by ${\mathfrak{u}}$), under a common smoothness structure among subspaces (for further details on SPOD weights, we refer the reader to \cite{Dea2014}). In contrast, the proposed POSD weights in \eqref{DHKM:posdeq} instead quantify the importance of each \textit{Fourier series coefficient} $\widehat{f}({\boldsymbol{k}})$ (indexed by ${\boldsymbol{k}}$), under a common smoothness structure among coefficients.
\subsection{Inferring POSD Weights from an Initial Sample}
Let ${\mathcal{C}}_{{\boldsymbol{\lambda}}}$ denote the cone of inputs defined in \eqref{DHKM:pilot_cone} by POSD weights ${\boldsymbol{\lambda}} = \bigl( \lambda_{{\boldsymbol{k}}} \bigr)_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}$. As mentioned above, our goal here is to infer ${\boldsymbol{\lambda}}$ from input function data. We start with an initial set of wavenumbers: \begin{equation} \label{DHKM:barKdef}
\bar{\calK}= \{ (0, \ldots, 0, k, 0, \ldots, 0): k = 0, \ldots, k_{\max}\}. \end{equation} The approximation to $f$ based on sampling the series coefficients for these wavenumbers is \begin{equation*}
f_{\text{app}} = \sum_{{\boldsymbol{k}} \in \bar{{\mathcal{K}}}} \widehat{f}({\boldsymbol{k}}) u_{{\boldsymbol{k}}}. \end{equation*} We choose the ${\mathcal{C}}_{{\boldsymbol{\lambda}}}$ that best fits $f$ by selecting ${\boldsymbol{\lambda}}$ to make the norm of $f_{\text{app}}$ small: \begin{multline} \bar{{\boldsymbol{\lambda}}} = {\boldsymbol{\lambda}}(\bar{{\boldsymbol{w}}}, \bar{{\boldsymbol{s}}}, {\boldsymbol{\Gamma}}), \\
\text{where } (\bar{{\boldsymbol{w}}}, \bar{{\boldsymbol{s}}} ) = \min \left\{ \argmin_{({\boldsymbol{w}},{\boldsymbol{s}}) \in {\mathcal{W}} \times {\mathcal{S}}} \left\|\left( \frac{\widehat{f}({\boldsymbol{k}})}{\lambda_{{\boldsymbol{k}}}({\boldsymbol{w}},{\boldsymbol{s}},{\boldsymbol{\Gamma}})} \right)_{{\boldsymbol{k}} \in \bar{{\mathcal{K}}}}\right\|_{\rho} \right \}. \label{DHKM:eq:inf} \end{multline} Here, ${\mathcal{W}}$ is a candidate set for coordinate weights, e.g., ${\mathcal{W}} = [0,w^*]^d$, and ${\mathcal{S}}$ is a candidate set for the smoothness weights, e.g., ${\mathcal{S}} = \{(1/k^r)_{k=1}^{\infty} \colon r > 0\}$. The inner minimization finds the $({\boldsymbol{w}},{\boldsymbol{s}})$ that minimizes the approximate norm of the input function. This minimizer may be non-unique, so the outer minimization chooses the smallest such $({\boldsymbol{w}},{\boldsymbol{s}})$. Making the coordinate and smoothness weights as small as possible helps enforce the principles of effect sparsity. The optimum, $(\bar{{\boldsymbol{w}}}, \bar{{\boldsymbol{s}}})$, then defines the \emph{data-inferred} POSD ${\boldsymbol{\lambda}}$, denoted $\bar{{\boldsymbol{\lambda}}}$.
The candidate sets ${\mathcal{W}}$ and ${\mathcal{S}}$ should be constructed such that the coordinate and smoothness weights have a priori upper bounds. Otherwise the inner minimization would choose huge values for ${\boldsymbol{w}}$ and ${\boldsymbol{s}}$ to maximize the $\lambda_{{\boldsymbol{k}}}({\boldsymbol{w}},{\boldsymbol{s}},{\boldsymbol{\Gamma}})$ and minimize the norm of $f_{\text{app}}$. The cardinality of the initial set of wavenumbers is $d k_{\max} + 1$. There is a trade-off between keeping $k_{\max}$ small enough to reducing cost and making $k_{\max}$ large enough to ensuring robustness.
For simplicity, we assume that order weights, ${\boldsymbol{\Gamma}}$, are fixed a priori. If desired, they too could be inferred as the next step. However, since we want to limit the size of the initial sample to ${\mathcal{O}}(d)$ we must sample judiciously the higher order interactions.
The optimization in \eqref{DHKM:eq:inf} is nontrivial to solve numerically. In practice, we iteratively optimize over ${\boldsymbol{w}}$ and then ${\boldsymbol{s}}$ until convergence is reached. At each step of the iteration $\|(\hat{f}({\boldsymbol{k}})/\lambda_{{\boldsymbol{k}}})_{{\boldsymbol{k}} \in \bar{\calK}}\|_{\rho}$ decreases.
Algorithm \ref{DHKM:InfPilotConeAlg} combines the construction of data-inferred POSD weights, $\bar{{\boldsymbol{\lambda}}}$, with Algorithm \ref{DHKM:PilotConeAlg} of Section \ref{DHKM:pilot_sec}. This algorithm succeeds for input functions in the cone \begin{equation} \label{DHKM:pilot_cone2}
\bar{{\mathcal{C}}} := \left \{ f \in {\mathcal{F}} : f \in {\mathcal{C}}_{\bar{{\boldsymbol{\lambda}}}} \text{ for } \bar{{\boldsymbol{\lambda}}} \text{ defined in \eqref{DHKM:eq:inf}} \right\}. \end{equation} The reason that $\bar{{\mathcal{C}}}$ is a cone is that the data-inferred $\bar{{\boldsymbol{\lambda}}}$ for the input function $f$ is exactly the same as for the input function $cf$, where $c$ is any constant.
\begin{algorithm}
\caption{Adaptive $\ALG$ Based on Data-Inferred POSD Weights \label{DHKM:InfPilotConeAlg}}
\begin{algorithmic}
\PARAM the bases $\{u_{{\boldsymbol{k}}}\}_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}$ and $\{v_{{\boldsymbol{k}}}\}_{{\boldsymbol{k}} \in {\mathbb{N}}_0^d}$; candidate sets ${\mathcal{W}}$ and ${\mathcal{S}}$; maximum smoothness degree, $k_{\max}$; order weights, $\boldsymbol{\Gamma}$; an inflation factor, $A > 1$; $\APP$ satisfying \eqref{DHKM:Refined_APP_err}
\INPUT a black-box function, $f$; an absolute error tolerance,
$\varepsilon>0$
\Ensure Error criterion \eqref{DHKM:err_crit} for the cone defined in \eqref{DHKM:pilot_cone2} \State Define the initial set of wavenumbers $\bar{\calK}$ defined in \eqref{DHKM:barKdef} \State Evaluate initial sample $\bigl\{\widehat{f}({\boldsymbol{k}})\bigr\}_{{\boldsymbol{k}} \in \bar{\calK}}$ \State Compute data-driven POSD weights, $\bar{{\boldsymbol{\lambda}}}$ according to \eqref{DHKM:eq:inf} \State Using these weights, $\bar{{\boldsymbol{\lambda}}}$, perform Algorithm \ref{DHKM:PilotConeAlg} to obtain $\ALG(f,\varepsilon)$ \RETURN $\ALG(f,\varepsilon)$ \end{algorithmic} \end{algorithm}
\subsection{Numerical Examples} \label{DHKM:numexamp_sec}
We now investigate the numerical performance of this adaptive algorithm using data-inferred POSD weights. For simplicity, only the case of $\rho = \infty$ and $\rho' = \tau = 1$ is considered in the following examples. Here, the basis functions $(u_{\boldsymbol{k}})_{{\boldsymbol{k}} \in \mathbb{N}_0^d}$ are Chebyshev polynomials in Section \ref{DHKM:secexamp}, and the solution operator is $\SOL (f) = f$ (i.e., function approximation). We note that $\|f - \ALG(f,\varepsilon)\|_\infty \le \|f - \ALG(f,\varepsilon)\|_{\mathcal{G}}$, so our error criterion \eqref{DHKM:err_crit} implies that $\|f - \ALG(f,\varepsilon)\|_\infty \le \varepsilon$.
The simulation set-up is as follows. The Fourier coefficients for input function $f$, $\{\hat{f}({\boldsymbol{k}})\}_{{\boldsymbol{k}} \in \mathbb{N}_0^d}$, are randomly sampled as: \begin{equation*}
\hat{f}({\boldsymbol{k}}) = Z_{{\boldsymbol{k}}} \, {\Gamma}_{\|{\boldsymbol{k}}\|_0}^{\rm tr} \prod_{\substack{\ell=1\\ k_\ell>0}}^d {w_\ell^{\rm tr}} {s}_{k_\ell}^{\rm tr}, \quad Z_{{\boldsymbol{k}}} \overset{i.i.d.}{\sim} \text{Unif}[-1, 1], \quad {\boldsymbol{k}} \in \mathbb{N}_0^d.
\end{equation*} Here, $(w_\ell^{\rm tr})_{\ell=1}^d = (1/L^2(\ell))_{l=1}^d$, $({\Gamma_k^{\rm tr}})_{k=1}^\infty \equiv 1$ and $(s_j^{\rm tr})_{j=1}^{k_{\rm max}} = (1/j^4)_{j=1}^4$ are the true coordinate, order, and smoothness weights, and $Z_{{\boldsymbol{k}}}$ randomly sets the magnitude and sign of each coefficient. Moreover, $\bigl(L(\ell)\bigr)_{\ell=1}^d$ is a random permutation of $1, \ldots, d$ to ensure that the order of input variables does not necessarily reflect their order of importance. We also set $\boldsymbol{\Gamma} = \boldsymbol{\Gamma}^{\rm tr}$ in Algorithm \ref{DHKM:InfPilotConeAlg} and use an inflation factor of $A = 1.1$.
Figures \ref{fig:four} (a) and (b) display the total required sample size from Algorithm \ref{DHKM:InfPilotConeAlg}, as a function of the error to tolerance ratio, $\|f - \ALG(f,\varepsilon)\|_\infty/\varepsilon$, in $d=4$ and $d=7$ dimensions, respectively. Each data point corresponds to a different error tolerance $\varepsilon$. A ratio $\|f - \ALG(f,\varepsilon)\|_\infty/\varepsilon$ close to, but not exceeding, one is desired, since this shows that our adaptive algorithm is successful. For $d=4$, $\|f - \ALG(f,\varepsilon)\|_\infty/\varepsilon$ fluctuates around 0.4 for all choices of $\varepsilon$; for $d=7$, this ratio begins at $\approx 0.1$ for $\varepsilon = 0.1$, then decreases to $\approx 0.014$ for $\varepsilon = 0.001$. This shows that our adaptive approximation algorithm works reasonably well. It appears slightly more effective in lower dimensions than in higher dimensions. A likely reason is that the underlying POSD structure can be more easily learned from a small pilot sample in lower dimensions than in higher dimensions.
\begin{figure}
\caption{$f$ is a $d=4$-dim. function with random Fourier coefficients.}
\label{fig:four1}
\caption{$f$ is a $d=7$-dim. function with random Fourier coefficients.}
\label{fig:four2}
\caption{Total required sample size as a function of error ratio $\|f - \ALG(f,\varepsilon)\|_{\infty}/\varepsilon$, with points colored by the absolute error tolerance level $\varepsilon$.}
\label{fig:four}
\end{figure}
\begin{small} \noindent\textbf{Authors' addresses:}\\
\noindent Yuhan Ding\\
Department of Mathematics\\
Misericordia University\\
301 Lake Street, Dallas, PA 18704 USA\\
\noindent Fred J. Hickernell\\
Department of Applied Mathematics\\
Illinois Institute of Technology\\
RE 220, 10 W.\ 32${}\text{nd}$ Street, Chicago, IL 60616 USA\\
\noindent Peter Kritzer\\
Johann Radon Institute for Computational and Applied Mathematics (RICAM)\\
Austrian Academy of Sciences\\
Altenbergerstr. 69, 4040 Linz, Austria\\
\noindent Simon Mak\\
H. Milton Stewart School of Industrial and Systems Engineering\\
Georgia Institute of Technology\\
755 Ferst Drive, Atlanta, GA 30332 USA\\
\end{small}
\end{document} | arXiv |
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Q5- Find
\(\lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x}\)
\(\frac{d \sin x}{d x}=\cos x\)
\(\frac{d \cos x}{d x}=-\sin x\)
\(\lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x}=\lim _{x \rightarrow \pi^{-}} \frac{\cos x}{-(-\sin x)}=\frac{\cos (\pi)}{\sin (\pi)}=\frac{-1}{0}=\infty\)
\(\lim _{x \rightarrow \pi^{-}} \frac{\sin (x)}{1-\cos (x)}=\frac{\sin (\pi)}{1-\cos (\pi)}=\frac{0}{1-(-1)}=\frac{0}{1+1}\)
\(=\frac{0}{2}=0\)
\(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}\)
\(\tan (0)=0\)
\(\sec (0)=1\)
\(\frac{d \tan (x)}{d x}=\sec ^{2}(x)\)
\(\frac{d}{d x} \sec ^{2}(x)=2 \sec ^{2} x \tan x\)
\(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}=\frac{\tan (0)-0}{0^{3}}= \frac{0-0}{0}=\frac{0}{0} \)$$
\(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}=\lim _{x \rightarrow 0} \frac{\sec ^{2} x-1}{3 x^{2}}\)
\(=\frac{\sec ^{2}(0)-1}{3(0)^{2}}=\frac{1-1}{0}=\frac{0}{0}\)
\(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}=\lim _{x \rightarrow 0} \frac{\sec ^{2} x-1}{3 x^{2}}=\lim _{x \rightarrow 0} \frac{2 \sec ^{2} x \tan x}{6 x}\)
\(=\frac{1}{3} \lim \frac{\sec ^{2} x \tan x}{x}\)
\(=\frac{1}{3} \lim _{x \rightarrow 0}\left(\sec ^{2} x\right) . \lim _{x \rightarrow 0}\left(\frac{\tan x}{x}\right)\)
\(\lim _{x \rightarrow 0} \sec ^{2}(x)=\sec ^{2}(0)=1\)
\(\lim _{x \rightarrow 0} \frac{\tan x}{x}=\frac{\tan \theta}{0}\)
\(=\frac{0}{0}\)
\(=\frac{1}{3} \cdot(1) \cdot \lim _{x \rightarrow 0} \frac{\sec ^{2} x}{1}=\frac{1}{3} \cdot \sec ^{2}(0)\)
\(=\frac{1}{3} \cdot(1)=\frac{1}{3}\)
\(\lim _{x \rightarrow 0^{+}} x \ln (x)\)
\(\ln \left(0^{+}\right)=-\infty\)
\(\frac{d \ln (x)}{d x}=\frac{1}{x}\)
\(\lim _{x \rightarrow 0^{+}} x \ln (x)=(0) \ln (0)=0 \cdot(-\infty)\)
\(\lim _{x \rightarrow 0^{+}} x \ln (x)=\lim _{x \rightarrow 0^{+}} \frac{\ln (x)}{\frac{1}{x}}=\frac{\ln (0)}{\frac{1}{0}}=\frac{-\infty}{\infty} \)
\(\frac{0}{0} , \frac{\infty}{\infty}\)
\(1^{\infty} , 0^{0} , \infty^{0}\)
\(\infty-\infty,0(-\infty)\)
\(\frac{a}{b}\)
\(\lim _{x \rightarrow 0^{+}} \frac{\ln (x)}{\frac{1}{x}}=\lim _{x \rightarrow 0^{+}} \frac{\frac{1}{x}}{\frac{-1}{x^{2}}}=\lim _{x \rightarrow 0^{+}} \frac{-x^{2}}{x}=\lim _{x \rightarrow 0^{+}}(-x)\)
\(=-(0)=0\)
\(\lim _{x \rightarrow \pi} \frac{\sin x}{\sqrt{x-\pi}}\)
\(\sin (\pi)=0\)
\(\frac{d \sin (x)}{d x}=\cos x\)
\(\lim _{x \rightarrow \pi^{+}} \frac{\sin x}{\sqrt{x-\pi}}=\frac{\sin (\pi)}{\sqrt{\pi-\pi}}=\frac{0}{\sqrt{0}}=\frac{0}{0}\)
\(\lim _{x \rightarrow \pi^{+}} \frac{\sin x}{\sqrt{x-\pi}}=\lim _{x \rightarrow \pi^{+}} \frac{\cos (x)}{\frac{1}{2 \sqrt{x_{-} \pi}}}=\lim _{x \rightarrow \pi^{+}} \frac{\cos (x)}{\frac{1}{2 \cdot(x-\pi)^{\frac{1}{2}}}}\)
\(\begin{array}{l}{=\lim _{x \rightarrow \pi^{+}} 2 \cdot \cos x \cdot(x-\pi)^{\frac{1}{2}}=2 \cos (\pi) \cdot(\pi-\pi)^{\frac{1}{2}}}\end{array} \)
\(=0\)
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\begin{document}
\title{Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields} \author{Yukiyoshi Nakkajima \date{}\thanks{2010 Mathematics subject classification number: 14F30, 14F40, 14J32. The first named author is supported from JSPS Grant-in-Aid for Scientific Research (C) (Grant No.~80287440). The second named author is supported by JSPS Fellow (Grant No.~15J05073).\endgraf}} \maketitle
$${\bf Abstract}$$ In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log points of finite fields.
\section{Introduction}\label{sec:int} In this article we discuss a new topic--rational points of the underlying schemes of log schemes in the sense of Fontaine-Illusie-Kato over the log point of a finite field--for interesting log schemes. First let us recall results on rational points of (proper smooth) schemes over a finite field. \par The following is famous Ax' and Katz' theorem:
\begin{theo}[{\bf \cite{ax}, \cite{kat}}]\label{theo:axka} Let ${\mab F}_q$ be the finite field with $q=p^e$-elements, where $p$ is a prime number. Let $n$ and $r$ be positive integers. Let $D_i$ $(1\leq i\leq r)$ be a hypersurface of ${\mab P}^n_{{\mab F}_q}$ of degree $d_i$. If $\sum_{i=1}^rd_i\leq n$, then $\# (\bigcap_{i=1}^rD_i)({\mab F}_{q^k})\equiv 1~{\rm mod}~q^k$. \end{theo}
\par In \cite{es} Esnault has proved the following theorem generalizing this theorem in the case where $\bigcap_{i=1}^rD_i$ is smooth over ${\mab F}_q$ and geometrically connected:
\begin{theo}[{\bf \cite[Corollary 1.3]{es}}]\label{theo:esn}
Let $X$ be a geometrically connected projective smooth scheme over ${\mab F}_q$.
If $X/{\mab F}_q$ is a Fano variety $($i.~e., the inverse of the canonical sheaf $\om_{X/{\mab F}_q}^{-1}$ of $X/{\mab F}_q$ is ample$)$, then $\# X({\mab F}_{q^k}) \equiv 1~{\rm mod}~ q^k$ $(k\in {\mab Z}_{\geq 1})$. \end{theo}
In \cite{ki} Kim has proved the following theorem and he has reproved Esnault's theorem as a corollary of his theorem by using the Lefschetz trace formula for the crystalline cohomology of $X/{\mab F}_q$:
\begin{theo}[{\bf \cite[Theorem 1]{ki}}]\label{theo:kim} Let $\kap$ be a perfect field of characteristic $p>0$. Set ${\cal W}:={\cal W}(\kap)$ and $K_0:={\rm Frac}({\cal W})$. Let $X$ be a projective smooth scheme over $\kap$.
If $X/\kap$ is a Fano variety, then $H^i(X,{\cal W}({\cal O}_X))\otimes_{\cal W}{K_0} =0$ for $i>0$. \end{theo}
In \cite{gir} Gongyo, Nakamura and Tanaka have proved the following theorem generalizing (\ref{theo:esn}) for the 3-dimensional case by using methods of MMP(=minimal model program) in characteristic $p\geq 7$:
\begin{theo}[{\bf \cite[Theorem (1.2), (1.3)]{gir}}]\label{theo:gnt} Let $\kap$ be as in {\rm (\ref{theo:kim})}. Assume that $p\geq 7$. Let $X$ be a geometrically connected proper variety over $\kap$. Let $\Del$ be an effective ${\mab Q}$-Cartier divisor on $X$. Assume that $(X,\Del)$ is klt$($=Kawamata log terminal$)$ pair over $\kap$ and that $-(K_X+\Del)$ is a ${\mab Q}$-Cartier ample divisor on $X$, where $K_X$ is the canonical divisor on $X$. Then the following hold$:$ \par $(1)$ $H^i(X,{\cal W}({\cal O}_{X}))\otimes_{\cal W}{K_0}=0$ for $i>0$.
\par $(2)$ Assume that $\kap={\mab F}_q$. Then $\# X({\mab F}_{q^k}) \equiv 1~{\rm mod}~ q^k$ $(k\in {\mab Z}_{\geq 1})$. \end{theo}
\parno See \cite{nt} for the case where $-(K_X+\Del)$ is nef and big and $(X,\Del)$ is log canonical. \par In this article we give other generalizations of the Theorems (\ref{theo:esn}) and (\ref{theo:kim}) under the assumption of certain finiteness: we give the definition of a log Fano variety and we prove a log and stronger version (\ref{theo:vc}) below of Kim's theorem under the assumption as a really immediate good application of a recent result: Nakkajima-Yobuko's Kodaira vanishing theorem for a quasi-$F$-split projective log smooth scheme of vertical type (\cite{ny}). In this vanishing theorem, we use theory of log structures due to Fontaine-Illusie-Kato (\cite{klog1}, \cite{klog2}) essentially. (See \S\ref{sec:psso} for the precise statement of this vanishing theorem.) As a corollary of (\ref{theo:vc}), we obtain the congruence of the cardinality of rational points of a log Fano variety over the log point of ${\mab F}_q$ ((\ref{coro:npf}) below).
\par To state our result (\ref{theo:vc}), we first recall the notion of the quasi-Frobenius splitting height due to Yobuko, which plays an important role for log Fano varieties in this article. \par
Let $Y$ be a scheme of characteristic $p>0$. Let $F_Y\col Y\lo Y$ be the Frobenius endomorphism of $Y$. Set $F:={\cal W}_n(F_Y^*)\col {\cal W}_n({\cal O}_Y)\lo F_{Y*}({\cal W}_n({\cal O}_Y))$.
This is a morphism of ${\cal W}_n({\cal O}_Y)$-modules. In \cite{y} Yobuko has introduced the notion of the quasi-Frobenius splitting height $h^F(Y)$ for $Y$. (In [loc.~cit.] he has denoted it by ${\rm ht}^S(Y)$.) It is the minimum of positive integers $n$'s such that there exists a morphism $\rho \col F_{Y*}({\cal W}_n({\cal O}_Y))\lo {\cal O}_Y$ of ${\cal W}_n({\cal O}_Y)$-modules such that $\rho \circ F\col {\cal W}_n({\cal O}_Y)\lo {\cal O}_Y$ is the natural projection. (If there does not exist such $n$, then we set $h^F(Y)=\infty$.) This is a highly nontrivial generalization of the notion of the Frobenius splitting by Mehta and Ramanathan in \cite{mr} because they have said that, for a scheme $Z$ of characteristic $p>0$, $Z$ is a Frobenius splitting(=$F$-split) scheme if $F\col {\cal O}_Z\lo F_{Z*}({\cal O}_Z)$ has a section of ${\cal O}_Z$-modules. Because the terminology ``quasi Frobenius splitting height'' is too long, we call this {\it Yobuko height}.
\par Let $\kap$ be a perfect field of characteristic $p>0$. Let $s$ be a log scheme whose underlying scheme is ${\rm Spec}(\kap)$ and whose log structure is associated to a morphism ${\mab N}\owns 1\lom a\in \kap$ for some $a\in \kap$. That is, $s$ is the log point of $\kap$ or $({\rm Spec}(\kap),\kap^*)$.
Let $X/s$ be a proper (not necessarily projective) log smooth scheme of pure dimension $d$ of vertical type with log structure $(M_X,\al \col M_X\lo {\cal O}_X)$.
Here ``vertical type'' means that $\al({\cal I}_{X/s}){\cal O}_X={\cal O}_X$, where ${\cal I}_{X/s}$ is Tsuji's ideal sheaf of the log structure $M_X$ of $X$ defined in \cite{tsp} and denoted by $I_f$ in [loc.~cit.], where $f\col X\lo s$ is the structural morphism. (In \S\ref{sec:psso} below we recall the definition of ${\cal I}_{X/s}$.) For example, the product of (locally) simple normal crossing log schemes over $s$ defined in \cite{nlk3}, \cite{ny} and \cite{nlw} is of vertical type. Let $\os{\circ}{X}$ be the underlying scheme of $X$. Let $\Om^i_{X/s}$ be the sheaf of log differential forms of degree $i$ on $\os{\circ}{X}$, which has been denoted by $\om^i_{X/s}$ in \cite{klog1}. Set $\om_{X/s}:=\Om^d_{X/s}$. We say that $X/s$ is a {\it log Fano scheme} if $\om_{X/s}^{-1}$ is ample. Moreover, if $\os{\circ}{X}$ is geometrically connected, then we say that $X/s$ a {\it log Fano variety}. \par In this article we prove the following:
\begin{theo}\label{theo:vc} Let $X/s$ be a log Fano scheme. Assume that $h^F(\os{\circ}{X})<\infty$. Then $H^i(X,{\cal W}_n({\cal O}_X))=0$ for $i>0$ and for $n>0$. Consequently $H^i(X,{\cal W}({\cal O}_X))=0$ for $i>0$. \end{theo}
As mentioned above, we obtain this theorem immediately by using Nakkajima-Yobuko's Kodaira vanishing theorem for a quasi-$F$-split projective log smooth scheme of vertical type (\cite{ny}). As a corollary of this theorem, we obtain the following:
\begin{coro}\label{coro:npf} Let $X/s$ be a log Fano variety. Assume that $\kap={\mab F}_q$ and
that $h^F(\os{\circ}{X})<\infty$. Then \begin{align*} \# \os{\circ}{X}({\mab F}_{q^k}) \equiv 1~{\rm mod}~ q^k \quad (k\in {\mab Z}_{\geq 1}). \tag{1.6.1}\label{ali:xfq} \end{align*} In particular $\os{\circ}{X}({\mab F}_q)\not= \emptyset$. \end{coro}
This is a generalization of Esnault's theorem (\ref{theo:esn}) under the assumption of the finiteness of the Yobuko height.
To derive (\ref{coro:npf}) from (\ref{theo:vc}), we use
\par (A): \'{E}tess-Le Stum's Lefschetz trace formula for rigid cohomology (with compact support) (\cite{el})
\parno and
\par (B) Berthelot-Bloch-Esnault's calculation of the slope $< 1$-part of the rigid cohomology (with compact support) via Witt sheaves (\cite{bbe})
\parno as in \cite{bbe}, \cite{gir} and \cite{nt}. However our proofs of (\ref{theo:vc}) and (\ref{coro:npf}) are very different from Esnault's, Kim's and Gongyo-Nakamura-Tanaka's proofs of (\ref{theo:esn}), (\ref{theo:kim}) and (\ref{theo:gnt}) in their articles because we do not use the rational connectedness of a Fano variety which has been used in them.
\par We guess that the assumption of the finiteness of the Yobuko height
is not a strong one for log Fano schemes.
However this assumption is not always satisfied for smooth Fano schemes because the Kodaira vanishing holds if the Yobuko height
is finite and because the Kodaira vanishing does not hold for certain Fano varieties (\cite{lr}, \cite{hl}, \cite{to}); the Yobuko heights
of them are infinity. Hence to calculate the Yobuko heights
of (log) Fano schemes is a very interesting problem. \par The conclusion of (\ref{coro:npf}) holds for a proper scheme $Y/{\mab F}_q$ such that $H^i(Y,{\cal O}_Y)=0$ $(i>0)$. H.~Tanaka has kindly told me that it is not known whether there exists an example of a smooth Fano variety over $\kap$ for which this vanishing of the cohomologies does not hold. (In \cite{j} Joshi has already pointed out this; Shepherd-Barron has already proved that this vanishing holds for a smooth Fano variety of dimension 3 (\cite[(1.5)]{sb})).
\par On the other hand, it is not clear at all that there is a precise rule as above about congruences of the cardinalities of the rational points of varieties except Fano varieties. One may think that there is no rule for them. In this article we show that this is not the case for
log Calabi-Yau varieties over $s$ of any dimension when $\os{\circ}{s}={\rm Spec}({\mab F}_q)$;
we are more interested in the cardinalities of the rational points of log Calabi-Yau varieties than those of log Fano varieties. \par First let us recall the following suggestive observation, which seems well-known (\cite{bth}). \par Let $E$ be an elliptic curve over ${\mab F}_p$.
It is well-known that $E$ is nonordinary if and only if
\begin{align*} \#E({\mab F}_p)=p+1 \tag{1.6.2}\label{ali:pp1} \end{align*} if $p\geq 5$.
By the purity of the weight for $E/{\mab F}_p$: \begin{align*} \vert \#E({\mab F}_p)- (p+1)\vert \leq 2\sqrt{p}, \tag{1.6.3}\label{ali:ppwt1} \end{align*} this equality is equivalent to a congruence \begin{align*} \#E({\mab F}_p)\equiv 1~{\rm mod}~p \tag{1.6.4}\label{ali:ppwcg} \end{align*} since $\sqrt{p}>2$. \par In this article we generalize the congruence (\ref{ali:ppwcg}) for higher dimensional (log) varieties as follows. (We also generalize (\ref{ali:pp1}) for for any nonordinary elliptic curve over ${\mab F}_q$ when $p\geq 5$.) \par Let $X/s$ be a proper (not necessarily projective) simple normal crossing log scheme of pure dimension $d$. Recall that, in \cite{ny}, we have said that $X/s$ is a log Calabi-Yau scheme of pure dimension $d$ if $H^i(X,{\cal O}_X)=0$ $(0< i< d)$ and $\om_{X/s}\simeq {\cal O}_X$. Moreover, if $\os{\circ}{X}$ is geometrically connected, then we say that $X/s$ is a log Calabi-Yau variety of pure dimension $d$. (This is a generalization of a log K3 surface defined in \cite{nlk3}.) Note that $H^d(X,{\cal O}_X) =H^d(X,\om_{X/s})\simeq \kap$. The last isomorphism is obtained by log Serre duality of Tsuji (\cite[(2.21)]{tsp}). More generally, we consider a proper scheme $Y$ of pure dimension $d$ satisfying only the following four conditions:
\par (a) $H^0(Y,{\cal O}_Y)={\kap}$, \par (b) $H^i(Y,{\cal W}({\cal O}_Y))_{K_0}=0$ for $0<i<d-1$, \par (c) $H^{d-1}(Y,{\cal O}_Y)=0$ if $d\geq 2$, \par (d) $H^d(Y,{\cal O}_Y)\simeq {\kap}$.
\parno Let $\Phi_{Y/\kap}$
be the Artin-Mazur formal group of $Y/\kap$ in degree $d$, that is, $\Phi_{Y/\kap}$ is the following functor:
$$\Phi_{Y/\kap}(A):=\Phi^d_{Y/\kap}(A):={\rm Ker}(H^d_{\rm et} (Y\otimes_{\kap}A,{\mab G}_m)\lo H^d_{\rm et}(Y,{\mab G}_m)) \in ({\rm Ab})$$
for artinian local $\kap$-algebras $A$'s with residue fields $\kap$. Then $\Phi_{Y/\kap}$ is pro-represented by a commutative formal Lie group over $\kap$ (\cite{am}). Denote the height of $\Phi_{Y/\kap}$ by $h(Y/\kap)$. We prove the following$:$
\begin{theo}\label{theo:xfh} Let $Y/\kap$ be as above.
Assume that $\kap ={\mab F}_q$. Set $h:=h(Y/{\mab F}_q)$. Then the following hold$:$ \par $(1)$ Assume that $h=\infty$. Then \begin{equation*} \# Y({\mab F}_{q^k}) \equiv 1~{\rm mod}~ q^k \quad (k\in {\mab Z}_{\geq 1}). \tag{1.7.1}\label{eqn:kfdintd} \end{equation*} In particular, $Y({\mab F}_{q})\not= \emptyset$.
\par $(2)$ Assume that $2\leq h<\infty$. Let $\lceil~\rceil$
be the ceiling function$:$
$\lceil x \rceil:=\min\{n\in {\mab Z}~\vert~x\leq n\}$.
Then \begin{equation*} \# Y({\mab F}_{q^{k}}) \equiv1~{\rm mod}~ p^{\lceil ek(1-h^{-1})\rceil} \quad (k\in {\mab Z}_{\geq 1}).
\tag{1.7.2}\label{eqn:kfd2td} \end{equation*} In particular, $Y({\mab F}_{q})\not= \emptyset$ $($recall that $e=\log_pq)$.
\par $(3)$ Assume that $h=1$. Then \begin{equation*} \# Y({\mab F}_{q^{k}}) \not\equiv1~{\rm mod}~ p \quad (k\in {\mab Z}_{\geq 1}).
\tag{1.7.3}\label{eqn:kfdbntd} \end{equation*} $($In particular $Y({\mab F}_{q^k})$ can be empty.$)$ \end{theo} To give the statement (\ref{theo:xfh}) is a highly nontrivial work. However the proof of (\ref{theo:xfh}) is not difficult. (It does not matter whether the proof is not difficult.) As far as we know, (\ref{theo:xfh}) even in the 2-dimensional trivial logarithmic and smooth case, i.~e., the case of K3 surfaces over finite fields, is a new result. Even in the case $d=1$,
$Y$ need not be assumed to be an elliptic curve over ${\mab F}_q$. \par The heights of Artin-Mazur formal groups describe the different phenomena about the congruences of rational points for schemes satisfying four conditions (a), (b), (c) and (d). \par By using (\ref{theo:xfh}), we raise an important problem how the certain supersingular prime ideals are distributed for a smooth Calabi-Yau variety of dimension less than or equal to $2$ over a number field. (I think that there is no relation with Sato-Tate conjecture in non-CM cases.) \par To obtain (\ref{theo:xfh}), we use the theorems (A) and (B) explained after (\ref{coro:npf}) again and the determination of the slopes of the Dieudonn\'{e} module $D(\Phi_{Y/\kap})$ of $\Phi_{Y/\kap}$.
\par The contents of this article are as follows. \par In \S\ref{sec:pss} we recall \'{E}tess-Le Stum's Lefschetz trace formula for rigid cohomology, Berthelot-Bloch-Esnault's theorem and the congruence of the cardinality of rational points of a separated scheme of finite type over a finite field.
\par In \S\ref{sec:psso} we prove (\ref{theo:vc}) and (\ref{coro:npf}).
\par In \S\ref{sec:cy} we prove (\ref{theo:xfh}). We also raise the important problem about the distribution of supersingular primes already mentioned.
\par In \S\ref{sec:tkz} we give the formulas of two kinds of zeta functions of a few projective SNCL(=simple normal crossing log) schemes over the log point of a finite field. One kind of them gives us examples of the conclusions of the congruences in (\ref{coro:npf}) and (\ref{theo:xfh}). \par In \S\ref{rema:arkg} we give a remark on Van der Geer and Katsura's characterization of the height $h(Y/\kap)$ (\cite{vgk}).
\par
\parno {\bf Acknowledgment.} I have begun this work after listening to Y.~Nakamaura's very clear talk in which the main theorem in \cite{nt} has been explained in the conference ``Higher dimensional algebraic geometry'' of Y.~Kawamata in March 2018 at Tokyo University. The talk of Y.~Gongyo in January 2017 at Tokyo Denki university for the explanation of the main theorem in \cite{gir} has given a very good influence to this article. Without their talks, I have not begun this work. I would like to express sincere gratitude to them. I would also like to express sincere thanks to H.~Tanaka and S.~Ejiri for their kindness for informing me of the articles \cite{lr}, \cite{hl}, \cite{to} and giving me an important remark.
\par\noindent {\bf Notations.} (1) For an element $a$ of a commutative ring $A$ with unit element and for an $A$-modules $M$, $M/a$ denotes $M/aM$. \par (2) For a finite field ${\mab F}_q$, $s_{{\mab F}_q}$ denotes the log point whose underlying scheme is ${\rm Spec}({\mab F}_q)$.
\section{Preliminaries}\label{sec:pss} In this section we recall \'{E}tess-Le Stum's Lefschetz trace formula for rigid cohomology with compact support (\cite{el}) and Berthelot-Bloch-Esnault's calculation of the slope $< 1$-part of the rigid cohomology with compact support via Witt sheaves with compact support (\cite{bbe}).
\par Let $K_0({\mab F}_q)$ be the fraction field of the Witt ring ${\cal W}({\mab F}_q)$ of ${\mab F}_q$. Let $Y$ be a separated scheme of finite type over ${\mab F}_q$ of dimension $d$. Let $F_q\col Y\lo Y$ be the $q$-th power Frobenius endomorphism of $Y$. The following is \'{E}tess-Le Stum's Lefschetz trace formula proved in \cite[Th\'{e}or\`{e}me II]{el}: \begin{align*} \# Y({\mab F}_{q})=\sum_{i=0}^{2d}(-1)^i{\rm Tr} (F^*_q\vert H^i_{\rm rig,c}(Y/K_0({\mab F}_q))). \tag{2.0.1}\label{ali:trfh} \end{align*}
Let $\{\alpha_{ij}\}_j$ be an eigenvalue of $F^*_q$ on $H^i_{\rm rig,c}(Y/K_0({\mab F}_q))$. Then \begin{align*} \# Y({\mab F}_q)=\sum_{i=0}^{2d}(-1)^i(\sum_j{\alpha}_{ij}). \tag{2.0.2}\label{ali:trfeh} \end{align*} By \cite[(3.1.2)]{clpe} (see also \cite[(17.2)]{nh3}), \begin{align*} \max \{0, i-d\}\leq {\rm ord}_q(\alpha_{ij})\leq \min \{i, d\}. \tag{2.0.3}\label{ali:trfodh} \end{align*} Henceforth we consider the equalities (\ref{ali:trfh}) and (\ref{ali:trfeh}) as the equalities in the integer ring $\ol{{\cal W}({\mab F}_q)}$ of an algebraic closure of $\ol{K_0({\mab F}_q)}$.
Let $\kap$ be a perfect field of characteristic $p>0$. Let $Y/\kap$ be a separated scheme of finite type. Let $K_0$ be the fraction field of the Witt ring ${\cal W}$ of $\kap$. Let $H^i_{\rm rig,c}(Y/K_0)_{[0,1)}$ be the slope $<1$-part of the rigid cohomology $H^i_{\rm rig,c}(Y/K_0)$ with compact support with respect to the absolute Frobenius endomorphism of $Y$. Let $H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0}))$ be the cohomology of the Witt sheaf with compact support of $Y/K_0$ defined by Berthelot, Bloch and Esnault in \cite{bbe}: \begin{align*} H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0})) :=H^i(Y,{\cal W}({\cal I}_{K_0})), \end{align*} where ${\cal W}({\cal I}_{K_0}) :={\rm Ker}({\cal W}({\cal O}_Z)_{K_0}\lo {\cal W}({\cal O}_Z/{\cal I})_{K_0})$ and ${\cal I}$ is a coherent ideal sheaf of ${\cal O}_Z$ for an open immersion $Y\os{\sus}{\lo} Z$ into a proper scheme over $\kap$ such that $V({\cal I})=Z\setminus Y$. They have proved that $H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0}))$ is independent of the choice of the closed immersion. By the definition of $H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0}))$, we have the following exact sequence \begin{align*} &H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0}))\lo H^i(Z,{\cal W}({\cal O}_{Z,K_0}))\lo H^i(Z,{\cal W}({\cal O}_Z/{\cal I})_{K_0})\tag{2.0.4}\label{ali:oyk}\\ &\lo H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0}))\lo \cdots . \end{align*} By replacing $Z$ by the closure of $Y$ in $Z$, we see that \begin{align*} H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0}))=0 \tag{2.0.5}\label{ali:oydk} \end{align*} if $i>d$. Then they have proved that there exists the following contravariantly functorial isomorphism \begin{align*} H^i_{\rm rig,c}(Y/K_0)_{[0,1)}\os{\sim}{\lo} H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0})) \tag{2.0.6}\label{ali:tk0} \end{align*} (\cite[Theorem (1.1)]{bbe}). \par Now let us come back to the case $\kap={\mab F}_q$. Since \begin{align*} H^i_{\rm rig,c}(Y/K_0({\mab F}_q))=\sum_{j=0}^{d-1} H^i_{\rm rig,c}(Y/K_0({\mab F}_q))_{[j,j+1)}\oplus H^i_{\rm rig,c}(Y/K_0({\mab F}_q))_{[d]}, \end{align*} \begin{align*} \# Y({\mab F}_{q})&=\sum_{i=0}^{2d}(-1)^i \sum_{j=0}^{d-1}{\rm Tr}(F^*_q\vert H^i_{\rm rig,c}(Y/K_0({\mab F}_q))_{[j,j+1)}) +\sum_{i=d}^{2d}(-1)^i{\rm Tr}(F^*_q\vert H^i_{\rm rig,c}(Y/K_0({\mab F}_q))_{[d]}). \tag{2.0.7}\label{ali:trslfh} \end{align*} Hence we have the following congruence by (\ref{ali:oydk}) and (\ref{ali:tk0}): \begin{align*} \# Y({\mab F}_{q})\equiv \sum_{i=0}^{d}(-1)^i{\rm Tr} (F^*_q\vert H^i_{\rm c}(Y,{\cal W}({\cal O}_{Y,K_0})))~\mod~q \tag{2.0.8}\label{ali:yftrys} \end{align*} in $\ol{{\cal W}({\mab F}_q)}$.
\begin{rema} In \cite[(1.4)]{bbe}, the following zeta function \begin{align*} Z^{\cal W}(Y/{\mab F}_q,t):=\prod_{i=0}^{d} {\rm det}(1-tF^*_q\vert H^i_{\rm c}(Y,{\cal W}({\cal O}_Y))_{K_0})^{(-1)^{i+1}} \end{align*} which is equal to the zeta function $$Z^{<1}(Y/{\mab F}_q,t) :=\prod_{{\rm ord}_q(\al_{ij})<1}(1-\al_{ij}t)^{(-1)^{i+1}}$$ has been considered. In this article we do not need this zeta function. We do not need Ax's theorem in \cite{ax} (see \cite[Proposition 6.3]{bbe}) either.
\end{rema}
\section{Proofs of (\ref{theo:vc}) and (\ref{coro:npf})}\label{sec:psso} It is well-known that the analogue of Kodaira's vanishing theorem for projective smooth schemes over a field of characteristic $0$ (\cite{ko}) do not hold in characteristic $p>0$ in general (\cite{raykv}). However, in \cite{ny}, we have proved the Kodaira vanishing theorem in characteristic $p>0$ under the assumption of the finiteness of the Yobuko height.
To state this theorem precisely, we recall the definition of the vertical type for a relative log scheme. \par
\par For a commutative monoid $P$ with unit element, an ideal is, by definition, a subset $I$ of $P$ such that $PI\subset I$. An ideal ${\mathfrak p}$ of $P$ is called a prime ideal if $P\setminus {\mathfrak p}$ is a submonoid of $P$ (\cite[(5.1)]{klog2}). For a prime ideal ${\mathfrak p}$ of $P$, the height ${\rm ht}({\mathfrak p})$ is the maximal length of sequence's ${\mathfrak p}\supsetneq {\mathfrak p}_1\supsetneq \cdots \supsetneq {\mathfrak p}_r$ of prime ideals of $P$. Let $h\col Q\lo P$ be a morphism of monoids. A prime ideal ${\mathfrak p}$ of $P$ is said to be horizontal with respect to $h$ if $h(Q)\subset P\setminus {\mathfrak p}$ (\cite[(2.4)]{tsp}). \par Let $Y\lo Z$ be a morphism of fs(=fine and saturated) log schemes. Let $h\col Q\lo P$ be a local chart of $g$ such that $P$ and $Q$ are saturated. Set $$I:=\{a\in P~\vert~a\in {\mathfrak p}~~\text{for any horizontal prime ideal of $P$ of height 1 with respect to}~h\}.$$
Let ${\cal I}_{Y/Z}$ be the ideal sheaf of $M_Y$ generated by ${\rm Im}(I\lo M_Y)$. In \cite[(2.6)]{tsp} Tsuji has proved that ${\cal I}_{Y/Z}$ is independent of the choice of the local chart $h$. Let ${\cal I}_{Y/Z}{\cal O}_Y$ be the ideal sheaf of ${\cal O}_Y$ generated by the image of ${\cal I}_{Y/Z}$.
\begin{defi}\label{defi:vt} We say that $Y/Z$ is {\it of vertical type} if ${\cal I}_{Y/Z}{\cal O}_Y={\cal O}_Y$. \end{defi}
In \cite[(1.9)]{ny} we have proved the following theorem:
\begin{theo}[{\bf Log Kodaira Vanishing theorem}]\label{theo:stk} Let $Y\lo s$ be a projective log smooth morphism of Cartier type of fs log schemes.
Assume that $\os{\circ}{Y}$ is of pure dimension $d$. Let ${\cal L}$ be an ample invertible sheaf on $\os{\circ}{Y}$.
Assume that $h^F(\os{\circ}{Y})<\infty$. Then $H^i(Y,{\cal I}_{Y/s}\om_{Y/s}\otimes_{{\cal O}_Y}{\cal L})=0$ for $i>0$. In particular, if $Y/s$ is of vertical type, then $H^i(Y,\om_{Y/s}\otimes_{{\cal O}_Y}{\cal L})=0$ for $i>0$. \end{theo}
\par Now let us prove (\ref{theo:vc}) and (\ref{coro:npf}) quickly. Let the notations be as in (\ref{coro:npf}). Since $\om_{X/s}^{-1}$ is ample, $H^i(X,{\cal O}_X)=0$ for $i>0$ by (\ref{theo:stk}). Hence, by the following exact sequence \begin{align*} 0\lo {\cal W}_{n-1}({\cal O}_X)\os{V}{\lo} {\cal W}_{n}({\cal O}_X)\lo {\cal O}_X\lo 0, \tag{3.2.1}\label{ali:yexyk} \end{align*} $H^i(X,{\cal W}_n({\cal O}_X))=0$ for $i>0$ and $n>0$. Hence \begin{align*} H^i(X,{\cal W}({\cal O}_X))= (\vpl_nH^i(X,{\cal W}_n({\cal O}_X)))=0. \tag{3.2.2}\label{ali:ywyk} \end{align*} Thus we have proved (\ref{theo:vc}).
\par Next let us prove (\ref{coro:npf}). It suffices to prove (\ref{coro:npf}) for the case $k=1$ by considering the base change $X\otimes_{{\mab F}_q}{\mab F}_{q^k}$. Because $H^0(X,{\cal W}({\cal O}_X))={\cal W}({\mab F}_q)$ and $F^*_q={\rm id}$ on $H^0(X,{\cal W}({\cal O}_X))$, we obtain the following by (\ref{ali:yftrys}): \begin{align*} \# \os{\circ}{X}({\mab F}_{q})\equiv 1~\mod~q \tag{3.2.3}\label{ali:yftrays} \end{align*} in $\ol{{\cal W}({\mab F}_q)}$. This shows (\ref{coro:npf}).
\begin{rema}\label{rema:fsa}
(1) If $X$ is a Fano variety over ${\mab Q}$, then the reduction ${\cal X}\mod p$ of a flat model ${\cal X}$ over ${\mab Z}$ of $X$ for $p\gg 0$ is a Fano variety and $F$-split (\cite[Exercise 1.6. E5]{bm}). In particular, $h^F({\cal X}~{\rm mod}~p)<\infty$ for $p\gg 0$.
\par (2) As pointed out in \cite[p.~58]{bm}, a Fano variety $X$ is not necessarily $F$-split. \par The Kodaira vanishing theorem does not hold for certain Fano varieties (\cite{lr}, \cite{hl}, \cite{to}). By (\ref{theo:stk}) we see that the Yobuko heights
of them are infinity. \par (3) Let $X/s$ be an SNCL Fano scheme of pure dimension $d$. Then any irreducible component of $\os{\circ}{X}_i$ of $\os{\circ}{X}$ is Fano. Indeed, since $\om^{-1}_{X/s}$ is ample, $\om^{-1}_{X/s}\otimes_{{\cal O}_X}{\cal O}_{X_i}= \om^{-1}_{\os{\circ}{X}_i/\kap}(-\sum_{j}\log D_j)$ is also ample. Here $\{D_j\}_j$ is the set of the double varieties in $\os{\circ}{X}_i$. Hence $-K_{\os{\circ}{X}_i}-\sum_j D_j$ is ample. Consequently $-K_{\os{\circ}{X}_i}$ is ample. \end{rema}
\begin{rema}\label{rema:ex} Let $X/{\mab F}_q$ be a separated scheme of finite type.
Assume that $X$ is geometrically connected. By the argument in this section, it is obvious that, if $H^i(X,{\cal O}_X)=0$ $(\forall i>0)$, then the congruence (\ref{ali:xfq}) holds for $X/{\mab F}_q$. In particular, if $d=2$, if $X/{\mab F}_q$ is smooth, if $H^1(X,{\cal O}_X)=0$ and if $H^0(X,\Om^2_{X/s})=0$, then the congruence (\ref{ali:xfq}) holds for $X/{\mab F}_q$. Such an example can be given by a proper smooth Godeaux surface. \par Other examples are given by proper smooth unirational threefolds
because $H^i(X,{\cal O}_X)=0$ $(\forall i>0)$ by \cite[Introduction, (2.5)]{nyn}.
\par
Let $X/s$ be an SNCL(=simple normal crossing log) classical Enriques surface $X/s$ for $p\not=2$, i.e., $(\Om^{2}_{X/s})^{\otimes 2}$ is trivial and the corresponding \'{e}tale covering $X'$ to $\Om^{2}_{X/s}$ is an SNCL $K3$ surface (In \cite[(7.1)]{nlk3} we have proved that $H^i(X,{\cal O}_X)=0$ for $i>0$.). Hence the congruence (\ref{ali:xfq}) also holds for $X/s_{{\mab F}_q}$. See (\ref{coro:en}) below for the zeta function of this example. By the formulas for the zeta function ((\ref{eqn:kfend}), (\ref{eqn:kenad})), we can easily verify that $\# \os{\circ}{X}({\mab F}_q)$ indeed satisfies the congruence (\ref{ali:xfq}).
\par More generally, if $H^i(X,{\cal W}({\cal O}_X))_{K_0}=0$ $(i>0)$, then the congruence (\ref{ali:xfq}) holds for $X/{\mab F}_q$ by the proof of (\ref{theo:vc}). By the main theorem of \cite{beru}, one obtains such examples which are special fibers of regular proper flat schemes over discrete valuation rings of mixed characteristics whose generic fibers are geometrically connected and of Hodge type $\geq 1$ in positive degrees. See also \cite{e} for a generalization of the main theorem in \cite{beru}. \end{rema}
\begin{exem}\label{exem:nNp} Let $n$ and $N$ be positive integers. Set ${\cal X}_1:={\mab P}^N_{{\cal W}(\kap)}$.
Blow up ${\cal X}_1$ along an $\kap$-rational hyperplane of ${\mab P}^N_{\kap}$ and let ${\cal X}_2$ be the resulting scheme. Let $\os{\circ}{X}_1$ and $\os{\circ}{X}_n$ be the irreducible components of the special fiber ${\cal X}_2$. Blow up ${\cal X}_2$ again along $\os{\circ}{X}_1\cap \os{\circ}{X}_n$ and let ${\cal X}_3$ be the resulting scheme. Let $\os{\circ}{X}_1$, $\os{\circ}{X}_n$ and $\os{\circ}{X}_{n-1}$ be the irreducible components of the special fiber ${\cal X}_3$. Blow up ${\cal X}_3$ again along $\os{\circ}{X}_1\cap \os{\circ}{X}_{n-1}$.
Continuing this process $(n-1)$-times, we have a projective semistable family ${\cal X}_n$ over ${\rm Spec}({\cal W}(\kap))$. Let $\os{\circ}{X}_i$ $(1\leq i\leq n)$ be the the irreducible components of the special fiber ${\cal X}_n$.
Let $X$ be the log special fiber of ${\cal X}_n$. Then $X$ is a projective SNCL scheme over $s$. Let $\os{\circ}{X}{}^{(i)}$ $(i=0,1)$ be the disjoint union of $(i+1)$-fold intersections of the irreducible components of $\os{\circ}{X}$. Then $\os{\circ}{X}{}^{(0)}={\mab P}^N_{\kap}\coprod \underset{n-1~{\rm times}} {\underbrace{({\mab P}^{N-1}_{\kap}\times_{\kap}{\mab P}^1_{\kap})\coprod \cdots \coprod({\mab P}^{N-1}_{\kap}\times_{\kap}{\mab P}^1_{\kap})}}$ and $\os{\circ}{X}{}^{(1)}=\underset{n-1~{\rm times}} {\underbrace{{\mab P}^{N-1}_{\kap}\coprod \cdots \coprod{\mab P}^{N-1}_{\kap}}}$. Using the following spectral sequence \begin{align*} E_1^{ij}=H^j(X^{(i)},{\cal O}_{X^{(i)}})\Lo H^{i+j}(X,{\cal O}_X) \tag{3.5.1}\label{ali:exi} \end{align*} and noting that the dual graph of $\os{\circ}{X}$ is a segment, we see that $H^i(X,{\cal O}_X)=0$ $(i>0)$. If $s=s_{{\mab F}_q}$, then it is easy to check that \begin{align*} \os{\circ}{X}({\mab F}_q)&=\dfrac{q^{N+1}-1}{q-1} +(n-1)\dfrac{q^{N}-1}{q-1}\dfrac{q^2-1}{q-1} -(n-1)\dfrac{q^{N}-1}{q-1}\\ &=\dfrac{q^{N+1}-1}{q-1}+ (n-1)q\dfrac{q^N-1}{q-1}. \end{align*} In particular, $\#\os{\circ}{X}({\mab F}_q)\equiv 1~{\rm mod}~q$. \par The restriction of $\om_{X/s}$ to $\os{\circ}{X}_i$ is isomorphic to ${\cal O}_{\os{\circ}{X}_i}(-N)$ for $i=0$, $N$ and ${\cal O}_{\os{\circ}{X}_i}(-(N-1))$ for $0< i<N$.
Hence $\om_{X/s}^{-1}$ is ample if $N\geq 2$. Since each $\os{\circ}{X}_i$ is $F$-split (the $F$-splitting is given by the ``$p^{-1}$-th power'' of the canonical coordinate of $\os{\circ}{X}_i$ (see \cite[(1.1.5)]{bm}) and because we have the following exact sequence \begin{align*} 0\lo {\cal O}_X\lo \bigoplus_{i=1}^{N}{\cal O}_{X_i}\lo \bigoplus_{i=1}^{N-1}{\cal O}_{X_{i}\cap X_{i+1}}, \end{align*} $\os{\circ}{X}$ is $F$-split.
\end{exem}
\section{Proof of (\ref{theo:xfh})}\label{sec:cy} Let the notations be as in (\ref{theo:xfh}).
In this section we prove (\ref{theo:xfh}). We may assume that $k=1$. \par Since $H^{d-1}(Y,{\cal O}_Y)=0$, we see that \begin{align*} H^{d-1}(Y,{\cal W}({\cal O}_Y))= \vpl_nH^i(Y,{\cal W}_n({\cal O}_Y))=0 \tag{4.0.1}\label{ali:yclk} \end{align*} by the same proof as that of (\ref{theo:vc}).
Set $\ol{Y}:=Y\otimes_{{\mab F}_q}\ol{\mab F}_{q}$ and $e:=\log_pq$. By \cite[II (4.3)]{am} the Dieudonn\'{e} module $M:=D(\Phi_{Y/\kap})$ of $\Phi_{Y/\kap}$ is equal to $H^d(Y,{\cal W}({\cal O}_Y))$. Let $h$ be the height of $\Phi_{Y/\kap}$. Hence $\Phi_{Y/\kap}$ is a commutative formal Lie group over $\kap$ of dimension 1 and the Dieudonn\'{e} module $M$ is a free ${\cal W}$-module of rank $h$ if $h< \infty$ (\cite[V (28.3.10)]{ha}).
Let $F\col M\lo M$ be the operator ``$F$'' on the Dieudonn\'{e} module $M$. By abuse of notation, we denote the induced morphism $M_{K_0}\lo M_{K_0}$ by $F$. By (\ref{ali:yftrys}) we have the following congruence \begin{align*} \# Y({\mab F}_{q})\equiv 1+{\rm Tr}(F^e\vert M_{K_0}) ~\mod~q \tag{4.0.2}\label{ali:yftrcys} \end{align*} in $\ol{{\cal W}({\mab F}_q)}$. Set $m:={\rm ord}_p({\rm Tr}(F^e\vert M_{K_0}))$. If $m\leq e={\rm ord}_p(q)$, then we obtain the following congruence by (\ref{ali:yftrcys}): \begin{align*} \# Y({\mab F}_{q})\equiv 1 ~\mod~p^{\lceil m \rceil} \tag{4.0.3}\label{ali:yftrcemys} \end{align*} in ${\mab Z}$.
\par First we give the proof of (\ref{theo:xfh}) (1).
\parno {\bf Proof of (\ref{theo:xfh}) (1).}
\par Assume that $h=\infty$. Then $D(\Phi_{\ol{Y}/\ol{\mab F}_q})$ is ${\cal W}(\ol{\mab F}_q)$-torsion. By \cite[II (4.3)]{am}, $H^d(\ol{Y},{\cal W}({\cal O}_{\ol{Y}}))_{K_0}= D(\Phi_{\ol{Y}/\ol{\mab F}_q})_{K_0}=0$. By \cite[I (1.9.2)]{idw}, ${\cal W}({\cal O}_{\ol{Y}}) ={\cal W}({\cal O}_Y)\otimes_{{\cal W}({\mab F}_q)}{\cal W}(\ol{\mab F}_q)$. Since $\os{\circ}{Y}$ is separated, we obtain the following equality $H^d(Y,{\cal W}({\cal O}_{\ol{Y}}))= H^d(Y,{\cal W}({\cal O}_Y))\otimes_{{\cal W}({\mab F}_q)}{\cal W}(\ol{\mab F}_q)$ by using \v{C}ech cohomologies. Hence $$H^d(Y,{\cal W}({\cal O}_Y))_{K_0({\mab F}_q)}=0.$$ (To obtain this vanishing, one may use the fact that the Dieudonn\'{e} module commutes with base change (cf.~the description of $D(\Phi_{Y/{\mab F}_q})$ in \cite[p.~309]{mub}.))
By (\ref{ali:yftrys}) this means the congruence (\ref{eqn:kfdintd}). \par Now assume that $h<\infty$.
Next we give the proof (\ref{theo:xfh}) (2).
\parno {\bf Proof of (\ref{theo:xfh}) (2).}
\par \par Let us recall the following well-known observation (\cite[Exercise 6.13]{li}):
\begin{prop}\label{prop:eobs} Let $G$ be
a commutative formal Lie group of dimension $1$ over a perfect field $\kap$ of characteristic $p>0$. Assume that the height $h$ of $G$ is finite. Then the slopes of the Dieudonn\'{e} module of $D(G)$ is $1-h^{-1}$. \end{prop} \begin{proof} Let $D(\kap)$ be the Cartier-Dieudonn\'{e} algebra over $\kap$. We may assume that $\kap$ is algebraically closed. In this case, the height is the only invariant which determines the isomorphism class of a 1-dimensional commutative formal group law over $\kap$ (\cite[(19.4.1)]{ha}). Hence $D(G)\simeq D(\kap)/D(\kap)(F-V^{h-1})$ (\cite[p.~266]{vgk}). Express $F(1,V, \cdots, V^{h-1})=(1,V, \cdots, V^{h-1})A$, where $A\in M_h({\cal W})$ (as if $F$ were ${\cal W}$-linear). Then ${\rm det}(tI-A)=t^h-p^{h-1}$. Hence the slopes of $D(G)$ is ${\rm ord}_p((p^{h-1})^{h^{-1}})=1-h^{-1}$. \end{proof}
\par By (\ref{prop:eobs}) and (\ref{ali:yftrcemys}), we obtain the following congruence \begin{align*} \# Y({\mab F}_{q})\equiv 1 ~\mod~p^{\lceil e(1-h^{-1})\rceil} \tag{4.1.1}\label{ali:yftzys} \end{align*} in ${\mab Z}$.
Lastly we give the proof of (\ref{theo:xfh}) (3) in the following.
\parno {\bf Proof of (\ref{theo:xfh}) (3).}
\par Let $\kap$ be a perfect field of characteristic $p>0$. Let $Y$ be a proper scheme over $\kap$ of pure dimension $d\geq 1$. $($We do not assume that $Y$ is smooth over $\kap$.$)$ Assume that $H^d(Y,{\cal O}_Y)\simeq \kap$ and that $H^{d-1}(Y,{\cal O}_Y)=0$ if $d\geq 2$. Then the following morphism \begin{align*} H^d(Y,{\cal W}({\cal O}_Y))/p\lo H^d(Y,{\cal O}_Y) \end{align*} is an isomorphism. Indeed, this is surjective and \begin{align*} {\rm dim}_{\kap}(H^d(Y,{\cal W}({\cal O}_Y))/p)= {\rm dim}_{\kap}(M/p)= 1={\rm dim}_{\kap}H^d(Y,{\cal O}_Y). \end{align*} Since $h=1$, $F$ on $H^d(Y,{\cal W}({\cal O}_Y))\otimes_{{\cal W}(\kap)}{\cal W}(\ol{\kap})$ is an isomorphism, Hence $F\col H^d(Y,{\cal O}_Y)\lo H^d(Y,{\cal O}_Y)$ is an isomorphism. Hence $\# Y({\mab F}_{q})\equiv 1+\al~{\rm mod}~q$ for a unit $\al \in {\cal W}({\mab F}_q)^*$. Now (\ref{theo:xfh}) (3) follows. \parno
\begin{rema}\label{rema:fe} (1) If $H^i(Y,{\cal O}_Y)=0$ for $0<i<d-1$ (this is stronger than (c) in the Introduction), then (\ref{theo:xfh}) (3) also follows from Fulton's trace formula (\cite{ftf}): \begin{align*} \# Y({\mab F}_{q})~{\rm mod}~p\equiv \sum_{i=0}^{d}(-1)^i{\rm Tr}(F^*_q\vert H^i(Y,{\cal O}_Y))\in {\mab F}_q \end{align*} (cf.~\cite[Proposition 5.6]{bth}). \par
(2) Let $X/s$ be a log Calabi-Yau scheme. In \cite[(10.1)]{ny} we have proved a fundamental equality $h^F(X/\kap)=h(X/\kap)$. Hence $X$ is quasi-$F$-split (resp.~$F$-split) if and only if $h(X/\kap)<\infty$ (resp.~$h(X/\kap)=1$).
\end{rema}
Though the following corollary immediately follows from \cite[(1.6)]{bbe}, we state it for the convenience of our remembrance.
\begin{coro}\label{coro:ht} Let $Y$ be as in {\rm (\ref{theo:xfh})}. Let $f\col Z_1\lo Z_2$ be a morphism of proper schemes over ${\mab F}_q$. Assume that $Z_1$ or $Z_2$ is isomorphic to $Y$ over ${\mab F}_q$. Assume that $\Phi_{Z_i/{\mab F}_q}$ $(i=1,2)$ is representable. If the pull-back $f^*\col H^i(Z_2,{\cal O}_{Z_2})\lo H^i(Z_1,{\cal O}_{Z_1})$ is an isomorphism, then the natural morphism $\Phi_{Z_2/{\mab F}_q}\lo \Phi_{Z_1/{\mab F}_q}$ is an isomorphism. In particular, $h(Z_1/{\mab F}_q)=h(Z_2/{\mab F}_q)$ and {\rm (\ref{theo:xfh})} for $\# Z_i({\mab F}_q)$ holds. \end{coro} \begin{proof} By the assumption, we have an isomorphism $f^*\col H^i(Z_2,{\cal W}({\cal O}_{Z_2}))\os{\sim}{\lo} H^i(Z_1,{\cal W}({\cal O}_{Z_1}))$. Hence the natural morphism $D(\Phi_{Z_2/{\mab F}_q})\lo D(\Phi_{Z_1/{\mab F}_q})$ is an isomorphism. By Cartier theory, the natural morphism $\Phi_{Z_2/{\mab F}_q}\lo \Phi_{Z_1/{\mab F}_q}$ is an isomorphism. This implies that $h(Z_2/{\mab F}_q)=h(Z_2/{\mab F}_q)$ and {\rm (\ref{theo:xfh})} for $\# Z_i({\mab F}_q)$ holds. \end{proof}
The following corollary immediately follows from the proof of \cite[(6.12)]{bbe}.
\begin{coro}\label{coro:eo} Let $Y$ be as in {\rm (\ref{theo:xfh})}. Let $G$ be a finite group acting on $Y/{\mab F}_q$ such that each orbit of $G$ is contained in an affine open subscheme of $Y$. If $\# G$ is prime to $p$ and the induced action on $H^d(Y,{\cal O}_Y)$ is trivial, then $h((Y/G)/{\mab F}_q)=h(Y/{\mab F}_q)$ and {\rm (\ref{theo:xfh})} for $\# (Y/G)({\mab F}_q)$ holds. \end{coro}
\begin{exem}\label{exem:gnt} We give examples of trivial logarithmic cases. \par (1) Let $E/{\mab F}_p$ be an elliptic curve. It is very well-known that $E/{\mab F}_p$ is supersingular if and only if $\# E({\mab F}_p)=p+1$ if $p\geq 5$ (\cite[V Exercises 5.9]{sil}). As observed in \cite[Example 5.11]{bth}, this also follows from the purity of the weight for an elliptic curve over ${\mab F}_p$: $\vert \# E({\mab F}_p)-(p+1)\vert \leq 2\sqrt{p}$ and Fulton's trace formula.
In fact, we can say more in (\ref{prop:cg}) below. \par (2) Let $d\geq 3$ be a positive integer such that $d\not\equiv 0~{\rm mod}~p$.
Consider a smooth Calabi-Yau variety ${\cal X}/{\cal W}({\mab F}_q)$ in ${\mab P}^{d-1}_{{\cal W}({\mab F}_q)}$ defined by the following equation: \begin{align*} a_0T^d_0+\cdots +a_{d-1}T^d_{d-1}=0 \quad (a_0, \ldots,a_{d-1}\in {\cal W}({\mab F}_q)^*). \end{align*} Set $a:=a_0\cdots a_{d-1}\in {\cal W}({\mab F}_q)$.
Let $X/{\mab F}_q$ be the reduction mod~$p$ of ${\cal X}/{\cal W}({\mab F}_q)$.
By \cite[Theorem 1]{stfg} (see also [loc.~cit., Example 4.13]), the logarithm $l(t)$ of $\Phi_{{\cal X}/{\cal W}({\mab F}_q)}$ is given by the following formula: \begin{align*} l(t)=\sum_{m=0}^{\infty}a^m\dfrac{(md)!}{(m!)^d}\dfrac{t^{md+1}}{md+1}. \end{align*} \par (a) If $p\equiv 1~{\rm mod}~d$, then $$pl(t)=pt+p\sum_{i=2}^{p-1}c_it^i +({\rm unit})t^p+({\rm higher~terms~than~}t^p)$$ for some $c_i\in {\cal W}({\mab F}_q)$ in ${\cal W}({\mab F}_q)[[t]]$. Hence $l^{-1}(pl(t))~{\rm mod}~p\equiv t^p+\cdots$ and the height of $\Phi_{X/{\mab F}_q}$ is equal to $1$. \par (b) If $p\not\equiv 1~{\rm mod}~d$, then $pl(t)\in p{\cal W}({\mab F}_q)[[t]]$. Hence the height of $\Phi_{X/{\mab F}_q}$ is equal to $\infty$. \par (a) and (b) above are much easier and much more direct proofs of \cite[Theorem 5.1]{vgkht}. \par (3) Especially consider the case $N=3$ in (2) and let $X/{\mab F}_p$ be a closed subscheme of ${\mab P}^N_{{\mab F}_p}$ defined by the following equation: \begin{align*} T^4_0+T^4_1+T^4_2+T^4_3=0. \end{align*}
\par (a) If $p=3$, then $\# X({\mab F}_3)\equiv 1~{\rm mod}~3$ by (\ref{theo:xfh}) (1). In fact, it is easy to see that $\# X({\mab F}_3)=4=1+3^2-3\times 2$. (This $X$ and $X$ in (c) are Tate's examples in \cite{ta} of a supersingular $K3$-surface (in the sense of T.~Shioda) over ${\mab F}_3$ and ${\mab F}_7$), respectively.) \par (b) If $p=5$, then $\# X({\mab F}_5)\not\equiv 1~{\rm mod}~5$ by (\ref{theo:xfh}) (3). In fact, it is easy to see that $\# X({\mab F}_5)=0$. More generally, for a power $q$ of a prime number $p$, let $X_q$ be a closed subscheme of ${\mab P}^{q-1}_{{\mab F}_q}$ defined by the following equation: \begin{align*} a_0T^{q-1}_0+\cdots +a_{q-2}T^{q-1}_{q-2}=0 \quad (a_0, \ldots,a_{q-2} \in {\mab F}_q^*,~(a_0, \ldots,a_{q-2})\not=(0,\ldots,0)), \end{align*} where $a_0, \ldots,a_{q-2}$ satisfying the following condition: for any nonempty set $I$ of $\{0, \ldots, q-2\}$, $\sum_{j\in I}a_j\not=0$ in ${\mab F}_q$. Then $\# X_q({\mab F}_q)=0$. \par (c) If $p=7$, then $\# X({\mab F}_7)\equiv 1~{\rm mod}~7$ by (\ref{theo:xfh}) (1). In fact, one can check that $\# X({\mab F}_7)=64=1+7^2+7\times 2$. In general, if $\Phi(X/{\mab F}_q)$ is supersingular, then $\# X({\mab F}_q)=1+q^2+q\al$ for some $\vert \al \vert \leq 22$ by the purity of the weight and by $b_2(\ol{X})= 22$. Here $b_2(\ol{X}/\ol{\mab F}_q)$ is the second Betti number of $\ol{X}/\ol{\mab F}_q$. (We do not know an example of the big $\vert a \vert$.) \par (4) See \cite[(4.8)]{yy} for explicit examples of $X/{\mab F}_q$'s such that $h(\Phi_{X/{\mab F}_q})=2$. See also \cite[\S6]{vgkht}. \end{exem}
\begin{exem} (1) Let $n$ be a positive integer. Let $X$ be an $n$-gon over ${\mab F}_q$. Then, by \cite[(6.7) (1)]{nlfc}, $X$ is $F$-split. In particular, $h^F(X)=h(X/{\mab F}_q)=1$. Then, by (\ref{theo:xfh}) (3), $\# X({\mab F}_q)\not\equiv 1~{\rm mod}~p$. In fact, it is easy to see that $\# X({\mab F}_q)=n(q+1)-n=q$. Compare this example with the example in (\ref{exem:nNp}). \par (2) Let $\kap$ be a perfect field of characteristic $p>0$. Let $X$ be an SNCL(=simple normal crossing log) $K3$-surface over $\kap$, that is, an SNCL Calabi-Yau variety of dimension $2$ (\cite{nlk3}). In \cite[(6.7) (2)]{nlfc} we have proved the following:
\par (a) If $\os{\circ}{X}$ is of Type II {\rm (\cite[\S3]{nlk3})}, then $X$ is $F$-split if and only if the isomorphic double elliptic curve is ordinary. In this case, $h(X/\kap)=1$. If this is not the case, $h(X/\kap)=2$. \par (b) If $\os{\circ}{X}$ is of Type III {\rm ([loc.~cit.])}, then $X$ is $F$-split and $h(X/\kap)=1$. \par See (\ref{coro:f}) below for the zeta function of these examples. By the formulas for the zeta function ((\ref{eqn:kfmdad}) and (\ref{eqn:kfdqad})), we can easily verify that $\# \os{\circ}{X}({\mab F}_q)$ indeed satisfies the congruences (\ref{eqn:kfdbntd}) and (\ref{eqn:kfd2td}). \end{exem}
\begin{rema}
(1) Let $X/{\mab F}_q$ and $X^*/{\mab F}_q$ be a strong mirror Calabi-Yau pair in the sense of Wan (\cite{wama}), whose strict definition has not been given. Then he conjectures that $\# X({\mab F}_q)\equiv \# X^*({\mab F}_q)~{\rm mod}~q$ (\cite[(1.3)]{wama}). Hence the following question seems natural: does the equality $h(\Phi_{X/{\mab F}_q})=h(\Phi_{X^*/{\mab F}_q})$ hold? If his conjecture is true, only one of $h(\Phi_{X/{\mab F}_q})$ and $h(\Phi_{X^*/{\mab F}_q})$ cannot be 1 by (\ref{theo:xfh}). This is compatible with Wan's generically ordinary conjecture in [loc.~cit., (8.3)]. \par (2) If $X$ satisfies the conditions (a), (c) and (d) in the Introduction and if $X$ is a special fiber of a regular proper flat scheme over a discrete valuation ring of mixed characteristics whose generic fibers are geometrically connected and of Hodge type $\geq 1$ in degrees in $[1,d-2]$, then we see that $X$ satisfies the condition (b) by \cite{beru}.
\end{rema}
We conclude this section by generalizing (\ref{exem:gnt}) (1) by using (\ref{theo:xfh}) and raise an important question:
\begin{prop}\label{prop:cg} Let $C$ be a proper smooth curve over ${\mab F}_{q}$ such that $H^0(C,{\cal O}_C)\simeq {\mab F}_{q}\simeq H^1(C,{\cal O}_C)$. Recall that $e=\log_pq$. Then the following hold$:$ \par $(1)$ Assume that $e$ is odd and $p\geq 5$. Then $h_{C/{\mab F}_q}=2$ if and only if $\#C({\mab F}_q)=1+q$. \par $(2)$ Assume that $e$ is odd and $p= 3$ or $2$. Then $h_{C/{\mab F}_q}=2$ if and only if $\#C({\mab F}_q)=1+q$ or $1+ q\pm p^{\frac{e+1}{2}}$. \par $(3)$ Assume that $e$ is even. Then $h_{C/{\mab F}_q}=2$ if and only if $\#C({\mab F}_q)=1+q+\al p^{\frac{e}{2}}$, where $\al \in {\mab N}$ and $\vert \al \vert \leq 2$. \end{prop} \begin{proof} By the purity of weight, we have the following inequality: \begin{align*} \vert \#C({\mab F}_{q})- (1+q)\vert \leq 2\sqrt q. \tag{4.8.1}\label{ali:caqq} \end{align*} \par (1): Assume that $h_{C/{\mab F}_q}=2$. By (\ref{eqn:kfd2td}), $\# C({\mab F}_q)\equiv 1~{\rm mod}~ p^{\lceil \frac{e}{2}\rceil} =1~{\rm mod}~ p^{\frac{e+1}{2}}$. Hence $\# C({\mab F}_{q})=1+m p^{\frac{e+1}{2}}$ for $m \in {\mab N}$. By (\ref{ali:caqq}) we have the following inequality: \begin{align*} p^{\frac{1}{2}}\vert m - p^{\frac{e-1}{2}}\vert \leq 2. \tag{4.8.2}\label{ali:cqq} \end{align*} Since $p\geq 5$, $m = p^{\frac{e-1}{2}}$. Hence $\# C({\mab F}_q)= 1+q$. \par Conversely, assume that $\# C({\mab F}_q)=1+q$. Then $C$ can be an elliptic curve over ${\mab F}_q$. Hence $h_{C/{\mab F}_q}=1$ or $2$ (\cite[IV (7.5)]{sil}). By (\ref{eqn:kfdbntd}) and (\ref{eqn:kfd2td}), $h_{C/{\mab F}_q}=2$. \par (2): Assume that $h_{C/{\mab F}_q}=2$. Then, by (\ref{ali:cqq}), $m=p^{\frac{e-1}{2}}$ or $m=\pm 1+p^{\frac{e-1}{2}}$. Hence $\# C({\mab F}_q)= 1+q$ or $\# C({\mab F}_q)=1+(\pm 1+p^{\frac{e-1}{2}})p^{\frac{e+1}{2}}=1+q\pm p^{\frac{e+1}{2}}$. \par The proof of the converse implication is the same as that in (1). \par (3): Assume that $h_{C/{\mab F}_q}=2$. By (\ref{eqn:kfd2td}), $\# C({\mab F}_q)\equiv 1~{\rm mod}~ p^{\frac{e}{2}}$. Hence $\# C({\mab F}_{q})=1+m p^{\frac{e}{2}}$ for $\al \in {\mab N}$. By (\ref{ali:caqq}), \begin{align*} \vert m - p^{\frac{e}{2}}\vert \leq 2. \tag{4.8.3}\label{ali:cbqq} \end{align*} Hence, by (\ref{ali:caqq}), $m=\al +p^{\frac{e}{2}}$ with $\vert \al \vert \leq 2$. Hence $\# C({\mab F}_q)=1+(\al+p^{\frac{e}{2}})p^{\frac{e}{2}} =1+q+\al p^{\frac{e}{2}}$. \par The proof of the converse implication is the same as that in (1). \end{proof}
\begin{rema} Assume that $e$ is even. By Honda-Tate's theorem for elliptic curves over finite fields (\ref{theo:pap}) below, the case $\vert \al \vert =1$ occurs only when $p\not\equiv 1~{\rm mod}~3$; the case $\al =0$ occurs only when $p\not\equiv 1~{\rm mod}~4$. \end{rema}
\begin{theo}[{\bf Honda-Tate's theorem for elliptic curves (\cite[(4.1)]{wat}, \cite[(4.8)]{pap})}]\label{theo:pap} For an elliptic curve $E/{\mab F}_q$, set $t_E:=1+q-\# E({\mab F}_q)$. Consider the following well-defined injective map$\!:$ \begin{align*} \{{\rm isogeny}~{\rm classes}~{\rm of}~ {\rm elliptic}~{\rm curves}~E/{\mab F}_q\}\owns E \lo t_E\in \{t\in {\mab Z}~\vert~\vert t \vert \leq 2\sqrt{q}\}. \end{align*} $($This map is indeed injective by Tate's theorem $(${\rm \cite[Main Theorem]{tae}}.$)$ The image of HT consists of the following values$:$ \par $(1)$ $t$ is coprime to $p$. \par $(2)$ $e$ is even and $t=\pm 2 \sqrt{q}$. \par $(3)$ $e$ is even and $p\not\equiv 1~{\rm mod}~3$ and $t=\pm \sqrt{q}$. \par $(4)$ $e$ is odd and $p=2$ or $3$ and $t=\pm p^{\frac{e+1}{2}}$. \par $(5)$ $e$ is odd, or $e$ is even and $p\not\equiv 1~{\rm mod}~4$ and $t=0$. \par The case $(1)$ arises from ordinary elliptic curves over ${\mab F}_q$. The case $(2)$ arises from supersingular elliptic curves over ${\mab F}_q$ having all their endomorphisms defined over ${\mab F}_q\!;$ the rest cases arises from supersingular elliptic curves over ${\mab F}_q$ not having all their endomorphisms defined over ${\mab F}_q$. \end{theo}
\begin{prob}\label{prob:dist} Let $K$ be an algebraic number field and ${\cal O}_K$ the integer ring of $K$. Let $x$ be a positive real number. \par (1) Consider the following set \begin{align*} {\cal P}(x) :=\{{\mathfrak p}\in {\rm Spec}({\cal O}_K)~\vert~ N_{K/{\mab Q}}({\mathfrak p})\leq x ~ {\rm and}~ \log_p(\# ({\cal O}_K/{\mathfrak p}))~{\rm is~even}\}, \end{align*} where $p={\rm ch}({\cal O}_K/{\mathfrak p})$. \par Assume that $p\geq 5$. Let $E/K$ be an elliptic curve. Let $\al$ be an integer such that $\vert \al \vert \leq 2$. Consider the following set \begin{align*} {\cal P}'(x;E/K,\al) :=\{{\mathfrak p}\in {\cal P}(x)~\vert~& E~{\rm has~a~good~reduction~}{\cal E}_0~{\rm at}~{\mathfrak p}\\ &{\rm and}~ \#{\cal E}_0({\mab F}_q)=1+q+\al \sqrt{q}\}. \end{align*} Set \begin{equation*} {\cal P}(x;E/K,\al) := \begin{cases} {\cal P}'(x;E/K,\al)& (\vert \al \vert =2), \\ \{{\mathfrak p}\in {\cal P}(x;E/K,\al)~\vert~p\not\equiv 1~{\rm mod}~3\} & (\vert \al \vert =1),\\ \{{\mathfrak p}\in {\cal P}(x;E/K,\al)~\vert~p\not\equiv 1~{\rm mod}~4\} & (\al =0). \end{cases} \label{eqn:kfptd} \end{equation*} Then, what is the function \begin{align*} x\lom \dfrac{\# {\cal P}(x;E/K,\al)}{\#{\cal P}(x)} \end{align*} when $x\rightarrow \infty$? (I do not know whether
$\lim_{x\rightarrow \infty}{\cal P}(x;E/K,\al)=\infty$ for each $\al$ such that $\vert \al \vert \leq 2$ for any non-CM elliptic curve over $K$ (see \cite[p.~185 Exercise 2.33 (a), (b)]{sila} for a CM elliptic curve over ${\mab Q}(\sqrt{-1})$: in this example, $\lim_{x\rightarrow \infty}{\cal P}(x;E/K,2)=\infty$, but ${\cal P}(x;E/K,\al)=0$ for $\al\not=2$ and for any $x$). If $[K:{\mab Q}]$ is odd or if $K$ has a real embedding, then $\lim_{x\rightarrow \infty} \sum_{\vert \al \vert \leq 2}{\cal P}(x;E/K,\al)=\infty$ by Elkies' theorems (\cite[Theorem 2]{elq}, \cite[Theorem]{elr}).) \par When $p=2$ or $3$, we can give a similar problem to the problem above by using (\ref{prop:cg}) (2). \par (2) Consider the following set \begin{align*} {\cal P}(x) :=\{{\mathfrak p}\in {\rm Spec}({\cal O}_K)~\vert~ N_{K/{\mab Q}}({\mathfrak p})\leq x\}. \end{align*} \par Let $S/K$ be a K3 surface. Let $\al$ be an integer such that $\vert \al \vert \leq 22$. Consider the following set \begin{align*} {\cal P}'(x;S/K,\al) :=\{{\mathfrak p}\in {\cal P}(x)~\vert~& S~{\rm has~a~good~reduction~}{\cal S}_0~{\rm at}~{\mathfrak p}\\ &{\rm and}~\#{\cal S}_0({\mab F}_q)=1+q^2+\al q\}. \end{align*} Then, what is the function \begin{align*} x\lom \dfrac{\# {\cal P}'(x;S/K,\al)}{\#{\cal P}(x)} \end{align*} when $x\rightarrow \infty$? (I do not know even whether $\lim_{x\rightarrow \infty} \sum_{\vert \al \vert \leq 22}{\cal P}'(x;S/K,\al)=\infty$.) \end{prob}
\section{Two kinds of zeta functions of degenerate SNCL schemes over the log point of ${\mab F}_q$}\label{sec:tkz} In this section we give a few examples of two kinds of local zeta functions of a separated scheme $Y$ of finite type over ${\mab F}_q$: one of them is defined by rational points of $Y$; the other is defined by the Kummer \'{e}tale cohomology of $Y$ when $Y$ is the underlying scheme of a proper log smooth scheme over the log point $s_{{\mab F}_q}$. \par First we introduce a Grothendieck group which is convenient in this section. \par Let $F$ be a field. Consider a Grothendieck group ${\cal K}(F)$ with the following generators and relations: the generators of ${\cal K}(F)$ are $[(V,\bet)]$'s, where $V$ is a finite-dimensional vector space over $F$ and $\bet$ is an endomorphism of $V$ over $F$. The relations are as follows: $[(V,\bet)] = [(U,\al)] +[(W,\gamma)]$ for a commutative diagram with exact rows \begin{equation*} \begin{CD} 0 @>>> U @>>> V @>>> W @>>> 0 \\ @. @V{\al}VV @V{\bet}VV @V{\gam}VV \\ 0 @>>> U @>>> V @>>> W @>>> 0. \end{CD} \end{equation*} \par Let $t$ be a variable. Note that ${\rm det}(1-t\bet \vert V)={\rm det}(1-t\al \vert U) {\rm det}(1-t\gamma \vert W)$. If $V=\{0\}$, we set ${\rm det}(1-t0 \vert V)=1$ $(1\in F)$. We have a natural map \begin{equation*} {\rm det}(1- t\bul \vert \bul) \col {\cal K}(F) \lo F(t)^*\cap (1+tF[[t]])^* \tag{5.0.1}\label{eqn:grdet} \end{equation*} of abelian groups. Here the intersection in the target of (\ref{eqn:grdet}) is considered in the ring of Laurent power series in one variable with coefficients in $F$. Set $Z((V,\al),t)={\rm det}(1- t\al \vert V)$. \par Let $Y$ be a separated scheme of finite type over ${\mab F}_q$. Set \begin{align*} [(E_p(Y/{\mab F}_q),F^*_q)]:= \sum_{i=0}^{\infty}(-1)^i[(H^i_{\rm rig,c}(Y/K_0({\mab F}_q)),F^*_q)] \in {\cal K}(K_0({\mab F}_q)), \tag{5.0.2}\label{ali:zet} \end{align*} where $E_p$ means the Euler-characteristic. Let \begin{align*} Z(Y/{\mab F}_q,t):={\rm exp} \left(\sum_{n=0}^{\inf}\dfrac{\# Y({\mab F}_{q^n})}{n}t^n\right) \tag{5.0.3}\label{ali:zcget} \end{align*} be the zeta function of $Y/{\mab F}_q$. We can reformulate (\ref{ali:trfh}) as the following formula: \begin{align*} Z(Y/{\mab F}_q,t)=Z([(E_p(Y/{\mab F}_q),F^*_q)])^{-1}. \tag{5.0.4}\label{ali:trfrgh} \end{align*}
\begin{prop}\label{prop:ykf} Let $Y$ be a proper SNC $($not necessarily log$)$ scheme over ${\mab F}_q$. Let $Y^{(i)}$ $(i\in {\mab Z}_{\geq 0})$ be the disjoint union of the $(i+1)$-fold intersections of the irreducible components of $Y$. Then \begin{align*} Z(Y/{\mab F}_q,t)=\prod_{i,j\geq 0} {\rm det}(1-tF^*_q\vert H^j_{\rm rig}(Y^{(i)}/K_0({\mab F}_q)))^{(-1)^{i+j+1}}. \tag{5.1.1}\label{ali:zytp} \end{align*} \end{prop} \begin{proof} \par Let $Y_{\bul}$ be the \v{C}ech diagram of an affine open covering of $Y$ by
finitely many affine open subschemes $U_j$'s of $Y$.
Set $U^{(i)}_j:=Y^{(i)}_j\cap U_j$, $Y^{(i)}_0:=\coprod_jU^{(i)}_j$ and $Y^{(i)}_n:={\rm cosk}_0^{Y^{(i)}}(Y^{(i)}_0)_n$ $(n\in {\mab N})$. Let $Y_{\bul}\os{\sus}{\lo} {\cal P}_{\bul}$ be a closed immersion into a formally smooth formal scheme over ${\rm Spf}({\cal W}({\mab F}_q))$. Then we have a closed immersion $Y^{(i)}_0 \os{\sus}{\lo} \coprod^{(i)} {\cal P}_0$, where $\coprod^{(i)} {\cal P}_0$ is a finite sum of ${\cal P}_0$ which depends on $i$. Let $\del_j \col Y_0^{(i+1)}\lo Y_0^{(i)}$ $(0\leq j\leq i)$ be the standard face morphism. Then we have a natural morphism $\Del_j\col \coprod^{(i+1)} {\cal P}_0\lo \coprod^{(i)} {\cal P}_0$ fitting into the following commutative diagram \begin{equation*} \begin{CD} Y_0^{(i+1)}@>{\del_j}>>Y_0^{(i)}\\ @V{\bigcap}VV @VV{\bigcap}V \\ \coprod^{(i+1)} {\cal P}_0@>{\Del_j}>>\coprod^{(i)} {\cal P}_0 \end{CD} \end{equation*} and satisfying the standard relations. Set ${\cal P}^{(i)}_{\bul}:= {\rm cosk}_0^{{\cal W}({\mab F}_q)}(\coprod^{(i)} {\cal P}_0)$.
Let ${\rm sp}\col ]Y^{(i)}_{\bul}[_{{\cal P}^{(i)}_{\bul}}\lo Y^{(i)}_{\bul}$ be the specialization map. Then, as in \cite[(2.3)]{cric}, the following sequence \begin{align*} 0\lo {\rm sp}_*(\Om^{\bul}_{]Y_{\bul}[_{{\cal P}_{\bul}}})\lo {\rm sp}_*(\Om^{\bul}_{]Y^{(0)}_{\bul}[_{{\cal P}^{(0)}_{\bul}}})\lo {\rm sp}_*(\Om^{\bul}_{]Y^{(1)}_{\bul}[_{{\cal P}^{(1)}_{\bul}}})\lo \cdots \end{align*} is exact. Hence we have the following spectral sequence \begin{align*} E_1^{ij}=H^j_{\rm rig}(Y^{(i)}/K_0({\mab F}_q))\Lo H^{i+j}_{\rm rig}(Y/K_0({\mab F}_q)). \tag{5.1.2}\label{ali:sps} \end{align*} By (\ref{ali:trfrgh}) and this spectral sequence, we obtain the following formula: \begin{align*} [(E_p(Y/{\mab F}_q),F^*_q)]= \sum_{i,j\geq 0}(-1)^{i+j}[H^j_{\rm rig}(Y^{(i)}/K_0({\mab F}_q)),F^*_q)] \in {\cal K}(K_0({\mab F}_q)). \tag{5.1.3}\label{ali:zeht} \end{align*} This formula implies (\ref{ali:zytp}). \end{proof}
\begin{coro}\label{coro:f} Let $X/{\mab F}_q$ be a non-smooth combinatorial $K3$ surface {\rm (\cite{kul}, \cite{fs}, \cite{nlk3})}. $($We do not assume that $X$ has a log structure of simple normal crossing type.$)$ Let $m$ be the summation of the times of the processes of blowing downs making all irreducible components relatively minimal. Let $M_1$ $($resp.~$M_2)$ be the cardinality of the irreducible components of $\os{\circ}{X}$ whose relatively minimal models are ${\mab P}^2_{{\mab F}_q}$ $($resp.~Hirzeburch surfaces =relatively minimal rational ruled surfaces$)$. Let $M$ be the cardinality of the irreducible components of $\os{\circ}{X}$. Then the following hold$:$ \par $(1)$ If $X$ is of ${\rm Type}$ ${\rm II}$ with double elliptic curve $E/{\mab F}_q$, then \begin{equation*} Z(\os{\circ}{X}/{\mab F}_q,t)= \dfrac{ {\rm det}(1-qtF^*_q \vert H^1_{\rm rig}(E/K_0({\mab F}_q)))^{M-2} }{(1-t){\rm det}(1-tF^*_q \vert H^1_{\rm rig}(E/K_0({\mab F}_q)))(1-qt)^{M_1+2M_2+M-3+m}(1-q^2t)^M}. \tag{5.2.1}\label{eqn:kfmdad} \end{equation*} \par $(2)$ Assume that $X$ is of ${\rm Type}$ ${\rm III}$. Let $d$ be the cardinality of the double curves of $\os{\circ}{X}$. Then \begin{equation*} Z(\os{\circ}{X}/{\mab F}_q,t)= \dfrac{1}{(1-t)^2(1-qt)^{M_1+2M_2+m-d}(1-q^2t)^M}. \tag{5.2.2}\label{eqn:kfdqad} \end{equation*} \end{coro} \begin{proof} First we give a remark on the rigid cohomology of a smooth projective rational surface $S$ over ${\mab F}_q$. Set $S_{{\mab F}_{q^{n}}}:=S\us{{\mab F}_q}{\otimes}{{\mab F}}_{q^{n}}$ and $S_{\ol{\mab F}_q}:=S\us{{\mab F}_q}{\otimes}{\ol{\mab F}_q}$. \par Let $\ol{S}_{\rm min}$ be a relatively minimal model of $S_{\ol{\mab F}_q}$. If $\ol{S}_{\rm min}\simeq {\mab P}^2_{\ol{\mab F}_q}$, we see that the motive $H(S_{\ol{\mab F}_{q}})$ is as follows by \cite[(6.12)]{dmi}: \begin{align*} H(S_{\ol{\mab F}_{q}})\simeq H({\mab P}^2_{{\ol{\mab F}_{q}}}) \oplus H(D)(-1), \end{align*} where $D$ is the disjoint sum of 0-dimensional points. Since $H^2({\mab P}^2_{\ol{\mab F}_q})$ is isomorphic to a Tate-twist, if a natural number $n$ is big enough, then $(F^*_q)^n$ on $H^2_{\rm rig}(S_{{\mab F}_{q^n}}/K_0({{\mab F}}_{q^{n}}))$ is ${\rm diag}(q^n, \ldots, q^n)$. Hence the eigenvalues of $(F^*_q)^n$ are $q^n$ $(n \gg 0)$ and thus the eigenvalues of $F^*_q$ are $q$. \par If $\ol{S}_{\rm min}$ is isomorphic to a relatively minimal ruled surface over a smooth curve $C$ over $\ol{\mab F}_q$, the motive $H(S_{\ol{\mab F}_{q}})$ is as follows by \cite[(6.10), (6.12)]{dmi}: \begin{align*} H(S_{\ol{\mab F}_q})\simeq H(C) \oplus H(C)(-1)\oplus H(D)(-1). \end{align*} Hence we see that $F^*_q$ on $H^2_{\rm rig}(S/K_0({\mab F}_q))$ is ${\rm diag}(q, \ldots, q)$ as above. \par (1): It is easy to check that \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(0)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^M& (i=0) \\ H^1_{\rm rig}(E/K_0({\mab F}_q))^{\oplus M-2} & (i=1)\\ K_0({\mab F}_q)(-1)^{M_1+2M_2+2(M-2)+m} & (i=2)\\ H^1_{\rm rig}(E/K_0({\mab F}_q))(-1)^{\oplus M-2}& (i=3)\\ K_0({\mab F}_q)(-2)^M& (i=4) \end{cases} \label{eqn:kfda0d} \end{equation*} and \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(1)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^{M-1}& (i=0) \\ H^1_{\rm rig}(E/K_0({\mab F}_q))^{\oplus M-1} & (i=1)\\ K_0({\mab F}_q)(-1)^{M-1}& (i=2). \end{cases} \label{eqn:kfdakd} \end{equation*} Now (\ref{eqn:kfmdad}) follows from (\ref{ali:zytp}). \par (2): Let $T$ be the cardinality of the triple points of $\os{\circ}{X}$. It is easy to check that \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(0)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^M& (i=0) \\ 0 & (i=1)\\ K_0({\mab F}_q)(-1)^{M_1+2M_2+m} & (i=2)\\ 0& (i=3)\\ K_0({\mab F}_q)(-2)^M& (i=4), \end{cases} \label{eqn:kafdad} \end{equation*} \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(1)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^d& (i=0) \\ 0 & (i=1)\\ K_0({\mab F}_q)(-1)^{d}& (i=2) \end{cases} \label{eqn:kfdabd} \end{equation*} and \begin{equation*} H^0_{\rm rig}(\os{\circ}{X}{}^{(2)}/K_0({\mab F}_q))= K_0({\mab F}_q)^T. \label{eqn:kfdiad} \end{equation*} Because the dual graph of $\os{\circ}{X}$ is a circle, $M-d+T=\chi({\mab S}^1)=2$. Now (\ref{eqn:kfdqad}) follows from (\ref{ali:zytp}) \end{proof}
\begin{rema}\label{rema:tri} If $p\not=2$, we can prove that $T$ is even (cf.~\cite{fs}). However we do not use this fact in this article. \end{rema}
\begin{coro}\label{coro:en} Assume that $p\not=2$. Let $X/{\mab F}_q$ be a non-smooth combinatorial classical Enriques surface {\rm (\cite{kul}, \cite{nlk3})}. Let $M_1$, $M_2$, $M$, $m$ and $d$ be as in {\rm (\ref{coro:f})}. Then the following hold$:$ \par $(1)$ If $X$ is of ${\rm Type}$ ${\rm II}$ with double elliptic curve $E/{\mab F}_q$, then \begin{equation*} Z(\os{\circ}{X}/{\mab F}_q,t)= \dfrac{{\rm det}(1-qtF^*_q \vert H^1_{\rm rig}(E/K_0({\mab F}_q)))^{M-1}}{(1-t)(1-qt)^{M_1+2M_2+M-1+m}(1-q^2t)^M}. \tag{5.4.1}\label{eqn:kfend} \end{equation*} \par $(2)$ Assume that $X$ is of ${\rm Type}$ ${\rm III}$. Let $d$ and $T$ be the cardinalities of the double curves of $\os{\circ}{X}$ and the triple points of $\os{\circ}{X}$, respectively. Then \begin{equation*} Z(\os{\circ}{X}/{\mab F}_q,t)= \dfrac{1}{(1-t)(1-qt)^{M_1+2M_2+m-d}(1-q^2t)^M}. \tag{5.4.2}\label{eqn:kenad} \end{equation*} \end{coro} \begin{proof} (1): It is easy to check that \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(0)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^M& (i=0) \\ H^1_{\rm rig}(E/K_0({\mab F}_q))^{\oplus M-1} & (i=1)\\ K_0({\mab F}_q)(-1)^{M_1+2M_2+2(M-1)+m} & (i=2)\\ H^1_{\rm rig}(E/K_0({\mab F}_q))(-1)^{\oplus M-1}& (i=3)\\ K_0({\mab F}_q)(-2)^M& (i=4) \end{cases} \label{eqn:kfdacd} \end{equation*} and \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(1)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^{M-1}& (i=0) \\ H^1_{\rm rig}(E/K_0({\mab F}_q))^{\oplus M-1} & (i=1)\\ K_0({\mab F}_q)(-1)^{M-1}& (i=2). \end{cases} \label{eqn:kfdadd} \end{equation*} \par (2): It is easy to check that \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(0)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^M& (i=0) \\ 0 & (i=1)\\ K_0({\mab F}_q)(-1)^{M_1+2M_2+m} & (i=2)\\ 0& (i=3)\\ K_0({\mab F}_q)(-2)^M& (i=4), \end{cases} \label{eqn:kfdaed} \end{equation*} \begin{equation*} H^i_{\rm rig}(\os{\circ}{X}{}^{(1)}/K_0({\mab F}_q))= \begin{cases} K_0({\mab F}_q)^d& (i=0), \\ 0 & (i=1)\\ K_0({\mab F}_q)(-1)^{d}& (i=2) \end{cases} \label{eqn:kfdafd} \end{equation*} and \begin{equation*} H^0_{\rm rig}(\os{\circ}{X}{}^{(2)}/K_0({\mab F}_q))= K_0({\mab F}_q)^T. \label{eqn:kfdiabd} \end{equation*} Because the dual graph of $\os{\circ}{X}$ is ${\mab P}^2({\mab R})$, $M-d+T=\chi({\mab P}^2({\mab R}))=1$. \end{proof}
\par Lastly we consider another type of local zeta functions. \par Let ${\cal V}$ be a complete discrete valuation ring of mixed characteristics with finite residue field ${\mab F}_q$ and let $K$ be the fraction field of ${\cal V}$. Let ${\mathfrak Y}$ be a proper smooth scheme over $K$ of dimension $d$ and let $I$ be the inertia group of the absolute Galois group ${\rm Gal}(\ol{K}/K)$. Then the zeta function of ${\mathfrak Y}$ is defined as follows: \begin{align*} Z({\mathfrak Y},t):=\Pi_{i=0}^{2d}{\rm det}(1-t\sigma \vert H^i_{\rm et}({\mathfrak Y}\us{K}{\otimes} \ol{K}, {\mab Q}_l)^I)^{(-1)^{h+1}}, \end{align*} where $\sigma \in {\rm Gal}(\ol{K}/K)$ is a lift of the geometric Frobenius of ${\rm Gal}(\ol{\mab F}_q/{\mab F}_q)$ and $l$ is a prime which is prime to $q$. If ${\mathfrak Y}$ is the generic fiber of a proper semistable family ${\cal Y}$ over ${\cal V}$ with special fiber $Y$, then the following formula holds by \cite{fkk} (\cite{ifkn}): \begin{align*} Z({\mathfrak Y},t)=\Pi_{h=0}^{2d}{\rm det}(1-t\sigma \vert H^h_{{\rm ket}}(\ol{Y}, {\mab Q}_l)^I)^{(-1)^{i+1}}. \end{align*} \par
Let ${\cal X}$ be a proper strict semistable family of surfaces over ${\cal V}$ with log special fiber $X$ over $s_{{\mab F}_q}$.
Then \cite[(6.3.3)]{msemi} tells us that $Z({\cal X}_K,t)$ can be described by the log crystalline cohomologies by the coincidence of the monodromy filtration and the weight filtration (\cite[(8.3)]{nlpi}, \cite[(6.2.4)]{msemi}; however see \cite[(11.15)]{ndw} and \cite[(7.1)]{nlpi}.): $$Z({\cal X}_K,t)=\Pi_{i=0}^{4}{\rm det} (1-tF^*_q\vert (H^i_{\rm crys}(X/ {\cal W}(s_{{\mab F}_q}))_{K_0({\mab F}_q)})^{N=0})^{(-1)^{i+1}},$$ where ${\cal W}(s_{{\mab F}_q})$ is the canonical lift of $s_{{\mab F}_q}$ over ${\cal W}({\mab F}_q)$, $H^i_{\rm crys}(X/{\cal W}(s_{{\mab F}_q}))$ is the $i$-th log crystalline cohomology of $X/{\cal W}({\mab F}_q)$ and $$N\col H^i_{\rm crys}(X/ {\cal W}(s_{{\mab F}_q}))_{K_0({\mab F}_q)}\lo H^i_{\rm crys}(X/ {\cal W}(s_{{\mab F}_q}))_{K_0({\mab F}_q)}(-1)$$ is the $p$-adic monodromy operator. More generally, for a proper SNCL scheme $Y/s_{{\mab F}_q}$ of pure dimension $d$, set \begin{align*} Z(H^i(Y/K_0(s_{{\mab F}_q})),t):={\rm det} (1-tF^*_q\vert (H^i_{\rm crys}(X/{\cal W}(s_{{\mab F}_q}))_{K_0({\mab F}_q)})^{N=0})^{(-1)^{i+1}} \tag{5.4.3}\label{ali:dhxw} \end{align*} and \begin{align*} Z(Y/s_{{\mab F}_q},t):=\prod_{i=0}^{2d}Z(H^i(Y/K_0(s_{{\mab F}_q})),t)^{(-1)^{i+1}} \tag{5.4.4}\label{ali:dhxzw}. \end{align*}
Let us recall the following result due to the author (\cite[(8.3)]{nmw}, (cf.~\cite[(2.2)]{may}, \cite[(6.4)]{cla})):
\begin{theo}[{\bf \cite[(8.3)]{nmw}}]\label{th:kumd} Let $\kap$ be a perfect field of characteristic $p>0$. Let $X/s$ be an SNCL $K3$ surface. Let $H^i_{{\rm log}}(X)$ $(i\in {\mab N})$ be the $i$-th log crystalline cohomology or the $i$-th Kummer \'{e}tale cohomology of $X/s$. Then the following hold$:$ \par $(1)$ The $\star$-adic $(\,\star=p,l\,)$ monodromy filtration and the weight one on $H^i_{{\rm log}}(X)$ coincide. \par $(2)$ The following hold$:$ \par $({\rm a})$ $X$ is of ${\rm Type}$ ${\rm I}$ if and only if $N=0$ on $H^2_{{\rm log}}(X)$. \par $({\rm b})$ $X$ is of ${\rm Type}$ ${\rm II}$ if and only if $N\not=0$ and $N^2=0$ on $H^2_{{\rm log}}(X)$. \par $({\rm c})$ $X$ is of ${\rm Type}$ ${\rm III}$ if and only if $N^2\not=0$ on $H^2_{{\rm log}}(X)$. \end{theo} \begin{proof} For the completeness of this article, we give the proof of (\ref{th:kumd}). \par We give the proof of this theorem in the $p$-adic case because the proof in the $l$-adic case is the same as that in the $p$-adic case. \par Recall the following weight spectral sequence (\cite[3.23]{msemi}, \cite[(2.0.1)]{ndw}): \begin{align*} E_{1}^{-k, i+k}&= \us{j\geq {\rm max}\{-k, 0\}} {\bigoplus} H^{i-2j-k}_{\rm rig} (\os{\circ}{X}{}^{(2j+k)}/{K_0})(-j-k)\Lo H^i_{\rm crys}(X/{\cal W}(s))_{K_0}. \tag{5.5.1}\label{eqn:pstwtsp} \end{align*} (See \cite{ndw} for the mistakes in \cite{msemi}.) Here we have used Berthelot's comparison isomorphism $H^{i}_{\rm crys}(Y/{\cal W})_{K_0}=H^{i}_{\rm rig}(Y/{K_0})$ $(i\in {\mab N})$ for a proper smooth scheme $Y$ over $\kap$. By \cite[(3.6)]{ndw} this spectral sequence degenerates at $E_2$. (The $l$-adic analogue of this spectral sequence also degenerates at $E_2$ by Nakayama's theorem (\cite[(2.1)]{nd}).) \par (1): We may assume that $\kap$ is algebraically closed. If $X/s$ is of Type I, there is nothing to prove. \par If $X$ is of Type III, the double curves and the irreducible components are rational, and hence $E_1^{0,1}=E_1^{1,1}=E_1^{0,3}=E_1^{-1,3}=0$. By \cite[(3.5) 3)]{nlk3}, $H^1_{{\rm log}{\textrm -}{\rm crys}}(X/{\cal W})=0$ and hence we have $E_2^{-1,2}=0$. (Note that we also have the similar vanishing for the first Kummer \'{e}tale cohomology of $X$ by the vanishing above and the existence of the ${\mab Q}$-structure of $E_2^{-1,2}$ (cf.~the proof of \cite[(8.3)]{nlpi}). By taking the duality in \cite[(10.5)]{ndw}, $E_2^{1,2}=0$. By \cite[6.2.1]{msemi} the $p$-adic monodromy operator $N\col H^2_{\rm crys}(X/{\cal W}(s)) \lo H^2_{\rm crys}(X/{\cal W}(s))(-1)$ induces an isomorphism $N^2 \col E_2^{-2,4}\os{\simeq}{\lo}E_2^{2,0}(-2)=K_0$. \par If $X$ is of Type II, $E_{1}^{-2,4}=E_{1}^{2,0}=0$. By \cite[(3.5) 3)]{nlk3} again, $H^1_{{\rm log}{\textrm -}{\rm crys}}(X/{\cal W})= H^3_{{\rm log}{\textrm -}{\rm crys}}(X/{\cal W})=0$. Hence $E_{2}^{ij}=0$ for $i+j=1,3$. Because $N\col H^2_{\rm crys}(X/{\cal W}(s)) \lo H^2_{\rm crys}(X/{\cal W}(s))(-1)$ induces an isomorphism $E_2^{-1,3} \os{\simeq}{\lo}E_2^{1,1}(-1)$ by \cite[6.2.2]{msemi}, we have proved (1). \parno (2): (2) follows from (1) and the non-vanishings of $E_2^{1,1}$ in the Type II case and $E_2^{2,0}$ in the Type III case, respectively. \end{proof}
\begin{rema} The author has found the theorem (\ref{th:kumd}) in December 1996 by using the $p$-adic weight spectral sequence (\ref{eqn:pstwtsp}). The key point of the proof is to notice to use the $p$-adic weight spectral sequence of $X/s$ instead of the Clemens-Schmid exact sequence used in Kulikov's article \cite{kul}. (In fact, the complex analogue (\ref{theo:cct}) below of (\ref{th:kumd}) holds; this is a generalization of Kulikov's theorem in [loc.~cit.] and the proof of (\ref{th:kumd}) is simpler than that in [loc.~cit.]. To my surprise, mathematicians who are working over ${\mab C}$ have not used the weight spectral sequence (\ref{eqn:dfwqtfi}).) The author has finished writing the preprint \cite{nmw} by 2000 at the latest (cf.~\cite[Remark 2.4 (3)]{nd}). However, after that, he has noticed that there are too many non-minor mistakes in theory of log de Rham-Witt complexes in Hyodo-Kato's article \cite{hk} and Mokrane's article \cite{msemi} as pointed out in \cite{ndw}. Because he has used Hyodo-Kato's and Mokrane's theory in \cite{nmw} heavily, he has to use their results in correct ways. However he has used his too much time for correcting their results in \cite{ndw}, he has no will to publish \cite{nmw} now (because \cite{nmw} is quite long and because he has to use more time for adding comments about Hyodo-Kato's and Mokrane's articles in \cite{nmw}). For example, $\nu$ is in \cite{msemi} is {\it not} a morphism of complexes, the left $N$ in the diagram in \cite[(2.2)]{may} is incorrect. \par
In \cite[(2.2)]{may} Matsumoto has proved (\ref{th:kumd}) for semistable algebraic spaces of $K3$-surfaces after looking at the proof in \cite{nmw}. (See ``Proof of $p$-adic case'' in the proof of \cite[Proposition 2.2]{may}.) \end{rema}
\begin{theo}[{\bf cf.~\cite{kul}}]\label{theo:cct} Let $s$ be the log point of ${\mab C}$. Let $X/s$ be an analytic SNCL $K3$ surface. Let $X_{\infty}$ be the base change of the Kato-Nakayama space $X^{\log}$ of $X$ {\rm (\cite{kn})} with respect to the morphism ${\mab R}\owns x\lom {\rm exp}(2\pi{\sqrt{-1}})\in {\mab S}^1$. Let $N\col H^i(X_{\infty},{\mab Q})\lo H^i(X_{\infty},{\mab Q})(-1)$ $(i\in {\mab N})$ be the monodromy operator constructed in {\rm \cite{fn}}. Then the following hold$:$ \par $(1)$ The weight filtration on $H^i(X_{\infty},{\mab Q})$ constructed in {\rm \cite{fn}} coincide with the monodromy filtration on $H^i(X_{\infty},{\mab Q})$ \par $(2)$ The following hold$:$ \par $({\rm a})$ $X$ is of ${\rm Type}$ ${\rm I}$ if and only if $N=0$ on $H^2(X_{\infty},{\mab Q})$. \par $({\rm b})$ $X$ is of ${\rm Type}$ ${\rm II}$ if and only if $N\not=0$ and $N^2=0$ on $H^2(X_{\infty},{\mab Q})$. \par $({\rm c})$ $X$ is of ${\rm Type}$ ${\rm III}$ if and only if $N^2\not=0$ on $H^2(X_{\infty},{\mab Q})$. \end{theo} \begin{proof} By \cite[(2.1.10)]{nlpi} we have the following weight spectral sequence: \begin{equation*} E_{1,\inf}^{-k, h+k}= \us{j\geq {\rm max}\{-k, 0\}}{\bigoplus} H^{h-2j-k}(\os{\circ}{X}{}^{(2j+k+1)},{\mab Q})(-j-k) \Lo H^h(X_{\infty},{\mab Q}). \tag{5.7.1}\label{eqn:dfwqtfi} \end{equation*} By \cite[(5.9)]{fr}, if $X$ is a combinatorial ${\rm Type}$ ${\rm II}$ or ${\rm Type}$ ${\rm III}$ $K3$ surface over ${\mab C}$, then $H^0(X, \Om_{X/{\mab C}}^{1})=0$. (Of course, if $X$ is of ${\rm Type}$ ${\rm I}$, then $H^0(X, \Om_{X/{\mab C}}^{1})=0$ by Hodge symmetry.) Hence $H^1(X_{\infty},{\mab C})=H^1_{\rm dR}(X/{\mab C}) =H^0(X,\Om^1_{X/s})\oplus H^1(X,{\cal O}_X)=0$. Here we have used the isomorphism between Steenbrink complexes $A_{\mab Q}\otimes_{\mab Q}{\mab C}$ and $A_{\mab C}$ of $X$ and the isomorphism between $A_{\mab C}$ and $\Om^{\bul}_{X/s}$ (\cite{fn}). By the duality of the $E_2$-terms of (\ref{eqn:dfwqtfi}) (\cite[(5.15) (2)]{nlpi}) and the degeneration at $E_2$ of (\ref{eqn:dfwqtfi}) (by Hodge theory), we obtain the vanishing of $H^3(X_{\infty},{\mab C})$. The rest of the proof is the same as that of (\ref{th:kumd}). \end{proof}
\begin{theo}[{\bf \cite[(15.1)]{nmw}}]\label{theo:zt} Let $X/s_{{\mab F}_q}$ be a projective SNCL $K3$ surface. Then the following hold$:$ \par $(1)$ \begin{equation*} Z(H^i(X/K_0(s_{{\mab F}_q})),t)= \begin{cases} 1-t& (i=0) \\ 1 & (i=1,3)\\ 1-q^2t& (i=4). \end{cases} \label{eqn:kfdagd} \end{equation*} \par $(2)$ If $X$ is of ${\rm Type}$ ${\rm II}$ with double elliptic curve $E$, then \begin{align*} Z(H^2(X/K_0(s_{{\mab F}_q})),t)={\rm det}(1-tF^*_q \vert H^1_{\rm rig}(E/K_0))(1-qt)^{18}. \end{align*} Consequently \begin{align*} Z(X/s_{{\mab F}_q},t)=\dfrac{1}{(1-t) {\rm det}(1-tF^*_q \vert H^1_{\rm rig}(E/K_0))(1-qt)^{18}(1-q^2t)}. \end{align*} \par $(3)$ If $X$ is of ${\rm Type}$ ${\rm III}$, then \begin{align*} Z(H^2(X/K_0(s_{{\mab F}_q})),t)=(1-t)(1-qt)^{19}. \end{align*} Consequently \begin{align*} Z(X/s_{{\mab F}_q},t)=\dfrac{1}{(1-t)^2(1-qt)^{19}(1-q^2t)}. \end{align*} \end{theo} \begin{proof} By \cite[(3.5)]{nlk3}, $H^i_{\rm crys}(X/{\cal W}(s_{{\mab F}_q}))=0$ $(i=1,3)$. Thus $Z(H^i(X/K_0(s_{{\mab F}_q})),t)=1$ $(i=1,3)$. By \cite[(6.9)]{nlk3}, $X$ is the log special fiber of a projective semistable family ${\cal X}$ over ${\rm Spec}(W({\mab F}_q))$. By \cite[(6.10)]{nlk3}, the generic fiber of ${\cal X}$ is a K3 surface. Hence, by Hyodo-Kato's isomorphism (\cite[(5.1)]{hk}) (however see \cite[\S7]{ndw} for incompleteness of the proof of Hyodo-Kato isomorphism), ${\rm dim}_{K_0({\mab F}_q)} H^2_{\rm crys}(X/{\cal W}(s_{{\mab F}_q}))_{K_0({\mab F}_q)}=22$. \par (1): In this case, by (\ref{th:kumd}), $N\not=0$, $N\col E_2^{-1,2}\lo E_2^{1,1}$ is an isomorphism, $N^2=0$ on $H^2_{\rm crys}(X/W(s_{{\mab F}_q}))_{K_0({\mab F}_q)}$ and $E_2^{-2,4}=E_2^{2,0}=0$. Hence we have the following exact sequence by (\ref{th:kumd}): $$ 0 \lo E_2^{1,1} \lo {\rm Ker}(N) \lo E_2^{0,2}\lo 0.$$ Because $E_2^{1,1}\simeq H^1_{\rm rig}(E/K_0({\mab F}_q))$, ${\rm det}(1-tF^*_q \vert E_2^{1,1})= {\rm det}(1-tF^*_q \vert H^1_{\rm rig}(E/K_0({\mab F}_q))$. On the other hand, $E_2^{0,2}$ is a subquotient of $$H^0_{\rm rig}(X^{(2)}/K_0({\mab F}_q))(-1) \oplus H^2_{\rm rig}(X^{(1)}/K_0({\mab F}_q)). $$ Hence $F^*_q$ on $E_2^{0,2}$ is ${\rm diag}(q, \ldots, q)$ as shown in the proof of (\ref{coro:f}). Since ${\rm Ker}(N)$ is 20-dimensional, we obtain (2). \parno (2): In this case, by (\ref{th:kumd}), $N^2\not=0$, $N^3=0$ and $E_2^{-1,3}=E_2^{1,1}=0$. Because $N^2 \col E_2^{-2,4} {\lo} E_2^{2,0}(-2)$ is an isomorphism,
$N \col E_2^{0,2} {\lo} E_2^{2,0}(-1)$ is surjective and hence the kernel of $N$ is $20$-dimensional. Obviously $F^*_q={\rm id}$ on $E_2^{2,0}$. As in (1), $F^*_q$ on $E_2^{0,2}$ is ${\rm diag}(q, \ldots, q)$. Hence we obtain (2). \end{proof}
\begin{theo}[{\bf \cite[(15.2)]{nmw}}]\label{theo:zeet} Let $X/s_{{\mab F}_q}$ be a projective non-smooth SNCL classical Enriques surface.
Then \begin{equation*} Z(H^i(X/K_0(s_{{\mab F}_q})),t)= \begin{cases} 1-t& (i=0), \\ 1& (i=1,3), \\ (1-qt)^{10} & (i=2)\\ 1-q^2t& (i=4). \end{cases} \label{eqn:kfdahd} \end{equation*} Consequently \begin{align*} Z(X/s_{{\mab F}_q},t)=\dfrac{1}{(1-t)(1-qt)^{10}(1-q^2t)}. \end{align*} \end{theo} \begin{proof} By \cite[(7.1)]{nlk3}, $H^i_{\rm crys}(X/{\cal W}(s_{{\mab F}_q}))=0$ $(i=1,3)$ and hence $Z(H^i(X/K_0(s_{{\mab F}_q})),t)=1$ $(i=1,3)$. By \cite[(7.1)]{nlk3} and the argument in \cite[(6.8), (6.11)]{nlk3}, $X$ is the log special fiber of a projective semistable family ${\cal X}$ over ${\cal W}({\mab F}_q)$ and the generic fiber of ${\cal X}$ is a classical Enriques surface. Hence ${\rm dim}_{K_0({\mab F}_q)} H^2_{\rm crys}(X/{\cal W}(s_{{\mab F}_q}))_{K_0 ({\mab F}_q)}=10$. The rest of the proof is the same as that of (\ref{theo:zt}) by noting that $0=E_2^{-1,3}=E_2^{11}=E_2^{20}=E_2^{-2,4}$, where $E_2^{\bul \bul}$'s are $E_2$-terms of the spectral sequence (\ref{eqn:pstwtsp}). \end{proof}
\parno \begin{center} {{\rm \Large{\bf Appendix}}} \end{center}
\section{A remark on Katsura and Van der Geer's result}\label{rema:arkg} In this section we generalize the argument in the proof of (\ref{theo:xfh}) (3). \par First we recall the following theorem in \cite{ny}. This is a generalization of Katsura and Van der Geer's theorem (\cite[(5.1), (5.2), (16.4)]{vgk}).
\begin{theo}[{\bf \cite[(2.3)]{ny}}]\label{prop:nex} Let $\kap$ be a perfect field of characteristic $p>0$. Let $Y$ be a proper scheme over $\kap$. $($We do not assume that $Y$ is smooth over $\kap$.$)$ Let $q$ be a nonnegative integer. Assume that $H^q(Y,{\cal O}_Y)\simeq \kap$, that $H^{q+1}(Y,{\cal O}_Y)=0$ and that $\Phi^q_{Y/\kap}$ is pro-representable. Assume also
that the Bockstein operator \begin{align*} \bet \col H^{q-1}(Y,{\cal O}_Y)\lo H^q(Y,{\cal W}_{n-1}({\cal O}_Y)) \end{align*} arising from the following exact sequence \begin{align*} 0\lo {\cal W}_{n-1}({\cal O}_Y)\os{V}{\lo} {\cal W}_n({\cal O}_Y) {\lo} {\cal O}_Y\lo 0 \end{align*} is zero for any $n\in {\mab Z}_{\geq 2}$. Let $V\col {\cal W}_{n-1}({\cal O}_Y) \lo {\cal W}_n({\cal O}_Y)$ be the Verschiebung morphism and let $F\col {\cal W}_{n}({\cal O}_Y) \lo {\cal W}_n({\cal O}_Y)$ be the induced morphism by the Frobenius endomorphism of ${\cal W}_{n}(Y)$. Let $n^q(Y)$ be the minimum of positive integers $n$'s such that the induced morphism $$F\col H^q(Y,{\cal W}_n({\cal O}_Y))\lo H^q(Y,{\cal W}_n({\cal O}_Y))$$ by the $F\col {\cal W}_{n}({\cal O}_Y) \lo {\cal W}_n({\cal O}_Y)$ is not zero. $($If $F=0$ for all $n$, then set $n^q(Y):=\infty.)$ Let $h^q(Y/\kap)$ be the height of the Artin-Mazur formal group $\Phi^q_{Y/\kap}$ of $Y/\kap$. Then $h^q(Y/\kap)=n^q(Y)$. \end{theo}
\begin{prop}\label{prop:nnz} Let the notations be as in {\rm (\ref{prop:nex})}.
Let $D(\kap)$ be the Cartier-Dieudonn\'{e} algebra over $\kap$. Then the following hold$:$ \par $(1)$ ${\rm length}_{\cal W} H^q(Y,{\cal W}_n({\cal O}_Y))=n$ $(n\in {\mab Z}_{\geq 1})$. \par $(2)$ Set $h:=h(\Phi^q_{Y/\kap})$. Assume that $h<\infty$. Let us consider the following natural surjective morphism
$H^q(Y,{\cal W}({\cal O}_Y))\lo H^q(Y,{\cal W}_h({\cal O}_Y))$.
Then this morphism induces the following isomorphism
\begin{align*} H^q(Y,{\cal W}({\cal O}_Y))/p\os{\sim}{\lo} H^q(Y,{\cal W}_h({\cal O}_Y)) \tag{6.2.1}\label{ali:oyyh} \end{align*} of $D(\kap)/p$-modules. \end{prop} \begin{proof} (1): By the assumptions we have the following exact sequence \begin{align*} 0\lo H^q(Y,{\cal W}_{n-1}({\cal O}_Y))\os{V}{\lo} H^q(Y,{\cal W}_n({\cal O}_Y))\lo H^q(Y,{\cal O}_Y)\lo 0. \end{align*} (1) immediately follows from this.
\par (2): First assume that $h=1$. Then $H^q(Y,{\cal W}({\cal O}_Y))\simeq {\cal W}$. In this case, (2) is obvious. \par Next assume that $1<h<\infty$. Set $M_n:=H^q(X,{\cal W}_n({\cal O}_X))$ $(n\in {\mab Z}_{\geq 1})$ and $M:=H^q(X,{\cal W}({\cal O}_X))$.
Consider the following exact sequence \begin{align*} 0\lo {\cal W}_{m-n}({\cal O}_Y)\os{V^n}{\lo} {\cal W}_{m}({\cal O}_Y)\lo {\cal W}_n({\cal O}_Y)\lo 0
\end{align*} for $m>n$. By the assumption and (\ref{ali:yexyk}) we see that $H^{q+1}(Y,{\cal W}_m({\cal O}_X))=0$ for any $m$. Hence the natural morphism $M_m\lo M_n$ is surjective and consequently the natural morphism $M\lo M_n$ is surjective. In particular, the natural morphism $M\lo M_h$ is surjective. Let $\eta$ be an element of $M_h$. We claim that $p\eta=0$. \par We have to distinguish the operator $F\col M_n\lo M_n$ and the operator $F\col M_n\lo M_{n-1}$. The latter ``$F$ is equal to $R_nF$, where $R_n\col M_n\lo M_{n-1}$ is the projection. We denote $R_nF$ by $F_n$ to distinguish two $F$'s. Since the following diagram \begin{equation*} \begin{CD} M_h@>{R_h}>> M_{h-1}\\ @V{F}VV @VV{F}V \\ M_h @>{R_h}>> M_{h-1} \end{CD} \tag{6.2.2}\label{cd:hhr1} \end{equation*} is commutative, we have the following: \begin{align*} p\eta=VF_h(\eta)=VR_hF(\eta)=VFR_h(\eta). \end{align*} Since $F=0$ on $M_{h-1}$ by (\ref{prop:nex}), the last term is equal to zero. Hence $p\eta=0$. Consequently the natural morphism $H^q(Y,{\cal W}({\cal O}_Y))\lo H^q(Y,{\cal W}_h({\cal O}_Y))$ factors through the projection $H^q(Y,{\cal W}({\cal O}_Y))\lo H^q(Y,{\cal W}({\cal O}_Y))/p$. Since the morphism (\ref{ali:oyyh}) is surjective and ${\dim}_{\kap}H^q(Y,{\cal W}({\cal O}_Y))/p=h= {\dim}_{\kap}H^q(Y,{\cal W}_h({\cal O}_Y))$ by (1), the morphism (\ref{ali:oyyh}) is an isomorphism. \end{proof}
\begin{rema} Let the notations be as in (\ref{theo:xfh}) (2). By using only (\ref{prop:nnz}) for the case $q=d$, we can prove that \begin{equation*} \# Y({\mab F}_{q^k}) \equiv1~{\rm mod}~ p^{[\frac{ke+1}{2}]} \quad (k\in {\mab Z}_{\geq 1}), \tag{6.3.1}\label{eqn:kfda2td} \end{equation*} where $[~]$ is the Gauss symbol. However the congruence (\ref{eqn:kfda2td}) is not sharper than (\ref{eqn:kfd2td}); only in the case $h=2$, (\ref{eqn:kfda2td}) is equivalent to (\ref{eqn:kfd2td}). \end{rema}
\parno Yukiyoshi Nakkajima \parno Department of Mathematics, Tokyo Denki University, 5 Asahi-cho Senju Adachi-ku, Tokyo 120--8551, Japan. \parno {\it E-mail address\/}: [email protected]
\end{document} | arXiv |
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Information flow (information theory)
Information flow in an information theoretical context is the transfer of information from a variable $x$ to a variable $y$ in a given process. Not all flows may be desirable; for example, a system should not leak any confidential information (partially or not) to public observers--as it is a violation of privacy on an individual level, or might cause major loss on a corporate level.
Introduction
Securing the data manipulated by computing systems has been a challenge in the past years. Several methods to limit the information disclosure exist today, such as access control lists, firewalls, and cryptography. However, although these methods do impose limits on the information that is released by a system, they provide no guarantees about information propagation.[1] For example, access control lists of file systems prevent unauthorized file access, but they do not control how the data is used afterwards. Similarly, cryptography provides a means to exchange information privately across a non-secure channel, but no guarantees about the confidentiality of the data are given once it is decrypted.
In low level information flow analysis, each variable is usually assigned a security level. The basic model comprises two distinct levels: low and high, meaning, respectively, publicly observable information, and secret information. To ensure confidentiality, flowing information from high to low variables should not be allowed. On the other hand, to ensure integrity, flows to high variables should be restricted.[1]
More generally, the security levels can be viewed as a lattice with information flowing only upwards in the lattice.[2]
For example, considering two security levels $L$ and $H$ (low and high), if $L\leq H$, flows from $L$ to $L$, from $H$ to $H$, and $L$ to $H$ would be allowed, while flows from $H$ to $L$ would not.[3]
Throughout this article, the following notation is used:
• variable $l\in L$ (low) shall denote a publicly observable variable
• variable $h\in H$ (high) shall denote a secret variable
Where $L$ and $H$ are the only two security levels in the lattice being considered.
Explicit flows and side channels
Information flows can be divided in two major categories. The simplest one is explicit flow, where some secret is explicitly leaked to a publicly observable variable. In the following example, the secret in the variable h flows into the publicly observable variable l.
var l, h
l := h
The other flows fall into the side channel category. For example, in the timing attack or in the power analysis attack, the system leaks information through, respectively, the time or power it takes to perform an action depending on a secret value.
In the following example, the attacker can deduce if the value of h is one or not by the time the program takes to finish:
var l, h
if h = 1 then
(* do some time-consuming work *)
l := 0
Another side channel flow is the implicit information flow, which consists in leakage of information through the program control flow. The following program (implicitly) discloses the value of the secret variable h to the variable l. In this case, since the h variable is boolean, all the bits of the variable of h is disclosed (at the end of the program, l will be 3 if h is true, and 42 otherwise).
var l, h
if h = true then
l := 3
else
l := 42
Non-interference
Non-interference is a policy that enforces that an attacker should not be able to distinguish two computations from their outputs if they only vary in their secret inputs. However, this policy is too strict to be usable in realistic programs.[4] The classic example is a password checker program that, in order to be useful, needs to disclose some secret information: whether the input password is correct or not (note that the information that an attacker learns in case the program rejects the password is that the attempted password is not the valid one).
Information flow control
A mechanism for information flow control is one that enforces information flow policies. Several methods to enforce information flow policies have been proposed. Run-time mechanisms that tag data with information flow labels have been employed at the operating system level and at the programming language level. Static program analyses have also been developed that ensure information flows within programs are in accordance with policies.
Both static and dynamic analysis for current programming languages have been developed. However, dynamic analysis techniques cannot observe all execution paths, and therefore cannot be both sound and precise. In order to guarantee noninterference, they either terminate executions that might release sensitive information[5] or they ignore updates that might leak information.[6]
A prominent way to enforce information flow policies in a program is through a security type system: that is, a type system that enforces security properties. In such a sound type system, if a program type-checks, it meets the flow policy and therefore contains no improper information flows.
Security type system
In a programming language augmented with a security type system every expression carries both a type (such as boolean, or integer) and a security label.
Following is a simple security type system from [1] that enforces non-interference. The notation $\;\vdash exp\;:\;\tau $ means that the expression $exp$ has type $\;\tau $. Similarly, $[sc]\vdash C$ means that the command $C$ is typable in the security context $sc$.
$[E1-2]\quad \vdash exp:high\qquad {\frac {h\notin Vars(exp)}{\vdash exp\;:\;low}}$
$[C1-3]\quad [sc]\vdash {\textbf {skip}}\qquad [sc]\vdash h\;:=\;exp\qquad {\frac {\vdash exp\;:\;low}{[low]\vdash l\;:=\;exp}}$
$[C4-5]\quad {\frac {[sc]\vdash C_{1}\quad [sc]\vdash C_{2}}{[sc]\vdash C_{1}\;;\;C_{2}}}\qquad {\frac {\vdash exp\;:\;sc\quad [sc]\vdash C}{[sc]\vdash {\textbf {while}}\ exp\ {\textbf {do}}\ C}}$
$[C6-7]\quad {\frac {\vdash exp\;:\;sc\quad [sc]\vdash C_{1}\quad [sc]\vdash C_{2}}{[sc]\vdash {\textbf {if}}\ exp\ {\textbf {then}}\ C_{1}\ {\textbf {else}}\ C_{2}}}\qquad {\frac {[high]\vdash C}{[low]\vdash C}}$
Well-typed commands include, for example,
$[low]\vdash \ {\textbf {if}}\ l=42\ {\textbf {then}}\ h\;:=\;3\ {\textbf {else}}\ l\;:=\;0$.
Conversely, the program
$l\;:=\;0\ ;\ {\textbf {while}}\ l<h\ {\textbf {do}}\ l\;:=\;l+1$ ;\ {\textbf {while}}\ l<h\ {\textbf {do}}\ l\;:=\;l+1}
is ill-typed, as it will disclose the value of variable $h$ into $l$.
Note that the rule $[C7]$ is a subsumption rule, which means that any command that is of security type $high$ can also be $low$. For example, $h:=1$ can be both $high$ and $low$. This is called polymorphism in type theory. Similarly, the type of an expression $exp$ that satisfies $h\notin Vars(exp)$ can be both $high$ and $low$ according to $[E1]$ and $[E2]$ respectively.
Declassification
As shown previously, non-interference policy is too strict for use in most real-world applications.[7] Therefore, several approaches to allow controlled releases of information have been devised. Such approaches are called information declassification.
Robust declassification requires that an active attacker may not manipulate the system in order to learn more secrets than what passive attackers already know.[4]
Information declassification constructs can be classified in four orthogonal dimensions: What information is released, Who is authorized to access the information, Where the information is released, and When is the information released.[4]
What
A what declassification policy controls which information (partial or not) may be released to a publicly observable variable.
The following code example shows a declassify construct from.[8] In this code, the value of the variable h is explicitly allowed by the programmer to flow into the publicly observable variable l.
var l, h
if l = 1 then
l := declassify(h)
Who
A who declassification policy controls which principals (i.e., who) can access a given piece of information. This kind of policy has been implemented in the Jif compiler.[9]
The following example allows Bob to share its secret contained in the variable b with Alice through the commonly accessible variable ab.
var ab (* {Alice, Bob} *)
var b (* {Bob} *)
if ab = 1 then
ab := declassify(b, {Alice, Bob}) (* {Alice, Bob} *)
Where
A where declassification policy regulates where the information can be released, for example, by controlling in which lines of the source code information can be released.
The following example makes use of the flow construct proposed in.[10] This construct takes a flow policy (in this case, variables in H are allowed to flow to variables in L) and a command, which is run under the given flow policy.
var l, h
flow H $\prec $ L in
l := h
When
A when declassification policy regulates when the information can be released. Policies of this kind can be used to verify programs that implement, for example, controlled release of secret information after payment, or encrypted secrets which should not be released in a certain time given polynomial computational power.
Declassification approaches for implicit flows
An implicit flow occurs when code whose conditional execution is based on private information updates a public variable. This is especially problematic when multiple executions are considered since an attacker could leverage the public variable to infer private information by observing how its value changes over time or with the input.
The naïve approach
The naïve approach consists on enforcing the confidentiality property on all variables whose value is affected by other variables. This method leads to partially leaked information due to on some instances of the application a variable is Low and in others High.
No sensitive upgrade
No sensitive upgrade halts the program whenever a High variable affects the value of a Low variable effectively preventing information leakage. Since it simply looks for expressions where an information leakage might happen without looking at the context it may halt a program that despite having potential information leakage it never actually leaks information.
In the following example x is High and y is Low.
var x, y
y := false
if x = true then
y := true
return true
In this case the program would be halted since it uses the value of a High variable to change a Low variable despite the program never leaking information.
Permissive upgrade
Permissive-upgrade introduces an extra security class P which will identify information leaking variables. When a High variable affects the value of a Low variable, the latter is labeled P. If a P labeled variable affects a Low variable the program would be halted. To prevent the halting the Low and P variables should be converted to High using a privatization function to ensure no information leakage can occur. On subsequent instances the program will run without interruption.
Privatization inference
Privatization inference extends permissive upgrade to automatically apply the privatization function to any variable that might leak information. This method should be used during testing where it will convert most variables. Once the program moves into production the permissive-upgrade should be used to halt the program in case of an information leakage and the privatization functions can be updated to prevent subsequent leaks.
Application in computer systems
Beyond applications to programming language, information flow control theories have been applied to OS,[11] Distributed Systems [12] and Cloud Computing.[13][14]
References
1. Andrei Sabelfeld and Andrew C. Myers. Language-Based Information-Flow Security. IEEE Journal on Selected Areas in Communications, 21(1), Jan. 2003.
2. Dorothy Denning. A lattice model of secure information flow. Communications of the ACM, 19(5):236-242, 1976.
3. Smith, Geoffrey (2007). "Principles of Secure Information Flow Analysis". Advances in Information Security. Vol. 27. Springer US. pp. 291–307.
4. Andrei Sabelfeld and David Sands. Dimensions and Principles of Declassification. In Proc. of the IEEE Computer Security Foundations Workshop, 2005.
5. Thomas H. Austin and Cormac Flanagan. Efficient purely-dynamic information flow analysis, Proc. of the ACM SIGPLAN Fourth Workshop on Programming Languages and Analysis for Security, ACM, 2009.
6. J. S. Fenton. Memoryless Subsystems, Comput. J. 17(2): 143-147 (1974)
7. S. Zdancewic. Challenges for information-flow security. In Workshop on the Programming Language Interference and Dependence (PLID’04) 2004.
8. A. Sabelfeld and A. C. Myers. A model for delimited information release. In Proc. of International Symposium on Software Security (ISSS) 2003.
9. Jif: Java information flow
10. A. Almeida Matos and G. Boudol. On declassification and the non-disclosure policy. In Proc. IEEE Computer Security Foundations Workshop 2005.
11. M. Krohn, A. Yip, M. Brodsky, N. Cliffer, M. Kaashoek, E. Kohler and R. Morris. Information flow control for standard OS abstractions. In ACM Special Interest Group on Operating Systems (SIGOPS) Symposium on Operating systems principles 2007.
12. N. Zeldovich, S. Boyd-Wickizer and D. Mazieres. Securing Distributed Systems with Information Flow Control. In USENIX Symposium on Networked Systems Design and Implementation 2008.
13. J. Bacon, D. Eyers, T. Pasquier, J. Singh, I. Papagiannis and P. Pietzuch. Information Flow Control for secure cloud computing. In IEEE Transactions on Network and Service Management 2014.
14. Pasquier, Thomas; Singh, Jatinder; Eyers, David; Bacon, Jean (2015). "CamFlow: Managed Data-sharing for Cloud Services". IEEE Transactions on Cloud Computing. 5 (3): 472–484. arXiv:1506.04391. Bibcode:2015arXiv150604391P. doi:10.1109/TCC.2015.2489211. S2CID 11537746.
| Wikipedia |
ddRADseq reveals determinants for temperature-dependent sex reversal in Nile tilapia on LG23
Stephan Wessels1,
Ina Krause1,
Claudia Floren2,
Ekkehard Schütz3,4,
Jule Beck4 &
Christoph Knorr2
In Nile tilapia sex determination is governed by a male heterogametic system XX/XY either on LG1 or LG23. The latter carries a Y-specific duplicate of the amh gene, which is a testis-determining factor. Allelic variants in the amh gene demonstrated to be major triggers for autosomal and temperature-dependent sex reversal. Further, QTL on LG23 and LG20 show a temperature-responsiveness with influence on the phenotypic sex relative to the sex chromosomes. Here we present a ddRADseq based approach to identify genomic regions that show unusual large differentiation in terms of fixation index (FST) between temperature-treated pseudomales and non-masculinized females using a comparative genome-scan. Genome-wide associations were identified for the temperature-dependent sex using a genetically all-female population devoid of amh-ΔY.
Twenty-two thousand three hundred ninety-two SNPs were interrogated for the comparison of temperature-treated pseudomales and females, which revealed the largest differentiation on LG23. Outlier FST-values (0.35–0.44) were determined for six SNPs in the genomic interval (9,190,077–11,065,693) harbouring the amh gene (9,602,693–9,605,808), exceeding the genome-wide low FST of 0.013. Association analysis with a set of 9104 selected SNPs confirmed that the same genomic region on LG23 exerts a significant effect on the temperature-dependent sex.
This study highlights the role of LG23 in sex determination, harbouring major determinants for temperature-dependent sex reversal in Nile tilapia. Furthermore FST outlier detection proves a powerful tool for detection of sex-determining regions in fish genomes.
Genotyping-by-sequencing is a cost-effective approach to interrogate a multiplicity of loci simultaneously in large numbers of individuals [1]. In fact, only a fraction of homologous regions in the genome is sequenced and genotyped for available Single Nucleotide Polymorphisms (SNPs). Such approaches have proven successful in non-model organisms to identify genomic regions that determine complex traits such as the sex-determining loci in some bony fish [2,3,4,5,6]. Fish display a series of sex-determining mechanisms. Commonly, genetic sex determination (GSD) is divided into male heterogametic XX/XY or female heterogametic ZZ/ZW systems [7]. Moreover, pure environmental sex determination (ESD) exists, as well as transitions between the latter and GSD [7, 8]. Thus, chromosomal sex can be overlaid by extreme environmental conditions, such as elevated temperatures (reviewed in [8,9,10]).
Nile tilapia is a perciform fish from the cichlid clade, native to the Middle East and Africa. On the one hand, it alleviates hunger and constitutes an export commodity in light of its ability to thrive under poor conditions and ease of culture [11]. On the other hand, it is an excellent model for the intriguing complexity of sex determination and early stage sex chromosome development. Sex determination in Nile tilapia is orchestrated by major and minor genetic effects in addition to temperature [10]. Recent advances in short-read sequencing technologies and reduced representation genotyping approaches, have enabled the identification of sex-determining regions on linkage groups LG1 and LG23 in different Tilapia populations [3, 12,13,14,15]. RADseq revealed a major QTL for sex determination on LG 1 [3]. Moreover, whole genome sequencing of DNA pools from male and female tilapia detected a strongly differentiated inversion of 8.8 Mbp width, surrounding the putative sex-determining factor on LG1 [14]. This candidate region might be identically to the recently characterized zfand3 gene. Though mapped in a hybrid of Mozambique tilapia and red tilapia, it showed a strict association between the phenotypic and genetic sex in the studied population [16]. The sex-determining factor on the sex chromosome LG23 is in fact a male specific duplication of the amh gene (amhy) [13]. Moreover, sex-skewing and temperature-dependent QTL reside on LG20, and LG23 [12, 13, 17, 18]. Allelic variants in the amh gene can lead to autosomal and temperature-dependent sex reversal [17].
The present study deals with a ddRADseq based methodology aimed at identifying regions of the genome with unusual large fixation index (FST). Temperature-treated pseudomales and females, lacking the LG23 sex-determining amhΔY, were compared in a genome-scan. Furthermore, a genome-wide association study for the temperature-induced pseudomale phenotype was pursued using an amhΔY-negative genetically all-female population.
Phenotypes and sex ratios
Three genetically female (XX) families and one genetically supermale (YY) family were raised. Supermales were used in order to verify the presence of amhΔY as the putative sex chromosome in the investigated population. As the present study explicitly aimed to further decipher determinants for temperature-dependent sex reversal, exclusion of autosomal genes or other sex-skewing modifiers was paramount. As only family 1 was devoid of males in the control group, initially this family was chosen for the ddRADseq approach (Fig. 1). Subsequently families 2 and 3, which showed some sex-reversal in the control groups reared at 28 °C, too, were additionally sequenced using ddRADseq for a case-control approach. The genetically female full sibs were divided into a control (28 °C) and a treatment group (36 °C, 10–20 days post fertilisation (dpf)). The average male percentage of all control groups was 6.1% with sex ratios of 0%, 13.5% and 4.7% in families 1, 2, and 3, respectively. The overall male percentage in the three corresponding temperature-treated groups was 73.7% and differed significantly from the ratio in the controls (p < 0.0001). Despite that, each family showed significant deviations of the male proportion compared to the control groups (p < 0.0001), there were large differences in the temperature-responsiveness among families. Family 1 showed a male proportion of 59.9% in the temperature-treatment group, whereas families 2 and 3 beared 92.1% and 69.0% of males, respectively.
Sex ratios, numbers of sexed individuals, and survival from 10 dpf until sexing in control groups (28 °C) and temperature-treatment groups (36 °C, 10–20 dpf) of a genetically female population. Graphics were produced by the author. Copyright permissions were not required for the use of these graphics
The population of YY-supermales contained no female offspring. Males were identified upon the presence of sperm. Further, some YY-supermales progeny were used as sires and were bred to normal females, resulting exclusively in all-male broods (data not shown).
Survey of proposed sex determination loci Oni23063, Oni28137, and amhΔY
Comparative sequencing of the putative LG1 sex determination loci Oni23063 and Oni28137 [3] in sires and dams of the genetically all-female population revealed genotype G/G for SNP Oni23063 and T/T for SNP Oni28137, respectively. The YY-individuals showed genotypes G/G at locus Oni23063, and T/T at locus Oni28137.
Furthermore, the genetic sex on LG23 was determined according to [19]. The 1252 bp PCR-fragment flanking exon 6 in amh was amplified in the 120 genetically female individuals and the 10 YY supermales. All YY supermales showed the expected cleavage pattern of amhΔY, resulting from the 5 bp insertion. Restriction analyses of the 120 genetically female individuals indicated no cleavage pattern, indicating that all individuals were devoid of the Y-specific amhΔY (Additional file 1: Fig. S1).
ddRAD-tags and genome mapping
In total three genetically female families with 20 males and 20 females each were sampled and sequenced (Additional file 2: Table S1). All DNA samples were individually barcoded and indexed to be subsequently sequenced in 4 lanes NextSeq 500 and 3 lanes HiSeq 2000. Eighty-eight percent (175,697,219 reads) of the initial 200,643,953 single end reads were mapped to the Tilapia genome version Orenil1.1 (http://www.ncbi.nlm.nih.gov/assembly/GCF_000188235.2). Subsequent SNP calling, genotyping of the individuals and population genomic statistical analysis was performed using the ref_map.pl script of the Stacks program Version 1.34 [20]. Stacks first groups the aligned ddRADseq reads into genomic loci based on their position on the reference genome. Then polymorphic sites in each individual are identified and the most likely genotypes are inferred by applying a bounded-error statistical model, which accounts for sequencing errors. The populations script was applied in order to calculate population genomic parameters, such as pairwise FST, kernel-smoothed FST, expected and observed heterozygosity (H) as well as nucleotide diversity (π). Independent but equally treated runs of the populations script were performed comprising the following data sets: 1) comparison of 20 temperature-treated males with 20 females in family 1; 2) comparison of 60 temperature-treated males (affected cases) and 60 temperature-treated but non-masculinized females (unaffected controls) from families 1, 2, and 3. The ddRADseq analysis led to the discovery of 22,392 shared SNPs among temperature-treated pseudomales and females in family 1 (data set 1). A total of 9104 SNPs (data set 2) were detected amongst the genetically affected cases/unaffected controls and 5716 SNPs were shared between the data set 1 and 2.
Genome-wide estimates of population differentiation between temperature-treated pseudomales and females
Using the obtained SNP genotype data, genome-wide differentiation between temperature-induced pseudomales and females for data set 1 was calculated. Average FST values of 0.0132 ± 0.0303 were obtained. Among the 22 linkage groups and unplaced scaffolds, LG23 showed the largest extent of differentiation between temperature-treated pseudomales and non-masculinized females, with a mean FST of 0.0519 ± 0.0787. Moreover, genome-wide extreme outlier loci exceeding the 99.9% quantile were observed exclusively on LG23 (Fig. 2, Additional file 3: Table S2). The most striking region on LG23 consisted of six SNP loci showing FST values larger than 0.3506 (Fig. 3). This region precisely corresponds with a region of both, reduced diversity (π) and heterozygosity (H) (Fig. 4), suggesting factors such as reduced recombination, low mutation rates or positive selection act on this genomic region. Within the corresponding 1875 Kbp interval of the Nile tilapia reference genome, 86 genes are located. The most striking candidate gene, namely the amh gene (ENSONIG00000004781, NC_022220: bp 9,602,068..9606447) has been flanked by three SNPs giving the highest observed FST values of 0.4322 at bp 9,280,012, of 0.4370 at bp 9,441,132, and of 0.41322 at bp 9,746,803. Amh and its Y-linked orthologues amhΔY and amhy have been identified as the sex-determining factor in some Nile tilapia populations [19]. Moreover, SNPs in the X-linked amh are known to trigger autosomal and temperature-dependent sex reversal [17]. Therefore, we aimed to confirm a consistently elevated FST in this genomic region and to test the hypothesis that it results from directional selection. Therefore a kernel-smoothed moving average (default parameter 3 σ base pairs to each side of the window) implemented in the Stacks software was applied. The FST as well as the kernel-smoothed FST both consistently peaked in the genomic region around the amh gene on LG23 (Figs. 3 and 4). Two more SNPs showed signs of directional selection. One of which was located at position 9,280,012 of LG23 in intron 14 of the protein unc-13 homolog A-like gene (LOC100702202), and one of which in an intron of the lingo3 (ENSONIG00000021358) gene. Figure 5 summarizes the genes residing in this genomic region (9,150,000..9,650,000), Additional file 4: Table S3 deals with all genes on LG23.
Boxplot of the genome- and chromosome-wide FST between temperature-induced males (XX) and non-temperature-reversed genetic females (XX). Centre lines show the medians; box limits indicate the 25th and 75th percentiles as determined by R software; whiskers extend 1.5 times the interquartile range from the 25th and 75th percentiles, outliers are represented by dots; Numbers right to y-axis correspond to the number of interrogated SNPs
Genome-wide FST (a) and kernel-smoothed FST (b) between temperature-induced pseudomales (XX) and non-masculinized genetic females (XX)
FST as well as kernel-smoothed FST (a), observed heterozygosity H (b), and kernel-smoothed nucleotide diversity pi (c) on LG23 between temperature-induced pseudomales and non-masculinized genetic females
Detailed view of the region of largest differentiation between temperature-induced pseudomales and temperature-treated but non-masculinized genetic females on LG23. The region from 9,150,000 to 9,650,000, harbours 20 genes and 3 most significant SNPs. Amh is located at position 9,602,693–9,605,808
Association analysis for the temperature-dependent sex using case-control approach
In order to identify loci associated with the temperature-dependent sex, a case-control design was pursued using data set 2. In total, 5716 (22.2%) SNPs were intersecting with SNPs detected in data set 1. Association analysis using 9104 SNPs in data set 2 revealed that only 10 SNPs showed at least a suggestive association (p < 0.001; Table 1). Six of them were located on LG23 in the vicinity of the amh gene (Table 1). Interestingly, the SNP showing the highest association with the temperature-dependent sex was the closest to amh (Δ = 161 Kbp; Fig. 6). Moreover, suggestively significant SNPs resided on LG5 (n = 2), LG 6 (n = 1) and on scaffold 19 (GL831288.1; bp = 1,470,141). SNP 34055582 on LG5 is located in intron 2 of the kif5a gene. Gene ontology annotations for kif5a point to a role in ATPase and microtubule motor activity. SNP 23554897 is only 3 Kbp apart from the currently uncharacterized locus called 102,079,995. So far, none of these QTL had been linked to sex determination in Nile tilapia. Nevertheless, Bonferroni correction led to a reduction of significantly associated SNPs: only the four SNPs positioned on LG23 retained a significant association. After correction, all SNPs on LG23 flanking the amh gene showed a significant association with the temperature-dependent sex, with SNP 9,546,272, only 56 Kbp apart from amh. The strongest Bonferroni corrected association (p < 0.01) was observed for the SNP at position 9,441,132. At this locus the largest effect on the phenotypic sex was observed for genotype CC. 85.3% of all genotyped individuals were males (Fig. 7). Amongst heterozygotes 36.8% of the individuals were males. Most interestingly all homozygous G-allele carriers were females.
Table 1 Significant SNPs from case-control association analysis for temperature-dependent sex in Nile tilapia
Manhattan plot of log-transformed p-values derived from basic association study between temperature-treated pseudomales (n = 60 affected cases) and non-masculinized genetic females (n = 60 unaffected controls). Asymptotic p-values were derived from basic allelic test chi-square with 1 degree of freedom (--assoc). All unplaced scaffolds are summarized as unknown (Unkn)
Relationship between SNP genotypes at position 9,441,132 on LG23 with temperature-dependent male proportion in Nile tilapia
The sex determination in tilapia underlies a complex multi-stage mechanism. Effects of temperature on the phenotypic sex have amply been proposed [9, 17, 21,22,23]. Genomic approaches identified several linkage groups of the tilapia genome harbouring QTL and candidate genes for sex determination [18, 24,25,26]. The application of next generation sequencing approaches recently fostered the amount and quality of data for this trait [3, 12,13,14], but the question of the causative genetic background is still not answered. At least in some populations of Nile tilapia a male specific duplication of the amh gene (amhy) on LG23 constitutes the sex chromosome [13]. Moreover, besides major sex-determining factors on LG1, sex-influencing genes and genomic locations exist at least on LG20, and LG23 [12, 13, 17] that show a temperature-responsiveness with influence on sexual fate which can override the action of the major sex factor.
Previous studies demonstrated that SNP ss831884014 within the amh-gene (on LG23) is a major cue for both autosomal and temperature-dependent sex reversal. Here, we provide further evidence that LG23 and a 1875 Kbp interval genomic region encompassing the amh gene shows the largest evidence of differentiation between temperature-induced pseudomales and temperature-treated non-sensitive females. The present study revealed sex-specific patterns of genetic variability and pseudomale specific alleles at SNPs in close vicinity of the amh gene, whose male specific duplication (amhy) acts as a sex-determining factor on the LG23 sex chromosome [13]. Here, we also provide evidence for the presence of the amhΔY as the putative Y-chromosome in this Nile tilapia population from Lake Manzala, Egypt.
The investigated all-female population was devoid of amhΔY, whereas fish derived from a genetic supermale YY-line, but originating from the same population as the genetically female (XX) line, were all harbouring the y-specific amhΔY. Nevertheless, LG1 genotypes of the all-female population exhibited homozygous allelic states at locus Oni23063 (G/G) as well as for SNP Oni28137 (T/T), respectively. YY-individuals were homozygous for the G-allele at locus Oni23063, as well as the T-allele at SNP Oni28137. Palaiokostas et al. similarly determined the G-allele in homozygous state in genetic females at Oni23063 and Oni28137, respectively [3]. Genetic males were heterozygous A/G and G/T at Oni23063 and Oni28137 [3]. Nevertheless, the presence of amhΔY, as wells as the GG-genotype at Oni23063 in the investigated XX- and especially YY-individuals might indicate that the former acts as sex-determining factor, and that LG23 might represent the sex chromosome in this population. Under this hypothesis, the large allelic heterogeneity between temperature-treated pseudomales and females on LG23, and more specifically in the genomic region encompassing the amh gene, might likely indicate that on this early stage sex chromosome, the recombination rate plays a crucial role. Besides, in light of the confined genomic region in which we find evidence for loci influencing temperature-dependent sex reversal, sex-skewing minor loci, as well as the sex-determining factor amhΔY it is tempting to postulate that LG23, sex chromosomes and temperature-dependent sex factors are co-evolving in the same genomic region. Perrin et al. stated in fact that since recombination rather depends on the phenotypic than the genetic sex, a rapid decay of proto-sex chromosomes might occur, especially when sex reversed XY females are frequent [27]. Sex-specific recombination had been observed earlier in sex-reversed pseudomale tilapia, which showed male-specific synaptonemal complexes [28]. One possible mechanism explaining the high degree of differentiation between temperature-induced pseudomales and non-masculinized females in vicinity of the amh gene might be the phenotypically male-specific recombination pattern observed in this mating between a temperature-induced pseudomale and female. Thus, also haplotypes fostering temperature-dependent sex reversal could thus be conserved more easily. Consequently, if genetic females carrying allelic variants that nurture temperature-responsiveness would mate with genetic males (XY) with subsequent exposition of fry to elevated temperature, under the assumption that the same loci also lead to XY sex reversal, then recombination between the X and Y chromosome could take place. According to [27] this might lead to the breakdown of the evolving Y-specific haplotypes, which in turn would diminish due to the natural selection pressure. Thus, in light of a hampered Muller's ratchet the homomorphism between X and Y persists. However, the data presented here is not sufficient to support this hypothesis entirely, but it should be one road of enquiry for future studies.
In Nile tilapia a rapid turnover of sex chromosomal systems has taken place. Depending on the strain of O. niloticus either a XY-system on LG1 or on LG23 dominates [14, 29]. Although a QTL for temperature-dependent sex was proposed for LG1 in an earlier work [18], we didn't find evidence of elevated genetic differentiation between temperature-treated pseudomales and non-masculinized females, as were found on LG23. More precisely, even in the genomic region harbouring the putative causative sex-determining factor in some O. niloticus strains, contained by loci Oni23063 and Oni28137, neither signs of genetic differentiation between temperature-induced pseudomales and non-masculinized females nor an association of loci with the temperature-dependent sex was found. However, sex-specific allelic patterns observed on LG23 might point to a rapid evolution of this young sex chromosome and possibly temperature-dependent sex reversal. Further, it confirms that LG23, the amh gene and modifications in the TGF-ß pathway play an integral role in evolution of sex-determining mechanisms in tilapia. Previously it was shown that allelic variants in the amh gene might lead to both temperature-dependent and spontaneous sex reversal, suggesting that at least partially the same genetic cascade is responsible, with more pronounced effects fostered by temperature treatment [17]. The LG23-wide elevated mean FST (Fig. 2) as well as the kernel-smoothed FST both consistently peaked in the genomic region around the amh gene indicating that it might in fact have been under directional selection (Fig. 4). Furthermore, another TGF-ß related gene pias4 (ENSONIG00000004799) is located in this genomic region, which might, if LD persists, be subjected to directional selection, too. However, no markers were found within the pias4 gene to prove this hypothesis. Moreover, another SNP, that showed signs of directional selection, was located at position 9,280,012 of LG 23 positioned in intron 14 of the protein unc-13 homolog A-like gene (LOC100702202). Furthermore, an intronic SNP in lingo3 (ENSONIG00000021358) was identified that is also suspected to be under directional selection. Lingo3 is a transmembrane protein localized in the plasma membrane, however its role is yet unknown, too.
The loci identified here exert a large effect on the formation of the temperature-dependent sex; too, however the connection to spontaneous sex reversal remains to be elucidated. Palaiokostas et al. reported a sex QTL on LG20 which plays an ambiguous role during temperature-dependent as well as spontaneous sex reversal [12]. The here presented approach combining population genomics and association mapping failed to recover the proposed QTL on LG20, despite the fact that the genetically female population investigated here showed a high average male proportion (73.7%) in the temperature-treatment group. Moreover, although all families were genetically female (XX) and were derived from temperature-treated families whose untreated full sibs were also all-female, two thirds of the control groups exhibited some male individuals (Fig. 1). Genetic drift could be one reason for the spontaneous presence of those males in control groups. Alternatively, the fact that both parents had been temperature-treated might also have led to transgenerational epigenetic effects, leading to masculinization of individuals reared under the control temperature of 28 °C. A comparative analysis of spontaneously sex-reversed males and genetic females in controls is further needed in order to shed more light on the intermingling nature of sex chromosomes, loci associated to temperature-dependent sex, as well as loci promoting spontaneous sex reversal.
The comparative genome-scan of temperature-treated pseudomales and non-masculinized females provides evidence that temperature-dependent sex reversal is evolving on a putative sex chromosome in Nile tilapia. Moreover, the present study indicates through the large extent of allelic variation in the genomic region harbouring the putative sex-determining gene amh/amhy, that LG23 might still be considered a very young and fairly homomorphic sex chromosome. Further research is needed in order to investigate the role of recombination and recombination suppression respective of the phenotypic sex, as well as to understand the co-evolution of sex chromosomes and temperature-dependent sex reversal. Additionally, research is needed to clarify the role of epigenetics and the life-history memory, as well as expression of gene networks during temperature-dependent sex reversal.
Animals and tissue collection
Breeding and rearing of genetically female XX and all-male YY Nile tilapia (Oreochromis niloticus, Lake Manzala, Egypt) was carried out at the warm water recirculation unit of the Division of Aquaculture and Water Ecology at the Department of Animal Sciences at the University of Goettingen. To obtain genetically all-female (XX) progenies, eggs were derived from temperature-treated F1-females from a cross between a weakly temperature-responsive (low-line) and strongly temperature-responsive (high-line) selection line [23, 30]. To obtain the selection lines, family selection was carried out upon the male percentages in treatment groups [23]. After three generations of divergent selection 50.4% males were observed after 36 °C treatment from 10 to 20 dpf in the low-line, whereas in the high-line 92.7% males were present [30]. Eggs from temperature-treated F1-females were fertilised with sperm from one of three high-line temperature-induced F1-pseudomales and incubated at 28 °C. Hatching occurred at ~4 dpf. After yolk sac absorption, larvae from each of the three families were randomly distributed into control and treatment groups (n ~ 110 larvae per tank, if available more fry were reared in separate tanks). Temperature in the control groups was permanently kept at 28 °C, whereas treatment groups were reared at an elevated temperature of 36 ± 0.5 °C from 10 to 20 dpf. Control and treatment groups were reared in 2 l plastic aquaria. Fry were fed ad libitum and were reared family-wise under a photoperiod of 12 h light and 12 h darkness. From 20 dpf onwards, the fish from control and treatment groups were raised at 28 °C in separate 80 l tanks for at least 3 months. Thereafter, fish from groups treated at high temperature and control groups were subjected to anaesthesia using essential oil of cloves (at 0.05 mL/L) and were immediately exanguated, dissected, and a fin clip was sampled and stored at −20 °C until DNA extraction. The sex of all fish was assessed based on microscopical inspection of squashed gonads according to [31], classifying them into either testes or ovaries.
For the production of YY-supermales, eggs were derived from a hormone-induced YY-pseudofemale and sperm was stripped from a single YY-male originating from a line developed by [32]. All YY-individuals were solely reared at 28 °C.
All procedures were in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the German Animal Welfare Act [33].
Genomic DNA was extracted from fin-clips using the DNeasy blood and tissue kit according to the manufacturer's protocol (Qiagen, Hilden, Germany). All DNA samples were RNase-treated according to the manufacturer's recommendations.
Genotyping sex determination regions on LG1 (Oni23063 and Oni28137) and LG23 (amh/amhΔY)
Flanking sequences to amplify SNP Oni23063 and Oni28137 were derived from Scaffold NC022199.1. The genomic sequence of the amh gene was derived from Scaffold GL831234.1 of the Nile tilapia genome sequence deposited in the Ensembl database (http://www.ensembl.org; Orenil1.0 GCA_000188235.1; location of amh: Scaffold GL831234.1, 1.688.687–1.691.779). Gene- and locus-specific primers were designed using the Primer3 software (see Additional file 5: Table S4). Primers were used to amplify a specific fragment of 561 bp harbouring SNP Oni23063 and Oni28137 as well as another 1252 bp long PCR fragment flanking exon 6 of amh. PCRs were carried out using 20 ng of genomic DNA, 1× PCR buffer containing MgCl2, 1× Q-solution, 10 pM of each primer, 10 mM dNTPs and 2 U FastStart Taq DNA polymerase in a final volume of 25 μl. All PCR components, except for primers (MWG-Biotech, Ebersberg, Germany) and the 1× Q-solution (Qiagen, Hilden, Germany) were purchased from Roche (Roche,Penzberg, Germany). PCR was performed using a Biometra T-3000 Thermocycler (Biometra,Goettingen, Germany) with an initial denaturation at 95 °C for 10 min, followed by 35 cycles of 92 °C for 30 s, 60 °C for 30s and 72 °C for 1 min with a final extension at 72 °C for 5 min. The fragment identity was controlled via gel-electrophoresis on 1.5% agarose gels. Primers for amh were specific for all three reported forms of amh, i.e. the X-linked amh, and the Y-specific amhy homolog as well as amhΔY [19]. First of all amh was screened for 120 genetically female individuals and the 10 YY supermales. AmhΔY features a 5 bp insertion in Exon VI (ATGTC), which contains a Taq αI specific recognition site. Therefore, all amplicons were Taq α I (NEB, Frankfurt, Germany) digested and separated on 2% agarose gels. The X-linked amh and the Y-linked amhY were represented by the amplicon of 1252 bp length, whereas, the Y-linked amhΔY was represented by cleaved fragments of 829 and 423 bp in length. In addition to the restriction digest, all results were further confirmed using Sanger sequencing. M13-tailed primers enabled direct bidirectional sequencing on an ABI-PRISM 3100® capillary analyser (Life Technologies, Darmstadt, Germany) using the Big Dye terminator Kit. PCR products were before purified with Exo-SAP-IT® (Thermo Fisher Scientific, Schwerte, Germany). The obtained sequences were trimmed, contigs were built, and SNPs were manually identified using the program software suite DNASTAR LasergeneTM6® (DNASTAR, Madison, WI, USA). In Nile tilapia populations with an LG1 sex chromosome, genetic females are supposed to carry genotypes G/G and T/T at loci Oni23063 and Oni28137, respectively [3]. The available genome reference sequence of O. niloticus exhibits G/G at locus Oni23063 and T/T at Oni28137, as it was derived from an individual of an XX homozygous isogenic line. Genetic males are supposed to be heterozygous showing genotypes A/G and G/T for loci Oni23063 and Oni28137, respectively [3].
ddRAD library preparation and sequencing
A modified approach of the ddRADseq protocol developed by [34] was pursued. DNA from each individual (500 ng) was digested using 0.5 μl EcoRI (20.000 U/ml) (specific for G|AATT|C recognition site) and 1 μl AluI (10.000 Units/ml) (specific for AG|CT recognition site) in 1× NEBuffer 2.1 (NEB, Frankfurt, Germany) at a final volume of 30 μl. Digestions were run at 37 °C for 3 h and at a holding period at 4 °C without heat inactivation. Individual digests were then purified using a 2× volume of AMPure XP beads. Custom double-stranded sequencing adapters were designed for sticky end ligation to the digested DNA. An EcoRI compatible 5′-phosphorylated overhang was added to the original Illumina P5 adapter sequence and the EcoRI sequence was followed by an individual 5 bp barcode sequence, which was read at the beginning of each P5 sequencing read. One unique barcode was used for library preparation of each individual animal. The Illumina P7 adapters were designed to carry an AluI compatible 5′-phosphorylated overhang and carried a biotin molecule (Additional file 2: Table S1). Both strands of each adapter were ordered as single stranded oligonucleotides (MWG-Biotech, Ebersberg, Germany) and were annealed in Ligase Buffer (NEB, Frankfurt, Germany) to form double-strands using the following temperature profile: 95 °C 1 min, 80 °C > 21 °C with 1°/sec, 21 °C 20 min. The resulting double-stranded adapters (375 nM each) were ligated to the digested DNA at room temperature for 30 min, using, 4 μL T4 DNA Ligase (USB/Affymetrix, Santa Clara, CA, USA) in 1× Ligase Buffer (NEB, Frankfort, Germany) and a total volume of 40 μL. After an additional purification with 2× volume of AMPure XP beads each sample was quantified using the Quant-it™ ds DNA Assay Kit (Thermo Fisher Scientific, Waltham, MA, USA) and a GENios Pro microplate reader (Tecan, Crailsheim, Germany). Samples were subsequently pooled in an equimolar way. The use of one biotinylated and one non-biotinylated adapter allowed the specific selection of proper P5-P7 library fragments by selective binding to streptavidin-coated beads and subsequent denaturing of the bound double-strands as described [35]. Binding and wash steps to 30 μL Dynabeads® MyOne (Thermo Fisher Scientific, Schwerte, Germany) were carried out according to manufacturer's protocol. P5-P7 strands were eluted by denaturing the double-stranded DNA with alkaline lysis buffer: 400 mM KOH, 10 mM EDTA, 80 mM DTT. Each eluted library pool was diluted 1:10 and 2 μL were used as template in the final library PCR containing 100 μM of each dNTP, 2 units FastStart Taq DNA Polymerase, 1 μL 50xEvaGreen© and 400 μM P5-Universal-Primer and 400 μM P7-Index-Primer (Additional file 2: Table S1). Finally, 6 pools were size-selected to ~200–500 bp long fragments using an E-Gel electrophoresis system (Thermo Fisher Scientific, Waltham, MA, USA) and quantified using a 2100 Bioanalyzer (Agilent Technologies, Santa Clara, CA, USA). Libraries were then sequenced either on the Illumina HiSeq 2000 or the Illumina NextSeq 500 system, yielding reads with maximum single end read length of 100 or 151 bp, respectively (Additional file 6: Table S5).
Raw read filtering, processing and population genomic analysis
The resulting raw single-end reads were demultiplexed using a custom Perl script allowing 1 mismatch in the adapter sequence. Reads shorter than 95 bp were filled with 'N'. All reads were then truncated to 95 base pairs, quality filtered, and merged into one file per individual. Reads were subsequently mapped onto the Tilapia genome version Orenil1.1.
(http://www.ncbi.nlm.nih.gov/assembly/GCF_000188235.2) using the programme package BWA [36]. All resulting BAM-files were further analysed using the Stacks program Version 1.34 [20]. The datasets generated and analysed during the current study are available in the NCBI Bioproject repository, (Accession: PRJNA354565;https://www.ncbi.nlm.nih.gov/bioproject/?term=PRJNA354565). First, ddRAD tags were analysed using the ref_map.pl script, with a minimum stack depth of 5 (−m 5), allowing 1 mismatch between loci when building the catalogue (−n 1). The file catalog.snps.tsv was screened for erroneous SNP alleles resulting from in silico elongation of reads to 95 bp. All loci containing the erroneous 'N'-allele were blacklisted using the –B option in subsequent analyses.
Secondly, the population script was applied in order to calculate population genetic parameters. As the present study explicitly aimed to further decipher determinants for temperature-dependent sex reversal, exclusion of autosomal genes or other sex-skewing modifiers was paramount. As only family 1 was devoid of males in the control group, initially this family was chosen for the ddRADseq approach. Subsequently families 2 and 3, which showed some sex-reversal in the controls, too, were additionally sequenced for a case-control approach. Independent runs of the populations script were performed comprising the following data sets: data set 1) comparison of 20 temperature-treated males with 20 females in family 1; 2) comparison of 60 temperature-treated males (affected cases) and 60 temperature-treated but non-masculinized females (unaffected controls) from families 1, 2, and 3. All data sets were filtered equally to a stack depth of >5 (−m 5), a minor allele frequency of >0.01 (--min_maf 0.01), and a minimum percentage of individuals in the population required to process the locus >70% (−r 0.7), ddRAD tags were requested to be present in all populations within a data set (−p 2). SNP and haplotype-based F-statistics were requested using the --fstats command, kernel smoothing of FST was enabled using the –k option applying a default window size of 3σ (150 Kbp). The fixation index FST was calculated with Stacks 1.34 [20] using an adapted formula, accounting for unequal sample size among populations by weighting [37]:
$$ {F}_{ST}=1-\frac{\sum_i\left(\begin{array}{c}\hfill {n}_j\hfill \\ {}\hfill 2\hfill \end{array}\right){\pi}_j}{\pi_{.}{\sum}_i\left(\begin{array}{c}\hfill {n}_j\hfill \\ {}\hfill 2\hfill \end{array}\right)} $$
where n j is the number of alleles sampled in population j, p j is the nucleotide diversity within population j, and π is the total nucleotide diversity across the pooled populations. Expected and observed heterozygosity (H) as well as nucleotide diversity (π) were calculated using the populations script.
Association analysis
Temperature-dependent sex loci were identified in a case-control study (data set 2). Hence, 60 temperature-treated males (affected cases) and 60 temperature-treated but non-masculinized females (unaffected controls) of the above characterized three families were included. Genotypes of 9104 loci were obtained using Stacks program (version 1.34) [20] after running the populations script, applying the filters as used for data set 2. Association analysis of segregating SNPs was carried out using the software package PLINK [38]. Sex was coded as a binary trait with temperature-treated but non-masculinized females as 1 (unaffected/ control) and temperature-treated males as 2 (affected/ case). A case-control association analysis (--assoc) was undertaken. Subsequently a Bonferroni correction (--assoc --adjust) was applied during association analysis.
Base pair
ddRADseq:
Double digest restriction site associated DNA sequencing
dNTP:
Deoxynucleotide
dpf:
Days post fertilisation
FST :
Fixation index
Heterozygosity
Kbp:
Kilo base pairs
LG:
Mbp:
Mega base pairs
Millimolar
ng:
Nanogram
nM:
Nanomolar
Picomolar
Quantitative trait loci
RADseq:
Restriction site associated DNA sequencing
Single nucleotide polymorphism
TGF-ß:
Transforming growth factor ß
μL:
Microliter
μM:
Micromolar
π:
Nucleotide diversity
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We are grateful to the support provided by Gabriele Hörstgen-Schwark. The authors thank Stefan Adeberg and Cord Bielke for bioinformatic support. We acknowledge the possibility to use the computing facilities provided by Prof. Henner Simianer (Animal Breeding Group at the University of Goettingen).
This work was funded by the German Research Foundation DFG (DFG WE4434/2–1). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
The datasets generated and analysed during the current study are available in the NCBI Bioproject repository, (Accession: PRJNA354565; https://www.ncbi.nlm.nih.gov/bioproject/?term=PRJNA354565).
Department of Animal Sciences, Division of Aquaculture and Water Ecology, University of Goettingen, Albrecht-Thaer-Weg 3, 37075, Goettingen, Germany
Stephan Wessels & Ina Krause
Department of Animal Sciences, Division of Livestock Biotechnology and Reproduction, University of Goettingen, Burckhardtweg 2, 37077, Goettingen, Germany
Claudia Floren & Christoph Knorr
Institute of Veterinary Medicine, University of Goettingen, 37077, Goettingen, Germany
Ekkehard Schütz
Chronix Biomedical GmbH, 37077, Goettingen, Germany
Ekkehard Schütz & Jule Beck
Stephan Wessels
Ina Krause
Claudia Floren
Jule Beck
Christoph Knorr
SW conceived, designed and analysed the experiments. SW, IK, and CF performed the experiments. JB and ES provided technical support and were responsible for sequencing. SW, JB, and CK wrote and/or edited the manuscript. All authors read and approved the final manuscript.
Correspondence to Stephan Wessels.
All procedures were in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the German Animal Welfare Act [33]. This study was approved by the Institutional Animal Care and Use Committee of Goettingen University. All fish used in the present study were derived from a Lake Manzala population of Nile tilapia (Oreochromis niloticus). Their use did not require a permit as fish were owned and managed by the author's division at University of Goettingen, Department of Animal Sciences, Division of Aquaculture and Water Ecology.
Amh and AmhΔY genotypings for 60 temperature-treated pseudomales (affected cases) and 60 non-masculinized genetic females (unaffected controls) as well as 10 YY supermales. (PDF 252 kb)
Overview of P5-Universal- and P7-Index-Primer sequences for 60 temperature-induced pseudomales and non-masculinized females. (XLSX 17 kb)
Number of temperature-treated pseudomales (affected cases) and non-masculinized genetic females (unaffected controls) in the region of LG23 showing the largest FST. (XLSX 29 kb)
Overview of refseq genes residing on LG23. (XLSX 234 kb)
Forward and reverse primers tailed with a universal M13 forward or reverse primer for bidirectional sequencing of the amh gene. Forward and reverse primers for sequencing LG1 loci Oni23063 and Oni28137. (XLSX 11 kb)
Overview of number of reads, mapped reads as well as unmapped reads per individual. (XLS 46 kb)
Wessels, S., Krause, I., Floren, C. et al. ddRADseq reveals determinants for temperature-dependent sex reversal in Nile tilapia on LG23. BMC Genomics 18, 531 (2017). https://doi.org/10.1186/s12864-017-3930-0
ddRADseq
Sex reversal
Nile tilapia
Sex chromosomes | CommonCrawl |
Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson–Cowan model
Benoit Duchet1,2,
Gihan Weerasinghe1,2,
Hayriye Cagnan1,2,3,
Peter Brown1,2,
Christian Bick4,5,6 &
Rafal Bogacz1,2
The Journal of Mathematical Neuroscience volume 10, Article number: 4 (2020) Cite this article
Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Constant stimulation can lead to side effects, it is therefore desirable to find ways to stimulate less while maintaining clinical efficacy. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor. To advance methods to optimise deep brain stimulation while providing insights into tremor circuits, we ask the question: can the effects of phase-locked stimulation be accounted for by a canonical Wilson–Cowan model? We first analyse patient data, and identify in half of the datasets significant dependence of the effects of stimulation on the phase at which stimulation is provided. The full nonlinear Wilson–Cowan model is fitted to datasets identified as statistically significant, and we show that in each case the model can fit to the dynamics of patient tremor as well as to the phase response curve. The vast majority of top fits are stable foci. The model provides satisfactory prediction of how patient tremor will react to phase-locked stimulation by predicting patient amplitude response curves although they were not explicitly fitted. We also approximate response curves of the significant datasets by providing analytical results for the linearisation of a stable focus model, a simplification of the Wilson–Cowan model in the stable focus regime. We report that the nonlinear Wilson–Cowan model is able to describe response to stimulation more precisely than the linearisation.
Essential tremor (ET) is the most common movement disorder, affecting 0.9% of the population [1]. It predominantly manifests as a tremor of the upper limbs, and can severely affect daily-life. When medications are ineffective or not tolerated, thalamic deep brain stimulation (DBS) is a well-established therapy option. Clinically available DBS continuously delivers high-frequency (about 130 Hz) electrical stimulation to deep structures within the brain via an electrode connected to a pulse generator implanted in the chest. There is no agreement in the research community on the mechanisms of action of high-frequency DBS [2], but it is believed there is room for improvement in terms of efficacy, decrease in power usage, avoidance of habituation, and most importantly reduction of side effects [3]. Reported side effects of high-frequency thalamic DBS include speech impairment, gait disorders, and abnormal dermal sensations [4].
Because side effects are the main clinical bottleneck, improving high-frequency DBS generally means stimulating less by closing the loop on a signal related to motor symptoms, while maintaining clinical efficacy. One example of closed-loop DBS is adaptive DBS, whereby stimulation is triggered in Parkinson's disease (PD) patients when pathological neural oscillation amplitude in the beta band is higher than a threshold. Compared to high-frequency DBS, it has been shown to improve motor performance, and reduce speech side effects in humans [5–7]. Another example is phase-dependent stimulation, which has been investigated in a computational model of PD [8], and in PD patients [9, 10].
Phase-locked DBS has recently been studied as a new therapy for ET [11]. Hand tremor is recorded, and the reduction in stimulation comes from stimulating with a burst of pulses according to the phase of tremor, only once per period of the tremor rather than continuously. In some patients, the strategy only requires half the energy delivered by high-frequency DBS for the same effect. Optimising phase-locked DBS requires a detailed understanding of the phase-dependence of the response across patients. However, data collection from phase-locked stimulation experiments has been restricted so far to small datasets because patients fatigue quickly. While direct analysis of the data has proven insightful [11], modelling phase-locked stimulation would allow predictions to be made from analytic and computational studies regarding the phase-dependence of the response to stimulation, and would open the door to supplement scarcely available patient data with synthetic data. The ability to easily generate large amounts of synthetic data could come in handy to help devise and test control algorithms. It could also be useful when trying to predict an effect that, because of noise in recordings, can only be deciphered when a large number of trials is available.
Tremulous hand movements are believed to be closely related to thalamic activity [12, 13], and it is believed that ET originates in the cerebellar–thalamic–cortical pathway [14]. However, detailed knowledge of how ET comes about is missing, which makes simple, canonical models natural candidates to study ET. Recently, phase-locked DBS was studied using Kuramoto phase oscillators which do not model interacting neural populations with distinct properties [15]. In the present work, we focus on a neural mass model, the Wilson–Cowan (WC) model, whose architecture can be mapped onto neural populations thought to be involved in the generation of ET, and allows for strong coupling between the populations. Additionally, stimulation can be delivered in the model to the most common stimulation site for ET, the ventral intermediate nucleus (VIM). The model describes the firing rates of an excitatory and an inhibitory population, and only has a few parameters, which makes it less prone to overfitting and significantly easier to constrain than more detailed models. The WC model has been shown to be adept at describing beta oscillations in PD [16, 17]. Moreover, the work presented in [18] provides evidence that the effects of high-frequency DBS for ET in a WC model are similar to the description given by conductance-based models. While the WC model has been used to design closed-loop strategies for PD [19, 20], whether a firing-rate model such as the WC can model the effects of phase-locked DBS has not been approached in the literature. Based on strong assumptions, Polina et al. reduced a WC model to a one-dimensional ordinary differential equation and looked at periodic forcing, but not in the context of DBS, and without attending to dependence on the phase of stimulation [21]. The present work will focus on reproducing the phase-dependent effects of phase-locked DBS measured in human data with a WC model.
Stimulation changes the phase and the amplitude of tremor and the dependence of these changes on the phase of stimulation can be quantified by the phase response curve (PRC, in this study change in tremor phase as a function of tremor phase) and the amplitude response curve (ARC, in this study change in tremor amplitude as a function of tremor phase). The ARC directly measures the change in tremor, hence the change in patient handicap, but both the ARC and the PRC are important to understand the effects of phase-locked DBS and potentially optimise the stimulation pattern. In mathematical neuroscience, PRCs and ARCs have been defined differently, mostly in the context of limit cycle models concerned with asymptotic response to infinitesimal perturbations; see for example [22–27]. In patients, DBS stimulation is not infinitesimal, and tremor data is very variable so stimulation happens in transient states. Therefore rather than considering an asymptotic description of the changes in phase and amplitude, we will be focussing on a close variant of the experimental response curve measurement methodology applied to blocks of stimulation in [11], which we will hereafter refer to as the "block method". It provides a finite time response to a finite perturbation and relies on the changes in the Hilbert phase and amplitude of the tremor signal following blocks of phase-locked stimulation (more details in Sect. 2.1). The only exception to this will be in analytical derivations (Sect. 4), where a first order measurement of the response curves (i.e. measurement at the end of the stimulation period) will be used for tractability, as a simplified first approach to the model. For coherence with the experimental response curve measurement methodology, the notion of phase and amplitude used throughout will be the Hilbert phase and amplitude or approximately equivalent. It should also be noted that we are considering population response curves and not single neuron response curves. The vast majority of best performing WC models in reproducing patient data are found in this work to give rise to stable foci, where tremor dynamics is being reproduced by adding noise to the system, so we restrict our analytical considerations to stable foci.
Starting with the data, the narrative will be guided by the following questions. How do patient responses to phase-locked deep brain stimulation depend on phase? How do patient phase and amplitude response curves relate to one other? Can patient response curves and their relationship be described analytically in a simple linear model? Can we model patient tremor and better model response to phase-locked deep brain stimulation with a nonlinear WC model? The main contributions of this work are as follows. We first focus on the data and analyse patient response curves, identify a subset of datasets passing appropriate statistical tests, and characterise the relationship between PRC and ARC in these patients (Sect. 2). Following the introduction of our biologically motivated WC model (Sect. 3), we derive approximate analytical expressions that delineate the response to stimulation of a 2D dynamical system described by a linearised focus, with the goals of better understanding the constraints built in the model and of providing a first level of description of the data (Sect. 4). The derived response curves are close to sinusoidal, and a relationship between them is found, revealing similarities in shape and phase shift with patients who have statistically significant PRCs and ARCs. We then show that for these patients, the WC model can be fitted to the data and can reproduce the dependence of the effects of stimulation on the phase of stimulation. The model is fitted to the PRC and can reasonably predict the ARC, and notably what is approximately the best phase to stimulate (Sect. 5). We then proceed to compare the relationship between response curves in the linearised and the full model and conclude that nonlinearity is important to better reproduce the relationship found in patients (Sect. 6). Finally a discussion is provided (Sect. 7).
Patient response curves and their phase relationship
In order to assess phase-dependence of the effects of DBS in patients, we extract PRCs and ARCs from patient's tremor data, provide a statistical analysis of the response curves, and analyse their phase relationship when applicable. This data characterisation will inform our modelling approaches of the next sections, and we also introduce relevant concepts.
Analysis method
We extract response curves from tremor acceleration data presented in [11]. The experimental paradigm in [11] is as follows. ET patients are fitted with an accelerometer to record their tremor acceleration, and DBS locked to the phase of tremor acceleration is provided in blocks of 5 s to the VIM of the thalamus, with 1 s without stimulation between blocks. An example of one such block of stimulation is shown in Fig. 1 in light blue, with the 1 s period without stimulation before the block highlighted in light orange (reference period). Each block targets a stimulation phase randomly selected out of 12 tremor phases (e.g. 120 degrees for the block shown in Fig. 1). Stimulation is delivered once per period at the target phase, in the form of a burst of four to six pulses at high frequency (130 Hz or higher). Details of the pulses making up a burst can be seen in the zoomed-in insert in Fig. 1. Tremor frequency being around 5 Hz and stimulation blocks lasting 5 s, there are about 25 bursts of stimulation at the same target phase per stimulation block. There are about 10 trials available per phase bin so about 120 stimulation blocks per patient (12 phase bins times around 10 trials per phase). The method described in [11] to obtain a patient's response curves was specifically developed for this type of data, and we closely follow it and provide additional statistical analysis of the phase-dependence. We refer to our version of the method as the "block method" and denote the response curves obtained by bPRC and bARC, "b" standing for block. More specifically, we define the bPRC and the bARC according to the following procedure.
Example showing the block method applied to a block of stimulation with a target stimulation phase of 120 degrees. The three panels have the same horizontal axis. The reference period without stimulation before the block is highlighted in light orange, and the stimulation block itself in light blue. The filtered tremor is shown in blue in the upper panel. Stimulation triggers are shown in black in the lower panel. The 25 bursts of stimulation are each composed of a number of individual pulses at high frequency as shown in the zoomed-in insert. As shown in the middle panel, the change in phase \(\Delta\phi_{i}\) due to the block of stimulation is obtained by comparing at the end of the block the actual Hilbert phase to a linear phase obtained by a straight line fit to the phase evolution 1 s before the block (reference period). The change in amplitude is given by the difference between the means \(\overline{\text{env}}^{\text{stim}}_{i}\) and \(\overline{\text{env}}^{\text{ref}}_{i}\) (top panel). Both the phase and amplitude responses are later normalised by the number of pulses in the block (not shown here)
The dominant axis tremor acceleration recordings are bandpass-filtered (4 Hz band encompassing the patient tremor frequency content) and z-scored. The filter used is a Butterworth second order filter, which provides a maximally flat response in the passband [28]. Because this study focuses on phase, we perform zero-phase filtering by applying our filter in the forward and backward directions to avoid phase distortions. Since the resulting signal is narrow-band, the instantaneous phase \(\phi(t)\) and amplitude \(\mathrm{env}(t)\) are obtained as the Hilbert phase and amplitude (also called Hilbert envelope) of the processed tremor acceleration. The Hilbert phase and amplitude are given by the phase and the modulus of the analytic signal, respectively. The analytic signal is complex valued, and its real part is the signal (here processed tremor acceleration), while its imaginary part is the Hilbert transform of the signal. In short, we have \(\operatorname{sig}(t) + \mathcal{H}(\operatorname{sig}(t)) = \mathrm{env}(t) e^{i \phi(t)}\), where \(\operatorname{sig}(t)\) is the processed tremor acceleration and \(\mathcal{H}\) denotes the Hilbert transform.
Obtaining the change in phase (bPRC)
For each block (we index blocks by the subscript i), a straight line \(\widehat{\phi}^{ \mathrm{ref}}_{i}(t)\) is fitted to the evolution of the Hilbert phase \(\phi_{i}(t)\) during the 1 s period without stimulation before the block (reference period; see middle panel in Fig. 1). The change in phase \(\Delta\phi_{i}\) due to block i is given by the difference between the actual Hilbert phase at the end of the block and the phase of the fitted reference line evaluated at the end of the block (see middle panel in Fig. 1), i.e.
$$ \Delta\phi_{i} = \phi_{i}\bigl(t^{\mathrm{end}}_{i} \bigr) - \widehat{\phi}^{ \mathrm{ref}}_{i}\bigl(t^{\mathrm{end}}_{i} \bigr), $$
where \(t^{\mathrm{end}}_{i}\) is the time of the end of block i. This phase response is divided by the number of pulses in blocks \(n_{ \mathrm{pulses}}\) (on the basis of four pulses per burst for patient 4R and 4L, and six pulses per burst for the rest), which gives an average response for one pulse. The target phase at which stimulation is supposed to occur is known for each block, but phase tracking not being perfect, the actual Hilbert phase at which stimulation occurred is determined for each burst of stimulation as the circular mean of the Hilbert phase during the burst (unlike in the original study [11] where target phase is directly used). We take the circular mean of these burst angles for a given block as the actual mean phase of stimulation for the block, and denote it \(\varPhi_{i}^{\mathrm{stim}}\) for block i. These values are then binned into 12 phases bins, and the change in phase is averaged within bins to obtain the bPRC. Put another way,
$$ \mathrm{bPRC}\bigl(\varPhi_{j}^{\mathrm{bin}}\bigr) = \frac{1}{n_{\mathrm{pulses}}n_{\mathrm{bin}_{j}}}\sum_{\varPhi_{i}^{ \mathrm{stim}}\in\mathrm{bin}_{j}} \Delta \phi_{i}, $$
where \(\varPhi_{j}^{\mathrm{bin}}\) is the center phase of bin j, and \(n_{\mathrm{bin}_{j}}\) is the number of blocks with \(\varPhi_{i}^{\mathrm{stim}}\) falling in \(\mathrm{bin}_{j}\).
Obtaining the change in amplitude (bARC)
For each block i, the change in amplitude \(\Delta\mathrm{env}_{i}\) is given by the difference between the mean of the Hilbert amplitude during the last second of the block \(\overline{\mathrm{env}}^{\mathrm{stim}}_{i}\) and the mean of the Hilbert amplitude during the one second without stimulation before the block \(\overline{\mathrm{env}}^{\mathrm{ref}}_{i}\) (see top panel in Fig. 1):
$$ \Delta\mathrm{env}_{i} = \overline{\mathrm{env}}^{\mathrm{stim}}_{i} - \overline{\mathrm{env}}^{\mathrm{ref}}_{i}. $$
Similarly to the change in phase, this amplitude response is divided by the number of pulses in the block, and averaged across blocks in the same phase bin to obtain the bARC. Explicitly, we have
$$ \mathrm{bARC}\bigl(\varPhi_{j}^{\mathrm{bin}}\bigr) = \frac{1}{n_{\mathrm{pulses}}n_{\mathrm{bin}_{j}}}\sum_{\varPhi_{i}^{ \mathrm{stim}}\in\mathrm{bin}_{j}} \Delta \mathrm{env}_{i}. $$
Measuring response curves significance and PRC-ARC phase shift
In order to identify significant patient's response curves, we performed two statistical analyses. First, bPRCs and bARCs were tested for a main effect of phase by means of a Kruskal–Wallis ANOVA (12 phase bins) to differentiate patients' response curves that may be dominated by noise (which could be due to a lack of phase-dependent response or our inability to measure it, possibly because of an insufficient amount of data). Second, since we are expecting response curves to have a dominant first harmonic, the cosine model \(y = c_{1} + \vert c_{2} \vert\cos(x+c_{3})\) was fitted to patients' phase and amplitude response curves. We assessed via F-tests whether the cosine model was better at describing the data than a horizontal line at the mean (\(y = c_{1}\), where \(c_{1}\) is the mean change in phase or the mean change in amplitude). Including the less specific ANOVA test allows for more generality, as we do not wish to exclude patients with significant, but non-sinusoidal response curves. On the other hand, the cosine test is more likely to detect phase-dependent effects of stimulation in patients which indeed have sinusoidal response curves. We therefore define the following criterion for selection of a patient for further study in the rest of the manuscript.
Significance criterion
Having both bPRC and bARC deemed significant under FDR control (see below) by at least one of the two tests—ANOVA test for a main effect of phase or cosine model F-test.
In both cases, we address the multiple testing problem by controlling the false discovery rate (FDR) at 5%, which guarantees that the expectation of the number of false positives over the total number of positives is less than 5%. Because of the high number of rejections of the null hypotheses compared to the number of tests (5 out of 12 for the ANOVA, 6 out of 12 for the F-test; see Table 1), the total number of tests is a very poor estimator of the number of true null hypotheses, which is needed when controlling the FDR. Instead, we used a better estimator \(\hat{m}_{0}\) of the number of true null hypothesis given by Story et al. [29], and applied an FDR control procedure based on this estimator (adaptive linear step-up procedure, reviewed in [30]).
Table 1 P-values of both statistical tests performed on patients' response curves: Kruskal–Wallis ANOVAs testing a main effect for phase in patients' response curves (third column), and cosine model F-tests (fourth column). P-values in bold are deemed significant with FDR control at the 5% level (separate FDR analyses per test type, \(\hat{m_{0}} \approx8.42\) for the ANOVAs and \(\hat{m_{0}} \approx7.37\) for the F-tests). Double stars indicate datasets satisfying our significance criterion as defined in Sect. 2.1
Additionally, in datasets where both bPRC and bARC are significant according to the cosine F-test, the relationship between bPRC and bARC is quantified by the shift in phase between the cosine model fits to the bPRC and the bARC. In these datasets, the PRC-ARC shift between the bPRC and bARC is calculated as
$$ \phi_{\mathrm{PRC}}-\phi_{\mathrm{ARC}} \equiv c_{3}^{\mathrm{PRC}}-c_{3}^{\mathrm{ARC}} \pmod{2\pi}, $$
with \(\phi_{\mathrm{PRC}}-\phi_{\mathrm{ARC}} \in [0,2\pi )\). Calculating a PRC-ARC shift in other cases is not meaningful. The PRC-ARC phase shift is an important quantity. Indeed, for PRCs and ARCs with a dominant first harmonic (close to sine curves), the ARC will be close to a scaled version of the PRC shifted in phase. The extent of the shift is given by the PRC-ARC phase shift. In other words, the minimum of the ARC (best phase to stimulate) will be at the minimum of the PRC plus the PRC-ARC shift. The shift highlights the difference in the phases of maximum sensitivity of the system in terms of its phase response and in terms of its amplitude response. As we will see later, the PRC-ARC shift will be a key differentiator between the nonlinear WC model and its linearisation in terms of their ability to describe the effects of phase-locked stimulation seen in data.
Results of the analysis
Analysing six datasets from the five patients included in [31] (datasets 4R and 4L are for the right and left upper limbs of the same patient) shows that half of the datasets satisfy our significance criterion. bPRCs and bARCs obtained are shown in Supplementary Fig. 1 in Appendix I, and results of the statistical tests are presented in Table 1. Based on the significance criterion defined in the previous section, patients 1, 5 and 6 are selected for further study, as both their bPRCs and their bARCs are found to be significant by the cosine F-test under FDR control. We note that patient 5 also has both his response curves deemed significant by the ANOVA test under FDR control. Datasets 3, 4R and 4L do not satisfy our selection criterion. In other words, for both tests, an effect of stimulation phase could not be found in at least one of their response curves (in most cases for both response curves, as seen in Table 1). In Fig. 2, the PRC-ARC shift \(\phi_{\mathrm{PRC}}-\phi_{\mathrm{ARC}}\) is plotted for patients for whom the cosine model was deemed significant in describing both their bPRC and bARC (which happens to be the same subset as patients satisfying our significance criterion). Figure 2 shows that the PRC-ARC shift in significant datasets is in \([\frac{\pi}{2},\pi ]\), patients 5 and 6 being quite close to \(\frac{\pi}{2}\).
PRC-ARC shift in patients. Only showing patients with significant cosine model F-test for bPRC and bARC under FDR control. The calculated PRC-ARC shifts are in \([\frac{\pi}{2},\pi ]\)
Implementation of the Wilson–Cowan model for essential tremor DBS
To model the experimental data described in the previous section, in particular the shape of the response curves and the PRC-ARC shift, we use a WC model that describes the interaction between an excitatory and an inhibitory population of neurons. Specifically, we map a two-population WC model without delays as described in [32] onto the anatomy of the thalamus (Fig. 3). The circuit we are about to describe is a good candidate, but not the only biologically plausible mapping of an excitatory/inhibitory loop in the context of tremor. In our candidate mapping, the VIM is modelled as an excitatory population, connected to an inhibitory population of the thalamus, the reticular nucleus (nRT). We model tremor by the activity of the excitatory population, and this is justified by the high coherence between ventral thalamic activity and electromyographic recordings of the contralateral wrist flexors [12, 13]. VIM and nRT are reciprocally connected (the excitatory projections from VIM to nRT are via Cortex). The VIM receives a constant input from the deep cerebellar nuclei (DCN) and is part of a self-excitatory loop via Cortex. nRT receives a constant cortical input. We add Gaussian white noise to this two-population WC, and the activity of the VIM, E, and the activity of the nRT, I, are described by the stochastic differential equations
$$ \textstyle\begin{cases} dE =F_{1}(E,I)\,dt + \zeta \,dW_{E}, \\ dI =F_{2}(E,I)\,dt + \zeta \,dW_{I}, \end{cases} $$
where \(dW_{E}\) and \(dW_{I}\) are Wiener processes, and ζ the noise standard deviation. We define
$$\begin{aligned} & F_{1}(E,I) =\frac{1}{\tau} \bigl(-E+f( \theta_{E}+w_{EE}E-w_{IE}I) \bigr), \\ & F_{2}(E,I) =\frac{1}{\tau} \bigl(-I+f( \theta_{I}+w_{EI}E) \bigr), \end{aligned}$$
with \(w_{\mathrm{PR}}\) the weight of the projection from population "P" to population "R", \(\theta_{P}\) the constant input to population "P", and τ a time constant (assumed to be the same for both populations). We use a sigmoid function,
$$\begin{aligned} & f(x) = \frac{1}{1+e^{-\beta(x-1)}}, \end{aligned}$$
parametrised by a steepness parameter β (same choice as in [32]). The VIM is the most common target of DBS for ET, which is why we model stimulation as a direct increase in E. Analytical expressions for response curves are out of reach for the full nonlinear model, which is why we study next a linearisation of a deterministic stable focus model to approximate the full model response and get a better understanding of the shape of its phase response curves and their relationship. This will provide a first level of description of the data.
The WC model can describe the populations thought to be involved in the generation of ET. The excitatory population E and the inhibitory population I model, respectively, the VIM and the nRT of the thalamus. Arrows denote excitatory connections or inputs, whereas circles denote inhibitory connections. The VIM is the target of DBS and also receives an input from the deep cerebellar nuclei (DCN). The self-excitatory loop of the VIM, as well as the excitatory connection from VIM to nRT are via cortex
Response curves and their relationship in a focus model
This section aims to provide a basis for understanding how the effects of stimulation on phase and amplitude are coupled in the WC model, and for comparison with experimental data. We therefore derive approximate analytic expressions for the first order phase and amplitude responses to one pulse of stimulation in the linearisation of a 2D dynamical system that is described by a (stable) focus. Such a linearisation can be applied to the deterministic WC model given by Eq. (6) with \(\zeta=0\) in the focus regime. We follow the previous section in modelling the tremor signal as the first coordinate of the dynamical system, and in providing stimulation pulses along the first dimension.
Linearisation of a focus
To distinguish scalars and vectors more easily, vectors will be denoted in bold. Let \(\dot{\mathbf {Z}}=F(\mathbf {Z})\) be a dynamical system, where \(\mathbf {Z}\in\mathbb{R}^{2}\) and F is differentiable. The Jacobian of F is
$$ J = \begin{bmatrix} \frac{\partial F_{1}}{\partial Z_{1}} & \frac{\partial F_{1}}{\partial Z_{2}} \\ \frac{\partial F_{2}}{\partial Z_{1}} & \frac{\partial F_{2}}{\partial Z_{2}} \end{bmatrix} . $$
Let \(\mathbf {Z^{*}}\) be a fixed point of F. If it is hyperbolic, the dynamics of \(\mathbf {X}=\mathbf {Z}-\mathbf {Z^{*}}\) are approximated in the vicinity of the equilibrium \(\mathbf {X} = \mathbf {0}\) by the linear equation
$$ \dot{\mathbf {X}} = J\bigl(\mathbf {Z^{*}}\bigr)\mathbf {X}, $$
where \(J(\mathbf {Z^{*}})\) is the Jacobian evaluated at the fixed point. We will treat the case of Jacobians having complex conjugate eigenvalues \(\lambda_{\pm}=\sigma\pm i\omega\). In particular, we are interested in stable hyperbolic foci, which imply \(\sigma< 0\) and \(\omega> 0\). The WC model can operate in that regime [32]. The nonhyperbolic case of the linearisation having purely imaginary complex conjugate eigenvalues will also be described for didactic purposes, although it is of little interest for patient fits. If \(\mathbf {k}=\mathbf {a}+i\mathbf {b}\) is the right eigenvector associated with \(\lambda_{+}\), and K and \(K'\) coefficients determined according to initial conditions, the general real valued solution of (8) reads
$$ \mathbf {X}(t)= \bigl\{ K (\mathbf {a}\cos{\omega t}-\mathbf {b}\sin{\omega t} ) +K' (\mathbf {a}\sin{\omega t}+\mathbf {b}\cos{\omega t} ) \bigr\} e^{\sigma t}. $$
We will be using the following notations for the coordinates of the eigenvector:
$$ \mathbf {k} = \begin{bmatrix} a_{1} + i b_{1} \\a_{2} + i b_{2} \end{bmatrix} . $$
Equation (9) and what follows are not valid in the case of real eigenvalues, which are of no interest for our purposes (no rotation).
Phase definition
The notion of phase is central to phase-locked stimulation, and in this section we define phase in a way that is approximately equivalent to the Hilbert phase, which is commonly used in the analysis of experimental data, and is used in the other sections of this manuscript. A typical signal only has one component, and the Hilbert transform provides a convenient way of reconstructing a phase from a single component. Despite being a protophase (see discussion section), the Hilbert phase is widely used to analyse experimental data (see for instance [9, 11, 33–35]), and this is the reason why we choose in our linearised system a phase definition approximately equivalent to it. We define a phase as \(\phi=\omega t\) with a zero-phase point defined as the maximum of \(X_{1}(t)\) (similarly to the Hilbert phase), which is therefore on the nullcline of the first coordinate. This phase definition is different from other common definitions such as the trajectory polar angle in the phase plane of a 2D system, or isochronal (asymptotic) phase. We demonstrate next that it is very close to the Hilbert phase of \(X_{1}\) for slow decay compared to the rotation (this condition is verified in patient fits presented in Sect. 5.2; see Supplementary Table 2). It should be noted that this is generally only true for the linearisation. As the Hilbert phase is also the phase definition used in the other sections of this manuscript, the following proof ensures consistency.
We now establish equivalence of our phase definition with the Hilbert phase of \(X_{1}\). Recall that we denote the Hilbert transform by \(\mathcal{H}\). The Hilbert phase of \(X_{1}\) is given by
$$ \phi^{\mathrm{Hilbert}}=\arctan{\frac{\mathcal{H}(X_{1}(t))}{X_{1}(t)}}. $$
A first step is to calculate the Hilbert transform of the signal \(X_{1}(t)\). The Hilbert transform is a linear operator, and \(X_{1}(t)\) is a linear combination of \(s(t)s_{c}(t)\) and \(s(t)s_{n}(t)\) with \(s(t)=e^{\sigma\vert t \vert}\), \(s_{c}(t)=\cos{\omega t}\), and \(s_{n}(t) = \sin{\omega t}\) (see Eq. (9)). We show in Appendix A that the Hilbert transform \(\mathcal{H}(s(t)s_{j}(t))\) can be approximated by \(s(t) \mathcal{H}(s_{j}(t))\) for \(j=c,n\). The Hilbert phase of \(X_{1}\) is therefore given by
$$ \phi^{\mathrm{Hilbert}}=\arctan{\frac{\mathcal{H}(X_{1}(t))}{X_{1}(t)}} \approx\arctan{ \frac{\sqrt{\alpha^{2}+\beta^{2}} \sin (\omega t - \arctan {\frac{\alpha}{\beta}} )}{\sqrt{\alpha^{2}+\beta^{2}}\sin (\omega t + \frac{\pi}{2}- \arctan{\frac{\alpha}{\beta }} )}}, $$
$$\begin{aligned}& \alpha = K' a_{1} - K b_{1}, \\& \beta = K a_{1} + K' b_{1}. \end{aligned}$$
Using trigonometric identities, we obtain
$$ \phi^{\mathrm{Hilbert}} \approx\omega t - \arctan{ \frac{\alpha }{\beta}}. $$
In our setting, trajectories start at \(t=0\) at the maximum of \(X_{1}(t)\), and we have \(\frac{\alpha}{\beta}=-\frac{\sigma}{\omega}\) (immediate with the coefficients of the reference trajectory \(K_{\mathrm{ref}}\) and \(K'_{\mathrm{ref}}\) introduced in Eq. (14) and given in Appendix B). Hence if \(\omega\gg\vert\sigma\vert\), Eq. (13) yields \(\phi^{\mathrm{Hilbert}} \approx\omega t\), which matches with our definition of phase ϕ (including our choice of zero-phase reference).
Reference trajectory and stimulated trajectory
In order to calculate first order response curves for our phase definition, we will consider a reference trajectory without stimulation, and a trajectory that underwent an instantaneous stimulation pulse \(\delta X_{1}\) at a stimulation phase \(\phi_{0}\). The effects of stimulation on phase and amplitude will be measured at the next maximum of \(X_{1}\) for both trajectories. We will denote these \(\mathrm{hPRC}^{(1)}\) and \(\mathrm{hARC}^{(1)}\) as they are first order responses based on a phase definition approximately equivalent to the Hilbert phase. A sketch of the method is provided in Fig. 4.
Illustration of the approach taken to derive expressions for the phase and amplitude responses in the linearisation of a 2D focus model. Top: phase plane, bottom: time series of \(X_{1}\). The tremor is modelled by \(X_{1}\), and the stimulation \(\delta X_{1}\) is applied to \(X_{1}\). The system shown corresponds to the linearised fit of patient 1 as described in Sect. 6.1
Expressions for the coefficients \(K_{\mathrm{ref}}\) and \(K'_{\mathrm{ref}}\) of the reference trajectory are derived in Appendix B. We want to study the effects of stimulating at phase \(\phi_{0}\). The point of stimulation \(\mathbf {X^{1^{-}}}\) at phase \(\phi_{0}\) is expressed as
$$ \mathbf {X^{1^{-}}}= \bigl\{ K_{\mathrm{ref}} (\mathbf {a} \cos{\phi _{0}}-\mathbf {b} \sin{\phi_{0}} ) +K'_{\mathrm{ref}} (\mathbf {a}\sin{\phi_{0}}+ \mathbf {b} \cos{\phi_{0}} ) \bigr\} e^{\sigma \frac{\phi_{0}}{\omega}}. $$
An instantaneous stimulation \(\delta X_{1}\) is applied at \(\mathbf {X^{1^{-}}}\) as
$$ \mathbf {X^{1^{+}}}=\bigl(X_{1}^{1^{+}},X_{2}^{1^{+}} \bigr)=\bigl(X_{1}^{1^{-}}+\delta X_{1},X_{2}^{1^{-}} \bigr). $$
The trajectory after stimulation is still constrained by the dynamics given by Eq. (9), which allows for expressions for the coefficients on this new trajectory \(K_{\mathrm{stim}}\) and \(K'_{\mathrm{stim}}\) to be found (see Appendix C). To measure the change in phase and amplitude between the next peaks of the stimulated trajectory and the reference trajectory, the phase \(\phi_{\mathrm{max}}\) of the next maximum of the first coordinate on the stimulated trajectory \(X_{1}^{\mathrm{stim}}\) is needed (the phase of the next maximum of \(X_{1}\) on the reference trajectory is 2π). A derivation for \(\phi_{\mathrm{max}}\) is provided in Appendix D.
The first order phase response curve can be calculated based on the reference trajectory period \(T_{0}\) and the stimulated trajectory period \(T_{\mathrm{stim}}\), which is given by the sum of the time spent on the reference trajectory before stimulation and the time spent on the new trajectory after stimulation:
$$\begin{aligned}& T_{0} = \frac{2\pi}{\omega}, \\& T_{\mathrm{stim}} = \frac{(\phi_{0} - 0)+(2\pi+ \phi_{\mathrm{max}} - \phi_{0})}{\omega } = \frac{2\pi+ \phi_{\mathrm{max}}}{\omega}. \end{aligned}$$
For a phase response curve in radians, we obtain
$$ \mathrm{hPRC}^{(1)}(\phi_{0}) = 2 \pi\frac{T_{0} - T_{\mathrm {stim}}}{T_{0}} = - \phi_{\mathrm{max}}{(\delta X_{1})}, $$
where the phase \(\phi_{\mathrm{max}}\) depends on the stimulation magnitude \(\delta X_{1}\) (see Eq. (15)). The dependency enters through \(K_{\mathrm{stim}}\) and \(K'_{\mathrm{stim}}\) (see Eq. (41) in Appendix D and Eqs. (37) and (39) in Appendix C). A Taylor expansion around \(\delta X_{1} = 0\) yields, to lowest order in \(\delta X_{1}\) (for weak stimulation),
$$ \mathrm{hPRC}^{(1)}(\phi_{0}) \approx\frac{\delta X_{1}}{X_{1}^{0}} (A\cos{\phi_{0}}-B\sin{ \phi_{0}} ) C e^{-\sigma \frac{\phi_{0}}{\omega}} $$
$$\begin{aligned}& A = (a_{1} a_{2} + b_{1} b_{2})\omega-(a_{1} b_{2} - a_{2} b_{1}) \sigma, \\& B = (a_{1} b_{2} - a_{2} b_{1})\omega+(a_{1} a_{2} + b_{1} b_{2}) \sigma, \\& C = \frac{\omega}{(\omega^{2}+\sigma^{2})(a_{1} b_{2} - a_{2} b_{1})}. \end{aligned}$$
Although we are omitting the amplitude dependence in our notations for convenience in Eqs. (16) and (17), the first order PRC is found to be proportional to the inverse of the peak amplitude of the oscillations at the beginning of the stimulation period \(X_{1}^{0}\). It is also directly proportional to the stimulation amplitude \(\delta X_{1}\), and directly depends on phase via sinusoidal functions and a factor related to the decay. But unlike in the cosine test (Sect. 2), no assumption was made on a dominant first harmonic in our derivation. The constants A, B, and C only depend on the real and imaginary parts of the eigenvalue \(\lambda_{+}\) (decay and rotation) and the associated eigenvector k.
Amplitude response
For our purposes we are interested in the amplitude of the first coordinate, and the first order ARC is obtained as the difference in first coordinates between the stimulated and the reference trajectories evaluated at their respective next peak after stimulation. It should be noted this is approximately equivalent to a first order change in Hilbert amplitude, at least for \(\omega\gg\vert\sigma\vert\). The first order ARC is calculated as
$$ \mathrm{hARC}^{(1)}(\phi_{0}) = X_{1}^{\mathrm{stim}} \biggl( \frac{2\pi+\phi_{\mathrm{max}}{(\delta X_{1})}}{\omega} \biggr)-X_{1}^{\mathrm{ref}} \biggl( \frac{2\pi}{\omega} \biggr). $$
A Taylor expansion around 0 yields, to lowest order in \(\delta X_{1}\),
$$ \mathrm{hARC}^{(1)}(\phi_{0}) { \approx} \delta X_{1} (\cos{\phi_{0}}+D \sin{ \phi_{0}} ) e^{-\sigma\frac{\phi_{0}-2\pi}{\omega}} $$
$$ D = \frac{a_{1} a_{2} + b_{1} b_{2}}{a_{1} b_{2} - a_{2} b_{1}}. $$
Interestingly, the first order ARC close to the fixed point does not depend on the amplitude of the oscillations \(X_{1}^{0}\). As expected, the first order ARC is directly proportional to the stimulation amplitude \(\delta X_{1}\). Similarly to the first order PRC, it directly depends on phase via sinusoidal functions and a factor related to the decay, and the constant D only depends on k. The obvious similarities between the first order PRC and ARC suggest there may be a relationship between the two.
Relationship between first order PRC and ARC
We seek a relationship involving the derivative of the first order PRC, which, based on Eq. (17), is given by
$$ -\frac{d\mathrm{hPRC}^{(1)}(\phi_{0})}{d\phi_{0}} \approx \frac{\delta X_{1}}{F X_{1}^{0}} ( \cos{ \phi_{0}} + G\sin{\phi_{0}}) e^{-\sigma\frac{\phi_{0}-2\pi}{\omega}} $$
$$\begin{aligned}& F = \frac{(a_{1} b_{2} - a_{2} b_{1}) (\omega^{2} +\sigma^{2} )}{ (a_{1} b_{2} - a_{2} b_{1}) (\omega^{2} -\sigma^{2} ) +2 (a_{1} a_{2} +b_{1} b_{2} )\omega\sigma} e^{\frac{2\pi\sigma}{\omega}}, \\& G = \frac{(a_{1} a_{2} + b_{1} b_{2})(\omega^{2}-\sigma^{2})-2(a_{1} b_{2} - a_{2} b_{1})\omega\sigma}{(a_{1} b_{2} - a_{2} b_{1})(\omega ^{2}-\sigma^{2})+2(a_{1} a_{2} + b_{1} b_{2})\omega\sigma}. \end{aligned}$$
For \(\omega\gg\vert\sigma\vert\), we have
$$\begin{aligned}& F = \frac{(a_{1} b_{2} - a_{2} b_{1}) (1+ (\frac{\sigma}{\omega } )^{2} )}{ (a_{1} b_{2} - a_{2} b_{1}) (1- (\frac{\sigma}{\omega} )^{2} ) +2 (a_{1} a_{2} +b_{1} b_{2} )\frac{\sigma}{\omega}} e^{\frac{2\pi\sigma}{\omega}} \approx1 - 2(D-\pi) \frac{\sigma}{\omega} \approx1, \\& G = \frac{(a_{1} a_{2} + b_{1} b_{2}) (1- (\frac{\sigma}{\omega } )^{2} )-2(a_{1} b_{2} - a_{2} b_{1}) (\frac{\sigma }{\omega} )}{(a_{1} b_{2} - a_{2} b_{1}) (1- (\frac {\sigma}{\omega} )^{2} )+2(a_{1} a_{2} + b_{1} b_{2}) (\frac{\sigma}{\omega} )} \approx D - 2\bigl(1+D^{2}\bigr) \frac{\sigma}{\omega} \approx D. \end{aligned}$$
Therefore in that case the first order ARC is approximately the opposite of the derivative of the first order PRC scaled by the peak amplitude at the beginning of the stimulation period (in general, the scaling factor is \(FX_{1}^{0}\)):
$$ - X_{1}^{0}\frac{d\mathrm{hPRC}^{(1)}(\phi_{0})}{d\phi_{0}} \approx \mathrm{hARC}^{(1)}(\phi_{0}). $$
For a slow decay compared to the rotation, the PRC-ARC shift in the linearisation of a focus will therefore be close to \(\frac{\pi}{2}\), which is the value observed for patient 5 (see Fig. 2). A detailed analysis of the PRC-ARC shift in the model is provided in Sect. 6.
Applications to simple systems
We turn to simple examples of linear systems to illustrate the results of the previous sections, in particular how the strength of the decay affects the sinusoidal character of the response curves and the PRC-ARC shift, and how a tilted ellipsoid flow impacts the response curves. Additionally, links to the WC model are provided when possible. In what follows, response curves are given for \(\delta X_{1} = 2 \times10^{-4}\) and \(X_{1}^{0} = 10^{-3}\), \(X_{1}^{0}\) being a maximum of \(X_{1}\) as a function of time (these only act as scaling factors of the response curves and will not change their shape).
Circular flow without decay
As an introductory example, let us consider a simple circular flow for which the J matrix is
$$ J_{\mathrm{circ}} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} . $$
The eigenvalues of \(J_{\mathrm{circ}}\) are ±i so the results of the previous sections can be applied. Equations (17) and (19) are plotted for this system with our choice of \(\delta X_{1}\) and \(X_{1}^{0}\). The result for the first order PRC is shown in Fig. 5, panel A2, and for the first order ARC in panel A3. For this system, \(\sigma= 0\), and \(\mathrm{hPRC}^{(1)}\) is simply the opposite of a sine, \(\mathrm{hARC}^{(1)}\) simply a cosine. Moreover, \(G = D\) (see Sect. 4.6) and Eq. (21) is exact, as exemplified in Fig. 5, panel A3. The amplitude response curve \(\mathrm{hARC}^{(1)}\) is obtained by only scaling the derivative of \(\mathrm{hPRC}^{(1)}\) by \(-X_{1}^{0}\) as \(a_{2} = b_{1} = 0\) and \(F = 1\). Note that WC parameters for which the system's Jacobian at the fixed point is \(J_{\mathrm{circ}}\) cannot be found as the second diagonal term cannot be 0, at least in the version of the WC model used in this work (see equation (44) in Appendix E).
Analytical results in simple systems (initial conditions as in the main text). First column: phase space. Second column: first order PRC as per Eq. (17) (scaling valid for the first cycle). Third column: first order ARC as per Eq. (19) and opposite of the derivative of the first order PRC scaled by \(FX_{1}^{0}\). Panel A corresponds to \(J_{\text{circ}}\) (circular flow, no decay), panel B to \(J_{\text{circ}}^{\text{slow}}\) (circular flow, slow decay), panel C to \(J_{\text{circ}}^{\text{fast}}\) (circular flow, fast decay), and panel D to \(J_{\text{ellip}}\) (tilted elliptic flow, no decay)
Circular flow with decay
We can introduce a slow decay (Fig. 5, panel B) and then a fast decay (Fig. 5, panel C) in the circular flow. We choose the J matrices
$$ J_{\mathrm{circ}}^{\mathrm{slow}} = \begin{bmatrix} -5{\times}10^{-3} & -1 \\ 1 & -5{\times}10^{-3} \end{bmatrix} , \qquad J_{\mathrm{circ}}^{\mathrm{fast}} = \begin{bmatrix} -2{\times}10^{-1} & -1 \\ 1 & -2{\times}10^{-1} \end{bmatrix} . $$
The slow decay leads to a scaling factor \(F \approx1\), and the approximation of Eq. (21) is very good, as \(\omega\gg\vert\sigma\vert\) (see Fig. 5, panel B3, close match of the ARC and the scaled derivative of the PRC, hence a shift close to \(\frac{\pi}{2}\)). The case of the fast decay corresponds to \(\omega= 5 \vert\sigma\vert\). The first order PRC and ARC no longer look like pure sinusoids and the approximation relating the response curves is less accurate (\(\omega= 200 \vert\sigma\vert\); see Fig. 5, panel C3), highlighting a shift different from \(\frac{\pi}{2}\). It is also more obvious that the first order response curves are not periodic due to the measurement of the changes in phase and amplitude at the end of the stimulation period. It is possible to find WC parameters for which the system's Jacobian at the fixed point is \(J_{\mathrm{circ}}^{\mathrm{slow}}\) or \(J_{\mathrm{circ}}^{\mathrm {fast}}\). How such parameters are found is explained in Appendix E, and the results are presented in Supplementary Table 1 in Appendix J. In both cases, \(w_{IE} = w_{IE}\), and \(w_{EE}=0\).
Tilted elliptic flow without decay
The tilted elliptic flow without decay of Fig. 5, panel D, corresponds to the J matrix
$$ J_{\mathrm{ellip}} = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} . $$
The first order PRC and ARC are sums of a sine and a cosine, which brings a horizontal shift in phase for both curves compared to a circular flow without decay. The eigenvalues are still purely imaginary, but F is no longer one. Because \(\sigma= 0\), the relationship of Eq. (21) is still exact (see Fig. 5, panel D3). It is possible to find WC parameters for which the system's Jacobian at the fixed point is \(J_{\mathrm{ellip}}\) (see Supplementary Table 1 in Appendix J). Patient fits fall in the category of (potentially tilted) elliptic flows with decay, and will be dealt with in Sect. 6.1.
For a slow decay compared to the rotation, the linearised stable focus model exhibits close to sinusoidal response curves and a PRC-ARC shift close to \(\frac{\pi}{2}\) as shown by Eq. (21). This is verified in Fig. 5, as the scaled first order PRC very closely match the ARC (except in panel C where the decay is fast). When contrasted with patient data (response curves passing the cosine model F test and PRC-ARC shifts in \([\frac{\pi}{2},\pi ]\) as shown in Fig. 2), these results already provide a good level of description of the data, but also a strong motivation to fit the more complex nonlinear WC model to data.
Fitting the full Wilson–Cowan model to patient data and response to phase-locked stimulation
With the insights on the linearised stable focus response curves given by the previous section in mind, and to provide a more accurate level of description of the data in particular in terms of PRC-ARC shift, we now turn to fitting our stochastic neural mass model introduced in Sect. 3 (Eq. (6)) to patient data. The model is fitted to features (also known as summary statistics) extracted from patient tremor recordings. The parameters we fit are shown in Table 2, and include model parameters, stimulation magnitude, and stimulation delay (time between when the stimulation trigger is recorded and when stimulation is actually provided to the E population, more about its interpretation in Sect. 7). Stimulation is implemented directly in the Euler update of our integration scheme. We aim at reproducing tremor dynamics and fit to three dynamical features: the power spectrum density (PSD) of the data, its Hilbert envelope probability density function (PDF), and its Hilbert envelope PSD. While the envelope PDF captures the range of amplitudes present in the tremor, the envelope PSD describes how quickly tremor amplitude changes. But we also aim at reproducing response to stimulation, and fit to the patient bPRC. The data dynamical features are obtained after filtering and z-scoring the data as described in Sect. 2.1. The data bPRC is obtained as described in Sect. 2.1.
Table 2 Best parameters for the three fitted patients
The fitting procedure is summarized in Fig. 6. Local optimisations are carried out using gradient free optimisation, specifically a direct search algorithm called the generalized pattern search algorithm (more details are given in Appendix F). In order to measure response to stimulation as in the data, each local optimisation step needs to simulate the model with phase-locked blocks of stimulation. This requires integrating the differential equations of the model while tracking the phase and providing stimulation at the right time, which is done by monitoring the zero-crossing phase alongside a Euler–Maruyama integration scheme. Appendix G details implementation of the simulator. The four features (PSD, envelope PDF, envelope PSD, bPRC) are computed on the model output at each optimisation step. The same method is used as for the data features, with three differences. The first is that for increased stability of the optimisation, the model bPRC is averaged over a much greater number of trials (600 trials), while the more robust dynamical features are obtained from nine trials only to reduce computation cost. The second is that the model output is not filtered to compute the dynamical features (only z-scored), as we want the model output to primarily generate the filtered tremor signal (a model generating mostly 1 Hz oscillations but reproducing patient tremor when filtered at 5 Hz would not be desirable). Computing the bPRC still requires filtering, as it relies on the Hilbert transform. The third difference is that the filtering window for the bPRC cannot be adjusted manually in optimisation steps, so a 4 Hz band centered on the model PSD peak is used. As for the data bPRC and bARC, the actual Hilbert phase at which stimulation occurred is used to compute response curves via the re-binning process described in Sect. 2.1, and the zero-crossing phase is only needed to enable phase-locked stimulation in the model. Phase-tracking performance is illustrated in Supplementary Fig. 2 in the Appendix.
The fitting procedure involves 2500 local optimisations for each patient. The simulation of the model at each optimisation step requires one to track the zero-crossing phase in order to provide stimulation at the right phase. The phase-tracking ability of the scheme is satisfactory when compared to the actual Hilbert phase (left, detailed in Supplementary Fig. 2 in Appendix I). The optimiser minimises a cost function that includes the comparison of three tremor dynamics features (tremor PSD, tremor envelope PSD, tremor envelope PDF) plus the bPRC against the data (middle). Response curves are obtained the same way for the data and the model. Following a second optimisation of the 20 best results with a finer time step, a best set of parameters comes out of the procedure, and the model bARC can be compared against the data bARC. More details on the fitting procedure are given in Appendix F
At each step, once the four features have been computed on the model output, the optimiser returns the cost
$$ c = \frac{1}{4}\sum_{n = 1}^{4} \biggl( \frac{\sum_{i = 1}^{N_{n}} (y_{n,i}^{\mathrm {data}}-y_{n,i}^{\mathrm{model}} )^{2}}{\sum_{i = 1}^{N_{n}} (y_{n,i}^{\mathrm{data}}-\overline{y_{n}^{\mathrm{data}}} )^{2}} \biggr), $$
with \(y_{n}, n \in\{1, 2, 3, 4\}\) being the four features considered, \(N_{n}\) the length of \(y_{n}\), and \(\overline{y_{n}^{\mathrm{data}}}\) the mean of data feature n. At the end of the procedure, the fit with the highest \(R^{2} = 1 - c\) for each patient is deemed the best fit. In the case of a tie (difference in mean costs lower than standard error of the mean), foci are preferred over limit cycles. The bifurcation structure of the original WC model has been studied in [36], but we simply differentiate between parameters giving rise to stable foci and limit cycles by forward simulating the model without noise, and exploring the region of phase space that is occupied by the system with noise.
Results of the fits
Patients with both of their response curves statistically significant (see significance criterion in Sect. 2), in other words with meaningful response curves, are fitted to. For these patients, namely patient 1, 5, and 6, we find that the model successfully reproduces tremor dynamics, including tremors with sudden bursts, and can fit to patient phase response to stimulation. The best fits obtained upon completion of the optimisation procedure are shown in Figs. 7, 8, and 9. In addition to reproducing tremor dynamics and being able to fit to patient bPRCs, the model seems to be able to reasonably predict patient bARCs (obtained as in Sect. 2.1, but not fitted to), and in particular which phases are approximately the best phases to stimulate, i.e. the phases at which the maximum decrease in tremor happens. Because of averaging across 600 trials, the model bPRC and bARC error bars are small compared to the data error bars (only about 10 trials per phase bin).
Best fit to patient 1. The four features that were included in the cost function are shown on the left, namely tremor PSD (A), tremor envelope PDF (B), tremor envelope PSD (C) and bPRC (D). The \(R^{2}\) for the model fit to these features is 0.795, and the model reasonably predicts the data bARC (E). The model phase plane is shown in (H), and the model tremor time series (F) is shown next to the patient tremor time series (G). The framed black bar in (H) indicates the fitted stimulation magnitude to scale
Best fit to patient 5. The four features that were included in the cost function are shown on the left, namely tremor PSD (A), tremor envelope PDF (B), tremor envelope PSD (C) and bPRC (D). The \(R^{2}\) for the model fit to these features is 0.823, and the model predicts the data bARC (E). The model phase plane is shown in (H), and the model tremor time series (F) is shown next to the patient tremor time series (G). The framed black bar in (H) indicates the fitted stimulation magnitude to scale
Validating fitted stimulation magnitude
As Cagnan et al. [11] report what the device settings are, and in particular the total electrical energy delivered (TEED) per unit time for each patient, we can validate fitted stimulation magnitudes against these values. We build an equivalent quantity Ξ for the model that we call "model effective stimulation per unit time", and that should scale with the TEED per unit time. We define Ξ as
$$ \varXi= \frac{\delta E}{E_{\sigma}}\overline{f_{E}}, $$
where \(E_{\sigma}\) is the standard deviation of the non z-scored first dimension of the model output, and \(\overline{f_{E}}\) is the mean frequency of the first dimension of the model output. Since stimulation in the model is a direct increase in E, δE should be scaled the same way, which is the purpose of the division by \(E_{\sigma}\). And since bursts are delivered once per period, the multiplication by \(\overline{f_{E}}\) ensures that Ξ is defined per unit time (the number of pulses per burst is the same for the three patients). Figure 10 shows the model effective stimulation per unit time for the 15 best performing fits against the TEED per unit time for each patient (correlation coefficient for fit averages \(r = 0.98\)). Under the assumption that patient intrinsic sensitivities to stimulation are somewhat similar, we can conclude from the correlation that the fitting procedure successfully captures the scale of stimulation across patients.
Model effective stimulation per unit time Ξ versus total electrical energy delivered per unit time by the device, for the three fitted patients. Showing the 15 best performing models for each patient, along with the mean and standard error of the mean error bars for each patient in black
PRC-ARC shift in WC synthetic data
The PRC-ARC shift is computed on WC synthetic data with phased-locked blocks of stimulation generated by the full model fitted to each patient. This time we can take full advantage of the model and compute bPRCs and bARCs from more trials than for patient data or model data in optimisation steps, and perform 10 repeats of 600 trials for the top 15 fits for each patient. The PRC-ARC shift is then measured as in Sect. 2.1 for each of the 10 repeats, and shown in Fig. 11. The large filled circles represent the mean of the 10 repeats for each patient fit. It appears that PRC-ARC shifts obtained for synthetic data of top patient fits mostly lie in the upper-left quadrant of the unit circle for all three patients \(( [\frac{\pi}{2},\pi ] )\), similarly to patient data. One fit to patient 6 is an outlier in terms of its shift, due to high model effective stimulation (defined in the previous section). While the nonlinear model can allow for a larger shift than \(\frac{\pi}{2}\), this is not the case for the linearised model, and the difference is the focus of the next section.
PRC-ARC shift in synthetic data (full WC model fitted to patients). For each patient, the shift for all 10 repeats of the top 15 fits is shown (smaller circles), as well as the repeat mean for each fit (larger circles). One repeat corresponds to 600 trials
PRC-ARC shift in the model
The analytical expressions for the linearised model make different predictions for patient response curves than synthetic data generated by the full model and analysed with the block method, in particular in terms of PRC-ARC shift. The present section will look at the deterministic linearisation of patient fits, and then contrast it with the full model with noise.
Relationship between analytic response curves in the linearised fitted WC models
The first order PRC and ARC expressions derived in Sect. 4 can be applied to the linearisation of the best WC models fitted to data from the three patients satisfying our significance criterion. The Jacobians at the fixed points are
$$\begin{aligned}& J_{1} = \begin{bmatrix} 11.9723 & -35.0323 \\ 34.9513 & -13.1953 \end{bmatrix} ,\qquad J_{5} = \begin{bmatrix} -0.2252 & -52.3293 \\ 23.2880 & -3.3351 \end{bmatrix} , \\& J_{6} = \begin{bmatrix} 2.8269 & -12.8784 \\ 101.6943 & -3.9789 \end{bmatrix} , \end{aligned}$$
where \(J_{i}\) corresponds to patient i. In the fits \(b_{1} = 0\) or \(b_{2} = 0\), which marginally simplifies Eqs. (17) and (19). The response curves obtained are shown in Fig. 12. The same values as in Sect. 4.7 are used for \(X_{1}^{0}\) and \(\delta X_{1}\). Note that the stimulation delay \(\Delta t_{\mathrm{stim}}\) is not shown—it affects both the PRC and the ARC and does not play a role in the PRC-ARC shift. More interestingly, we observe that \(\omega\gg\vert\sigma\vert\) in the 3 fits (see Supplementary Table 2 in Appendix J), suggesting that the response curves' relationship described by Eq. (21) should approximately hold. This is indeed the case as shown in the third column of Fig. 12, which tells us that the PRC-ARC shift should be close to \(\frac{\pi}{2}\). The decay is higher for patient 5 (about 5% of the rotation versus less than 2% for the other two patients) and as expected, the approximation is slightly worse for this patient (panel B3 in Fig. 12). For small stimulation and close to the fixed point, the deterministic picture with patient parameters is that the PRC-ARC shift should be close to \(\frac{\pi}{2}\). In what follows, we investigate the difference between this idealised picture and what is observed in synthetic data.
Analytical results for linearised patient fits (initial conditions as in the main text). First column: phase space. Second column: first order PRC as per Eq. (17) (scaling valid for the first cycle). Third column: first order ARC as per Eq. (19) and opposite of the derivative of the first order PRC scaled by \(FX_{1}^{0}\). Panel A, B, and C correspond to patient 1, 5, and 6, respectively
Accounting for the difference in shift between focus model analytic expressions and WC synthetic data
Four factors could account for the difference in PRC-ARC shift between the idealised picture given by analytic response curves with patient parameters (previous section) and what is observed in WC synthetic data (Sect. 5.2). First, the stimulation may be large enough that the Taylor expansions used to derive the analytic PRC and ARC expressions are no longer approximately valid. Second, tremor in patient fits may correspond to a regime where trajectories are not so close to the fixed point, compromising the linearisation validity. Third, the introduction of noise in the model may result in effects on the PRC-ARC shift that do not average out to zero. Fourth, in synthetic data, the response to stimulation is measured by the block method, which differs from the first order approach taken in our derivations. We next show that for the three best fits considered, nonlinearity is the main driver.
Ten repeats of 600 trials of synthetic data are generated for the linearisation of the best fits to each patient. The integration scheme with live phase tracking and stimulation is the same as described in Sect. 5.1, only the stochastic differential equations are now
$$ \begin{bmatrix} dE \\dI \end{bmatrix} = J \begin{bmatrix} E - E^{*} \\I - I^{*} \end{bmatrix} \,dt + \zeta \begin{bmatrix} dW_{E} \\dW_{I} \end{bmatrix} , $$
where \(dW_{E}\) and \(dW_{I}\) are Wiener processes, ζ the noise standard deviation (same values as in the nonlinear case), \(E^{*}\) and \(I^{*}\) are the coordinates of the fixed point, and J is the Jacobian at the fixed point of the patient fit. The same values as in the nonlinear case are used for the stimulation magnitude and delay, with the exception of patient 5, for whom the stimulation magnitude is set to a fifth of its value in the nonlinear case, as higher values were seen to cause a breakdown of phase tracking, and result in unreliable response curves.
For each patient and for each of the 10 repeats, bPRCs and bARCs are obtained, and the PRC-ARC shift is then measured as in Sect. 2.1. The results are shown in Fig. 13 (middle), alongside the shifts measured from the response curves presented in Sect. 6.1 (left), and the shifts measured in the full WC model (right). It can be seen that going from the analytic response curves to the linearised model (i.e. adding noise, measuring the response to stimulation via the block method and not a first order method, and using a finite stimulation magnitude rather than a infinitesimal stimulation), does not affect the shift much (compare the left and middle panels of Fig. 13). However, a substantial increase in the shift is obtained by introducing the nonlinearity (compare the middle and right panels of Fig. 13), which brings the shift in the upper-left quadrant, where patient data lie. The PRC-ARC shift can be modulated in the nonlinear model in a way that is not available in the linearisation.
Nonlinearity accounts for most of the difference in PRC-ARC shift seen in synthetic data (middle and right), when compared to the PRC-ARC shift derived in the focus model (left). When computed from synthetic data, the PRC-ARC shift of all 10 repeats is shown (smaller circles), as well as the repeat mean (larger circles). One repeat corresponds to 600 trials, only showing the best fit for each patient
We showed that in a 2D linearised stable focus model, the first order PRC and ARC based on a phase definition approximately equivalent to the Hilbert phase are close to sinusoidal for small decay. Moreover, the PRC-ARC shift is close to \(\frac{\pi}{2}\). Half of the patients in our dataset had significant sinusoidal bPRCs and bARCs (an effect of stimulation phase could not be found in other patients in at least one of their response curves), and the significant patients have a PRC-ARC shift in the interval \([\frac{\pi}{2},\pi ]\). A full WC model can be fitted to tremor dynamics features and to the bPRC for these patients, and as hinted at by the similarities seen in the linearised focus model and the data, the best fits—a vast majority of stable foci—can reproduce the dependence of the effects of stimulation on the phase of stimulation. The best fits also reasonably predict the bARC, and notably what is approximately the best phase to stimulate. Compared to the 2D linearised focus, the nonlinearities of the full WC model allow for a better reproduction of the phase-dependence found in patient data, in particular as far as the PRC-ARC shift is concerned. Our full model can capture the behaviour of neural populations plausibly involved in the generation of tremor, which, together with its success in reproducing phase response and predicting amplitude response in patients, makes it a strong candidate for further study of phase-locked DBS.
While asymptotic phase definitions are common in theoretical studies, experimental studies tend to favour instantaneous phase definitions such as the Hilbert phase. To reproduce the data, an instantaneous phase seems more appropriate than an asymptotic phase, as there is no indication of stimulation happening on or close to an attractor. It has been shown recently in [37] how an operational definition of the phase can describe transient spiking, when an asymptotic phase does not capture the phase-dependence of transients. Moreover, stimulation is assumed to be small in our analytical expressions (Sect. 4), but not in the full model, contrary to standard asymptotic phase reduction strategies.
In this study, our phase definition is the Hilbert phase of the tremor data or approximately equivalent. It is therefore referenced to the maximum of the tremor oscillations (represented by the first coordinate of the dynamical system in our models), and does not require a limit cycle. The Hilbert phase is an angle in the analytic signal space, it does not generally grow linearly with time, and is a protophase [38]. This is not a concern from the perspective of describing patient data, as this is the observable choice we are making for both the data and the model. Commonly used with data, the Hilbert transform has also been proposed as a robust method to measure steady state PRCs in single neuron models [39].
Linearisation
The response curves derived for the linearisation of a 2D focus in Sect. 4 can be related to previously published expressions. In particular, the infinitesimal PRC for radial isochron clocks has been derived in [40], and has been recently included in [41] under the larger umbrella of general radial isochron clocks. The radial clock case (\(K(\phi)=\omega\) in [41]) perturbed along the first dimension agrees with our Eq. (17) for the case of a circular flow (see Sect. 4.7). For this simple system, the asymptotic phase response is the same as the first order Hilbert phase response.
Moreover, we demonstrated that in the linearisation of a 2D focus, the best phase to stimulate (i.e. the minimum of the ARC), corresponds to the maximum positive slope of the first order PRC (see Eq. (21)). This is valid for small decay compared to the rotation, for the phase and amplitude definitions given in Sect. 4.2 (phase approximately equivalent to the Hilbert phase, amplitude defined as the first coordinate), and for a small stimulation magnitude. In fact, the first order ARC is simply a scaled version of the opposite of the first order PRC derivative. A similar relationship has been first reported in a theoretical study in the context of an individual oscillator [42], and more recently in [15] in the context of population responses arising from the individual responses of coupled phase oscillators, whose time evolution follows Kuramoto equations, and where the level of synchrony takes the role of amplitude. The results in [15] also assume particular distributions of oscillator frequencies. It is noteworthy that we found a similar result with very few assumptions on the dynamics: our result is valid for the linearisation of any 2D focus with slow decay, i.e. any linearisation obeying Eq. (9) with slow decay. This applies in particular for the linearisation of the WC model, another popular neuroscience model very different in essence from coupled oscillator models. In the thermodynamic limit and under certain assumptions about the distribution of oscillator frequencies, the Kuramoto model can be reduced to a two-dimensional system [43, 44]. Our results are applicable to the linearisation of a fully desynchronised reduced Kuramoto model observed through \(X_{1} = \rho\cos{\theta}\) where \(\mathbf {r} = \rho e^{i\theta}\) is the order parameter (ρ is the modulus and θ the angle in the complex plane). Such a system therefore satisfies Eq. (21) as well (for small decay).
Our derivations do not assume proximity to a limit cycle, and this allows the study of the dependence of the response to stimulation on the amplitude of the oscillations for a given model (limit cycles do not have an amplitude variable in the case of infinitesimal perturbations). In the linearisation, the PRC is found to be inversely proportional to the amplitude of the oscillations before stimulation (see \(X_{1}^{0}\) term in Eq. (17)), while the ARC does not depend on it.
Because the block method phase and amplitude response used in the rest of paper are normalised by the number of pulses and blocks are only about 25 period long, it seems legitimate to think that, although they are different objects, the first order response to a single pulse (\(\mathrm{hPRC}^{(1)}\) and \(\mathrm{hARC}^{(1)}\)) and the block method response (bPRC and bARC) could be related, and in particular that they might have similar PRC-ARC relationships. Part of the connection hinges on our proof that the phase definition in the linearisation of the focus model matches with the Hilbert phase when the decay is small compared to the rotation (Sect. 4.2). And indeed, the PRC-ARC shift predicted by our expressions derived for the first order response to one pulse of stimulation in a linearised focus is very close to the shift obtained by the block method on linearised WC synthetic data (compare the left and middle panels of Fig. 13). Our analytical derivations provide a rationale to fit the full WC model to data and an intuition for why the model can predict patient ARC, but do not offer an exact analytic treatment of the block method. Specifically, individual pulses in a block may have different effects depending on where they are located in the block and depending on stimulation history within the block [11].
To the best of our knowledge, there is no simple way of getting analytical PRCs and ARCs based on Hilbert phase and amplitude or equivalent in the nonlinear system, making the analytical expressions for the linearisation more valuable. It is also of interest to understand what can be achieved with a simple, linear model before adding more complexity. In fact, realising that the linear model can explain already the data to some extent is a motivation to fit the nonlinear model, which is an expansive endeavour.
Fits of the nonlinear WC model were performed using the generalized pattern search algorithm on many sets of random initial parameters. This approach was chosen for its robustness and computational efficiency in a non-smooth, non-convex landscape with four nonlinear features and 10 parameters, despite requiring the use of a supercomputer. In particular it has been deemed superior to the simplex algorithm in finding better fits. The implementation used also has the additional benefit of being able to handle failed simulations (which occasionally happen as response curves with 12 phase bins cannot be obtained for some parameter sets with noise values too high compared to the vector field). However, the fitting procedure results in many "good" local optima. What these "good" sets of parameters have in common and what they can tell us about the patients we are fitting to is not easily addressed with our current fitting strategy. Even real biological networks may have redundancies, and may exhibit the same behavior under different network configurations. Approximate Bayesian computation [45, 46] allows one to approximate the posterior distribution over parameters for intractable likelihoods, hence to answer the question what is the space of parameters consistent with the data. Whether approximate Bayesian computation methods could successfully tackle a complicated landscape and provide more meaningful insight on fitted model parameters in the setting of the present work is an interesting avenue for further research. A limitation of our fitting method is related to the integration scheme: to reduce computation cost, the Euler step used in the first optimisation process is 1 ms. The top 20 best fits are then re-optimised based on a Euler step of 0.1 ms, and results are produced with this finer time step, as dynamics can be qualitatively different (further reduction in the Euler step has not been seen to change the dynamics). While the need to track the phase at each integration step to decide if stimulation has to be applied precludes the direct use of built-in, powerful integration schemes, a more advanced custom event-based stochastic integration scheme could remove the need for a second optimisation while keeping the computation cost down. The performance of our simple phase-tracking strategy is good for patient 1 and 6 and satisfactory for patient 5 (see Supplementary Fig. 2 in the Appendix). Response curves are obtained based on the actual Hilbert phase of stimulation in a post hoc manner, which makes up for the reduced performance observed for patient 5. Still, more accurate algorithms could be explored. Our simple live phase estimation strategy is based on a linear phase evolution between zero-crossings of E (details in Appendix G), and it would benefit from a better frequency estimate for the current period (currently simply based on the duration of the previous period) and more robustness to noise. Even better live estimates of the Hilbert phase could be obtained thanks to autoregressive forward prediction [47], but at the expense of a higher computational cost, and of a need to adjust hyper parameters for each time series.
Nonlinear WC model
The fitting procedure discussed above was applied to fit to data the full WC model with Gaussian white noise (Eq. (6)). The best performing fits are stable foci for all three patients, and very few limit cycles are found in the top 15 fits for all three patients. One is found for patient 1 (shares the 1st place with a stable focus—distance between mean costs only 30% of the standard error of the mean), one for patient 5, and none for patient 6. In the stable focus regime, noise brings the system away from the stable fixed point, and the interaction of the noise with the dynamics of the system makes the reproduction of patient tremor possible. In our study, noise corresponds to contributions that are not modelled by either the E/I populations or the inputs to these populations. We are considering that these contributions have no explanatory power, and model them with uncorrelated noise. While in the absence of noise, the system would converge to the stable fixed point and no tremor would be generated, Gaussian white noise cannot generate realistic tremor time series. Symptoms in the model depend just as much on the noise as on the other parameters of the model. This is shown in Appendix H where an expression is obtained for the stationary standard deviation of the linearisation of the WC model. The standard deviation is dependent on the noise, but also on the other parameters of the model via the Jacobian at the fixed point. A limitation of our approach is that comparison of the fitted weights or fitted inputs across patients may be difficult when noise levels are not comparable. Enforcing a constant level of noise in the fits or limiting noise to the minimum level required to reproduce the data may address this point. Instead of noise, tremor-like activity may be obtained by exploiting chaotic dynamics arising from coupling several WC models together [48], but this would significantly increase the complexity of the model (more on increasing complexity in the last part of this section).
Contrary to weights, stimulation delays can more easily be compared across patients, and the fitted values obtained deserve some discussion. In fitting our thalamic model to tremor acceleration, we are assuming thalamic activity and tremor are directly related as mentioned before (see Sect. 5.1). Tremor activity is, however, expected to lag thalamic activity due to conduction delays. The accelerometer used to measure tremor is also expected to introduce an electromechanical coupling delay. In the model, we allow for a stimulation delay \(\Delta t_{\mathrm{stim}}\) between the stimulation trigger and the time when stimulation is actually delivered to the excitatory population. This parameter is fitted to the data, and gives the model the ability to shift its bPRC in phase. Fitted stimulation delays are hundreds of milliseconds, and conduction and accelerometer delays (tens of milliseconds) only account for a small part. The higher fitted values are required by the model to match data bPRCs. With our candidate VIM/nRT mapping in mind, the higher fitted values remain unexplained on the biology side, although as mentioned before tremor generation and ET DBS are not fully understood. It is interesting to note that the stimulation delay of the best performing model for patient 5 is longer than one period (see Table 2). This is found consistently in the top three best fits, and reducing the delay to its value modulo the average period substantially reduces the quality of the bPRC fit. Besides this short-term delay, our model does not include medium- or long-term plasticity effects, which are not expected to be strongly present in the recordings as stimulation is only delivered for periods of 5 seconds in a row. In our model, stimulation is provided to the E population via a direct increase in the population activity. While stimulation is provided via the sigmoid function of the excitatory population in other studies [18], we found this approach too restrictive due to sigmoid saturation, and inadequate to reproduce the full extent of the response to phase-locked DBS in some patients. As a reminder, the choice of stimulating the excitatory population rather than the inhibitory population is made for biological consistency, as the VIM is the most common stimulation target in ET DBS.
The success of the nonlinear WC model in predicting patient ARCs when fitted to their PRCs is partially explained by its ability to modulate the PRC-ARC shift. The PRC-ARC shift in the full model can reach the range found in patients while the linearised version of the WC is limited to the close vicinity of \(\frac{\pi}{2}\). The response curves of the full WC model are also better at reproducing the data and can deviate from pure sinusoids. However, there is still some room for improvement in reproducing the shift, in particular as far as patient 1 is concerned (patient shift quite a bit larger than the model). The model can allow for a larger shift as shown by a fit hand-picked in the top 15 shown in Supplementary Fig. 3 in Appendix I. While the troughs of the model bPRCs are roughly aligned with the troughs of the data bPRCs in Supplementary Fig. 3 and in our best fit in Fig. 7, it is apparent that the peaks of the model bARCs are closer to the peaks of the data bARCs in Supplementary Fig. 3 than in Fig. 7. This highlights that the PRC-ARC shift of the model is closer to that of the data in Supplementary Fig. 3 than in Fig. 7. The PRC-ARC shift could be selected as an additional feature to fit to in order to improve ARC reproduction.
In its two-population version, the suggested mapping of the excitatory and inhibitory populations (VIM and nRT) is not the only possibility. Other candidates include antidromically stimulated structures at the cerebellar level or below, such as DCN as the inhibitory population, and the inferior olive as the excitatory population. The model could also be extended by including more populations. With our current mapping in mind, the cortex and the DCN could be turned into populations of their own, which would make the model four-dimensional. As suggested in [18], the inferior olive which provides input to the DCN could also be modelled, and the spatial extent of the VIM could be accounted for by splitting it in two populations or more. Increasing the number of populations would, however, increase the number of parameters of the model, and make the optimisation process more computationally intensive, and the model more prone to over-fitting. In contrast, the incorporation of additional loops in the model architecture may help explain the inertia in stimulation effects discussed above. Nevertheless, the model seems to be able to reproduce the data in its current state, which suggests an increase in complexity is not warranted. It is remarkable that one excitatory/inhibitory loop seems to be enough to model the phase-dependent effects of ET DBS in the datasets available with statistically significant response curves. It gives some support to the hypothesis that sub-circuits of the central tremor network may behave as individual oscillators entraining each other [49].
The nonlinear focus WC model with noise can reproduce the phase-dependence of the response to phase-locked DBS in ET patient data with statistically significant response curves, as well as predict tremor reduction in response to phase-locked stimulation. Phase-locked stimulation promises less stimulation, hence less side effects for the same clinical benefits, which would be highly desirable for patients. Our study positions the WC as a strong candidate to model the effects of phase-locked DBS. Its ability to describe all patients with both response curves statistically significant in at least one of our tests should be re-assessed as more data becomes available, both in terms of number of patients and recording length. Phase-dependent activity is thought to play a central role in physiological information processing [50, 51], and in our analytical derivations, the phase of the linearised model was defined in a way that does not depend on modelling oscillations by a limit cycle, and that for small decay approximately matches with a phase definition widely used in experiments, the Hilbert phase. Finally, as far as ET generation is concerned, we showed that a single excitatory/inhibitory loop is enough to reproduce both the dynamics of the tremor and the phase-dependent effects of stimulation, however, it should be nonlinear.
We include here technicalities on approximating the Hilbert phase in the linearisation (Appendix A), details of the derivations leading to response curves analytical expressions in the linearised system (Appendices B to D), and the procedure used to obtain WC parameters from a given Jacobian (Appendix E). We also present details of the two-step optimisation used for fitting to patient data (Appendix F), the implementation of live-phase tracking and stimulation (Appendix G), as well as an analytical expression for the standard deviation of the tremor for the stationary linearised model (Appendix H). Supplementary figures and supplementary tables make up Appendix I and Appendix J, respectively.
amplitude response curve
DBS:
DCN:
deep cerebellar nuclei
FDR:
false discovery rate
nRT:
reticular nucleus
probability density function
PRC:
phase response curve
PSD:
power spectrum density
TEED:
total electrical energy delivered
VIM:
ventral intermediate nucleus
Wilson–Cowan
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The authors would also like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work http://dx.doi.org/10.5281/zenodo.22558.
The datasets analysed in the current study are available online [31].
This work was supported by Medical Research Council grant MC_UU_12024/5.
Nuffield Department of Clinical Neuroscience, University of Oxford, Oxford, UK
Benoit Duchet, Gihan Weerasinghe, Hayriye Cagnan, Peter Brown & Rafal Bogacz
MRC Brain Network Dynamics Unit, University of Oxford, Oxford, UK
Wellcome Centre for Human Neuroimaging, UCL Institute of Neurology, London, UK
Hayriye Cagnan
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford, UK
Christian Bick
Centre for Systems, Dynamics, and Control and Department of Mathematics, University of Exeter, Exeter, UK
EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
Benoit Duchet
Gihan Weerasinghe
Rafal Bogacz
BD was involved in the conceptualisation of the study, carried out the statistical, analytical and computational studies, and wrote the paper. GW provided guidance for the optimisation work. CB and RB provided supervision and were involved in the conceptualisation of the study. HC and PB collected the original tremor data. All authors were involved in the discussion of the results and edited the paper. All authors read and approved the final manuscript.
Correspondence to Benoit Duchet.
Appendix 1: Hilbert transforms of sine and cosine exponential decays with error terms
The goal here is to show that \(\mathcal{H}(s(t)s_{j}(t)) \approx s(t) \mathcal{H}(s_{j}(t))\) for \(j=c,n\), with \(s(t)=e^{\sigma\vert t \vert}\), \(s_{c}(t)=\cos{\omega t}\), and \(s_{n}(t) = \sin{\omega t}\). The Bedrosian identity [52] states that the Hilbert transform of the product of a low-pass and a high-pass signal with non-overlapping spectra is the product of the low-pass signal and the Hilbert transform of the high-pass signal. The spectrum support of s is \(\mathbb{R}\) and therefore overlaps with the spectra of \(s_{c}\) and \(s_{n}\), but for low decay compared to the rotation, the spectrum of s is very small where it overlaps with the spectra of \(s_{c}\) or \(s_{n}\). Because of the overlaps, the equality given by the Bedrosian identity is not exact and turns into an approximation, and inspired by the proof in [52], we can calculate error terms. Let S and \(S_{c}\) be the Fourier transforms of s and \(s_{c}\), respectively:
$$\begin{aligned}& s(t)s_{c}(t) = \frac{1}{(2\pi)^{2}} \int_{-\infty}^{\infty} \int_{- \infty}^{\infty}S(u)S_{c}(v)e^{i(u+v)t}\,du\,dv, \end{aligned}$$
$$\begin{aligned}& \mathcal{H}\bigl(s(t)s_{c}(t)\bigr) = \frac{1}{(2\pi)^{2}} \int_{-\infty}^{ \infty} \int_{-\infty}^{\infty}S(u)S_{c}(v)i \operatorname {sgn}(u+v)e^{i(u+v)t}\,du\,dv. \end{aligned}$$
The Fourier transform of \(s_{c}\) is given by \(S_{c}(v) = \pi [\delta(v-\omega)+\delta(v+\omega) ]\), so
$$ \mathcal{H}\bigl(s(t)s_{c}(t)\bigr) = \frac{1}{(2\pi)^{2}} \int_{-\infty}^{ \infty}S(u)e^{iut}\varGamma(u)\,du, $$
where \(\varGamma(u) = \frac{\pi}{i} [\operatorname {sgn}(u+\omega)e^{i\omega t} + \operatorname {sgn}(u-\omega)e^{-i\omega t} ]\). This can be simplified as
$$ \varGamma(u) = 2\pi\sin{\omega t} + \textstyle\begin{cases} 0, & \vert u \vert < \omega, \\ - \frac{2\pi}{i}e^{i\omega t}, & u < -\omega, \\ \frac{2\pi}{i}e^{-i\omega t}, & u > \omega. \end{cases} $$
The Fourier transform \(S(u)=\frac{2\sigma}{\sigma^{2}+u^{2}}\) is even, therefore
$$\begin{aligned}& \mathcal{H}\bigl(s(t)s_{c}(t)\bigr) = \frac{\sin(\omega t)}{(2\pi)^{2}} \int_{-\infty}^{\infty}S(u)e^{iut}\,du + \frac{1}{2\pi i} \int_{ \omega}^{\infty}S(u) \bigl(e^{i(u-\omega)t}-e^{-i(u-\omega)t} \bigr)\,du, \end{aligned}$$
$$\begin{aligned}& \mathcal{H}\bigl(s(t)s_{c}(t)\bigr) = s(t)\mathcal{H} \bigl(s_{c}(t)\bigr) + \mathcal{I}_{s_{c}}, \end{aligned}$$
$$\begin{aligned}& \mathcal{H}\bigl(s_{c}(t)\bigr) = \sin{\omega t}, \\& \mathcal{I}_{s_{c}} = \frac{2}{\pi} \int_{\omega}^{\infty} \frac{\sigma}{\sigma^{2}+u^{2}}\sin{(u- \omega)t}\,du. \end{aligned}$$
A similar derivation provides
$$ \mathcal{H}\bigl(s(t)s_{n}(t)\bigr) = s(t)\mathcal{H} \bigl(s_{n}(t)\bigr) + \mathcal{I}_{s_{n}} $$
$$\begin{aligned}& \mathcal{H}\bigl(s_{n}(t)\bigr) = -\cos{\omega t}, \\& \mathcal{I}_{s_{n}} = \frac{2}{\pi} \int_{\omega}^{\infty} \frac{\sigma}{\sigma^{2}+u^{2}}\cos{(u- \omega)t}\,du. \end{aligned}$$
Numerical integration demonstrates that for \(\omega\gg\vert\sigma\vert\), and in particular in the case of the patients we are interested in, \(\mathcal{I}_{s_{c}}\) and \(\mathcal{I}_{s_{n}}\) are under 5% of the signal scale for about 12 periods (see Fig. 14). This is more than enough for our purposes as only one period is needed to derive response curves. It is therefore reasonable to ignore \(\mathcal{I}_{s_{c}}\) and \(\mathcal{I}_{s_{n}}\).
Relative error made across patients in estimating \(\mathcal{H}(s(t)s_{c}(t))\) by \(s(t)\mathcal{H}(s_{c}(t))\) (solid lines) and \(\mathcal{H}(s(t)s_{n}(t))\) by \(s(t)\mathcal{H}(s_{n}(t))\) (dashed lines). The error is calculated as the ratio of \(\mathcal{I}_{s_{c}}\) (respectively, \(\mathcal{I}_{s_{n}}\)) over the modulus of the numerical Hilbert transform of the signal, which is the envelope of the signal. The relative error is under 5% in all cases for at least 12 periods
Appendix 2: Reference trajectory without stimulation
Let us find the coefficients \(K_{\mathrm{ref}}\) and \(K'_{\mathrm{ref}}\) of the trajectory starting at \(t=0\) at a maximum of the first coordinate \(X_{1}=X_{1}^{0}>0\). With the choice \(\phi= \omega t\), this will ensure we are referencing the phase to the maximum of \(X_{1}\). It should be noted at this point that we are not using the Jacobian in what follows as we are interested in the dependence of the response on the rotation ω and the decay σ. From the initial condition at \(t=0\),
$$ K_{\mathrm{ref}} a_{1} + K'_{\mathrm{ref}} b_{1} = X_{1}^{0}. $$
Additionally, \(X_{1}^{0}\) being a maximum requires that \(\frac{d X_{1}}{dt}=0\) at \(t=0\), therefore
$$\begin{aligned} \frac{d X_{1}}{dt} =& e^{\sigma t} \bigl[ -\omega\bigl(K_{\mathrm {ref}}a_{1}+K'_{\mathrm{ref}}b_{1} \bigr) \sin{\omega t} + \omega\bigl(-K_{\mathrm{ref}}b_{1}+K'_{\mathrm {ref}}a_{1} \bigr)\cos{\omega t} \\ &{}+\sigma \bigl\{ \bigl(K_{\mathrm{ref}}a_{1}+K'_{\mathrm {ref}}b_{1} \bigr)\cos{\omega t}+\bigl(-K_{\mathrm{ref}}b_{1}+K'_{\mathrm{ref}}a_{1} \bigr) \sin{\omega t} \bigr\} \bigr]. \end{aligned}$$
Using the condition at \(t=0\),
$$\begin{aligned}& K_{\mathrm{ref}}(\sigma a_{1} - \omega b_{1})+K'_{\mathrm{ref}}( \sigma b_{1} + \omega a_{1}) = 0 , \end{aligned}$$
$$\begin{aligned}& \text{(31)} + \text{(33)}\quad \implies\quad \begin{aligned} &K_{\mathrm{ref}} = \frac{\sigma b_{1} + \omega a_{1}}{\omega( a_{1}^{2} + b_{1}^{2})}X_{1}^{0}, \\ &K'_{\mathrm{ref}} = \frac{-\sigma a_{1} + \omega b_{1}}{\omega( a_{1}^{2} + b_{1}^{2})}X_{1}^{0}. \end{aligned} \end{aligned}$$
We are excluding the case where the denominator in (34) is equal to zero, which corresponds to both \(a_{1}\) and \(b_{1}\) being zero, which would imply \(X_{1}(t) = 0\). Also note that by picking a positive \(X_{1}^{0}\), we are ensuring that the null derivative corresponds to a maximum of \(X_{1}\) rather than a minimum.
Appendix 3: Trajectory with stimulation
Let us determine what the coefficients \(K_{\mathrm{stim}}\) and \(K'_{\mathrm{stim}}\) are for the stimulated trajectory (still constrained by the dynamics of Eq. (9)). We have
$$ \mathbf {X^{1^{+}}}= \bigl\{ K_{\mathrm{stim}} (\mathbf {a}\cos{\phi _{0}}-\mathbf {b} \sin{\phi_{0}} ) +K'_{\mathrm{stim}} (\mathbf {a}\sin{\phi_{0}}+ \mathbf {b}\cos{\phi_{0}} ) \bigr\} e^{\sigma \frac{\phi_{0}}{\omega}}. $$
Solving for \(K_{\mathrm{stim}}\) gives
$$ K_{\mathrm{stim}}= \frac{X_{2}^{1^{-}}(a_{1}\sin{\phi_{0}}+b_{1}\cos{\phi _{0}})-(X_{1}^{1^{-}}+\delta X_{1})(a_{Y} \sin{\phi_{0}} + b_{2} \cos {\phi_{0}})}{a_{2} b_{1} - a_{1} b_{2}} e^{-\sigma\frac{\phi _{0}}{\omega}}. $$
Plugging in \(X_{1}^{1^{-}}\), \(X_{2}^{1^{-}}\), and the expressions for \(K_{\mathrm{ref}}\) and \(K'_{\mathrm{ref}}\) yields
$$ K_{\mathrm{stim}}= \frac{\omega a_{1} + \sigma b_{1}}{\omega( a_{1}^{2} + b_{1}^{2} )}X_{1}^{0} - \frac{a_{2}\sin{\phi_{0}}+b_{2}\cos{\phi_{0}}}{a_{2} b_{1} - a_{1} b_{2}} \delta X_{1} e^{-\sigma \frac{\phi_{0}}{\omega}}. $$
Similarly for \(K'_{\mathrm{stim}}\), using the previous result,
$$\begin{aligned}& K'_{\mathrm{stim}}= \frac{X_{2}^{1-} e^{-\sigma\frac{\phi_{0}}{\omega}} - K_{\mathrm {stim}}(a_{1}\cos{\phi_{0}}-b_{1}\sin{\phi_{0}})}{a_{1}\sin{\phi _{0}}+b_{1}\cos{\phi0}}, \end{aligned}$$
$$\begin{aligned}& K'_{\mathrm{stim}}= \frac{-\sigma a_{1} + \omega b_{1}}{\omega( a_{1}^{2} + b_{1}^{2} )}X_{1}^{0} + \frac{a_{2}\cos{\phi_{0}}-b_{2}\sin{\phi_{0}}}{a_{2} b_{1} - a_{1} b_{2}} \delta X_{1} e^{-\sigma \frac{\phi_{0}}{\omega}}. \end{aligned}$$
Appendix 4: Phase at the next maximum of \(X_{1}\) on the stimulated trajectory
We are looking for \(\phi_{\mathrm{max}}\) such that \(\frac{d X_{1}^{\mathrm{stim}}}{dt} = 0 \) at \(\omega t = \phi_{\mathrm {max}}\). This give us
$$\begin{aligned}& e^{\sigma\frac{\phi_{\mathrm{max}}}{\omega}} \bigl[ -\omega \bigl(K_{\mathrm{stim}}a_{1}+K'_{\mathrm{stim}}b_{1} \bigr) \sin{\phi_{\mathrm{max}}} + \omega\bigl(-K_{\mathrm {stim}}b_{1}+K'_{\mathrm{stim}}a_{1} \bigr)\cos{\phi_{\mathrm{max}}} \\& \quad {}+\sigma \bigl\{ \bigl(K_{\mathrm{stim}}a_{1}+K'_{\mathrm {stim}}b_{1} \bigr)\cos{\phi_{\mathrm{max}}}+\bigl(-K_{\mathrm {stim}}b_{1}+K'_{\mathrm{stim}}a_{1} \bigr) \sin{\phi_{\mathrm{max}}} \bigr\} \bigr]=0, \end{aligned}$$
$$\begin{aligned}& \tan{\phi_{\mathrm{max}}}= \frac{K_{\mathrm{stim}}(\sigma a_{1} - \omega b_{1}) + K'_{\mathrm {stim}}(\sigma b_{1} + \omega a_{1})}{K_{\mathrm{stim}}(\sigma b_{1} + \omega a_{1}) + K'_{\mathrm{stim}}(-\sigma a_{1} + \omega b_{1})}. \end{aligned}$$
The phase \(\phi_{\mathrm{max}}\) is returned by the arctan function in \((-\frac{\pi}{2},\frac{\pi}{2} )\), and corresponds to the previous peak on the stimulated trajectory extended backwards. The next peak has the same \(\mathrm{phase} \pmod{2\pi}\) as the expression in square brackets in Eq. (40) is 2π-periodic.
Appendix 5: Finding WC parameters corresponding to a given Jacobian
The Jacobian of (6) evaluated at \((E^{*},I^{*})\) can be simplified by making use of \(f'(x)=\beta f(x)(1-f(x))\). We also have
$$\begin{aligned}& f(\varTheta_{1}) = E^{*} , \end{aligned}$$
$$\begin{aligned}& f(\varTheta_{2}) = I^{*} , \end{aligned}$$
$$\begin{aligned}& \varTheta_{1} = w_{EE}E^{*} - w_{IE}I^{*} + \theta_{E}, \\& \varTheta_{2} = w_{EI}E^{*} + \theta_{I}. \end{aligned}$$
The Jacobian of (6) evaluated at \((E^{*},I^{*})\) is therefore given by
$$\begin{aligned} J_{\mathrm{WC}} =& \frac{1}{\tau} \begin{bmatrix} w_{EE}f'(\varTheta_{1}) - 1 & -w_{IE}f'(\varTheta_{1}) \\w_{EI}f'(\varTheta_{2}) & -1 \end{bmatrix} \\ =& {\frac{1}{\tau}} \begin{bmatrix} w_{EE}\beta E^{*}(1-E^{*}) - 1 & -w_{IE}\beta I^{*}(1-I^{*}) \\ w_{EI}\beta E^{*}(1-E^{*}) & -1 \end{bmatrix}. \end{aligned}$$
We are interested in finding WC parameters so that the linearisation of the WC model at the fixed point will be characterised by a given Jacobian matrix
$$ J = \begin{bmatrix} J_{11} & J_{12} \\J_{21} & J_{22} \end{bmatrix} . $$
If we pick values for β, \(E^{*}\) and \(I^{*}\), the remaining parameters can be obtained by equating (44) and (45), and by re-arranging Eqs. (42) and (43). The parameters in Supplementary Table 1 were obtained using this method, which yields
$$\begin{aligned}& \tau = - \frac{1}{J_{22}}, \\& w_{EE} = \frac{\tau J_{11} + 1}{\beta E^{*} (1-E^{*}),} \\& w_{IE} = - \frac{\tau J_{12}}{\beta E^{*} (1-E^{*})}, \\& w_{EI} = \frac{\tau J_{21}}{\beta I^{*} (1-I^{*})}, \\& \theta_{E} = 1 - \frac{1}{\beta}\ln \biggl( \frac{1}{E^{*}}-1 \biggr) - w_{EE}E^{*} + w_{IE}I^{*}, \\& \theta_{I} = 1 - \frac{1}{\beta}\ln \biggl( \frac{1}{I^{*}}-1 \biggr) - w_{EI}E^{*}. \end{aligned}$$
Appendix 6: Two-step optimisation
The optimisation procedure is as follows. For each patient, random sets of parameters are picked from uniform distributions (bounds in Supplementary Table 3). To improve the efficiency of the optimisation, we accept parameters only if the PSD peak of the corresponding model (without stimulation) is within 1 Hz and 25% in magnitude of the data PSD peak. Once 2500 parameter sets have been accepted, we put them through local optimisations. Local optimisations are carried out using a direct search algorithm called the generalized pattern search algorithm [53, 54]. The pattern is a set of fixed vectors in parameter space. At each step, points to be polled (the mesh) are generated by adding a scaled version of the pattern to the current best point. If a point with a lower value of the objective function is found, this point becomes the new best point, and a scaled up version of the pattern is used next. If not, a scaled down version of the pattern is used next. Parameters supplied to pattern search are put on a similar scale to improve search robustness, and hard limits are given to the optimiser (see Supplementary Table 3 in Appendix J). Optimisations are performed in parallel on a supercomputer. A time step of 1 ms is used for the fits (a period is about 200 ms). At the end of this process, the 20 best performing sets of parameters were put through more local optimisations with a finer time step of 0.1 ms and stop criteria leaving room for more steps. The finer time step is also used to produce the results shown in Sect. 5.2).
The implementation of the generalized pattern search algorithm used is Matlab's patternsearch optimiser with the poll method "positive basis 2N" and the following stop criteria:
main optimisation (time step of 1 ms): mesh size of 10−4, function call budget of 800,
second optimisation (time step of 0.1 ms): mesh size of 10−5, function call budget of 1000.
Appendix 7: Live phase tracking and stimulation
One simulation consists of 600 trials with 12 blocks of phase-locked stimulation each. As in the experimental paradigm, blocks last 5 s, and inter-block intervals are 1 s. Inter-trial intervals are 5 s, and the first trial starts after about 200 periods. During this initial time, the mean of E and the standard deviation of E, \(\sigma_{\mathrm{sim}}\), are obtained from about 20 periods after a ramp-up of about 40 periods. Phase-tracking subsequently starts: E is centered and a threshold \(T = 0.2\sigma_{\mathrm{sim}}\) is used to track positive zero-crossings. We define a positive zero-crossing as happening when
$$ \textstyle\begin{cases} E(n) < -T, \\ E(p) > T, \\ p>n, \\ \forall i \in \{ n+1,\ldots,p-1 \} ,\quad E(i) \in[-T,T]. \end{cases} $$
These conditions are constantly monitored, and if found true, a positive zero-crossing is declared to have happened at time step \(\chi=\frac{n+p}{2}\). The threshold T was found critical to handle the noise included in the model, as it prevents situations where a negative zero-crossing very closely follows a positive zero-crossing (or vice versa) from interfering. We evolve the zero-crossing phase according to a frequency based on the previous period, and if \(\chi_{k}\) is the last positive zero-crossing to have occurred, the current value of the zero-crossing phase is given by
$$ \varphi= \frac{2\pi}{t_{\chi_{k}}-t_{\chi_{k-1}}}(t-t_{\chi_{k}}). $$
If the value of 2π is reached, the phase value is set to 0 until the next positive zero-crossing is detected. Stimulation is provided after φ reaches the target phase for the block, and the stimulation trigger is recorded \(\Delta t_{\mathrm{stim}}\) before stimulation occurs. If the zero-crossing phase has not reached the target stimulation phase yet when the next positive zero-crossing is detected, stimulation is provided right then. As in [11], a pulse of stimulation consists of six quick bursts at 130 Hz.
Appendix 8: Stationary standard deviation of the first coordinate in the linearised model
In the absence of stimulation, the stationary covariance matrix \(P^{\infty}\) of the linearised model (Eq. (24)) must satisfy (see [55])
$$ JP^{\infty}+ P^{\infty}J^{\mathrm{T}} + \begin{bmatrix} \zeta^{2} & 0 \\ 0 & \zeta^{2} \end{bmatrix} = 0. $$
Solving for \(P^{\infty}\) and following the notation in Eq. (45), we can write the stationary standard deviation of the first coordinate in the linearised model as
$$ \sqrt{P^{\infty}_{11}} = \zeta\sqrt{ \frac{J_{12}^{2} + J_{22}^{2} + J_{11}J_{22} - J_{12}J_{21}}{2(J_{11}+J_{22})(J_{12}J_{21}-J_{11}J_{22})}}. $$
Appendix 9: Supplementary figures
Supplementary Figure 1
Patients' bPRCs (first column) and bARCs (second column) obtained as described in Sect. 2.1. Datasets with both response curves significant according to at least one of our statistical tests under FDR control are highlighted with green rectangles
Phase tracking illustrated in the three fitted patients by histograms of the pair (target stimulation phase for the stimulation block, average of actual Hilbert phase at stimulation for the stimulation block). The actual Hilbert phase is obtained post hoc after filtering. A block average includes averaging across bursts and within the block. Averages are obtained using circular means. The effect of the stimulation delay was removed, and phases are reference to positive zero-crossings. Phase tracking is satisfactory for all patients, although tracking is less precise for later phases in patient 5
Fit to patient 1 showing the best PRC-ARC shift. The four features that were included in the cost function are shown on the left, namely tremor PSD (A), tremor envelope PDF (B), tremor envelope PSD (C) and bPRC (D). The model better predicts the data bARC (E) thanks to a PRC-ARC shift closer to that of the data (equivalent PRC alignment between the model and the data, but better ARC alignment). The model phase plane is shown in (H), and the model tremor time series (F) is shown next to the patient tremor time series (G). The framed black bar in (H) indicates the fitted stimulation magnitude to scale
Appendix 10: Supplementary tables
Supplementary Table 1 WC parameters corresponding to the Jacobians presented in Sect. 4.7. The steepness parameter β was set to 4, \(E^{*}\) and \(I^{*}\) to 0.5, and parameters were determined according the method presented in Appendix E
Supplementary Table 2 \(\vert\sigma\vert/\omega\) ratios in the linearisation of patient fits
Supplementary Table 3 Lower and upper bounds of parameters uniform distributions used to generate initial parameters for fitting, and hard limits enforced by pattern search during the optimisation process
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Duchet, B., Weerasinghe, G., Cagnan, H. et al. Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson–Cowan model. J. Math. Neurosc. 10, 4 (2020). https://doi.org/10.1186/s13408-020-00081-0
Received: 08 April 2019
Phase-locked stimulation
Wilson Cowan model
Focus model | CommonCrawl |
Computer Aided Verification
International Conference on Computer Aided Verification
CAV 2019: Computer Aided Verification pp 591-608 | Cite as
Flexible Computational Pipelines for Robust Abstraction-Based Control Synthesis
Eric S. Kim
Murat Arcak
Sanjit A. Seshia
Successfully synthesizing controllers for complex dynamical systems and specifications often requires leveraging domain knowledge as well as making difficult computational or mathematical tradeoffs. This paper presents a flexible and extensible framework for constructing robust control synthesis algorithms and applies this to the traditional abstraction-based control synthesis pipeline. It is grounded in the theory of relational interfaces and provides a principled methodology to seamlessly combine different techniques (such as dynamic precision grids, refining abstractions while synthesizing, or decomposed control predecessors) or create custom procedures to exploit an application's intrinsic structural properties. A Dubins vehicle is used as a motivating example to showcase memory and runtime improvements.
Control synthesis Finite abstraction Relational interface
The authors were funded in part by AFOSR FA9550-18-1-0253, DARPA Assured Autonomy project, iCyPhy, Berkeley Deep Drive, and NSF grant CNS-1739816.
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A control synthesizer's high level goal is to automatically construct control software that enables a closed loop system to satisfy a desired specification. A vast and rich literature contains results that mathematically characterize solutions to different classes of problems and specifications, such as the Hamilton-Jacobi-Isaacs PDE for differential games [3], Lyapunov theory for stabilization [8], and fixed-points for temporal logic specifications [11, 17]. While many control synthesis problems have elegant mathematical solutions, there is often a gap between a solution's theoretical characterization and the algorithms used to compute it. What data structures are used to represent the dynamics and constraints? What operations should those data structures support? How should the control synthesis algorithm be structured? Implementing solutions to the questions above can require substantial time. This problem is especially critical for computationally challenging problems, where it is often necessary to let the user rapidly identify and exploit structure through analysis or experimentation.
1.1 Bottlenecks in Abstraction-Based Control Synthesis
This paper's goal is to enable a framework to develop extensible tools for robust controller synthesis. It was inspired in part by computational bottlenecks encountered in control synthesizers that construct finite abstractions of continuous systems, which we use as a target use case. A traditional abstraction-based control synthesis pipeline consists of three distinct stages:
Abstracting the continuous state system into a finite automaton whose underlying transitions faithfully mimic the original dynamics [21, 23].
Synthesizing a discrete controller by leveraging data structures and symbolic reasoning algorithms to mitigate combinatorial state explosion.
Refining the discrete controller into a continuous one. Feasibility of this step is ensured through the abstraction step.
This pipeline appears in tools PESSOA [12] and SCOTS [19], which can exhibit acute computational bottlenecks for high dimensional and nonlinear system dynamics. A common method to mitigate these bottlenecks is to exploit a specific dynamical system's topological and algebraic properties. In MASCOT [7] and CoSyMA [14], multi-scale grids and hierarchical models capture notions of state-space locality. One could incrementally construct an abstraction of the system dynamics while performing the control synthesis step [10, 15] as implemented in tools ROCS [9] and ARCS [4]. The abstraction overhead can also be reduced by representing systems as a collection of components composed in parallel [6, 13]. These have been developed in isolation and were not previously interoperable.
1.2 Methodology
Figure 1 depicts this paper's methodology and organization. The existing control synthesis formalism does not readily lend itself to algorithmic modifications that reflect and exploit structural properties in the system and specification. We use the theory of relational interfaces [22] as a foundation and augment it to express control synthesis pipelines. Interfaces are used to represent both system models and constraints. A small collection of atomic operators manipulates interfaces and is powerful enough to reconstruct many existing control synthesis pipelines.
By expressing many different techniques within a common framework, users are able to rapidly develop methods to exploit system structure in controller synthesis.
One may also add new composite operators to encode desirable heuristics that exploit structural properties in the system and specifications. The last three sections encode the techniques for abstraction-based control synthesis from Sect. 1.1 within the relational interfaces framework. By deliberately deconstructing those techniques, then reconstructing them within a compositional framework it was possible to identify implicit or unnecessary assumptions then generalize or remove them. It also makes the aforementioned techniques interoperable amongst themselves as well as future techniques.
Interfaces come equipped with a refinement partial order that formalizes when one interface abstracts another. This paper focuses on preserving the refinement relation and sufficient conditions to refine discrete controllers back to concrete ones. Additional guarantees regarding completeness, termination, precision, or decomposability can be encoded, but impose additional requirements on the control synthesis algorithm and are beyond the scope of this paper.
1.3 Contributions
To our knowledge, the application of relational interfaces to robust abstraction-based control synthesis is new. The framework's building blocks consist of a collection of small, well understood operators that are nonetheless powerful enough to express many prior techniques. Encoding these techniques as relational interface operations forced us to simplify, formalize, or remove implicit assumptions in existing tools. The framework also exhibits numerous desirable features.
It enables compositional tools for control synthesis by leveraging a theoretical foundation with compositionality built into it. This paper showcases a principled methodology to seamlessly combine the methods in Sect. 1.1, as well as construct new techniques.
It enables a declarative approach to control synthesis by enforcing a strict separation between the high level algorithm from its low level implementation. We rely on the availability of an underlying data structure to encode and manipulate predicates. Low level predicate operations, while powerful, make it easy to inadvertently violate the refinement property. Conforming to the relational interface operations minimizes this danger.
This paper's first half is domain agnostic and applicable to general robust control synthesis problems. The second half applies those insights to the finite abstraction approach to control synthesis. A smaller Dubins vehicle example is used to showcase and evaluate different techniques and their computational gains, compared to the unoptimized problem. In an extended version of this paper available at [1], a 6D lunar lander example leverages all techniques in this paper and introduces a few new ones.
1.4 Notation
Let \(=\) be an assertion that two objects are mathematically equivalent; as a special case '\(\equiv \)' is used when those two objects are sets. In contrast, the operator '\(==\)' checks whether two objects are equivalent, returning true if they are and false otherwise. A special instance of '\(==\)' is logical equivalence '\(\Leftrightarrow \)'.
Variables are denoted by lower case letters. Each variable v is associated with a domain of values \(\mathcal {D}(v)\) that is analogous to the variable's type. A composite variable is a set of variables and is analogous to a bundle of wrapped wires. From a collection of variables \(v_1, \ldots , v_M\) a composite variable \(v\) can be constructed by taking the union \(v\equiv v_1 \cup \ldots \cup v_M\) and the domain \(\mathcal {D}(v) \equiv \prod _{i=1}^M \mathcal {D}(v_i)\). Note that the variables \(v_1, \ldots , v_M\) above may themselves be composite. As an example if v is associated with a M-dimensional Euclidean space \(\mathbb {R}^M\), then it is a composite variable that can be broken apart into a collection of atomic variables \(v_1, \ldots , v_M\) where \(\mathcal {D}(v_i) \equiv \mathbb {R}\) for all \(i \in \{1,\ldots , M\}\). The technical results herein do not distinguish between composite and atomic variables.
Predicates are functions that map variable assignments to a Boolean value. Predicates that stand in for expressions/formulas are denoted with capital letters. Predicates P and Q are logically equivalent (denoted by \(P \Leftrightarrow Q\)) if and only if \(P \Rightarrow Q\) and \(Q \Rightarrow P\) are true for all variable assignments. The universal and existential quantifiers \(\forall \) and \(\exists \) eliminate variables and yield new predicates. Predicates \({\exists w}P\) and \({\forall w}P\) do not depend on \(w\). If \(w\) is a composite variable \(w\equiv w_1 \cup \ldots \cup w_N\) then \({\exists w}P\) is simply a shorthand for \({\exists w_1}\ldots {\exists w_N} P\).
2 Control Synthesis for a Motivating Example
As a simple, instructive example consider a planar Dubins vehicle that is tasked with reaching a desired location. Let \(x = \{p_x, p_y, \theta \}\) be the collection of state variables, \(u = \{v, \omega \}\) be a collection input variables to be controlled, \(x^+ = \{p_x^+, p_y^+, \theta ^+\}\) represent state variables at a subsequent time step, and \(L = 1.4\) be a constant representing the vehicle length. The constraints
characterize the discrete time dynamics. The continuous state domain is \(\mathcal {D}(x) \equiv [-2,2] \times [-2,2] \times [-\pi ,\pi )\), where the last component is periodic so \(-\pi \) and \(\pi \) are identical values. The input domains are \(\mathcal {D}(v) \equiv \{0.25, 0.5\}\) and \(\mathcal {D}(\omega ) \equiv \{-1.5, 0, 1.5\}\)
Let predicate \(F = F_x \wedge F_y \wedge F_\theta \) represent the monolithic system dynamics. Predicate T depends only on x and represents the target set \([-0.4,0.4] \times [-0.4,0.4] \times [-\pi ,\pi )\), encoding that the vehicle's position must reach a square with any orientation. Let Z be a predicate that depends on variable \(x^+\) that encodes a collection of states at a future time step. Equation (1) characterizes the robust controlled predecessor, which takes Z and computes the set of states from which there exists a non-blocking assignment to u that guarantees \(x^+\) will satisfy Z, despite any non-determinism contained in F. The term \({\exists x^+}F\) prevents state-control pairs from blocking, while \({\forall x^+}(F \Rightarrow Z)\) encodes the state-control pairs that guarantee satisfaction of Z.
$$\begin{aligned} {{\texttt {cpre}}}(F, Z) = {\exists u} ( {\exists x^+} F \wedge {\forall x^+}(F \Rightarrow Z)). \end{aligned}$$
The controlled predecessor is used to solve safety and reach games. We can solve for a region for which the target T (respectively, safe set S) can be reached (made invariant) via an iteration of an appropriate \({\texttt {reach}}\) (\({\texttt {safe}}\)) operator. Both iterations are given by:
$$\begin{aligned} \text {Reach Iter:} \qquad Z_0&= \bot \qquad Z_{i+1} = {\texttt {reach}}(F,Z_i,T) = {{\texttt {cpre}}}(F, Z_{i}) \vee T. \end{aligned}$$
$$\begin{aligned} \text {Safety Iter:}\qquad Z_0&= S \qquad Z_{i+1} = {\texttt {safe}}(F, Z_i, S)= {{\texttt {cpre}}}(F, Z_{i}) \wedge S. \end{aligned}$$
Approximate solution to the Dubins vehicle reach game visualized as a subset of the state space.
The above iterations are not guaranteed to reach a fixed point in a finite number of iterations, except under certain technical conditions [21]. Figure 2 depicts an approximate region where the controller can force the Dubins vehicle to enter T. We showcase different improvements relative to a base line script used to generate Fig. 2. A toolbox that adopts this paper's framework is being actively developed and is open sourced at [2]. It is written in python 3.6 and uses the dd package as an interface to CUDD [20], a library in C/C++ for constructing and manipulating binary decision diagrams (BDD). All experiments were run on a single core of a 2013 Macbook Pro with 2.4 GHz Intel Core i7 and 8 GB of RAM.
The following section uses relational interfaces to represent the controlled predecessor \({{\texttt {cpre}}}(\cdot )\) and iterations (2) and (3) as a computational pipeline. Subsequent sections show how modifying this pipeline leads to favorable theoretical properties and computational gains.
3 Relational Interfaces
Relational interfaces are predicates augmented with annotations about each variable's role as an input or output1. They abstract away a component's internal implementation and only encode an input-output relation.
(Relational Interface [22]). An interface \(M(i,o)\) consists of a predicate M over a set of input variables \(i\) and output variables \(o\).
For an interface M(i, o), we call (i, o) its input-output signature. An interface is a sink if it contains no outputs and has signature like \((i, {\varnothing })\), and a source if it contains no inputs like \(({\varnothing }, o)\). Sinks and source interfaces can be interpreted as sets whereas input-output interfaces are relations. Interfaces encode relations through their predicates and can capture features such as non-deterministic outputs or blocking (i.e., disallowed, error) inputs. A system blocks for an input assignment if there does not exist a corresponding output assignment that satisfies the interface relation. Blocking is a critical property used to declare requirements; sink interfaces impose constraints by modeling constrain violations as blocking inputs. Outputs on the other hand exhibit non-determinism, which is treated as an adversary. When one interface's outputs are connected to another's inputs, the outputs seek to cause blocking whenever possible.
3.1 Atomic and Composite Operators
Operators are used to manipulate interfaces by taking interfaces and variables as inputs and yielding another interface. We will show how the controlled predecessor \({{\texttt {cpre}}}(\cdot )\) in (1) can be constructed by composing operators appearing in [22] and one additional one. The first, output hiding, removes interface outputs.
(Output Hiding [22]). Output hiding operator \({\texttt {\textit{ohide}}}(w, F)\) over interface \(F(i, o)\) and outputs w yields an interface with signature \((i,o\setminus w)\).
$$\begin{aligned} {\texttt {\textit{ohide}}}(w, F) = {\exists w}F \end{aligned}$$
Existentially quantifying out w ensures that the input-output behavior over the unhidden variables is still consistent with potential assignments to w. The operator \({\texttt {nb}}(\cdot )\) is a special variant of \({\texttt {\textit{ohide}}}(\cdot )\) that hides all outputs, yielding a sink encoding all non-blocking inputs to the original interface.
(Nonblocking Inputs Sink). Given an interface \(F(i,o)\), the nonblocking operation nb(F) yields a sink interface with signature \((i, {\varnothing })\) and predicate \({\texttt {nb}}(F) = {\exists o}F\). If \(F(i, {\varnothing })\) is a sink interface, then \({\texttt {nb}}(F) = F\) yields itself. If \(F({\varnothing }, o)\) is a source interface, then \({\texttt {nb}}(F) = \bot \) if and only if \(F \Leftrightarrow \bot \); otherwise \({\texttt {nb}}(F) = \top \).
The interface composition operator takes multiple interfaces and "collapses" them into a single input-output interface. It can be viewed as a generalization of function composition in the special case where each interface encodes a total function (i.e., deterministic output and inputs never block).
(Interface Composition [22]). Let \(F_1(i_1, o_1)\) and \(F_2(i_2, o_2)\) be interfaces with disjoint output variables \(o_1 \cap o_2 \equiv {\varnothing }\) and \(i_1 \cap o_2 \equiv {\varnothing }\) which signifies that \(F_2\)'s outputs may not be fed back into \(F_1\)'s inputs. Define new composite variables
$$\begin{aligned} io_{12}&\equiv o_1 \cap i_2 \end{aligned}$$
$$\begin{aligned} i_{12}&\equiv (i_1 \cup i_2) \setminus io_{12} \end{aligned}$$
$$\begin{aligned} o_{12}&\equiv o_1 \cup o_2. \end{aligned}$$
Composition \({\texttt {comp}}(F_1, F_2)\) is an interface with signature \((i_{12}, o_{12})\) and predicate
$$\begin{aligned} F_1 \wedge F_2 \wedge {\forall o_{12}}(F_1 \Rightarrow {\texttt {nb}}(F_2)). \end{aligned}$$
Interface subscripts may be swapped if instead \(F_2\)'s outputs are fed into \(F_1\).
Interfaces \(F_1\) and \(F_2\) are composed in parallel if \(io_{21} \equiv {\varnothing }\) holds in addition to \(io_{12} \equiv {\varnothing }\). Equation (8) under parallel composition reduces to \(F_1 \wedge F_2\) (Lemma 6.4 in [22]) and \({\texttt {comp}}(\cdot )\) is commutative and associative. If \(io_{12} \not \equiv {\varnothing }\), then they are composed in series and the composition operator is only associative. Any acyclic interconnection can be composed into a single interface by systematically applying Definition 4's binary composition operator. Non-deterministic outputs are interpreted to be adversarial. Series composition of interfaces has a built-in notion of robustness to account for \(F_1\)'s non-deterministic outputs and blocking inputs to \(F_2\) over the shared variables \(io_{12}\). The term \({\forall o_{12}}(F_1 \Rightarrow {\texttt {nb}}(F_2))\) in Eq. (8) is a predicate over the composition's input set \(i_{12}\). It ensures that if a potential output of \(F_1\) may cause \(F_2\) to block, then \({\texttt {comp}}(F_1, F_2)\) must preemptively block.
The final atomic operator is input hiding, which may only be applied to sinks. If the sink is viewed as a constraint, an input variable is "hidden" by an angelic environment that chooses an input assignment to satisfy the constraint. This operator is analogous to projecting a set into a lower dimensional space.
(Hiding Sink Inputs). Input hiding operator \({\texttt {ihide}}(w,F)\) over sink interface \(F(i, {\varnothing })\) and inputs w yields an interface with signature \((i \setminus w, {\varnothing })\).
$$\begin{aligned} {\texttt {ihide}}(w, F) = {\exists w} F \end{aligned}$$
Unlike the composition and output hiding operators, this operator is not included in the standard theory of relational interfaces [22] and was added to encode a controller predecessor introduced subsequently in Eq. (10).
3.2 Constructing Control Synthesis Pipelines
The robust controlled predecessor (1) can be expressed through operator composition.
The controlled predecessor operator (10) yields a sink interface with signature \((x, {\varnothing })\) and predicate equivalent to the predicate in (1).
$$\begin{aligned} {{\texttt {cpre}}}(F, Z)&= {\texttt {ihide}}(u, {\texttt {\textit{ohide}}}(x^+, {\texttt {comp}}(F,Z))). \end{aligned}$$
The simple proof is provided in the extended version at [1]. Proposition 1 signifies that controlled predecessors can be interpreted as an instance of robust composition of interfaces, followed by variable hiding. It can be shown that \({\texttt {safe}}(F,Z, S) = {\texttt {comp}}({{\texttt {cpre}}}(F, Z), S)\) because \(S(x, {\varnothing })\) and \({{\texttt {cpre}}}(F, Z)\) would be composed in parallel.2 Figure. 3 shows a visualization of the safety game's fixed point iteration from the point of view of relational interfaces. Starting from the right-most sink interface S (equivalent to \(Z_0\)) the iteration (3) constructs a sequence of sink interfaces \(Z_1, Z_2,...\) encoding relevant subsets of the state space. The numerous \(S(x, {\varnothing })\) interfaces impose constraints and can be interpreted as monitors that raise errors if the safety constraint is violated.
Safety control synthesis iteration (3) depicted as a sequence of sink interfaces.
3.3 Modifying the Control Synthesis Pipeline
Equation (10)'s definition of \({{\texttt {cpre}}}(\cdot )\) is oblivious to the domains of variables x, u, and \(x^+\). This generality is useful for describing a problem and serving as a blank template. Whenever problem structure exists, pipeline modifications refine the general algorithm into a form that reflects the specific problem instance. They also allow a user to inject implicit preferences into a problem and reduce computational bottlenecks or to refine a solution. The subsequent sections apply this philosophy to the abstraction-based control techniques from Sect. 1.1:
Sect. 4: Coarsening interfaces reduces the computational complexity of a problem by throwing away fine grain information. The synthesis result is conservative but the degree of conservatism can be modified.
Sect. 5: Refining interfaces decreases result conservatism. Refinement in combination with coarsening allows one to dynamically modulate the complexity of the problem as a function of multiple criteria such as the result granularity or minimizing computational resources.
Sect. 6: If the dynamics or specifications are decomposable then the control predecessor operator can be broken apart to refect that decomposition.
These sections do more than simply reconstruct existing techniques in the language of relational interfaces. They uncover some implicit assumptions in existing tools and either remove them or make them explicit. Minimizing the number of assumptions ensures applicability to a diverse collection of systems and specifications and compatibility with future algorithmic modifications.
4 Interface Abstraction via Quantization
A key motivator behind abstraction-based control synthesis is that computing the game iterations from Eqs. (2) and (3) exactly is often intractable for high-dimensional nonlinear dynamics. Termination is also not guaranteed. Quantizing (or "abstracting") continuous interfaces into a finite counterpart ensures that each predicate operation of the game terminates in finite time but at the cost of the solution's precision. Finer quantization incurs a smaller loss of precision but can cause the memory and computational requirements to store and manipulate the symbolic representation to exceed machine resources.
This section first introduces the notion of interface abstraction as a refinement relation. We define the notion of a quantizer and show how it is a simple generalization of many existing quantizers in the abstraction-based control literature. Finally, we show how one can inject these quantizers anywhere in the control synthesis pipeline to reduce computational bottlenecks.
4.1 Theory of Abstract Interfaces
While a controller synthesis algorithm can analyze a simpler model of the dynamics, the results have no meaning unless they can be extrapolated back to the original system dynamics. The following interface refinement condition formalizes a condition when this extrapolation can occur.
(Interface Refinement [22]). Let F(i, o) and \(\hat{F}(\hat{i}, \hat{o})\) be interfaces. \(\hat{F}\) is an abstraction of F if and only if \(i \equiv \hat{i}\), \(o \equiv \hat{o}\), and
$$\begin{aligned} {\texttt {nb}}(\hat{F})&\Rightarrow {\texttt {nb}}(F) \end{aligned}$$
$$\begin{aligned} \left( {\texttt {nb}}(\hat{F}) \wedge F\right)&\Rightarrow \hat{F} \end{aligned}$$
are valid formulas. This relationship is denoted by \(\hat{F} \preceq F\).
Definition 6 imposes two main requirements between a concrete and abstract interface. Equation (11) encodes the condition where if \(\hat{F}\) accepts an input, then F must also accept it; that is, the abstract component is more aggressive with rejecting invalid inputs. Second, if both systems accept the input then the abstract output set is a superset of the concrete function's output set. The abstract interface is a conservative representation of the concrete interface because the abstraction accepts fewer inputs and exhibits more non-deterministic outputs. If both the interfaces are sink interfaces, then \(\hat{F} \preceq F\) reduces down to \(\hat{F} \subseteq F\) when \(F, \hat{F}\) are interpreted as sets. If both are source interfaces then the set containment direction is flipped and \(\hat{F} \preceq F\) reduces down to \(F \subseteq \hat{F}\).
Example depiction of the refinement partial order. Each small plot on the depicts input-output pairs that satisfy an interface's predicate. Inputs (outputs) vary along the horizontal (vertical) axis. Because B blocks on some inputs but A accepts all inputs \(B \preceq A\). Interface C exhibits more output non-determinism than A so \(C \preceq A\). Similarly \(D \preceq B\), \(D \preceq C\), \(\top \preceq C\), etc. Note that B and C are incomparable because C exhibits more output non-determinism and B blocks for more inputs. The false interface \(\bot \) is a universal abstraction, while \(\top \) is incomparable with B and D.
The refinement relation satisfies the required reflexivity, transitivity, and antisymmetry properties to be a partial order [22] and is depicted in Fig. 4. This order has a bottom element \(\bot \) which is a universal abstraction. Conveniently, the bottom element \(\bot \) signifies both boolean false and the bottom of the partial order. This interface blocks for every potential input. In contrast, Boolean \(\top \) plays no special role in the partial order. While \(\top \) exhibits totally non-deterministic outputs, it also accepts inputs. A blocking input is considered "worse" than non-deterministic outputs in the refinement order. The refinement relation \(\preceq \) encodes a direction of conservatism such that any reasoning done over the abstract models is sound and can be generalized to the concrete model.
Theorem 1
(Informal Substitutability Result [22]). For any input that is allowed for the abstract model, the output behaviors exhibited by an abstract model contains the output behaviors exhibited by the concrete model.
If a property on outputs has been established for an abstract interface, then it still holds if the abstract interface is replaced with the concrete one. Informally, the abstract interface is more conservative so if a property holds with the abstraction then it must also hold for the true system. All aforementioned interface operators preserve the properties of the refinement relation of Definition 6, in the sense that they are monotone with respect to the refinement partial order.
(Composition Preserves Refinement [22]). Let \(\hat{A} \preceq A\) and \(\hat{B} \preceq B \). If the composition is well defined, then \({\texttt {comp}}(\hat{A}, \hat{B}) \preceq {\texttt {comp}}(A,B)\).
(Output Hiding Preserves Refinement [22]). If \(A \preceq B\), then \({\texttt {\textit{ohide}}}(w,A) \preceq {\texttt {\textit{ohide}}}(w,B)\) for any variable w.
(Input Hiding Preserves Refinement). If A, B are both sink interfaces and \(A \preceq B\), then \({\texttt {ihide}}(w, A) \preceq {\texttt {ihide}}(w, B)\) for any variable w.
Proofs for Theorems 2 and 3 are provided in [22]. Theorem 4's proof is simple and is omitted. One can think of using interface composition and variable hiding to horizontally (with respect to the refinement order) navigate the space of all interfaces. The synthesis pipeline encodes one navigated path and monotonicity of these operators yields guarantees about the path's end point. Composite operators such as \({{\texttt {cpre}}}(\cdot )\) chain together multiple incremental steps. Furthermore since the composition of monotone operators is itself a monotone operator, any composite constructed from these parts is also monotone. In contrast, the coarsening and refinement operators introduced later in Definitions 8 and 10 respectively are used to move vertically and construct abstractions. The "direction" of new composite operators can easily be established through simple reasoning about the cumulative directions of their constituent operators.
4.2 Dynamically Coarsening Interfaces
In practice, the sequence of interfaces \(Z_i\) generated during synthesis grows in complexity. This occurs even if the dynamics F and the target/safe sets have compact representations (i.e., fewer nodes if using BDDs). Coarsening F and \(Z_i\) combats this growth in complexity by effectively reducing the amount of information sent between iterations of the fixed point procedure.
Spatial discretization or coarsening is achieved by use of a quantizer interface that implicitly aggregates points in a space into a partition or cover.
A quantizer Q(i, o) is any interface that abstracts the identity interface \((i == o)\) associated with the signature (i, o).
Quantizers decrease the complexity of the system representation and make synthesis more computationally tractable. A coarsening operator abstracts an interface by connecting it in series with a quantizer. Coarsening reduces the number of non-blocking inputs and increases the output non-determinism.
(Input/Output Coarsening). Given an interface F(i, o) and input quantizer \(Q_{}(\hat{i}, i)\), input coarsening yields an interface with signature \((\hat{i}, o)\).
$$\begin{aligned} {\texttt {icoarsen}}(F, Q(\hat{i}, i))&= {\texttt {\textit{ohide}}}(i, {\texttt {comp}}(Q(\hat{i}, i), F) ) \end{aligned}$$
Similarly, given an output quantizer \(Q_{}(o, \hat{o})\), output coarsening yields an interface with signature \((i, \hat{o})\).
$$\begin{aligned} {\texttt {ocoarsen}}(F, Q(o, \hat{o}))&= {\texttt {\textit{ohide}}}(o, {\texttt {comp}}( F, Q(o, \hat{o}))) \end{aligned}$$
Coarsening of the \(F_x\) interface to \(2^3, 2^4\) and \(2^5\) bins along each dimension for a fixed v assignment. Interfaces are coarsened within milliseconds for BDDs but the runtime depends on the finite abstraction's data structure representation.
Figure 5 depicts how coarsening reduces the information required to encode a finite interface. It leverages a variable precision quantizer, whose implementation is described in the extended version at [1].
The corollary below shows that quantizers can be seamlessly integrated into the synthesis pipeline while preserving the refinement order. It readily follows from Theorems 2, 3, and the quantizer definition.
Input and output coarsening operations (13) and (14) are monotone operations with respect to the interface refinement order \(\preceq \).
Number of BDD nodes (red) and number of states in reach basin (blue) with respect to the reach game iteration with a greedy quantization. The solid lines result from the unmodified game with no coarsening heuristic. The dashed lines result from greedy coarsening whenever the winning region exceeds 3000 BDD nodes. (Color figure online)
It is difficult to know a priori where a specific problem instance lies along the spectrum between mathematical precision and computational efficiency. It is then desirable to coarsen dynamically in response to runtime conditions rather than statically beforehand. Coarsening heuristics for reach games include:
Downsampling with progress [7]: Initially use coarser system dynamics to rapidly identify a coarse reach basin. Finer dynamics are used to construct a more granular set whenever the coarse iteration "stalls". In [7] only the \(Z_i\) are coarsened during synthesis. We enable the dynamics F to be as well.
Greedy quantization: Selectively coarsening along certain dimensions by checking at runtime which dimension, when coarsened, would cause \(Z_i\) to shrink the least. This reward function can be leveraged in practice because coarsening is computationally cheaper than composition. For BDDs, the winning region can be coarsened until the number of nodes reduces below a desired threshold. Figure 6 shows this heuristic being applied to reduce memory usage at the expense of answer fidelity. A fixed point is not guaranteed as long as quantizers can be dynamically inserted into the synthesis pipeline, but is once quantizers are always inserted at a fixed precision.
The most common quantizer in the literature never blocks and only increases non-determinism (such quantizers are called "strict" in [18, 19]). If a quantizer is interpreted as a partition or cover, this requirement means that the union must be equal to an entire space. Definition 7 relaxes that requirement so the union can be a subset instead. It also hints at other variants such as interfaces that don't increase output non-determinism but instead block for more inputs.
5 Refining System Dynamics
Shared refinement [22] is an operation that takes two interfaces and merges them into a single interface. In contrast to coarsening, it makes interfaces more precise. Many tools construct system abstractions by starting from the universal abstraction \(\bot \), then iteratively refining it with a collection of smaller interfaces that represent input-output samples. This approach is especially useful if the canonical concrete system is a black box function, Simulink model, or source code file. These representations do not readily lend themselves to the predicate operations or be coarsened directly. We will describe later how other tools implement a restrictive form of refinement that introduces unnecessary dependencies.
Interfaces can be successfully merged whenever they do not contain contradictory information. The shared refinability condition below formalizes when such a contradiction does not exist.
(Shared Refinability [22]). Interfaces \(F_1(i,o)\) and \(F_2(i,o)\) with identical signatures are shared refinable if
$$\begin{aligned} \left( {\texttt {nb}}(F_1) \wedge {\texttt {nb}}(F_2) \right) \Rightarrow {\exists o} (F_1 \wedge F_2) \end{aligned}$$
For any inputs that do not block for all interfaces, the corresponding output sets must have a non-empty intersection. If multiple shared refinable interfaces, then they can be combined into a single one that encapsulates all of their information.
Definition 10
(Shared Refinement Operation [22]). The shared refinement operation combines two shared refinable interfaces \(F_1\) and \(F_2\), yielding a new identical signature interface corresponding to the predicate
$$\begin{aligned} {\texttt {refine}}(F_1, F_2) = ({\texttt {nb}}(F_1) \vee {\texttt {nb}}(F_2)) \wedge ({\texttt {nb}}(F_1) \Rightarrow F_1) \wedge ({\texttt {nb}}(F_2) \Rightarrow F_2). \end{aligned}$$
The left term expands the set of accepted inputs. The right term signifies that if an input was accepted by multiple interfaces, the output must be consistent with each of them. The shared refinement operation reduces to disjunction for sink interfaces and to conjunction for source interfaces.
Shared refinement's effect is to move up the refinement order by combining interfaces. Given a collection of shared refinable interfaces, the shared refinement operation yields the least upper bound with respect to the refinement partial order in Definition 6. Violation of (15) can be detected if the interfaces fed into \({\texttt {refine}}(\cdot )\) are not abstractions of the resulting interface.
5.1 Constructing Finite Interfaces Through Shared Refinement
A common method to construct finite abstractions is through simulation and overapproximation of forward reachable sets. This technique appears in tools such as PESSOA [12], SCOTS [19], MASCOT [7], ROCS [9] and ARCS [4]. By covering a sufficiently large portion of the interface input space, one can construct larger composite interfaces from smaller ones via shared refinement.
(Left) Result of sample and coarsen operations for control system interface \(F(x \cup u, x^+)\). The I and \(\hat{I}\) interfaces encode the same predicate, but play different roles as sink and source. (Right) Visualization of finite abstraction as traversing the refinement partial order. Nodes represent interfaces and edges signify data dependencies for interface manipulation operators. Multiple refine edges point to a single node because refinement combines multiple interfaces. Input-output (IO) sample and coarsening are unary operations so the resulting nodes only have one incoming edge. The concrete interface F refines all others, and the final result is an abstraction \(\hat{F}\).
Smaller interfaces are constructed by sampling regions of the input space and constructing an input-output pair. In Fig. 7's left half, a sink interface \(I(x \cup u, {\varnothing })\) acts as a filter. The source interface \(\hat{I}({\varnothing }, x \cup u)\) composed with \(F(x \cup u, x^+)\) prunes any information that is outside the relevant input region. The original interface refines any sampled interface. To make samples finite, interface inputs and outputs are coarsened. An individual sampled abstraction is not useful for synthesis because it is restricted to a local portion of the interface input space. After sampling many finite interfaces are merged through shared refinement. The assumption \(\hat{I}_i \Rightarrow {\texttt {nb}}(F)\) encodes that the dynamics won't raise an error when simulated and is often made implicitly. Figure 7's right half depicts the sample, coarsen, and refine operations as methods to vertically traverse the interface refinement order.
Critically, \({\texttt {refine}}(\cdot )\) can be called within the synthesis pipeline and does not assume that the sampled interfaces are disjoint. Figure 8 shows the results from refining the dynamics with a collection of state-control hyper-rectangles that are randomly generated via uniformly sampling their widths and offsets along each dimension. These hyper-rectangles may overlap. If the same collection of hyper-rectangles were used in MASCOT, SCOTS, ARCS, or ROCS then this would yield a much more conservative abstraction of the dynamics because their implementations are not robust to overlapping or misaligned samples. PESSOA and SCOTS circumvent this issue altogether by enforcing disjointness with an exhaustive traversal of the state-control space, at the cost of unnecessarily coupling the refinement and sampling procedures. The lunar lander in the extended version [1] embraces overlapping and uses two mis-aligned grids to construct a grid partition with \(p^N\) elements with only \(p^N (\frac{1}{2})^{N-1}\) samples (where p is the number of bins along each dimension and N is the interface input dimension). This technique introduces a small degree of conservatism but its computational savings typically outweigh this cost.
The number of states in the computed reach basin grows with the number of random samples. The vertical axis is lower bounded by the number of states in the target 131k and upper bounded by 631k, the number of states using an exhaustive traversal. Naive implementations of the exhaustive traversal would require 12 million samples. The right shows basins for 3000 (top) and 6000 samples (bottom).
6 Decomposed Control Predecessor
A decomposed control predecessor is available whenever the system state space consists of a Cartesian product and the dynamics are decomposed component-wise such as \(F_x, F_y\), and \(F_\theta \) for the Dubins vehicle. This property is common for continuous control systems over Euclidean spaces. While one may construct F directly via the abstraction sampling approach, it is often intractable for larger dimensional systems. A more sophisticated approach abstracts the lower dimensional components \(F_x, F_y\), and \(F_\theta \) individually, computes \(F = {\texttt {comp}}(F_x, F_y, F_\theta )\), then feeds it to the monolithic \({{\texttt {cpre}}}(\cdot )\) from Proposition 1. This section's approach is to avoid computing F at all and decompose the monolithic \({{\texttt {cpre}}}(\cdot )\). It operates by breaking apart the term \({\texttt {\textit{ohide}}}(x^+, {\texttt {comp}}(F,Z))\) in such a way that it respects the decomposition structure. For the Dubins vehicle example \({\texttt {\textit{ohide}}}(x^+, {\texttt {comp}}(F,Z))\) is replaced with
$$\begin{aligned} {\texttt {\textit{ohide}}}(p_x^+, {\texttt {comp}}(F_x, {\texttt {\textit{ohide}}}(p_y^+, {\texttt {comp}}(F_y, {\texttt {\textit{ohide}}}(\theta ^+, {\texttt {comp}}(F_\theta , Z)))))) \end{aligned}$$
yielding a sink interface with inputs \(p_x, p_y, v, \theta \), and \(\omega \). This representation and the original \({\texttt {\textit{ohide}}}(x^+, {\texttt {comp}}(F,Z))\) are equivalent because \({\texttt {comp}}(\cdot )\) is associative and interfaces do not share outputs \(x^+ \equiv \{p_x^+, p_y^+, \theta ^+\}\). Figure 9 shows multiple variants of \({{\texttt {cpre}}}(\cdot )\) and improved runtimes when one avoids preemptively constructing the monolithic interface. The decomposed \({{\texttt {cpre}}}(\cdot )\) resembles techniques to exploit partitioned transition relations in symbolic model checking [5].
No tools from Sect. 1.1 natively support decomposed control predecessors. We've shown a decomposed abstraction for components composed in parallel but this can also be generalized to series composition to capture, for example, a system where multiple components have different temporal sampling periods.
A monolithic \({{\texttt {cpre}}}(\cdot )\) incurs unnecessary pre-processing and synthesis runtime costs for the Dubins vehicle reach game. Each variant of \({{\texttt {cpre}}}(\cdot )\) above composes the interfaces \(F_x, F_y\) and \(F_\theta \) in different permutations. For example, \(F_{xy}\) represents \( {\texttt {comp}}(F_x, F_y)\) and F represents \({\texttt {comp}}(F_x, F_y, F_\theta )\).
Tackling difficult control synthesis problems will require exploiting all available structure in a system with tools that can flexibly adapt to an individual problem's idiosyncrasies. This paper lays a foundation for developing an extensible suite of interoperable techniques and demonstrates the potential computational gains in an application to controller synthesis with finite abstractions. Adhering to a simple yet powerful set of well-understood primitives also constitutes a disciplined methodology for algorithm development, which is especially necessary if one wants to develop concurrent or distributed algorithms for synthesis.
Relational interfaces closely resemble assume-guarantee contracts [16]; we opt to use relational interfaces because inputs and outputs play a more prominent role.
Disjunctions over sinks are required to encode \({\texttt {reach}}(\cdot )\). This will be enabled by the shared refinement operator defined in Definition 10.
http://arxiv.org/abs/1905.09503
https://github.com/ericskim/redax/tree/CAV19
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\begin{document}
\author[C.~Croke]{Christopher Croke$^+$} \address{ Department ofbb: Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 USA} \email{[email protected]} \thanks{Supported in part by NSF grant DMS 10-03679 and an Eisenbud Professorship at M.S.R.I.}
\title[Scattering rigidity] {Scattering rigidity with trapped geodesics }
\keywords{Scattering rigidity, Lens rigidity, trapped geodesics}
\begin{abstract} We prove that the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$.
The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to $\gamma'_V(T_0)$ where $\gamma_V$ is the unit speed geodesic determined by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that $\gamma_V(t)$ again lies in the boundary.
We show that any other Riemannian manifold $(M,\partial M,g)$ with boundary $\partial M$ isometric to $\partial(D^n\times S^1)$ and with the same scattering data must be isometric to $D^n\times S^1$.
This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M,\partial M,g)$ have measure 0 in the unit tangent bundle.
\end{abstract}
\maketitle
\section{Introduction} In this paper we prove scattering rigidity (see below) for a number of compact Riemannian manifolds with boundary that have trapped geodesics. A geodesic $\gamma(t)$ in a manifold with boundary is trapped if its domain of definition is unbounded.
Consider a compact Riemannian manifold $(M,\partial M,g)$ with boundary $\partial M$ and metric $g$. We will let $U^+\partial M$ represent the space of inwardly pointing unit vectors at the boundary. That is $V\in U^+\partial M$ means that $V$ is a unit vector based at a boundary point and $\langle V,\eta^+\rangle \geq 0$ where $\eta^+$ is the unit vector of $M$ normal to $\partial M$ and pointing inward. Similarly we let $U^-\partial M$ represent the outward vectors. Note that $U^+\partial M\cap U^-\partial M=U(\partial M)$ the unit tangent bundle of $\partial M$.
For $V\in U^+\partial M$ let $\gamma_V(t)$ be the geodesic with $\gamma'(0)=V$. We let $TT(V)\in [0,\infty]$ (the travel time) be the first time $t>0$ when $\gamma_V(t)$ hits the boundary again. If $\gamma_V(t)$ never hits the boundary again then $TT(V)=\infty$ (i.e $\gamma_V$ is trapped) while if either $\gamma_V(t)$ does not exist for any $t>0$ or there are arbitrarily small values of $t>0$ such that $\gamma(t)\in \partial M$ then we let $TT(V)=0$. Note that $TT(V)=0$ implies that $V\in U(\partial M)$.
The scattering map ${\emph{S}}:U^+\partial M\to U^-\partial M$ takes a vector $V\in U^+\partial M$ to the vector $\gamma'(TT(V))\in U^-\partial M$. It will not be defined when $TT(V)=\infty$ and will be $V$ itself when $TT(V)=0$. If another manifold $(M_1,\partial M_1,g_1)$ has isometric boundary to $(M,\partial M,g)$ in the sense that $(\partial M,g)$ (g restricted to $\partial M$) is isometric to $(\partial M_1,g_1)$ then we can identify $U^+\partial M_1$ with $U^+\partial M$ and $U^-\partial M_1$ with $U^-\partial M$. We say that $(M,\partial M,g)$ and $(M_1,\partial M_1,g_1)$ have the same scattering data if they have isometric boundaries and under the identifications given by the isometry they have the same scattering map. If in addition the travel times $TT(V)$ coincide then they are said to have the same lens data.
A compact manifold $(M,\partial M,g)$ is said to be scattering (resp. lens) rigid if for any other manifold $(M_1,\partial M_1,g_1)$ with the same scattering (resp. lens) data there is an isometry from $M_1$ to $M$ that agrees with the given isometry of the boundaries.
\begin{theorem} \label{3+dims} For any $n\geq 2$ the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is a ball in $\R^n$. \end{theorem}
The fact that not all manifolds are scattering rigid was pointed out in \cite{Cr91}. For $\frac 1 4 >\epsilon>0$ let $h(t)$ be a small smooth bump function which is 0 outside $(-\epsilon ,\epsilon)$ and positive in $(-\epsilon,\epsilon)$. For $s\in (-1+2\epsilon,1-2\epsilon)$ consider surfaces of revolution $g_s$ with smooth generating functions $F_s(t)=1+h(s+t)$ for $t\in [-1,1]$. These surfaces of revolution look like flat cylinders with bumps on them that are shifted depending on $s$ but otherwise look the same (see figure \ref{cylinders}). The Clairaut relations show that, independent of $s$, geodesics entering one side with a given initial condition exit out the other side after the same distance at the same point with the same angle. Hence all metrics have the same scattering data (and in fact lens data) but are not isometric. A much larger class of examples was given in section 6 of \cite{Cr-Kl94}. All of the examples have in common that there are trapped geodesics.
\begin{figure}
\caption{not isometric but same scattering and lens data}
\label{cylinders}
\end{figure}
The scattering and lens rigidity problems are closely related to other inverse problems. In particular the boundary rigidity problem is equivalent to the lens rigidity question in the Simple and SGM cases. See \cite{Cr91} and \cite{Cr04} for definitions and relations to some other problems. There is a vast literature on these problems (see for example \cite{Be83,Bu-Iv06,Cr91,Cr90,Gr83,Mi81,Mu77,Ot90,Pe-Sh88,Pe-Ul05,St-Uh09}). Most of the results in these papers concern manifolds with no trapped geodesics. An exception is \cite{St-Uh09} where they prove local scattering rigidity (i.e. if the two metrics are in a particular $C^k$ neighborhood they must be isometric) for a class of Riemannian manifolds that includes those discussed in this paper. However, to date all of the global rigidity results concern manifolds without trapped geodesics. The results in this paper constitute the first examples of (global) scattering rigid manifolds that have trapped geodesics.
The key difficulty in our case is to show that the set of unit vectors tangent to trapped geodesic rays in the metric $g_1$ has measure 0 in the unit tangent bundle. This allows us (with an application of Santal\'o's formula) to conclude that $g$ and $g_1$ have the same volumes. Since the metric $g$ has a real factor (i.e. $D^n\subset \R\times \R^{n-1}$) we can use a result from \cite{Cr-Kl98} to complete the argument. In fact, the argument in Theorem \ref{3+dims} extends (see section \ref{gens}) to the case where $D^n$ above is replaced by a ball in $\R\times N^{n-1}$ where $N$ is a complete simply connected Riemannian manifold with nonpositive curvature. (In fact with more work one could extend this to the case of no conjugate points but we chose not to give the slightly different arguments here.)
One case that was not dealt with in Theorem \ref{3+dims} is the two dimensional case, namely the flat cylinder $[-1,1]\times S^1$ and the M\"obius strip. There are ways in which this case is easier and ways in which it is harder. The major differences are that the scattering data does not determine the lens data and we cannot conclude that the $C^\infty$ jets of the metrics agree at the boundary. The problem of lens rigidity in the two dimensional case will be taken up in a future paper with Pilar Herreros. In particular, it turns out that the M\"obius strip is not scattering rigid if $(M_1,\partial M_1,g_1)$ is allowed to be $C^1$.
The author would like to thank Gunther Uhlmann who first posed the problem of the rigidity of $D^2 \times S^1$ to him some years ago, to Haomin Wen for pointing out the Eaton Lens example, and Pilar Herreros for a careful reading of earlier drafts.
\section{The $D^n\times S^1$ for $n\geq 2$ case} \label{highdim}
In this section we prove Theorem \ref{3+dims}. We consider generalizations in Section \ref{gens}. Throughout the section $n\geq 2$ and $g$ will be the standard flat product metric on $M=D^n\times S^1$. For concreteness we will take $D^n$ to be the unit ball and $S^1$ to have length $2\pi$. $(M_1,S^{n-1}\times S^1,g_1)$ will be another Riemannian metric on a manifold $M_1$ whose boundary is isometric to that of $M$. We use this isometry to identify the two boundaries. We assume that $g_1$ has the same scattering data as $g$.
We do not a-priori assume that $M_1$ is diffeomorphic to $M$. For each $p\in \partial M=\partial M_1$ we let $\tau_p\subset \partial M=\partial M_1$ be the closed curve in vertical (i.e. $S^1$) direction.
\begin{lemma} \label{lensdata} $g_1$ has the same lens data as $g$. \end{lemma}
{\bf Proof:} This is an application of the first variation formula. For $V\in U^+\partial M$ let $G(V)= L(\gamma_{1 V})-L(\gamma_{V})$. We need to show that $G(V)=0$ for all $V$. A smooth curve of initial conditions $s\mapsto V(s)$ in the interior of $U^+\partial M$ gives rise to smooth variations $\gamma_{V(s)}$ through unit speed geodesics in $M$ and $\gamma_{1V(s)}$ through unit speed geodesics in $M_1$ whose initial and final tangents agree. (Note that this uses the convexity of the boundary since for more general manifolds with boundary there may be a discontinuous jump in the endpoints of geodesics.) The first variation formula (along with the fact that the metrics agree at the boundary) tells us that $\frac d{ds} L(\gamma_{V(s)})= \frac d{ds} L(\gamma_{1V(s)})$ . Hence $G(V(s))= L(\gamma_{1V(s)})-L(\gamma_{V(s)})$ is independent of $s$. Since $U^+\partial M$ is connected, G is a constant $C$. Further when $V$ approaches a non vertical vector (i.e. one not tangent to the $S^1$ factor) in $\partial(U^+\partial M)=U\partial M$ then $L(\gamma_V)$ approaches $0$ and hence $C>0$. If we knew $L(\gamma_{1 V})$ approached $0$ for this or any sequence then $C=0$ and the lemma would follow. We now show that this must happen.
If this is not the case (i.e. if $C>0$) then the boundary must be {\em concave} everywhere and further when $V_i$ approaches a non vertical vector $V$ in $\partial(U^+\partial M)=U\partial M$ as above then $L(\gamma_{1 V})$ approaches $C$ and the limiting geodesic $\gamma_{1V}$ must be a closed geodesic of length $C$ with initial (and final) tangent vector $V$. By taking limits this is also true for the vertical vectors. (This certainly looks unlikely to happen. However, there is an example \cite{H-H-L06} - an Eaton lens - of a manifold with a singularity having the same scattering data as the flat 2-disc but different lens data. In that case the boundary is a closed geodesic.)
Note that in our case (since the dimension of the boundary is at least 2) all of these closed geodesics $\gamma_{1V}$ for $V$ based at a boundary point $p$ are homotopic to each other since there is a curve of initial tangent vectors $V_t$ between any two initial tangents (i.e. the homotopy is through the curves $\gamma_{1V_t}$). In particular, they are all homotopic to their negatives (i.e. running around in the opposite direction). Thus going twice around such a geodesic is a contractible curve.
In fact, the closed geodesic $\gamma_V$ of length $C$ tangent to vertical direction $V$ at $p\in \partial M$ is a multiple of $\tau_p$. To see this let $x=\tau_p(t)$ be a point on $\tau_p$ close to $p$ (say $t<\frac \pi 2$) and $x_i$ a sequence of points on $\partial M$ approaching $x$ but not on $\tau_p$. Let $\gamma_{V_i}$ be the minimal $g$ geodesics from $p$ to $x_i$ with $x_i=\gamma_{V_i}(t_i)$. We see that $V_i$ approaches $V$ and $t_i$ approaches $t$. Looking at the other metric we see $x_i=\gamma_{1 V_i}(t_i+C)$ and hence by taking limits $x=\gamma_{1 V}(t+C)$. Since this is true for all $p$ and $x=\tau_p(t)$ for $t<\frac \pi 2$ $\frac \pi 2$ we see that $\tau_p(t)= \gamma_{1 V_i}(t+C)$ and hence $\tau_p$ is the $g_1$ geodesic with initial tangent $V$. Thus $\gamma_{1 V_i}$ simply goes around $\tau_p$ a number of times (i.e. $C$ is an integer multiple of $2\pi$). Thus we know that going twice as many times around $\tau_p$ yields a contractible curve. Thus the next sublemma \ref{noncontractible} gives the desired contradiction. \qed
\begin{sublemma} \label{noncontractible} No multiple of $\tau_p$ is contractible in $M_1$. \end{sublemma}
{\bf Proof:} This argument is an oriented intersection number argument. We begin by seeing that when $C>0$ then $M_1$ is orientable. Fix $x$ on the boundary. Then for every element of $\pi_1$ the shortest loop representing this class is either a geodesic loop in $M_1$ or partly runs along the boundary. By the scattering data assumption the only geodesic loops at $p$ are those that start tangent to the boundary (i.e. the closed geodesics we are discussing). All of these are homotopic to a multiple of $\tau_p$. If the minimizing path runs along the boundary some of the time then (since when it leaves the boundary it must be tangent) the only parts not on the boundary are the closed geodesic loops again. Hence every element of $\pi_1$ has a representative that lies in the boundary. Since running along such curves does not change the orientation, $M_1$ must be orientable.
Chose an orientation reversing diffeomorphism $F:S^{n-1}\to S^{n-1}$. This induces a map $H:S^{n-1}\times [0,1] \to M_1$ by $$H(x,0)=(x,0)\in \partial M_1=S^{n-1}\times S^1\ \ \ \ \ \ \ H(x,1)=(F(x),\frac \pi {10}).$$ For fixed $x$, $H(x,t)$ is the $g_1$ geodesic (parameterized proportional to arclength) with the same initial tangent as the minimizing $g$ geodesic from $(x,0)$ to $(F(x),\frac \pi {10})$ (and hence has length $C$ longer). Note that if $F(x)=x$ then the geodesic $H(x,t)$ may wrap around $\tau_{(x,0)}$ many times before ending at $(x,\frac \pi {10})$.)
Although $H$ may not be transverse to the boundary that is easy to fix. For some small $\epsilon>0$ we can parameterize the $\epsilon$ neighborhood $N_\epsilon$ in $M_1$ of $\partial M_1$ as $[0,\epsilon)\times S^{n-1}\times S^1$ (where $\partial M_1=\{0\}\times S^{n-1}\times S^1$). Further we have a diffeomorphism $D_\epsilon:M_1\to M_1-N_\epsilon$ such that $D_\epsilon(0,x,\theta)=(\epsilon,x,\theta)$. Thus we can take $\tilde H:S^{n-1}\times [-\epsilon,1+\epsilon]\to M_1$ by letting $\tilde H (x,t)= (t,x,0)$ for $t\leq 0$, $\tilde H(x,t)=D_\epsilon(H(x,t))$ for $0\leq t \leq 1$, and let $\tilde H(x,t)=(1+\epsilon-t,F(x),\frac \pi {10})$ for $t\geq 1$. This is now transverse to $\partial M_1$ and we can tweak this to make it smooth. We can double this picture in the manifold double $M_1\times M_1$ to get a smooth map $\bar H: S^{n-1}\times S^1\to M_1\cup M_1$ which is transverse to the curve $\tau_p$. $\tau_p$ passes through each of $S^{n-1}\times \{0\}\in \partial M_1$ and $S^{n-1}\times \{\frac \pi {10}\}\in \partial M_1$ once each time around and our choice of orientations guarantees that the two contributions to the intersection number of $\bar H$ with $\tau_p$ have the same sign. Thus $\bar H$ has a nonzero intersection with any multiple of $\tau_p$. Thus the homology class of any multiple of $\tau_p$ is non zero in $M_1\cup M_1$ and hence in $M_1$.
\qed
Since the lens data and hence distances between nearby boundary points agree, the $C^\infty$ jets of $g$ and $g_1$ also must agree at the boundary. This follows from \cite{L-S-U03},\cite{Uh-Wa03}, or \cite{Zh11} since, for the flat metric $g$, the second fundamental form of the boundary has a positive eigenvalue at every point. (Note that this argument wont work in the two dimensional case $n=1$.) This in particular means that we can glue $(\R^n-D^n)\times S^1$ along the boundary of $M_1$ to yield a smooth metric $M_1^{ext}$ which is isometric to $\R^n\times S^1$ outside of $M_1$.
\begin{lemma} \label{fundgroup} $\pi_1(M_1)=\mathbb Z$ and the generator is represented by the $S^1$ factor of the boundary. \end{lemma}
{\bf Proof:} To see this fix a base point $p\in \partial M_1$. The lens data being the same tells us that the only geodesic loops at a $p$ are the multiples of $\tau_p$. Further the convexity of the boundary guarantees that there is at least one geodesic loop in each homotopy class (the shortest curve in that class). In particular we see as before that $M_1$ is orientable. The lemma now follows from the proof of sublemma \ref{noncontractible}.
\qed
Thus the Riemannian universal cover $\widetilde{M_1}$ of $M_1$ sits naturally in $\widetilde{M_1^{ext}}$ the universal cover of $M_1^{ext}$ and further $\widetilde{M_1^{ext}} - \widetilde{M_1}$ is isometric to $(R^n-D^n)\times \R$. Also $\widetilde M = D^n\times \R$ has the same scattering data as $\widetilde M_1$. We will slightly abuse notation and call the metrics on the universal covers $g$ and $g_1$ as well.
\begin{lemma} \label{geosminimize} An $M_1$ geodesic $\gamma$ between boundary points is the shortest path in its homotopy class (rel boundary points). \end{lemma}
{\bf Proof:} This is the same as saying that such geodesics in the universal cover are the minimizing paths between the endpoints. This is true for $\widetilde M$ where there is a unique geodesic between any two boundary points. Thus there is also a unique geodesic between boundary points in $\widetilde M_1$ which must thus be the minimizing geodesic.
\qed
In fact, this implies that all geodesic segments in $\widetilde{M_1^{ext}}$ are minimizing except possibly in the case that they are segments of geodesics trapped in $\widetilde{M_1}$. If $p$ and $q$ are points in $\widetilde{M_1^{ext}}-\widetilde{M_1}=\R^{n+1}-D^n\times \R$ then this implies that $d_1(p,q)=d(p,q)$. In particular, for $p\in \widetilde{M_1^{ext}}-\widetilde{M_1}$ all geodesics from $p$ minimize. Hence the exponential map is a diffeomorphism which allows us to conclude that not only are the fundamental groups the same but $M_1$ is diffeomorphic to $M$.
A geodesic $\gamma_{1V}$ will either be trapped or coincide with an oriented Euclidean line $L_V$ outside $\widetilde{M_1}$. By the direction of $L_V$ we mean the oriented line through the origin parallel to $L_V$. $L_V$ will be called ``positive'' if it makes a (strictly) positive inner product with the upward vertical direction.
There are two cases that are exceptional. These are trapped geodesics and vertical geodesics (i.e. $\{x\}\times \R$ for $x\in R^n-D^n$). We will exclude both these cases by the phrase ``$L_v$ is not vertical''. For $p\in \widetilde{M_1^{ext}}$ we let ${\bf A}(p)=\{V\in U_p\widetilde{M_1^{ext}}|L_V \textrm{ is not vertical}\}$. Note that for $p\in \widetilde{M_1^{ext}}-\widetilde{M_1}$ we have ${\bf A} (p)$ is just the unit sphere with the north and south pole removed. {$\bf A$ will represent the union of the ${\bf A}(p)$.
\begin{lemma} \label{vertical} If $V_i\in {\bf A}(p)$ converges to a vector $V\in U_p-{\bf A}(p)$ then the directions of the lines $L_{V_i}$ become vertical. \end{lemma}
{\bf Proof:} Assume this is not the case. Then there is a subsequence of the $V_i$ (which we will still call $V_i$) such that the directions of the lines $L_{V_i}$ converge to a non vertical direction $L$. We claim that a subsequence of these $L_{V_i}$ converge to a line $L_W$. To see this we only need to note that the lines $L_{V_i}$ intersect a common compact set. Now the length of $\gamma_{V_i}\cap\widetilde M_1$ is the same as the length of $L_{V_i}\cap D^n\times \R\subset \R^{n+1}$ which is uniformly bounded above (say by $B$) since the directions of the $L_{V_i}$ converge to $L$ which is not vertical. Thus all the $L_{V_i}$ intersect the boundary of $\widetilde M_1$ inside the compact ball about $p$ of radius $B$. This means that the geodesics $\gamma_{V_i}$ (for the subsequence) converge to $L_V$ outside $\widetilde M_1$ but they converge to $\gamma_V$ which is supposed to be trapped. This yields the desired contradiction.
\qed
We next see that even though $\widetilde{M_1^{ext}}$ might a-priori have conjugate points (along geodesics trapped in $\widetilde{M_1}$), Busemann functions along rays where $L_V$ is not vertical behave like those in manifolds without conjugate points. In particular they are $C^{1,1}$ smooth, $|\nabla b_{1V}|=1$, and the Lipshitz constant for $\nabla b_{1V}$ is uniformly bounded.
For $V\in U\widetilde{M_1^{ext}}$ such that $\gamma_V(t)$ minimizes for all positive $t$, let $b_{1V}:\widetilde{M_1^{ext}}\to \R$ be the Busemann function defined by $V$, i.e. $$b_{1V}(p)=lim_{t\to\infty} d_1(p,\gamma_v(t))-t.$$
Since $d_1(p,q)=d(p,q)$ when $p$ and $q$ are points in $\widetilde{M_1^{ext}}-\widetilde{M_1}$, $b_{1V}$ coincides with the Euclidean $b_V$ outside $\widetilde{M_1}$ as long as $\gamma_{1V}$ is not trapped. That is $b_{1V}$ will coincide with the height function (up to a constant) in the direction $L_V$.
For all reals $s$ we will let $H_V(s)=\{p\in \widetilde{M_1^{ext}}| b_{1V}(p)=s \}$ be the $s$ level set of $b_{1V}$. Of course, outside of $\widetilde{M_1}$, $H_V(s)$ is just a hyperplane perpendicular to $L_V$.
\begin{lemma} \label{busemannsmooth} For all $V\in {\bf A}$, $b_{1V}$ is $C^{1,1}$ and the Lipshitz constant of $\nabla b_{1V}$ is bounded by a constant independent of $V$.
\end{lemma}
{\bf Proof:} The proof is the usual proof that such a statement holds on manifolds without conjugate points. This is done by showing that the approximating functions, $f_t(p)=d_1(p,\gamma_{1V}(t))-t$, are $C^\infty$ have $|\nabla f_t|=1$ and have uniformly bounded Hessian. If $\gamma_{1V}(t)\in \widetilde{M_1^{ext}}-\widetilde{M_1}$ then all geodesics from $\gamma_{1V}(t)$ minimize so the distance function from $\gamma_{1V}(t)$ is $C^\infty$ for large $t$. The fact that $|\nabla f_t|=1$ is clear. The uniform control on the Hessian is also the same as we will see. Fix a number $r$ less than the convexity radius of ${M_1^{ext}}$ - which exists since $M_1^{ext}-M_1$ is flat. Since there is a compact set $K\subset M_1^{ext}$ of base points such that for $q\notin K$ the ball $B(q,r)$ is flat we conclude that the eigenvalues of the second fundamental forms of the boundaries of $B(q,r)$ are uniformly bounded above and below independent of $q$. This same bound applies to balls in the universal cover $\widetilde{M_1^{ext}}$. Now to bound the Hessian of $f_t$ at $q\in \widetilde{M_1^{ext}}$ let $\tau(s)$ be the (unique) geodesic from $\gamma_{1V}(t)$ to $q=\tau(s_0)$ (we can assume $s_0\gg r$ since we will be taking the limit as $t\to \infty$). Then by the triangle inequality the level set of $f_t$ at $q$ (i.e. $\partial B(\gamma_{1V}(t),s_0)$) lies outside both $B(\tau(s_0-r),r)$ and $B(\tau(s_0+r),r)$ which are tangent to the level set at $q$. Hence the second fundamental forms of the level sets are uniformly bounded and hence so is the Hessian. Thus the lemma follows.
\qed
The usual properties of Busemann functions (see \cite{Es77} for basic properties of Busemann functions) tell us that if $W(p)=\nabla b_{1V}(p)$ then $\gamma_{1W}'(t) = \nabla b_{1V}(\gamma_{1W}(t))$ for all $t$. Hence if $\gamma_{1W}$ is not trapped then $L_W$ will be parallel to $L_V$. A straightforward open and closed argument shows that for all $p$, $\gamma_{1W(p)}$ is not trapped.
\begin{lemma} \label{levelsets} Let $V$ and $W$ in ${\bf A}$ be such that $L_V$ and $L_W$ are positive and not parallel to each other. Then for any given $s$ the maximum and minimum values of $b_{1V}$ on the compact $H_W(s)\cap \widetilde{M_1}$ are achieved on the boundary of $H_W(s)\cap \widetilde{M_1}$.
\end{lemma}
{\bf Proof:} We first note that $H_W(s)\cap \widetilde{M_1}$ is compact. Indeed, if $D$ is the diameter of $M_1$ then for every $p\in H_W(s)\cap \widetilde{M_1}$ there is a point $q\in \partial(\widetilde{M_1})$ with $d(p,q)\leq D$. Since $b_{1W}$ has Lipshitz constant 1, we know that $s-D \leq b_{1w}(q)\leq s+D$ and hence $p$ lies in the (compact) set of points that are at distance $\leq D$ from the compact (since $W$ is not horizontal since it is positive) set of boundary points $\{q\in \partial \widetilde{M_1}| s-D\leq b_{1W}(q)\leq s+D\}=\{q\in \partial (B^{n-1}\times \R)| s-D\leq b_{W}(q)\leq s+D\}$.
If the maximum (or minimum) value of $b_{1V}$ occurs in the interior then $\nabla b_{1V}$ must be perpendicular to $H_W(s)$ there and hence coincides with $\pm \nabla b_{1W}$ at that point. However this contradicts the condition that $L_V$ and $L_W$ are positive and not parallel.
\qed
We now see that if $V$ and $W$ in ${\bf A}$ are such that $L_V$ and $L_W$ are parallel then $b_{1V}-b_{1W}$ is constant. Since they agree with the height functions outside $\widetilde{M_1}$, $b_{1V}-b_{1W}=C$ outside $\widetilde{M_1}$. But then they must also differ by $C$ along any geodesic whose corresponding line is parallel to $L_V$ and $L_W$. But since such a geodesic passes though every point $p\in \widetilde{M_1}$ (i.e. take the geodesic in the direction of $\nabla b_{1V}(p)$) this says $b_{1V}-b_{1W}=C$ everywhere. In particular for every $p$ there is a unique geodesic passing through $p$ and parallel to a given line. This gives a natural identification of ${\bf A}(p)$ with the space of non vertical directions.
\begin{proposition} \label{mainhigher} Through every $p\in \widetilde M_1$ there is exactly one trapped geodesic. \end{proposition}
{\bf Proof:} This is equivalent to showing that for every $p\in \widetilde{M_1^{ext}}$, ${\bf A}(p)$ consists of the unit sphere $U_p\widetilde{M_1^{ext}}$ minus a pair of antipodal points. Assume that $p$ is a point with more trapped geodesics. Note that if there is a trapped half geodesic at $p$ then the other half must also be trapped by the assumption that the scattering data coincides with the flat case. Thus the tangent directions to trapped geodesics come in antipodal pairs. Of course there is at least one trapped geodesic through $p$ since one could take the limit of a subsequence of geodesics from $p$ to a boundary points $q_i$ where $q_i$ runs off to infinity. We only need to consider $p$ in the interior of $\widetilde{M_1}$.
A limiting half geodesic of a sequence of trapped half geodesics starting at $p$ will be a half geodesic starting at $p$ that stays in $\widetilde M_1$. (In fact it stays in the interior since the only half geodesics in $\widetilde M_1$ that are tangent to the boundary are the vertical ones hence stay in the boundary.) Thus the set of tangent directions to trapped geodesics (i.e. $U_p-{\bf A}(p)$) is closed in $U_p$ and thus ${\bf A}(p)$ is open and nonempty (by the correspondence with non vertical directions). The set of boundary points of $U_p-{\bf A}(p)$ is thus non empty and if it consisted of a single antipodal pair then $U_p-{\bf A}(p)$ would also be a single antipodal pair. Thus there is a pair of distinct unit vectors $V$ and $W$ in $U_p-{\bf A}(p)$ such that $\langle V,W\rangle =C$ with $-1 < C < 1$ (one could take $C\geq 0$) and such that there exists sequences $V_i\in {\bf A}(p)$ and $W_i\in {\bf A}(p)$ such that $V_i$ converges to $V$ and $W_i$ converges to $W$. We extend $V_i$ and $W_i$ to vector fields by letting $V_i(q)=\nabla b_{1V_i}(q)$ and $W_i(q)=\nabla b_{1W_i}(q)$. By the uniform bound on Lipshitz constants on Busemann functions, i.e. Lemma \ref{busemannsmooth}, there is an $\epsilon>0$ (depending only on $C$ and the Lipshitz constant of the busemann functions but not on $i$) such that for all $q\in B_p(\epsilon)$ (the $\epsilon$ ball about $p$) and all sufficiently large $i$ we have $$-\frac{1+C}{2} < \langle V_i(q),W_i(q)\rangle < \frac{1+C}{2}.$$ This holds since for large $i$ we have $\langle V_i(p),W_i(p)\rangle $ is approximately $C$ and then, with respect to a parallel frame along geodesics of length $\epsilon$, the change of $V_i$ and $W_i$ is uniformly bounded by the Lipshitz constant and $\epsilon$. We can further take $\epsilon$ less than the distance from $p$ to $\partial \widetilde{M_1}$.
Now consider the Busemann function $b_{1V_i}$ on the $0$ level set $H_{W_i}(0)$ of $b_{1W_i}$ (i.e. the level set through $p$). By the inner product condition above we can find unit speed differentiable curves $\tau_1$ and $\tau_2$ in $H_{W_i}(0)$ starting at $p$ of length $\epsilon$ (and hence in $H_{W_i}(0)\cap B_p(\epsilon)$) such that $$\langle \tau_1'(s),V_i(\tau_1(s))\rangle > \bar C=\sqrt{1-\Big (\frac{1+C}{2}\Big )^2}\ $$ and $$\langle \tau_2'(s),V_i(\tau_2(s))\rangle < -\bar C=-\sqrt{1-\Big (\frac{1+C}{2}\Big )^2}\ .$$ Thus for every sufficiently large $i$ there are points $z^1_i,z^2_i \in H_{W_i}(0)\cap B_p(\epsilon)$ such that $b_{1V_i}(z^1_i) > \bar C \epsilon$ and $b_{1V_i}(z^2_i) < -\bar C \epsilon$. Thus by lemma \ref{levelsets} for every sufficiently large $i$ there are points $y^1_i$ and $y^2_i$ on the boundary of $H_{W_i}(0)\cap \widetilde{M_1}$ with $b_{1V_i}(y^1_i) > \bar C \epsilon$ and $b_{1V_i}(y^2_i) <-\bar C \epsilon$.
Since $V_i$ and $W_i$ converge to trapped geodesics, Lemma \ref{vertical} says that as $i\to \infty$ the lines $L_{V_i}$ and $L_{W_i}$ converge to vertical. But this means that the the Busemann functions $b_{1V_i}$ and $b_{1W_i}$ (which are height functions outside $\widetilde M_1$) approximate the vertical height function. In particular, for $i$ large enough the values of $b_{1V_i}$ on the boundary of $H_{W_i}(0)\cap \widetilde{M_1}$ vary by less than $\bar C\epsilon$. This contradicts the simultaneous existence of $y^1_i$ and $y^2_i$ for large $i$.
\qed
The first consequence of this proposition is that the trapped geodesics are also minimizing (as limits of minimizing geodesics) and hence $g$ has no conjugate points. Start with a large enough flat $n+1$ torus $T^n\times S^1$ so that $D^n$ sits isometrically in $T^n$. Now if we replace $D^n\times S^1$ with $(M_1,\partial M_1,g)$ we get an $n+1$ torus since $M_1$ is diffeomorphic to $M$ (as was pointed out after the proof of Lemma \ref{geosminimize}). Further it has no conjugate points. Then the theorem of Burago-Ivanov \cite{Bu-Iv94} proving the E. Hopf conjecture says that the metric is flat. This gives a proof of Theorem \ref{3+dims}.
However the above proof does not generalize very far. In the next section we give an alternative proof that does generalize.
\section{generalizations} \label{gens}
In the previous section we considered only flat metrics so as to make the the proof more transparent. However the arguments extend almost without change to give
\begin{proposition} \label{mainhighergen}
Let $D^n$ be a ball in a complete simply connected manifold $N^n$ with nonpositive curvature, and $(M_1,\partial M_1,g_1)$ a Riemannian manifold with boundary that has the same scattering data as $(D\times S^1,\partial D\times S^1,g)$ where $g$ is the product metric. Then through every point of $M_1$ there is exactly one trapped geodesic. \end{proposition}
The main change that affects the proof is that geodesics in $\widetilde{M_1^{ext}}$ are not lines outside $\widetilde M_1$ but geodesics in $N$. Oriented geodesic rays in $N$ are thought of as parallel if they have the same limit point at infinity (which means that they stay a bounded distance from each other). The notion of ``positive'' geodesics also makes sense. This allows us to relate ${\bf A}(p)$ to ${\bf A}(q)$. Most of the arguments go through exactly as before. In particular Lemmas \ref{fundgroup}, \ref{lensdata}, and \ref{geosminimize} go through as is. Lemma \ref{vertical} also goes through as before where convergence of directions needs to be interpreted as the endpoints at infinity converging (where the topology of infinity is the cone topology - hence homeomorphic to the standard $n-1$ sphere). The argument in Lemma \ref{levelsets} needs to be viewed carefully since there may be many oriented geodesics corresponding to $-\nabla b_{1W}$ as the point on $H_W(s)$ varies. However, none of these will be positive so again the proof goes through. The only part of Proposition \ref{mainhigher} that needs to be noted is that in nonpositive curvature the level sets of Busemann functions vary continuously with the initial vector.
\qed
\begin{remark} The above arguments likely can be modified to cover manifolds without conjugate points. One first needs to deal with the fact that balls may not be convex. This would seem to give us problems with the differentiability of the metric on $\widetilde{M_1^{ext}}$ at the boundary of $\widetilde M_1$. However (except possibly in the case where the boundary of $D$ contains a region that is totally geodesic) since there are no conjugate points Theorem 1 of \cite{St-Uh09} will still tell us that the metric will be smooth. In fact very little of the argument really needs the boundary to be convex (or the metric to be smooth for that mater) but extending the arguments would look somewhat different from the above. Also one has to worry about relating ${\bf A}(p)$ and ${\bf A}(q)$. This can be done by fixing a base point $x_0\in \widetilde{M_1^{ext}}-\widetilde M_1$ and looking only at Busemann functions defined by vectors in $U_{x_0}$. Also Busemann functions are not as well behaved. In any event, the arguments would look very different and we wont pursue them here. \end{remark}
We now want to generalize Theorem \ref{3+dims} to the nonflat case. The first point to note is that for $g$ and $g_1$ as in Proposition \ref{mainhighergen} we have $$Vol(g_1)=Vol(g).$$
Let ${\bf T_1}\subset UM_1$ be the set of unit vectors tangent to trapped geodesic rays. Similarly define ${\bf T}\subset UM$ (here $M=D\times S^1$). Using the fact that the metrics are lens equivalent we consider the standard measure preserving map $F:UM_1-{\bf T}\to UM-{\bf T}$ which assigns to each vector $V\in UM_1-{\bf T}$ the unique vector $W\in UM_1-{\bf T}$ such that $V=\gamma'_1(t)$ and $W=\gamma'(t)$ where $\gamma$ and $\gamma_1$ are geodesics with corresponding initial conditions on the boundary. I.e. $\gamma(0)\in \partial M$ and $\gamma_1(0) \in \partial M_1$ are corresponding points while $\gamma'(0)$ and $\gamma'_1(0)$ are corresponding inwardly pointing unit vectors. That $F$ (which conjugates the geodesic flows) is measure preserving is a standard fact which follows for example from Santal\'o's formula (see for example \cite{Cr91}). The fact that ${\bf T}$ and ${\bf T_1}$ have measure 0 tells us that the unit tangent bundles (and hence the manifolds) have the same volume.
The generalization of Theorem \ref{3+dims} is
\begin{theorem} \label{generalizations}
Let $D^n$ be a ball in $N^{n-1}\times \R$ where $N$ is a complete simply connected manifold with nonpositive curvature. Then $D\times S^1$ is scattering rigid.
\end{theorem}
{\bf Proof:} One proves this via Proposition 2.2 of \cite{Cr-Kl98}. That result compares two metrics $g$ and $g_1$ on manifolds without conjugate points with an additional condition on $g$ that it contain a real factor (which is satisfied by our assumption on $D$). Under a weak lens equivalency condition (which is satisfied since the metrics are lens equivalent) one concludes that $Vol(g_1)\geq Vol(g)$ with equality holding if and only if $g_1$ is isometric to $g$. This (along with the above fact that $Vol(g_1)=Vol(g)$) proves the Theorem.
\qed
Note that we do not claim the the result for $D^n$ a ball in a complete simply connected manifold with nonpositive curvature. $D^n$ is a ball in $N^{n-1}\times \R$. This means that in the universal cover of $D$ (and hence of $M=D\times S^1$) there is a bounded $\R$ factor (as opposed to the unbounded $\R$ factor coming from the $S^1$ factor in $M$). This is because Proposition 2.2 of \cite{Cr-Kl98} requires a set whose $\R$ component in the universal cover is bounded.
\end{document} | arXiv |
Normal basis
In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.
Normal basis theorem
Let $F\subset K$ be a Galois extension with Galois group $G$. The classical normal basis theorem states that there is an element $\beta \in K$ such that $\{g(\beta ):g\in G\}$ forms a basis of K, considered as a vector space over F. That is, any element $\alpha \in K$ can be written uniquely as $ \alpha =\sum _{g\in G}a_{g}\,g(\beta )$ for some elements $a_{g}\in F.$
A normal basis contrasts with a primitive element basis of the form $\{1,\beta ,\beta ^{2},\ldots ,\beta ^{n-1}\}$, where $\beta \in K$ is an element whose minimal polynomial has degree $n=[K:F]$.
Group representation point of view
A field extension K / F with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself. Representations of G over the field F can be viewed as left modules for the group algebra F[G]. Every homomorphism of left F[G]-modules $\phi :F[G]\rightarrow K$ is of form $\phi (r)=r\beta $ for some $\beta \in K$. Since $\{1\cdot \sigma |\sigma \in G\}$ is a linear basis of F[G] over F, it follows easily that $\phi $ is bijective iff $\beta $ generates a normal basis of K over F. The normal basis theorem therefore amounts to the statement saying that if K / F is finite Galois extension, then $K\cong F[G]$ as left $F[G]$-module. In terms of representations of G over F, this means that K is isomorphic to the regular representation.
Case of finite fields
For finite fields this can be stated as follows:[1] Let $F=\mathrm {GF} (q)=\mathbb {F} _{q}$ denote the field of q elements, where q = pm is a prime power, and let $K=\mathrm {GF} (q^{n})=\mathbb {F} _{q^{n}}$ denote its extension field of degree n ≥ 1. Here the Galois group is $G={\text{Gal}}(K/F)=\{1,\Phi ,\Phi ^{2},\ldots ,\Phi ^{n-1}\}$ with $\Phi ^{n}=1,$ a cyclic group generated by the q-power Frobenius automorphism $\Phi (\alpha )=\alpha ^{q},$with $\Phi ^{n}=1=\mathrm {Id} _{K}.$ Then there exists an element β ∈ K such that
$\{\beta ,\Phi (\beta ),\Phi ^{2}(\beta ),\ldots ,\Phi ^{n-1}(\beta )\}\ =\ \{\beta ,\beta ^{q},\beta ^{q^{2}},\ldots ,\beta ^{q^{n-1}}\!\}$
is a basis of K over F.
Proof for finite fields
In case the Galois group is cyclic as above, generated by $\Phi $ with $\Phi ^{n}=1,$ the normal basis theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character is a mapping χ from a group H to a field K satisfying $\chi (h_{1}h_{2})=\chi (h_{1})\chi (h_{2})$; then any distinct characters $\chi _{1},\chi _{2},\ldots $ are linearly independent in the K-vector space of mappings. We apply this to the Galois group automorphisms $\chi _{i}=\Phi ^{i}:K\to K,$ thought of as mappings from the multiplicative group $H=K^{\times }$. Now $K\cong F^{n}$as an F-vector space, so we may consider $\Phi :F^{n}\to F^{n}$ as an element of the matrix algebra Mn(F); since its powers $1,\Phi ,\ldots ,\Phi ^{n-1}$ are linearly independent (over K and a fortiori over F), its minimal polynomial must have degree at least n, i.e. it must be $X^{n}-1$.
The second basic fact is the classification of finitely generated modules over a PID such as $F[X]$. Every such module M can be represented as $ M\cong \bigoplus _{i=1}^{k}{\frac {F[X]}{(f_{i}(X))}}$, where $f_{i}(X)$ may be chosen so that they are monic polynomials or zero and $f_{i+1}(X)$ is a multiple of $f_{i}(X)$. $f_{k}(X)$ is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case $ \dim _{F}M=\sum _{i=1}^{k}\deg f_{i}$, in the second case $\dim _{F}M=\infty $. In our case of cyclic G of size n generated by $\Phi $ we have an F-algebra isomorphism $ F[G]\cong {\frac {F[X]}{(X^{n}-1)}}$ where X corresponds to $1\cdot \Phi $, so every $F[G]$-module may be viewed as an $F[X]$-module with multiplication by X being multiplication by $1\cdot \Phi $. In case of K this means $X\alpha =\Phi (\alpha )$, so the monic polynomial of smallest degree annihilating K is the minimal polynomial of $\Phi $. Since K is a finite dimensional F-space, the representation above is possible with $f_{k}(X)=X^{n}-1$. Since $\dim _{F}(K)=n,$ we can only have $k=1$, and $ K\cong {\frac {F[X]}{(X^{n}{-}\,1)}}$ as F[X]-modules. (Note this is an isomorphism of F-linear spaces, but not of rings or F-algebras.) This gives isomorphism of $F[G]$-modules $K\cong F[G]$ that we talked about above, and under it the basis $\{1,X,X^{2},\ldots ,X^{n-1}\}$ on the right side corresponds to a normal basis $\{\beta ,\Phi (\beta ),\Phi ^{2}(\beta ),\ldots ,\Phi ^{n-1}(\beta )\}$ of K on the left.
Note that this proof would also apply in the case of a cyclic Kummer extension.
Example
Consider the field $K=\mathrm {GF} (2^{3})=\mathbb {F} _{8}$ over $F=\mathrm {GF} (2)=\mathbb {F} _{2}$, with Frobenius automorphism $\Phi (\alpha )=\alpha ^{2}$. The proof above clarifies the choice of normal bases in terms of the structure of K as a representation of G (or F[G]-module). The irreducible factorization
$X^{n}-1\ =\ X^{3}-1\ =\ (X{+}1)(X^{2}{+}X{+}1)\ \in \ F[X]$
means we have a direct sum of F[G]-modules (by the Chinese remainder theorem):
$K\ \cong \ {\frac {F[X]}{(X^{3}{-}\,1)}}\ \cong \ {\frac {F[X]}{(X{+}1)}}\oplus {\frac {F[X]}{(X^{2}{+}X{+}1)}}.$
The first component is just $F\subset K$, while the second is isomorphic as an F[G]-module to $\mathbb {F} _{2^{2}}\cong \mathbb {F} _{2}[X]/(X^{2}{+}X{+}1)$ under the action $\Phi \cdot X^{i}=X^{i+1}.$ (Thus $K\cong \mathbb {F} _{2}\oplus \mathbb {F} _{4}$ as F[G]-modules, but not as F-algebras.)
The elements $\beta \in K$ which can be used for a normal basis are precisely those outside either of the submodules, so that $(\Phi {+}1)(\beta )\neq 0$ and $(\Phi ^{2}{+}\Phi {+}1)(\beta )\neq 0$. In terms of the G-orbits of K, which correspond to the irreducible factors of:
$t^{2^{3}}-t\ =\ t(t{+}1)\left(t^{3}+t+1\right)\left(t^{3}+t^{2}+1\right)\ \in \ F[t],$
the elements of $F=\mathbb {F} _{2}$ are the roots of $t(t{+}1)$, the nonzero elements of the submodule $\mathbb {F} _{4}$ are the roots of $t^{3}+t+1$, while the normal basis, which in this case is unique, is given by the roots of the remaining factor $t^{3}{+}t^{2}{+}1$.
By contrast, for the extension field $L=\mathrm {GF} (2^{4})=\mathbb {F} _{16}$ in which n = 4 is divisible by p = 2, we have the F[G]-module isomorphism
$L\ \cong \ \mathbb {F} _{2}[X]/(X^{4}{-}1)\ =\ \mathbb {F} _{2}[X]/(X{+}1)^{4}.$
Here the operator $\Phi \cong X$ is not diagonalizable, the module L has nested submodules given by generalized eigenspaces of $\Phi $, and the normal basis elements β are those outside the largest proper generalized eigenspace, the elements with $(\Phi {+}1)^{3}(\beta )\neq 0$.
Application to cryptography
The normal basis is frequently used in cryptographic applications based on the discrete logarithm problem, such as elliptic curve cryptography, since arithmetic using a normal basis is typically more computationally efficient than using other bases.
For example, in the field $K=\mathrm {GF} (2^{3})=\mathbb {F} _{8}$ above, we may represent elements as bit-strings:
$\alpha \ =\ (a_{2},a_{1},a_{0})\ =\ a_{2}\Phi ^{2}(\beta )+a_{1}\Phi (\beta )+a_{0}\beta \ =\ a_{2}\beta ^{4}+a_{1}\beta ^{2}+a_{0}\beta ,$
where the coefficients are bits $a_{i}\in \mathrm {GF} (2)=\{0,1\}.$ Now we can square elements by doing a left circular shift, $\alpha ^{2}=\Phi (a_{2},a_{1},a_{0})=(a_{1},a_{0},a_{2})$, since squaring β4 gives β8 = β. This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.
Proof for the case of infinite fields
Suppose $K/F$ is a finite Galois extension of the infinite field F. Let [K : F] = n, ${\text{Gal}}(K/F)=G=\{\sigma _{1}...\sigma _{n}\}$, where $\sigma _{1}={\text{Id}}$. By the primitive element theorem there exists $\alpha \in K$ such $i\neq j\implies \sigma _{i}(\alpha )\neq \sigma _{j}(\alpha )$ and $K=F[\alpha ]$. Let us write $\alpha _{i}=\sigma _{i}(\alpha )$. $\alpha $'s (monic) minimal polynomial f over K is the irreducible degree n polynomial given by the formula
${\begin{aligned}f(X)&=\prod _{i=1}^{n}(X-\alpha _{i})\end{aligned}}$
Since f is separable (it has simple roots) we may define
${\begin{aligned}g(X)&=\ {\frac {f(X)}{(X-\alpha )f'(\alpha )}}\\g_{i}(X)&=\ {\frac {f(X)}{(X-\alpha _{i})f'(\alpha _{i})}}=\ \sigma _{i}(g(X)).\end{aligned}}$
In other words,
${\begin{aligned}g_{i}(X)&=\prod _{\begin{array}{c}1\leq j\leq n\\j\neq i\end{array}}{\frac {X-\alpha _{j}}{\alpha _{i}-\alpha _{j}}}\\g(X)&=g_{1}(X).\end{aligned}}$
Note that $g(\alpha )=1$ and $g_{i}(\alpha )=0$ for $i\neq 1$. Next, define an $n\times n$ matrix A of polynomials over K and a polynomial D by
${\begin{aligned}A_{ij}(X)&=\sigma _{i}(\sigma _{j}(g(X))=\sigma _{i}(g_{j}(X))\\D(X)&=\det A(X).\end{aligned}}$
Observe that $A_{ij}(X)=g_{k}(X)$, where k is determined by $\sigma _{k}=\sigma _{i}\cdot \sigma _{j}$; in particular $k=1$ iff $\sigma _{i}=\sigma _{j}^{-1}$. It follows that $A(\alpha )$ is the permutation matrix corresponding to the permutation of G which sends each $\sigma _{i}$ to $\sigma _{i}^{-1}$. (We denote by $A(\alpha )$ the matrix obtained by evaluating $A(X)$ at $x=\alpha $.) Therefore, $D(\alpha )=\det A(\alpha )=\pm 1$. We see that D is a non-zero polynomial, and therefore it has only a finite number of roots. Since we assumed F is infinite, we can find $a\in F$ such that $D(a)\neq 0$. Define
${\begin{aligned}\beta &=g(a)\\\beta _{i}&=g_{i}(a)=\sigma _{i}(\beta ).\end{aligned}}$
We claim that $\{\beta _{1},\ldots ,\beta _{n}\}$ is a normal basis. We only have to show that $\beta _{1},\ldots ,\beta _{n}$ are linearly independent over F, so suppose $ \sum _{j=1}^{n}x_{j}\beta _{j}=0$ for some $x_{1}...x_{n}\in F$. Applying the automorphism $\sigma _{i}$ yields $ \sum _{j=1}^{n}x_{j}\sigma _{i}(g_{j}(a))=0$ for all i. In other words, $A(a)\cdot {\overline {x}}={\overline {0}}$. Since $\det A(a)=D(a)\neq 0$, we conclude that ${\overline {x}}={\overline {0}}$, which completes the proof.
It is tempting to take $a=\alpha $ because $D(\alpha )\neq 0$. But this is impermissible because we used the fact that $a\in F$ to conclude that for any F-automorphism $\sigma $ and polynomial $h(X)$ over $K$ the value of the polynomial $\sigma (h(X))$ at a equals $\sigma (h(a))$.
Primitive normal basis
A primitive normal basis of an extension of finite fields E / F is a normal basis for E / F that is generated by a primitive element of E, that is a generator of the multiplicative group K×. (Note that this is a more restrictive definition of primitive element than that mentioned above after the general normal basis theorem: one requires powers of the element to produce every non-zero element of K, not merely a basis.) Lenstra and Schoof (1987) proved that every finite field extension possesses a primitive normal basis, the case when F is a prime field having been settled by Harold Davenport.
Free elements
If K / F is a Galois extension and x in K generates a normal basis over F, then x is free in K / F. If x has the property that for every subgroup H of the Galois group G, with fixed field KH, x is free for K / KH, then x is said to be completely free in K / F. Every Galois extension has a completely free element.[2]
See also
• Dual basis in a field extension
• Polynomial basis
• Zech's logarithm
References
1. Nader H. Bshouty; Gadiel Seroussi (1989), Generalizations of the normal basis theorem of finite fields (PDF), p. 1; SIAM J. Discrete Math. 3 (1990), no. 3, 330–337.
2. Dirk Hachenberger, Completely free elements, in Cohen & Niederreiter (1996) pp. 97–107 Zbl 0864.11066
• Cohen, S.; Niederreiter, H., eds. (1996). Finite Fields and Applications. Proceedings of the 3rd international conference, Glasgow, UK, July 11–14, 1995. London Mathematical Society Lecture Note Series. Vol. 233. Cambridge University Press. ISBN 978-0-521-56736-7. Zbl 0851.00052.
• Lenstra, H.W., jr; Schoof, R.J. (1987). "Primitive normal bases for finite fields". Mathematics of Computation. 48 (177): 217–231. doi:10.2307/2007886. JSTOR 2007886. Zbl 0615.12023.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Menezes, Alfred J., ed. (1993). Applications of finite fields. The Kluwer International Series in Engineering and Computer Science. Vol. 199. Boston: Kluwer Academic Publishers. ISBN 978-0792392828. Zbl 0779.11059.
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| Wikipedia |
M. Lothaire
M. Lothaire is the pseudonym of a group of mathematicians, many of whom were students of Marcel-Paul Schützenberger. The name is used as the author of several of their joint books about combinatorics on words. The group is named for Lothair I.[1]
Members
Mathematicians in the group have included Jean-Paul Allouche,[2] Jean Berstel,[3][4] Valérie Berthé,[2] Véronique Bruyère,[3] Julien Cassaigne,[3] Christian Choffrut,[4] Robert Cori,[4] Maxime Crochemore[2] Jacques Desarmenien,[3] Volker Diekert,[3] Dominique Foata,[3][4] Christiane Frougny,[3] Guo-Niu Han,[3] Tero Harju,[3] Philippe Jacquet,[2] Juhani Karhumäki,[3] Roman Kolpakov,[2] Gregory Koucherov,[2] Eric Laporte,[2] Alain Lascoux,[3] Bernard Leclerc,[3] Aldo De Luca,[3] Filippo Mignosi,[3] Mehryar Mohri,[2] Dominique Perrin,[3][4] Jean-Éric Pin,[4] Giuseppe Pirillo,[4] Nadia Pisanti,[2] Wojciech Plandowski,[3] Dominique Poulalhon,[2] Gesine Reinert,[2] Antonio Restivo,[3] Christophe Reutenauer,[3][4] Marie-France Sagot,[2] Jacques Sakarovitch,[4] Gilles Schaeffer,[2] Sophie Schbath,[2] Marcel-Paul Schützenberger,[4] Patrice Séébold,[3] Imre Simon,[4] Wojciech Szpankowski,[2] Jean-Yves Thibon,[3] Stefano Varricchio,[3] and Michael Waterman.[2]
See also
• Séminaire Lotharingien de Combinatoire
Publications
• Lothaire, M. (1983), Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 17, Addison-Wesley Publishing Co., Reading, Mass., doi:10.1017/CBO9780511566097, ISBN 978-0-201-13516-9, MR 0675953
• Lothaire, M. (2002), Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, ISBN 978-0-521-81220-7, MR 1905123
• Lothaire, M. (2005), Applied Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge University Press, ISBN 978-0-521-84802-2, MR 2165687
References
1. Lothaire, M. (1997), Combinatorics on Words, Cambridge Mathematical Library, Cambridge University Press, p. xvi, ISBN 9780521599245.
2. MR2165687
3. MR1905123.
4. MR0675953.
External links
• Lothaire's books
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\begin{document}
\begin{singlespace}
\if11 { \title{Recursive Nearest Neighbor Co-Kriging Models for Big Multiple Fidelity Spatial Data Sets } \author[1]{Si Cheng}
\author[1]{Bledar A. Konomi \thanks{Corresponding author:Bledar A. Konomi ([email protected])} }
\author[2]{Georgios Karagiannis} \author[1]{Emily L. Kang} \affil[1]{Division of Statistics and Data Sciences, Department of Mathematical Sciences, University of Cincinnati, USA}
\affil[2]{Mathematical Sciences, Durham University, UK} \maketitle } \fi \end{singlespace}
\begin{abstract} \begin{singlespace} Large datasets are daily gathered from different remote sensing platforms and statistical models are usually used to combine them by accounting for spatially varying bias corrections. The statistical inference of these models is usually based on Markov chain Monte Carlo (MCMC) samplers which involve updating a high-dimensional random effect vector and hence present slow mixing and convergence. To overcome this and enable fast inference in big spatial data problems, we propose the recursive nearest neighbor co-kriging (RNNC) model and use it as a framework which allows us to develop two computationally efficient inferential procedures: a) the collapsed RNNC that reduces the posterior sampling space by integrating out the latent processes, and b) the conjugate RNNC which is an MCMC free inference that significantly reduces the computational time without sacrificing prediction accuracy. The good computational and predictive performance of our proposed algorithms are demonstrated on benchmark examples and the analysis of the High-resolution Infrared Radiation Sounder data gathered from two NOAA polar orbiting satellites in which we managed to reduce the computational time from multiple hours to just a few minutes. \end{singlespace} \end{abstract} \begin{singlespace} Keywords: Recursive Co-kriging Model, Nearest neighbor Gaussian process, Remote Sensing \end{singlespace}
\section{Introduction}
Global geophysical information is measured daily by numerous satellite sensors. Due to aging and exposure to the harsh environment of space the satellite sensors degrades over time causing a decrease on performance reliability. This decrease on performance may result in inaccuracy of the data measurements \citep{goldberg2011}. In addition, newer satellites with technologically more advanced sensors provide information of higher fidelity than older sensors. These discrepancies in sensors performances have created the need to develop efficient methods to analyse daily global remote sensing measurements with varying fidelity. Our work in this manuscript is motivated by data products from the high-resolution infrared radiation sounder (HIRS) which provides daily hundred of thousands of measurements from multiple satellite platforms.
Multiple methods in remote sensing have been developed to assess satellite sensor performance and consistency \citep{chander2013, xiong2010, nrc2004}. These methods do not account for spatial correlation and oversimplify the relationship between sensors. On the other hand statistical methods to analyse these data sets and accounting for spatial correlation are also challenging due to the multifideltiy presence as well as the size and computationally intensive procedures. \citet{nguyen2012spatial,nguyen2017multivariate} have proposed data fusion techniques to model multivariate spatial data at potentially different spatial resolutions based on fixed ranked kriging \citep{cressie2008fixed}. The accuracy of this approach relies on the number of basis functions and can only capture large scale variation of the covariance function. When the data sets are dense, strongly correlated, and the noise effect is sufficiently small, the low rank kriging techniques have difficulty to account for small scale variation \citep{stein2014limitations}.
Autoregressive co-kriging models \citep{kennedy2000predicting,ZhiguangConnerJanetWu2005,le2013bayesian}, which have been originally built for computer simulation problems, can also be used for the analysis of multiple fidelity remote sensing observations when they have spatially nested structure. \cite{konomikaragiannisABTCK2019, ma2019multifidelity} relaxed the nested design requirements by using imputation ideas and properly augmenting the data. However, the aforesaid methods rely on Gaussian process models and are computationally impossible for big data problems. For cases when the observed space can be expressed as a tensor product \citet{Konomi2022_JABES} uses a separable covariance function within the co-kriging model to improve the computational efficiency. For more general cases of the observed space, \citet{Si_etall2020_NNCGP} proposed the nearest neighbour co-kriging Gaussian process (NNCGP) which embeds nearest neighbor Gaussian process \citep[NNGP;][]{datta2016hierarchical} into an autoregressive co-kriging model to make computations possible. The proposed NNCGP uses imputation ideas into the latent variables to construct a nested reference set of multiple NNGP levels. Despite the fact that the NNCGP makes the analysis of big multi-fidelity data sets computationally possible, its computational speed depends on an expensive iterative procedure which makes it impractical for analysing daily large data sets.
To overcome the iterative MCMC procedure, we propose a recursive formulation based on the latent variable of the NNCGP model following similar ideas with \citet{le2014recursive} who proposed the recursive formulation directly in the observations. Based on this new formulation, which we call recursive nearest neighbors co-kriging (RNNC), we are able to build a nearest neighbors co-kriging model with $T$ levels by building $T$ conditionally independent NNGPs. This enables the development of two alternative inferential procedures which aim to reduce high-dimensional parametric space, improve convergence, and reduce computational time in comparison to the NNCGP. Both proposed procedures are able to address applications for large non-nested and irregular spatial data sets from different platforms and with varying quality. In the first proposed alternative procedure that we call Collapsed RNNC, we reduce the MCMC posterior sampling space by integrating out the spatial latent variables. Based on the RNNC, we propose an MCMC free procedure to achieve Bayesian inference. We build an algorithm which sequentially decomposes the parametric space into conditionally independent parts for every fidelity level so that we are able to use a combination of closed form conditional means and K-fold optimization method. Each sequential step can be viewed as collapsed NNGP \citep{finley2019JCGS} where the bases component of the mean is determined at the previous step. We name this second inferential procedure conjugate RNNC. We note that the MCMC free procedure proposed in \citep{finley2019JCGS} cannot be applied directly in the NNCGP because the computational complexity of the $K$-fold cross-validation method depends on the dimension of the parametric space. Based on our simulation study and the analysis of the HERS data sets, we show that the proposed conjugate RNNC procedure reduces the computational time notably without significantly sacrificing prediction accuracy over the existing NNCGP approach.
The layout of the paper is as follows. In Section~\ref{sec: data}, we introduce the high-resolution infrared radiation sounder data studied in this work. In Section 3, we review the autoregressive co-kriging and NNCGP model. In Section 4, we introduce the proposed RNNC model. In Section 4.1, we integrate out the latent variables from the model and design an MCMC algorithm for this model. In Section 4.2, we design an MCMC free approach tailored to the proposed RNNC model that facilitates parametric and predictive inference. In Section 6, we investigate the performance of the proposed procedure on a simulation example. In Section 7, we implement the proposed method for the analysis of data sets from two satellites, NOAA-14 and NOAA-15. Finally, we conclude in Section 8.
\section{High-resolution Infrared Radiation Sounder Data}\label{sec: data}
Satellite soundings have been providing measurements of the Earth’s atmosphere, oceans, land, and ice since the 1970s to support the study of global climate system dynamics. Long term observations from past and current environmental satellites are widely used in developing climate data records (CDR) \citep{nrc2004}. HIRS mission objectives include observations of atmospheric temperature, water vapor, specific humidity, sea surface temperature, cloud cover, and total column ozone. The HIRS instrument is comprised of twenty channels, including twelve longwave channels, seven shortwave channels, and one visible channel. The dataset being considered in this study is limb-corrected HIRS swath data as brightness temperatures \citep{jackson2003}. The data is stored as daily files, where each daily file records approximately 120,000 geolocated observations. The current archive includes data from NOAA-5 through NOAA-17 along with Metop-02, covering the time period of 1978-2017. In all, this data archive is more than 2 TB, with an average daily file size of about 82 MB. The HIRS CRD faces some common challenges regarding the consistency and accuracy over time, due to degradation of sensors and intersatellite discrepancies. Furthermore, there is missing information caused by atmospheric conditions such as thick cloud cover.
\begin{figure}\label{fig:sfig1-1}
\label{fig:sfig2-1}
\label{fig:fig-1}
\end{figure}
We examine HIRS Channel 5 observations from a single day, March 1, 2001, as illustrated in Figure \ref{fig:fig-1}. On this day, we may exploit a period of temporal overlap in the NOAA POES series where two satellites captured measurements: NOAA-14 and NOAA-15. The HIRS sensors on these two satellites have similar technical designs which allow us to ignore the spectral and spatial footprint differences. NOAA-14 became operational in December 1994 while NOAA-15 became operational in October 1998. The spatial resolution footprint for both satellites is approximately 10 km at nadir. Given the sensor age difference, it is reasonable to consider that the instruments on-board NOAA-15 are in better condition than those of NOAA-14. Therefore, we treat observations from NOAA-14 as a dataset of low fidelity level, and those from NOAA-15 as a dataset of high fidelity level.
\section{Nearest Neighbor Co-kriging Gaussian Process}\label{sec:sequentialNNCGP} Let $y_{t}(s)$ denote the output function at the spatial location $s$ at fidelity level $t=1,...,T$ in a system with fidelity $T$ levels. The fidelity level index $t$ runs from the least accurate to the most accurate one. Let $z_{t}(\mathbf s)$ denote the observed output at location $s$. We specify an autoregressive co-kriging model as \setlength{\belowdisplayskip}{5pt}\setlength{\belowdisplayshortskip}{5pt} \setlength{\abovedisplayskip}{5pt}\setlength{\abovedisplayshortskip}{5pt} \begin{align}
& z_{t}(\mathbf s)=y_{t}(\mathbf s)+\epsilon_{t},\nonumber \\
& y_{t}(\mathbf s)=\zeta_{t-1}(\mathbf s)y_{t-1}(\mathbf s)+\delta_{t}(\mathbf s),\label{eq:davdsgdaf}\\
& \delta_{t}(\mathbf s)=\mathbf{h}^T_{t}(\mathbf s)\boldsymbol \beta_{t}+w_{t}(\mathbf s), \nonumber \end{align} where $z_{t}(\mathbf s)$ is contaminated by additive random noise $\epsilon_{t}\sim N(0,\tau_{t}^2)$ for $t=2,\ldots,T$, and $y_{1}(\mathbf s)=\mathbf{h}^T_{1}(\mathbf s)\boldsymbol{\beta_{1}}+w_{1}(\mathbf s)$ is the noiseless output. Here, $\zeta_{t-1}(\mathbf s)$ and $\delta_{t}(\mathbf s)$ represent the scale and additive discrepancies between systems with fidelity levels $t$ and $t-1$, $\mathbf{h}_{t}(\cdot)$ is a design matrix, and $\boldsymbol \beta_{t}$ is a vector of coefficients at fidelity level $t$. We model the latent function $w_{t}(\mathbf s)$ as a Gaussian processes, mutually independent for different $t$; i.e. $w_{t}(\cdot)\sim GP(0,C_{t}(\cdot,\cdot;\boldsymbol \theta_{t}))$ where $C_{t}(\cdot,\cdot;\boldsymbol \theta_{t})$ is a cross-covariance function with covariance parameters $\boldsymbol \theta_{t}$ at fidelity
level $t$. Any well defined covariance function can be used $C_t(\mathbf s,\mathbf s'|\boldsymbol{\theta}_t)=\sigma_t^{2}R(s,s'|\boldsymbol{\phi}_t)$, where $\boldsymbol{\theta}_t=\{\sigma_t^2,\boldsymbol{\phi}_t\}$.
This indicates that discrepancy term $\delta_{t}(\mathbf s)$ given $\mathbf{h}^T_{t}(\mathbf s)\boldsymbol \beta_{t}$ is a Gaussian process.
The unknown scale discrepancy function $\zeta_{t-1}(s)$ is modeled as a basis expansion $\zeta_{t-1}(\mathbf s|\boldsymbol \gamma_{t-1})=\mathbf{g}_{t-1}(s)^{T}\boldsymbol \gamma_{t-1}$ (usually low degree), where $\mathbf{g}_{t}(\mathbf s)$ is a vector of polynomial basis functions and $\{\boldsymbol \gamma_{t-1}\}$ is a vector of random coefficients, for $t=2,\dots,T$.
Let us assume the system is observed at $n_t$ locations at fidelity level $t$. Let $\mathbf S_{t}=\{s_{t,1},\dots,s_{t,n_t}\}$ be the set of $n_t$ observed locations, let $\mathbf w_t={w}_{t}(\mathbf S_t)=\{w_{t}(s_{t,1}),\dots,w_{t}(s_{t,n_{t}})\}$ the latent spatial random effect vector at fidelity level $t$, and let $\mathbf Z_{t}={z}_{t}(\mathbf S_t)=\{z_{t}(s_{t,1}),\dots,z_{t}(s_{t,n_{t}})\}$ represent the observed output at fidelity level $t$. If data $\{\mathbf Z_{t}\}$ are observed in non-nested locations across the fidelity levels, the calculation of the likelihood requires $\mathcal{O}((\sum_{t=1}^{T}n_t)^3)$ flops to invert the covariance matrix of the observations (which we denote by $\boldsymbol{\Lambda}$) and additional $\mathcal{O}((\sum_{t=1}^{T}n_t)^2)$ memory to store it as explained in \cite{konomikaragiannisABTCK2019}. To reduce the computational complexity, \citet{Si_etall2020_NNCGP} proposed NNCGP which assigns independent nested NNGP priors within a nested reference set. For Bayesian inference, \citet{Si_etall2020_NNCGP} proposed a Gibbs sampler which is able to take advantage of the nearest neighbour structure at each level. However, this sampler is based on updating a conditionally independent high dimensional latent variable which could cause slow convergence and high autocorrelation \citep{LiuEtAl1994}. The slow convergence can significantly increase the number of the Gibbs sampler iterations $I$ and has an overall computational cost of $\mathcal{O}(I\times\tilde{n}_{1}m^{2})$ flops.
For a fixed computational budget, the produced Monte Carlo estimates may be sensitive to the initial values of the MCMC sampler. Despite reducing the computational complexity to linear for every MCMC iteration, the number of the iterations can significantly increase computational cost. This simple observation makes the NNCGP too expensive for the vast majority of real remote sensing applications.
\section{Recursive Nearest Neighbor Co-kriging Model}
Improvement of MCMC convergence can be achieved by integrating out the latent variable $\mathbf w=(\mathbf w_1, \dots, \mathbf w_{S})$ from the Bayesian hierarchical NNCGP model which allows the reduction of the dimensionality of the involved posterior distributions. However, integrating out the latent variables $\mathbf w$ in the NNCGP model is not infeasible for non-nested observations. This is because the posterior distribution of latent variable of the lower fidelity is affected by the likelihood of higher fidelity. To make possible the integration of latent variables $\mathbf w$, and relax the dependencies of the latent variables, we propose a recursive formulation for the co-kriging model by using ideas from \citep{le2014recursive} and then coupling it with strategies similar to \citet{finley2019efficient} with purpose to integrate out the latent variable $\mathbf w$. Precisely, our proposed recursive nearest neighbors co-kriging (RNNC) has the following hierarchical structure:
\begin{align}
z_{t}(\mathbf s)&=y_{t}(\mathbf s)+\epsilon_{t}\nonumber \\
y_{t}(\mathbf s)& =\zeta_{t-1}(\mathbf s)\hat{y}_{t-1}(\mathbf s)+\delta_{t}(\mathbf s),\label{eq:recursive}\\
\delta_{t}(\mathbf s) & =\mathbf{h}^T_{t}(\mathbf s)\boldsymbol \beta_{t}+w_{t}(\mathbf s), \nonumber \end{align}
where $\delta_{t}(\mathbf s)$ is a Gaussian process as before and $\hat{y}_{t-1}(\mathbf s)$ is a Gaussian process with distribution $[\hat{y}_{t-1}(\mathbf s)|\mathbf Z_{t-1},\hat{y}^{t-2}(\mathbf s), \boldsymbol \theta_{t-1}]$. Basically, we express $y_t(\mathbf s)$ (the Gaussian process modeling the response at level $t$) as a function of the Gaussian process $y_{t-1}(x)$ conditioned by the values $\mathbf Z^{(t-1)} = (\mathbf Z_1,\dots, \mathbf Z_{t-1})$. For computational efficiency, we assume NNGP independent priors for $w_{t}(\mathbf s)$, $t=1,\dots, T$. Based on the NNGP priors, the conditional distribution can be computed for all types of reference sets. So, based on the recursive representation we can relax the nested condition on the NNCGP nested reference set.
\begin{align}
\hat{y}_{t-1}(\mathbf s)|\cdot \sim & N(\zeta_{t-2}\hat{y}_{t-2}(\mathbf s)+\mathbf{h}^T_{t-1}(\mathbf s)\boldsymbol \beta_{t-1}+V_{t-1,\mathbf s}\mu_{t-1,\mathbf s},V_{t-1,\mathbf s}), \label{response_nncgp_yimputation}
\end{align}
with, $\mu_{t-1,\mathbf s}= V_{t-1,\mathbf s}^{-1}\mathbf{B}_{t-1,\mathbf s}\biggl[z_{t-1}(N_{t-1}(\mathbf s))-\mathbf{h}^T_{t-1}(N_{t-1}(\mathbf s))\boldsymbol \beta_{t-1}-\zeta_{t-2}(N_{t-1}(\mathbf s))\circ \hat{y}_{t-2}(N_{t-1}(\mathbf s))$, $\mathbf{B}_{t,\mathbf s} = C_{\mathbf s,N_t(\mathbf s)}^TC_{N_t{\mathbf s}}^{-1}$, and $V_{t,\mathbf s} = C(\mathbf s,\mathbf s)-C_{\mathbf s,N_t(\mathbf s)}^TC_{N_t{\mathbf s}}^{-1}C_{\mathbf s,N_t(\mathbf s)}$.
Using the Markovian property of the co-kriging model \citep{o1998markov}, the joint likelihood of the proposed model in (\ref{eq:recursive}) can be factorized as a product of likelihoods from different fidelity levels conditional on $\hat{y}_{t-1}(\mathbf S_{t})$ for $t=2,\dots,T$ and prior $\mathbf w_{t}$ for $t=1,...,T$, i.e.:
\begin{align}
L(\boldsymbol{Z}_{1:T}|\cdot) & =p(\mathbf Z_{1}|\mathbf w_{1},\boldsymbol \beta_{1},\tau_{1})\prod_{t=2}^{T}p(\mathbf Z_{t}|\mathbf w_{t},\boldsymbol \beta_{t},\hat{y}_{t-1}(\mathbf S_{t}),\boldsymbol \gamma_{t-1},\tau_{t})\nonumber \\
& =N(\mathbf Z_{1}|\mathbf{h}_{1}(\mathbf S_{1})\boldsymbol \beta_{1}+\mathbf w_{1},\tau_{1}\mathbf{I})\prod_{t=2}^{T}N(\mathbf Z_{t}|\zeta_{t-1}(\mathbf S_{t})\circ \hat{y}_{t-1}(\mathbf S_{t})+\mathbf{h}_{t}(\mathbf S_{t})\boldsymbol \beta_{t}+\mathbf w_{t},\tau_{t}\mathbf{I}),\label{conditional_ind} \end{align} where $\circ$ is the Hadamard production symbol. This representation makes it possible to integrate out the latent variable $\mathbf w_t$ for $t=1\dots T$ independently for each fidelity level.
\subsection{Collapsed Recursive Nearest Neighbor Co-kriging Model}
We represent the multivariate Gaussian latent variable $\mathbf w_{t}(\mathbf S_{t})$ as a linear model: \begin{align*}
w_{t}(\mathbf s_{t,1}) & =0+\eta_{t,1}, \\
w_{t}(\mathbf s_{t,i}) & =a_{t,i,1}w_{t}(\mathbf s_{t,1})+a_{t,i,2}w_{t}(\mathbf s_{t,2})+\dots+a_{t,i,i-1}w_{t}(\mathbf s_{t,i-1})+\eta_{t,i}, \text{for $i=2,\dots, n_t$} \end{align*}
for $t=1,\dots,T$. We set $\eta_{t,i} \sim N(0,d_{t,i,i})$ independently for all $t,i$, $d_{t,1,1}= \var(w_{t,1})$ and $d_{t,i,i}= \var(w_{t,i}|\{w_{t,j}; j<i \})$ for $i=2,\dots,n_t$ and $t=1,\dots,T$. In a matrix form we can write $\mathbf w_{t}(\mathbf S_{t}) =\mathbf A_{t}\mathbf w_{t}(\mathbf S_{t}) +\boldsymbol \eta_{t}$, where $\mathbf A_{t}$ is an $n\times n$ strictly lower-triangular matrix and $\boldsymbol \eta_{t}\sim N(0,\mathbf D)$ and $\mathbf D$ is diagonal. Based on the structure of $\mathbf A_t$, we can write the covariance of each level as $\mathbf C_t(\boldsymbol \theta_t) = (\mathbf{I}_t - \mathbf A_t)^{-1}\mathbf D_t(\mathbf{I}_t - \mathbf A_t)^{-T}$. The NNGP prior constructs a sparse strictly lower triangular matrix $\mathbf A$ with no more than $m \ (\ll n)$ non-zero entries in each row resulting in an approximation of the covariance matrix $\mathbf C_t$. So the approximated inverse $\tilde{\mathbf C}_t^{-1}(\boldsymbol \theta_t)= (\mathbf{I}_t - \mathbf A_t)\mathbf D_t^{-1}(\mathbf{I}_t - \mathbf A_t)^{T}$ is a sparse matrix and can be computed based on $O(n m^2)$ operations.
We call the integrated version of the above model collapsed RNNC model. Specifically, after integrating out ${\bf w}_t$ the proposed RNNC model can be written as: \begin{align}
z_1(\mathbf{S}_1)|\cdot & \sim N(\mathbf{h}_1^T(\mathbf{S}_1)\boldsymbol \beta_1, \tilde{\Lambda}_1(\mathbf S_1,\boldsymbol \theta_1,\tau_1)), \nonumber \\
z_t(\mathbf{S}_t)|\cdot & \sim N(\zeta_t(\mathbf S_t)\circ \hat{y}_{t-1}(\mathbf S_t)+\mathbf{h}_t^T(\mathbf{S}_t)\boldsymbol \beta_t,\tilde{\boldsymbol \Lambda}_t), \label{collapse_structure_NNCGP_AR} \end{align} for $t=2,\ldots,T$, where $\tilde{\Lambda}_t(\boldsymbol \theta_t,\tau_t) = \tilde{C}_t(\boldsymbol \theta_t)+\tau_t^2\mathbf{I}=\sigma_{t}^2\tilde{\mathbf{R}}_{t}(\boldsymbol \phi_t)+\tau_t^2\mathbf{I}$ is the covariance matrix of the observations, $\tilde{C}_t(\boldsymbol \theta_t)$ is the sparse covariance matrix with parameters $\boldsymbol{\theta}_t=\{\sigma_t^2,\boldsymbol{\phi}_t\}$ and $\tau_t^2$ is the variance of the error $\epsilon_t$ at level $t$. By applying Sherman-Morrison-Woodbury formula, the inverse and determinant of $\tilde{\boldsymbol{\Lambda}}$ get the computationally convenient form \begin{align*} \tilde{\boldsymbol{\Lambda}}^{-1}_t = \tau^{-2}_t\mathbf{I}-\tau^{-4}_t(\tilde{\mathbf C}_t(\boldsymbol \theta_t)^{-1}+\tau^{-2}_t\mathbf{I})^{-1},\\ \text{det}(\tilde{\boldsymbol{\Lambda}}_t)=\tau^{2n}_t\text{det}(\tilde{\mathbf C}_t(\boldsymbol \theta_t))\text{det}(\tilde{\mathbf C}_t(\boldsymbol \theta_t)^{-1}+\tau^{-2}_t\mathbf{I}). \end{align*}
Here, $y_{t-1}(\mathbf{S}_{t-1})$ is the latent noiseless output at fidelity level $t-1$ with a space covariance matrix defined by the nearest neighbors. To reduce the dimensionality of $\hat{y}$ variables, for nested location $s_u \in \mathbf S_{t-1}$, we use an empirical approach to estimate $\hat{y}_{t-1}(s_u)$ by $ z_{t-1}(s_u)$. Based on this representation, the joint posterior distribution is approximated as: \begin{align} \small
p(\boldsymbol \Theta_{1:T}| \mathbf Z_{1:T}) & \approx p(\boldsymbol \Theta_{1}) \tilde{L}(\mathbf Z_{1}|\boldsymbol \Theta_{1}) \prod_{t=2}^{T} \int p(\boldsymbol \Theta_{t})\tilde{L}(\mathbf Z_{t}|\boldsymbol \Theta_{t},\hat{\mathbf y}_{t-1}(\mathbf S_{t}))
\tilde{p}(\hat{\mathbf y}_{t-1}(\mathbf S_{t}^*)|\cdot)\,d\hat{\mathbf y}_{t-1}(\mathbf S_{t}^*), \label{eq:dsvsgsdbsbg} \end{align}
where $\tilde{L}(\mathbf Z_{t}|\boldsymbol \Theta_{t},\hat{\mathbf y}_{t-1}(\mathbf S_{t}))$ is the approximated likelihood using the space representation and $\tilde{p}(\hat{\mathbf y}_{t-1}(\mathbf S_{t}^*)|\cdot)$ the nearest neighbor Gaussian process prediction at locations $\mathbf S_{t}^{*}=\bigcup\limits _{i=t+1}^{T}\mathbf S_{i}\backslash\mathbf S_{t}=\{s_{t,1}^{*},\ldots,s_{t,n_{t}^{*}}^{*}\}$ as a set of knots of fidelity level $t$. This contains the observed locations that are not in the $t^{th}$ level but in the higher fidelity levels. Note that the prediction probability can be excluded for cases with hierarchically nested structure for the spatial locations. For hierarchical nested structure for the spatial locations, it can be proven that the prediction distribution mean and variance of level $T$ is the same as the mean and variance of the prediction distribution of the NNCGP. The proof is similar to \citet{le2014recursive} where we substitute the GP priors with the NNGP priors and adding the nugget effect in each level.
By assigning independent conjugate prior $\boldsymbol \beta_t \sim N(\boldsymbol \mu_{\boldsymbol \beta_t},\mathbf{V}_{\boldsymbol \beta_t})$ and $\gamma_t \sim N(\boldsymbol \mu_{\gamma_t},\mathbf{V}_{\gamma_{t}})$,
the conditional distribution for parameters $\boldsymbol \beta_t$ and $\gamma_{t}$ is: \begin{align}
&\boldsymbol \beta_t|\cdot \sim N(\mathbf{V}_{\beta_t}^*\boldsymbol \mu_{\beta_t}^*,\mathbf{V}_{\beta_t}^*), \label{collapse_NNCGP_beta} \\
&\boldsymbol \gamma_t|\cdot \sim N(\mathbf{V}_{\gamma_{t}}^*\boldsymbol \mu_{\gamma_{t}}^*,\mathbf{V}_{\gamma_{t}}^*),\label{collapse_NNCGP_gamma} \end{align}
where $\mathbf{V}_{\beta_t}^*\boldsymbol \mu_{\beta_t}^*, \mathbf{V}_{\gamma_{t}}^*,\boldsymbol \mu_{\gamma_{t}}^*$ are give in Appendix C. The posterior density functions of $\tau_t^2,\boldsymbol \theta_t$ is not given in closed form because they appear in the covariance matrix $\tilde{\boldsymbol \Lambda}_t$. A Metropolis-Hastings (MH) algorithm targeting the distribution $p(\boldsymbol \theta_t,\tau_t^2|\mathbf Z_t,\mathbf y_{t-1}(\mathbf S_{t}),\boldsymbol \beta_t)$ can be used for the Monte Carlo estimation of these parameters.
\subsection{Conjugate Recursive Nearest Neighbor Co-kriging Model} Both NNCGP and collapsed RNNC models are fully Bayesian approaches which are able to provide the joint posterior distributions for multi-fidelity systems. They both rely on the MCMC inference which can be practically prohibitive when analyzing thousands or millions of spatial data sets. We design an MCMC free procedure to achieve exact Bayesian inference at a more practical time based on \cite{finley2019JCGS} estimation strategies. Because the computational efficiency of the estimation procedure in \cite{finley2019JCGS} is sensitive to the number of parameters, it cannot be applied directly to our model. We utilize the RNNC model conditionally independent posterior representation to decompose the parametric space into smaller different groups based on the fidelity levels. To make MCMC free inference possible, we re-paramterize the covariance function of the collapsed recursive co-kriging model as $\tilde{\boldsymbol \Lambda}_t(\boldsymbol \theta_t,\tilde{\tau}_t^2)= \sigma_t^2 \tilde{\boldsymbol \Sigma}_t$, where $\tilde{\boldsymbol \Sigma}_t = \tilde{\mathbf{R}}_{t}+\tilde{\tau}_t^2\mathbf{I}$ , $\tilde{\mathbf{R}}_{t}$ is the nearest-neighbor approximation correlation matrix, and $\tilde{\tau}_t^2 = \frac{\tau_t^2}{\sigma_t^2}$. To avoid the computational bottleneck due to the MCMC, we propose to make fast estimation $(\boldsymbol \phi_t,\tilde{\tau}_t^2)$ through a cross-validation approach independently for each level as well as using the prediction means of $\mathbf y_{t-1}(\mathbf S_{t})$ based on the estimated values. We estimate $\hat{y}_{t}(\mathbf S_t^*)$ by the posterior mean $\bar{\hat{y}}_t(\mathbf S_t^*) = \mathbf{1}_{t>1}(t)\mathbf{g}_{t-1}^T(\mathbf S_t^*)\hat{\boldsymbol \gamma}_{t-1}\hat{y}_{t-1}(\mathbf S_t^*)+\mathbf{h}_{t}^T(\mathbf S_t^*)\hat{\boldsymbol \beta}_t + V_{t,\mathbf S_t^*}\mu_{t,\mathbf S_t^*}$. In the case that we have some points nested, $y_t(s_u)$ for a location $s_u \in \mathbf S_{t-1}$ is estimated with an empirical approach $\bar{\hat{y}}_{t-1}(s_u)$ as $z_{t-1}(s_u)$ and its variance is equal to the variance of the nugget effect.
If we fix the values of $\boldsymbol \phi_t,\tilde{\tau}_t^2$ and $\hat{y}_{t}(\mathbf S_t)$ first, the convariance matrix $\tilde{\boldsymbol \Sigma}_t$ can be calculated by $\boldsymbol \phi_t$ and $\tilde{\tau}_t^2$. We assign an independent conjugate prior for the parameters of each level $p(\boldsymbol \beta_t,\sigma_t^2,\boldsymbol \gamma_t)=p(\boldsymbol \beta_t)p(\sigma_t^2)p(\boldsymbol \gamma_t)$ such as $\boldsymbol \beta_t \sim N(\boldsymbol \mu_{\boldsymbol \beta_t},\sigma_t^2\mathbf{V}_{\boldsymbol \beta_t})$, $\sigma_t^2 \sim IG(a_t,b_t)$, and $\boldsymbol \gamma_t \sim N(\boldsymbol \mu_{\boldsymbol \gamma_{t}},\sigma_{t+1}^2\mathbf{V}_{\boldsymbol \gamma_{t}})$.
Based on this specifications, the posterior density function can be separated for each level $t$ as: \begin{align}
p(\boldsymbol \beta_t,\boldsymbol \gamma_{t-1},\sigma_t^2 &|\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t)) \propto IG(\sigma_t^2|a_t,b_t)N(\boldsymbol \beta_t|\boldsymbol \mu_{\boldsymbol \beta_t},\sigma_t^2\mathbf{V}_{\boldsymbol \beta_t})N(\boldsymbol \gamma_{t-1}|\boldsymbol \mu_{\boldsymbol \gamma_{t-1}},\sigma_t^2\mathbf{V}_{\boldsymbol \gamma_{t-1}}) \nonumber \\
& \times N(\mathbf Z_t|\zeta_{t-1}(\mathbf S_t)\circ \hat{y}_{t-1}(\mathbf S_t)+\mathbf{h}^T_t\boldsymbol \beta_t,\sigma_t^2\tilde{\boldsymbol \Sigma}_t). \end{align}
We can compute the full conditional density function of $\boldsymbol \gamma_{t-1},\boldsymbol \beta_t$, and $\sigma_t^2$ as \begin{align}
\boldsymbol \gamma_{t-1}|\boldsymbol \beta_t,\sigma_t^2,\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t)) & \sim N(\boldsymbol \gamma_{t-1}|\tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}}\tilde{\boldsymbol \mu}_{\boldsymbol \gamma_{t-1}},\sigma^2\tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}}), \label{conjugate_gamma}\\
\boldsymbol \beta_{t}|\sigma_t^2,\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t) & \sim N(\boldsymbol \beta_t|\tilde{\mathbf{V}}_{\beta_t}\tilde{\boldsymbol \mu}_{\beta_t}, \sigma_t^2\tilde{\mathbf{V}}_{\beta_t}) \label{conjugate_beta}\\
\sigma_t^2|\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t) & \sim IG(\sigma_t^2|a_t^*, b_t^*) \label{conjugate_sigma} \end{align} were $\tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}},\tilde{\boldsymbol \mu}_{\boldsymbol \gamma_{t-1}},\tilde{\mathbf{V}}_{\beta_t}\tilde{\boldsymbol \mu}_{\beta_t}, \sigma_t^2\tilde{\mathbf{V}}_{\beta_t}, a_t^*$, and $b_t^*$ are given analytically in Appendix D. Note that for $t = 1$, $\boldsymbol \gamma_{0}$ and $y_{0}(\mathbf S_t)$ do not exist. Also the conditional posterior density function of $\boldsymbol \beta_1$ and $\sigma_1^2$ are slightly different as explained in Appendix D.
\allowdisplaybreaks \begin{algorithm} \begin{description} \item [{step~1}] Start from fidelity level 1($t=1$) , construct a set $L_{t}$ that contains $l_{t}$ numbers of candidates of parameters $\boldsymbol \phi_{t}$ and $\tilde{\tau}_{t}^{2}$. \item [{step~2}] Choose a $(\phi_{t},\tilde{\tau_{t}}^{2})$ from $L_{t}$. Split the data set of fidelity level t into $K$ folds.
\item [{step~3}] Remove $k^{th}$ fold of data set $\mathbf S_{t}$, denote as $\mathbf S_{t,k}$, then estimate $\sigma_{t}^{2}|\mathbf Z_{t},\hat{y}_{t-1}(\mathbf S_{t})$ by posterior mean $\hat{\sigma}_{t}^{2}=\frac{b_{t}^{*}}{a_{t}^{*}-1}$
by \eqref{conjugate_sigma},\eqref{conjugate_sigma_4}. Estimate $\boldsymbol \beta_{t}|\sigma_{t}^{2},\mathbf Z_{t},\hat{y}_{t-1}(\mathbf S_{t})$ by posterior mean $\hat{\boldsymbol \beta}_{t}=\tilde{\mathbf{V}}_{\beta_{t}}\tilde{\boldsymbol \mu}_{\beta_{t}}$
\eqref{conjugate_beta},\eqref{conjugate_beta1}. Estimate $\boldsymbol \gamma_{t-1}|\boldsymbol \beta_{t},\sigma_{t}^{2},\mathbf Z_{t},\hat{y}_{t-1}(\mathbf S_{t})$ by posterior mean $\hat{\boldsymbol \gamma}_{t-1}=\tilde{\mathbf{V}}_{\gamma_{t-1}}\tilde{\boldsymbol \mu}_{\gamma_{t-1}}$\eqref{conjugate_gamma}. \item [{step~4}] Predict test data set $z_{t}(\mathbf S_{t,k})$ by posterior mean $$\hat{z}_{t}(\mathbf S_{t,k})=\mathbf{1}_{t>1}(t)\mathbf{g}_{t-1}^{T}(\mathbf S_{t,k})\hat{\boldsymbol \gamma}_{t-1}\hat{y}_{t-1}(\mathbf S_{t,k})+\mathbf{h}_{t}^{T}(\mathbf S_{t,k})\hat{\boldsymbol \beta}_{t}+V_{t,\mathbf S_{t,k}}\mu_{t,\mathbf S_{t,k}}.$$ \item [{step~5}] Repeat steps 3-4 over all $K$ folds, calculate the average root mean square prediction error(RMSPE) by \[ \text{RMSPE}=\frac{\sum_{k=1}^{K}\biggl[\sum_{\mathbf s=\mathbf S_{t,k}}(z_{t}(\mathbf s)-\hat{z}_{t}(\mathbf s))^{2}/n_{k}\biggr]}{K}. \] \item [{step~6}] Repeats steps 2-5 over all values in candidate set $L_{t}$, choose the value of $\hat{\boldsymbol \phi}_{t}$ and $\hat{\tilde{\tau}}_{t}^{2}$ that minimizes the RMSPE. Repeat step 3 on full data set $\mathbf S_{t}$ by fixing $\boldsymbol \phi_{t}=\hat{\boldsymbol \phi}_{t}$, $\boldsymbol \sigma_{t}^{2}=\hat{\boldsymbol \sigma}_{t}^{2}$. Estimate $\hat{y}_{t}(\mathbf S_{t}^{*})$ by posterior mean $$\bar{\hat{y}}_{t}(\mathbf S_{t}^{*})=\mathbf{1}_{t>1}(t)\mathbf{g}_{t-1}^{T}(\mathbf S_{t}^{*})\hat{\boldsymbol \gamma}_{t-1}\hat{y}_{t-1}(\mathbf S_{t}^{*})+\mathbf{h}_{t}^{T}(\mathbf S_{t}^{*})\hat{\boldsymbol \beta}_{t}+V_{t,\mathbf S_{t}^{*}}\mu_{t,\mathbf S_{t}^{*}}$$ \item [{step~7}] For a new input location $\mathbf s_{p}$, predict $y_{t}(\mathbf s_{p})$
by posterior: $$\hat{y}_{t-1}(\mathbf s)|\cdot\sim N(\zeta_{t-2}\hat{y}_{t-2}(\mathbf s)+\mathbf{h}_{t-1}^{T}(\mathbf s)\boldsymbol \beta_{t-1}+V_{t-1,\mathbf s}\mu_{t-1,\mathbf s},V_{t-1,\mathbf s})$$ Find a confidence interval based on the quantiles of the above distributions.
\item [{step~8}] Repeat steps 1-7 over all $T$ fidelity levels. \end{description} {\footnotesize{}\caption{The Algorithm steps for the MCMC free conjugate RNNC procedure. MCMC free posterior sampling for multi-fidelity level system with $T$ levels. \label{step:conjugate_nncgp-1}} }{\footnotesize\par} \end{algorithm} A K-fold cross-validation method is used for the selection of optimal values for the parameters $\boldsymbol \phi_t$ and $\tilde{\tau}_t^2$ at level $t$ that provide best prediction performance for the model, from a group of candidates. The criteria of choosing $\boldsymbol \phi_t$ and $\tilde{\tau}_t^2$ can be the root mean square prediction error (RMSPE) over the $K$ folds of data set. The geolocated observations of the $t$ fidelity are partitioned into $K$ equal size subsets. Then, one of the subsets is used as a test set and the others are used for training. The procedure is repeated $K$ times such that each subset is used once as a test set. The computational complexity of these procedures is reduced significantly from the use of the nearest neighbors Gaussian process priors in the recursive co-kriging model. The estimation, tuning and prediction procedure of conjugate RNNC model are given in Algorithm \ref{step:conjugate_nncgp-1}. Similar to other NNCGP models, the conjugate RNNC model analyzes the data set of each fidelity level sequentially from the lowest level to the highest. For each single fidelity level $t$, the conjugate RNNC model is able to run in parallel for tuning the parameter $\boldsymbol \phi_t$ and $\tau_t^2$ and $K$ fold cross validation procedure. Thus, the computational complexity is only $\mathcal{O}(\tilde{n}_1m^3)$ for each run in a parallel computing environment, which makes the conjugate RNNC model an extremely fast algorithm. Compared to the MCMC based NNCGP models, it is obvious that the conjugate RNNC model provides an empirical estimation of spatial effect parameter $\boldsymbol \phi_t$ and noise parameter $\tilde{\tau_t}^2$ with a crude resolution, meanwhile, the prediction performance of conjugate RNNC model relies on the choice of the candidates of $\boldsymbol \phi_t$ and $\tilde{\tau}_t^2$. For applications that require very precise estimation of covariance parameters, such as emulating some physical science and engineering computer models, the conjugate RNNC model may not be the first choice compared to other RNNC models. However, for large data applications whose main focus is on providing moderate predictions in a limited time and there is less interest in the inference about the covariance, the conjugate RNNC model provides satisfying outputs with extremely fast speed. We note that the proposed MCMC free inference can be viewed as a sequential optimization technique which splits the parametric space into several lower dimension components where we can apply conditional independent conjugate NNGP models.
\section{Synthetic Data Example and Real Data Analysis}
We study the performance of our proposed procedures, the conjugate RNNC and the collapsed RNNC, as well as compare their performances with that of the sequential NNCGP. The empirical study is based on a synthetic data set example once with nested and once with non-nested input data sets. and based on a real satellite data set application. As measures of performance we use the root mean squared prediction errors (RMSPE), coverage probability of the $95\%$ equal tail credible interval (CVG($95\%$)), average length of the $95\%$ equal tail credible interval (ALCI($95\%$)), and continuous rank probability score (CRPS) \citep{GneitingRaftery2007}. The simulations were performed in MATLAB R2018a, on a personal computer with specifications (intelR i7-3770 3.4GHz Processor, RAM 8.00GB, MS Windows 64bit).
\subsection{Simulation Study}
We consider a two-fidelity level system represented by the hierarchical statistical model (3.1) defined on a two dimensional unit square domain with univariate observation data sets for both $\mathbf Z_{1}$ and $\mathbf Z_{2}$. Let the design matrix be $\mathbf{h}(\mathbf S_{t})=\mathbbm{1}$, the autoregressive coefficient function be an unknown constant $\zeta_{1}(s) = \gamma_1$, and exponential covariance functions. We generate two synthetic data sets for the above statistical model. The true values of the parameters are listed in Table \ref{tab:Univariate-two-fildelity-nested-chapter3} and Table \ref{tab:Univariate-two-fildelity-nonnested-chapter3}. The data sets on the nested spatial locations consists of observations $\mathbf Z_{1}$ and $\mathbf Z_{2}$ from $100\times100$ grids $\mathbf S_{1}$ and $\mathbf S_{2}$, respectively. The data sets, shown in Figures \ref{fig:full_gp_nest_prediction_gp} and \ref{fig:full_gp_nest_prediction_nngp} are based on a fully non-nested input where the low fidelity observations $\mathbf Z_{1}$ and the high fidelity observations $\mathbf Z_{2}$ are generated at irregularly located at point in sets $\mathbf S_{1}$ and $\mathbf S_{2}$ of size $5000$, while $\mathbf S_{1}\cap\mathbf S_{2}=\emptyset$. In all data sets, a few small square regions from $\mathbf Z_{2}$ are treated as a testing data-set, and the rest of $\mathbf Z_{2}$ and $\mathbf Z_{1}$ are treated as training data sets. The testing regions for the non-nested input can be seen as white boxes in Figure 2(b).
Regarding the Bayesian inference, we compared the sequential NNCGP model, with the proposed recursive collapsed RNNC model and that with the recursive conjugate RNNC model, on both nested and non-nested data sets. We assigned similar non-informative priors for all the four models. We assign independent conjugate prior on parameters $\beta_{1}\sim N(0,1000)$, $\beta_2\sim N(0,1000)$, and scale parameter $\gamma_1$. We assign independent inverse gamma prior on spatial variance parameters $\sigma_1^2\sim IG(2,1)$, and $\sigma_2^2\sim IG(2,1)$ and on the noise parameters $\tau_1^2\sim IG(2,1)$, $\tau_2^2 \sim IG(2,1)$. We also assign uniform prior on the range parameters $\phi_1\sim U(0,100)$, and $\phi_2\sim U(0,100)$. For the collapsed RNNC model, we run Markov chain Monte Carlo (MCMC) samplers for $35000$ iterations where the first $5000$ iterations are discarded as a burn-in, and convergence of the MCMC sampler was diagnosed from the individual trace plots. The RMSPE with a 5-fold cross-validation was used for the conjugate RNNC model, we select $\phi_t$ from the range $[0.1 , 20]$, and $\tilde{\tau}_t^2$ from the range $[0.005 , 0.5]$. No significant differences were observed when we used 3-fold cross-validation and 7-fold cross-validation appraoch.
\begin{table}[ht] \centering {\footnotesize{}{}\centering}
\begin{tabular}{c|c|cc|cc|c} \hline \multirow{1}{*}{} & \multirow{1}{*}{{\footnotesize{}{}True}} & \multicolumn{5}{c}{{\footnotesize{}{}Nested data-set}} \tabularnewline
& {\footnotesize{}{}values} & \multicolumn{2}{c}{{\footnotesize{}{}Sequential NNCGP}} & \multicolumn{2}{c}{{\footnotesize{}{}Collapsed RNNC}} & \multicolumn{1}{c}{{\footnotesize{}{}Conjugate RNNC}}\tabularnewline \hline {\footnotesize{}{}$\beta_{1}$} & {\footnotesize{}{}10} & {\footnotesize{}{}10.29 } & {\footnotesize{}{}(9.93,10.57)} & {\footnotesize{}{}9.96 } & {\footnotesize{}{}(9.60,10.32)} & {\footnotesize{}{}10.02} \tabularnewline {\footnotesize{}{}$\beta_{2}$} & {\footnotesize{}{}1} & {\footnotesize{}{}0.77} & {\footnotesize{}{}(0.59,1.04)} & {\footnotesize{}{}0.87} & {\footnotesize{}{}(0.59,1.13)} & {\footnotesize{}{}0.82} \tabularnewline {\footnotesize{}{}$\sigma_{1}^{2}$} & {\footnotesize{}{}4}& {\footnotesize{}{}3.55} & {\footnotesize{}{}(2.77,4.38)} & {\footnotesize{}{}3.46} & {\footnotesize{}{}(2.96,4.27)} & {\footnotesize{}{}3.15} \tabularnewline {\footnotesize{}{}$\sigma_{2}^{2}$} & {\footnotesize{}{}1}& {\footnotesize{}{}0.81} & {\footnotesize{}{}(0.27, 2.05)} & {\footnotesize{}{}0.98} & {\footnotesize{}{}(0.43, 1.88)} & {\footnotesize{}{}0.79} \tabularnewline {\footnotesize{}{}$1/\phi_{1}$} & {\footnotesize{}{}10} & {\footnotesize{}{}10.42} & {\footnotesize{}{}(8.15,13.47)} & {\footnotesize{}{}10.50} & {\footnotesize{}{}(8.59,13.90)} & {\footnotesize{}{}12.1} \tabularnewline {\footnotesize{}{}$1/\phi_{2}$} & {\footnotesize{}{}10} & {\footnotesize{}{}14.96} & {\footnotesize{}{}(3.37, 20.29)} & {\footnotesize{}{}15.69} & {\footnotesize{}{}(5.92, 19.98)} & {\footnotesize{}{}19.6} \tabularnewline {\footnotesize{}{}$\gamma_{1}$} & {\footnotesize{}{}1} & {\footnotesize{}{}0.991} & {\footnotesize{}{}(0.982,1.003)} & {\footnotesize{}{}0.991} & {\footnotesize{}{}(0.976,1.004)} & {\footnotesize{}{}0.992} \tabularnewline {\footnotesize{}{}$\tau_{1}^{2}$} & {\footnotesize{}{}0.1} & {\footnotesize{}{}0.121} & {\footnotesize{}{}(0.098,0.138)} & {\footnotesize{}{}0.153} & {\footnotesize{}{}(0.099,0.195)} & {\footnotesize{}{}0.125} \tabularnewline {\footnotesize{}{}$\tau_{2}^{2}$} & {\footnotesize{}{}0.05} & {\footnotesize{}{}0.067} & {\footnotesize{}{}(0.033,0.106)} & {\footnotesize{}{}0.099} & {\footnotesize{}{}(0.041,0.194)} & \footnotesize{}{}0.158{\footnotesize{} } \tabularnewline {\footnotesize{}{}$m$} & {\footnotesize{}{}10} & {\footnotesize{}{}-} & {\footnotesize{}{}-} & {\footnotesize{}{}-} & - & {\footnotesize{}{}-} \tabularnewline \hline \end{tabular}{\footnotesize{}{}\caption{The estimation of parameters in nested input dataset, using sequential NNCGP, collapsed RNNC and conjugate RNNC models. \label{tab:Univariate-two-fildelity-nested-chapter3}} } \end{table}
\begin{table}[ht] \centering
\begin{tabular}{c|ccc} \hline \multirow{1}{*}{} & \multicolumn{3}{c}{{\footnotesize{}{}Nested data-set}} \tabularnewline
& {\footnotesize{}{}Sequential NNCGP} & {\footnotesize{}{}Collapsed RNNC} & {\footnotesize{}{}Conjugate RNNC} \tabularnewline \hline {\footnotesize{}{}RMSPE} & {\footnotesize{}{}0.6278} & {\footnotesize{}{}0.6875}& {\footnotesize{}{}0.7217} \tabularnewline {\footnotesize{}{}NSME } & {\footnotesize{}{}0.7664} & {\footnotesize{}{}0.7547 }& {\footnotesize{}{}0.7081} \tabularnewline {\footnotesize{}{}CRPS} & {\footnotesize{}{}0.4453} & {\footnotesize{}{}0.4417 }& {\footnotesize{}{}0.4102} \tabularnewline {\footnotesize{}{}CVG(95\%) } & {\footnotesize{}{}0.9054} & {\footnotesize{}{}0.8761 } & {\footnotesize{}{}0.9284} \tabularnewline {\footnotesize{}{}ALCI(95\%)} & {\footnotesize{}{}1.9284} & {\footnotesize{}{}1.9209 }& {\footnotesize{}{}2.5377} \tabularnewline \hline {\footnotesize{}{}Time(Hour) } & {\footnotesize{}{}4.5} & {\footnotesize{}{}4.7 }& {\footnotesize{}{}0.08} \tabularnewline \hline \end{tabular}{\footnotesize{}{} \caption{Performance measures for the predictive ability of the sequential NNCGP model, collapsed RNNC model and conjugate RNNC model. \label{tab:efficient_nested_metrics_nested_chapter3}} } \end{table}
In Tables \ref{tab:Univariate-two-fildelity-nested-chapter3} and \ref{tab:Univariate-two-fildelity-nonnested-chapter3}, we report the Monte Carlo estimates of the posterior means and the associated $95\%$ marginal credible intervals of the unknown parameters using the two different NNCGP based procedures: sequential NNCGP, collapsed RNNC, along with the posterior mean and tuned values of parameters using conjugate RNNC, with $m=10$. There is no significant difference in the estimation of parameters for all MCMC based models (NNCGP and collapsed RNNC) and the true values of the parameters are successfully included in the $95\%$ marginal credible intervals. The introduction of latent interpolants may have caused a small overestimation of $\tau_{2}^2$ for all models. Instead, the conjugate RNNC is underestimating the variance of the nugget for the second fidelity level. This underestimation seems to be associated with the relatively better approximate the CVG but also with the large ALC. The uncertainty in the parameter estimations can be improved with a semi-nested or nested structure between the observed locations of the fidelity levels, and it is also shown for the auto-regressive co-kriging model in \citet{konomikaragiannisABTCK2019}. Results also showed that when select $\phi_t$ and $\tau_t^2$ corresponding to minimized RMSPE, the MCMC free conjugate RNNC model has similar estimation of other parameters comparing to NNCGP and RNNC models. We also observe that the conjugate RNNC model prefers larger value of $\tau_2^2$ when minimizing RMSPE, which is consistent with the previous discussion on NNCGP models.
\begin{table} \centering {\footnotesize{}{}\centering}
\begin{tabular}{c|c|cc|cc|c} \hline \multirow{1}{*}{} & \multirow{1}{*}{{\footnotesize{}{}True}} & \multicolumn{5}{c}{{\footnotesize{}{} Non-nested data-set}} \tabularnewline
& {\footnotesize{}{}values} & \multicolumn{2}{c}{{\footnotesize{}{}Sequential NNCGP}} & \multicolumn{2}{c}{{\footnotesize{}{}Collapsed RNNC}} & \multicolumn{1}{c}{{\footnotesize{}{}Conjugate RNNC}}\tabularnewline \hline {\footnotesize{}{}$\beta_{1}$} & {\footnotesize{}{}10} & {\footnotesize{}{}9.71} & {\footnotesize{}{}(9.36, 10.16)} & {\footnotesize{}{}9.97 } & {\footnotesize{}{}(9.52,10.41)} & {\footnotesize{}{}9.71} \tabularnewline {\footnotesize{}{}$\beta_{2}$} & {\footnotesize{}{}1} & {\footnotesize{}{}0.87} & {\footnotesize{}{}(0.39,1.36)} & {\footnotesize{}{}1.23} & {\footnotesize{}{}(0.24,2.19)} & {\footnotesize{}{}1.27} \tabularnewline {\footnotesize{}{}$\sigma_{1}^{2}$} & {\footnotesize{}{}4}& {\footnotesize{}{}3.51} & {\footnotesize{}{}(2.71,4.52)} & {\footnotesize{}{}3.28} & {\footnotesize{}{}(3.02,3.72)} & {\footnotesize{}{}3.84} \tabularnewline {\footnotesize{}{}$\sigma_{2}^{2}$} & {\footnotesize{}{}1}& {\footnotesize{}{}1.05} & {\footnotesize{}{}(0.18,2.31)} & {\footnotesize{}{}1.00} & {\footnotesize{}{}(0.64, 1.49)} & {\footnotesize{}{}1.19} \tabularnewline {\footnotesize{}{}$1/\phi_{1}$} & {\footnotesize{}{}10} & {\footnotesize{}{}10.77} & {\footnotesize{}{}(8.07,13.91)} & {\footnotesize{}{}13.23} & {\footnotesize{}{}(9.93,15.85)} & {\footnotesize{}{}5} \tabularnewline {\footnotesize{}{}$1/\phi_{2}$} & {\footnotesize{}{}10} & {\footnotesize{}{}12.61} & {\footnotesize{}{}(3.93,24.07)} & {\footnotesize{}{}16.01} & {\footnotesize{}{}(12.84, 19.92)} & {\footnotesize{}{}20} \tabularnewline {\footnotesize{}{}$\gamma_{1}$} & {\footnotesize{}{}1} & {\footnotesize{}{}0.995} & {\footnotesize{}{}(0.983,1.051)} & {\footnotesize{}{}0.967} & {\footnotesize{}{}(0.942,0.994)} & {\footnotesize{}{}0.959} \tabularnewline {\footnotesize{}{}$\tau_{1}^{2}$} & {\footnotesize{}{}0.1} & {\footnotesize{}{}0.125} & {\footnotesize{}{}(0.097,0.148)} & {\footnotesize{}{}0.102} & {\footnotesize{}{}(0.072,0.152)} & {\footnotesize{}{}0.11} \tabularnewline {\footnotesize{}{}$\tau_{2}^{2}$} & {\footnotesize{}{}0.05} & {\footnotesize{}{}0.158} & \footnotesize{}{}(0.041,0.232) & {\footnotesize{}{}0.182} & {\footnotesize{}{}(0.032,0.292)} & \footnotesize{}{}0.01{\footnotesize{} } \tabularnewline {\footnotesize{}{}$m$} & {\footnotesize{}{}10} & {\footnotesize{}{}-} & {\footnotesize{}{}-} & {\footnotesize{}{}-} & - & {\footnotesize{}{}-} \tabularnewline \hline \end{tabular}{\footnotesize{}{}\caption{The estimation of parameters in non-nested input dataset using sequential NNCGP, collapsed RNNC and conjugate RNNC models. \label{tab:Univariate-two-fildelity-nonnested-chapter3}} } \end{table}
\begin{table} \center
\begin{tabular}{c|ccc} \hline \multirow{1}{*}{} & \multicolumn{3}{c}{{\footnotesize{}{}Non-nested data-set}} \tabularnewline
& {\footnotesize{}{}Sequential NNCGP} & {\footnotesize{}{}Collapsed RNNC} & {\footnotesize{}{}Conjugate RNNC} \tabularnewline \hline {\footnotesize{}{}RMSPE} & {\footnotesize{}{}0.9319} & {\footnotesize{}{}0.9241}& {\footnotesize{}{}1.0680} \tabularnewline {\footnotesize{}{}NSME } & {\footnotesize{}{}0.7489} & {\footnotesize{}{}0.7827 }& {\footnotesize{}{}0.7480} \tabularnewline {\footnotesize{}{}CRPS} & {\footnotesize{}{}0.6109} & {\footnotesize{}{}0.6110 }& {\footnotesize{}{}0.6516} \tabularnewline {\footnotesize{}{}CVG(95\%) } & {\footnotesize{}{}0.9286} & {\footnotesize{}{}0.9714 }& {\footnotesize{}{} 0.9538} \tabularnewline {\footnotesize{}{}ALCI(95\%)} & {\footnotesize{}{}1.4855} & {\footnotesize{}{}1.7612 }& {\footnotesize{}{}2.4885} \tabularnewline \hline {\footnotesize{}{}Time(Hour) } & {\footnotesize{}{}4.1} & {\footnotesize{}{}4.3 } & {\footnotesize{}{}0.05} \tabularnewline \hline \end{tabular}{\footnotesize{}{} \caption{Performance measures for the predictive ability of the Sequential NNCGP model, collapsed RNNC model and conjugate RNNC model, in non-nested input. \label{tab:efficient_nonnested_metrics_nonnested_chapter3}} } \end{table}
In Table \ref{tab:efficient_nested_metrics_nested_chapter3} and Table \ref{tab:efficient_nonnested_metrics_nonnested_chapter3}, we report standard performance measures (defined in the Appendix) for the sequential NNCGP, collapsed RNNC and conjugate RNNC with $m=10$ number of neighbours. All performance measures indicate that the collapsed RNNC model has similar predictive ability with the sequential NNCGP model. The conjugate RNNC model produced RMSPE value that is 10\% larger than other NNCGP and collapsed RNNC models, but it is still significantly smaller than the RMSPE values from single level NNGP and combined NNGP. The tables also shows that the running time of collapsed RNNC model is not different from sequential NNCGP model, this is consistent with our previous discussion since the two procedures have the same computational complexity. We observe that the conjugate RNNC model has extremely smaller running time compared to sequential NNCGP models, since the inference for conjugate RNNC model requires the same amount of running time as one iteration in collapsed RNNC model. It is worth pointing out that the cross validation process and tuning process in Algorithm \ref{step:conjugate_nncgp-1} are independent from each other, which makes conjugate RNNC model benefit from parallel computation environments and greatly reduce computational time.
\begin{figure}\label{fig:full_gp_nest_prediction_gp}
\label{fig:full_gp_nest_prediction_nngp}
\label{fig:nested_and_non-nested_data_chapter3}
\end{figure}
\begin{figure}\label{fig:large_nonnest_prediction_test3}
\label{fig:large_nonnest_prediction_nncgp3a}
\label{fig:large_nonnest_prediction_nncgp3b}
\label{fig:large_nonnest_prediction_nngp3}
\label{fig:large_nonnest_prediction_nncgp3c}
\label{fig:large_nonnest_prediction_nngp3d}
\label{fig:large_nonnest_prediction_conjugate_plots}
\end{figure}
Figure \ref{fig:nested_and_non-nested_data_chapter3} and \ref{fig:large_nonnest_prediction_conjugate_plots} provide the non-nested synthetic observations and the prediction plots from sequential NNCGP, collapsed RNNC, conjugate RNNC, combined NNGP and single level NNGP models. We observed that for the testing regions the NNCGP models provides similar prediction surfaces and all NNCGP models has the better presentation of patterns in prediction surface comparing to single level NNGP model.
\subsection{Application to High-resolution Infrared Radiation Sounder data}
We model our data based on the two-fidelity level conjugate RNNC model and on the two-fidelity level sequential NNCGP model. Moreover, we provide comparisons with the single level NNGP model and combined NNGP model. We consider a linear model for the mean of the Gaussian processes, in $y_1(\cdot)$ and $\delta_{2}(\cdot)$, with linear basis function representation $\mathbf{h}(s_{t})$ and coefficients $\boldsymbol{\beta}_{t}=\{\beta_{0,t},\beta_{1,t},\beta_{2,t}\}^{T}$. We consider the scalar discrepancy $\zeta(s)$ to be unknown constant and equal to $\gamma$. The number of nearest neighbors $m$ is set to 10, and the spatial process $\mathbf w_{t}$ is consider to have a diagonal anisotropic exponential covariance function.
\begin{table}[H] \begin{adjustbox}{width=1\textwidth} {}{}\centering
\begin{tabular}{c|ccccc} \hline \multirow{1}{*}{{}{} } & \multicolumn{5}{c}{{}{}Model}\tabularnewline
& {}{}Sequential NNCGP & {}{}Single level NNGP & {}{}Combined NNGP & {}{}Collapsed RNNC & {}{}Conjugate RNNC \tabularnewline \hline {}{}RMSPE & {}{}1.2044 & {}{}1.8153 & {}{}1.6772 & {}{}1.2113 & {}{}1.3606 \tabularnewline {}{}NSME & {}{}0.8439 & {}{}0.5499 &0.6726 & {}{}0.8487 & {}{}0.8198 \tabularnewline {}{}CRPS & {}{}0.7023 & {}{}1.6498 &0.9274 & {}{}0.6767 & {}{}0.7475 \tabularnewline {}{}CVG(95\%) & {}{}0.9255 & {}{}0.8350 &0.9197 & {}{}0.9398 & {}{} 0.9443 \tabularnewline {}{}ALCI(95\%) & {}{}3.094 & {}{}4.214 &5.778 & {}{}3.162 & {}{} 4.392 \tabularnewline \hline {}{}Time(Hour) & {}{}38 & {}{}20 & 32 & {}{}40 & {}{}0.3 \tabularnewline \hline \end{tabular} \end{adjustbox} \caption{{\large{}{}{}{}{}{}{}\label{real_data_table_conjugate}}Performance measures for the predictive ability of sequential NNCGP, single level NNGP, combined NNGP, collapsed RNNC and conjugate RNNC models in NOAA 14 and NOAA 15 HIRS instrument data analysis.} \end{table}
We assign independent normal distribution priors with zero mean and large variances for $\beta_{0,t},\beta_{1,t},\beta_{2,t}$ and $\gamma$. We assign independent uniform prior distributions $U(0,1000)$ to the range correlation parameters $({\phi}_{t,1},{\phi}_{t,2})$ for $t=1,2$. Also, we assign independent $IG(2,1)$ prior distributions for the variance parameters $\sigma_{t}^{2}$ and $\tau_{t}^{2}$. For the Bayesian inference of the sequential NNCGP, we run the MCMC sampler with of $35,000$ iterations where the first $5,000$ iterations are discarded as a burn-in. For the Bayesian inference of conjugate RNNC, we consider using posterior means as the estimated values for parameters $\beta_{0,t},\beta_{1,t},\beta_{2,t}$, $\sigma_t^2$ and $\gamma$; we also use posterior means as the imputation values for latent process $\tilde{\mathbf y}_t$ and for the prediction values of $z(s_p)$ at location $s_p\not\in\tilde{\mathbf S}_t$.
\begin{figure}
\caption{Predictions of NOAA-15 Brightness Temperatures(K) testing data-set by (b) sequential NNCGP, (c) collapsed RNNC, (d) conjugate RNNC, (e) single level NNGP and (f) combined NNGP models. }
\label{fig:HIRS_conjugate_test}
\label{fig:HIRS_conjugate_sequentiala}
\label{fig:HIRS_conjugate_sequentialb}
\label{fig:HIRS_conjugate_conjugate}
\label{fig:HIRS_conjugate_single}
\label{fig:HIRS_conjugate_combine}
\label{fig:Hirs_conjugate_model}
\end{figure}
The prediction performance metrics of the four different methods are given in Table \ref{real_data_table_conjugate}. Compared to the single level NNGP model and combined NNGP model, the sequential NNCGP model
and conjugate RNNC model produced a 20-30\% smaller RMSPE and their NSME is closer to 1. The sequential NNCGP model and collapsed RNNC model also produced larger CVG and smaller ALCI than the single level NNGP model and combined NNGP model. The result suggests that the NNCGP and RNNC models have a substantial improvement in terms of predictive accuracy in real data analysis too. In the prediction plots (Figure \ref{fig:Hirs_conjugate_model}) of the testing data of NOAA-15, we observe that RNNC models are more capable of capturing the pattern of the testing data than single level NNGP model and combined NNGP model. This is reasonable because the observations from NOAA-14 have provided information of the testing region, and comparing to combined NNGP model, the NNCGP and RNNC models are capable of modeling the discrepancy of observations from different satellites. In the non-nested structure, the computational complexity of the single level NNGP model is $\mathcal{O}(n_{2}m^{3})$ and that of NNCGP model is $\mathcal{O}((n_{1}+n_{2})m^{3})$, for an MCMC iteration. However, the whole computational complexity of the conjugate RNNC model is $\mathcal{O}((n_{1}+n_{2})m^{3})$ with parallel computational environment, which makes it remarkably computationally efficient without losing significant prediction accuracy. This is consistent with the running times of the models shown in Table \ref{real_data_table_conjugate}.
\begin{figure}\label{fig:sfig2-2-1}
\label{fig:fig-7}
\end{figure}
We apply the MCMC free conjugate RNNC model for gap-filling predictions based upon a discrete global grid. We chose to use $1^{\circ}$ latitude by $1.25^{\circ}$ longitude ($1^{\circ}\times1.25^{\circ}$) pixels as grids with global spatial coverage from $-70^{\circ}$ to $70^{\circ}$N. By applying the NNCGP model, we predict gridded NOAA-15 brightness temperature data on the center of the grids, based on the NOAA-14 and NOAA-15 swath-based spatial support. The prediction plot (Figure \ref{fig:fig-7}) illustrates the ability of the MCMC free conjugate RNNC model to handle large irregularly spaced data sets and produce a gap-filled composite gridded dataset. The resulting global image of the brightness temperature is practically the same as the sequential NNCGP.
\section{Summary and conclusions}
We have proposed a new computationally efficient co-kriging method, the recursive nearest neighbor Autoregressive Co-Kriging (RNNC) model, for the analysis of large and multi-fidelity spatial data sets. In particular, we proposed two computationally efficient inferential versions of it the collapsed RNNC and the conjugate RNNC. For the collapsed RNNC, we integrate out the latent variables of the RNNC model which enables the factorisation of the likelihood into terms involving smaller and sparse covariance matrices within each level. Then, a prediction focused approximation is applied on the aforesaid model to further speed up the computation. The idea is that cross-validation using grid search on a two or three dimensional space is a computationally feasible method to estimate the hyperparameters. The proposed conjugate RNNC is MCMC free and at most computationally linear in the total number of all spatial locations of all fidelity levels. Both of the proposed procedures, have similarities to modularization approach in Bayesian statistics \cite{Bayarri_module} where the analysis is done in steps rather than jointly. We compared the proposed collapsed RNNC and conjugate RNNC with NNCGP in a simulation study and a real data application of intersatellite calibration. We observed that similar to NNCGP, the collapsed and conjugate RNNC were also able to improve the accuracy of the prediction for the HIRS brightness temperatures from the NOAA-15 polar-orbiting satellite by incorporating information from an older version of the same HIRS sensor on board the polar orbiting satellite NOAA-14.
The proposed procedures can be used for a variety of large multi-fidelity data sets in remote sensing with overlapping areas of observed locations. One can use more complex Vechias approximations \citep{vecchia1988estimation,stein2004approximating,Guinness2018,Katzfuss_2020}, similar to the NNGP priors, where the ordering of the data can be more complicated but can result in a better approximation. These Vechias approximation techniques of ordering can be applied naturally in the proposed RNNC model, however, they are out of the scope of this paper and can be investigated in a future work. As a future research, the proposed method can be also extended in the multivariate setting by using ideas from parallel partial autoregressive co-kriging \citep{ma2019multifidelity} and NNGP spatial factor models \citep{taylor2018spatial}. One way to introduce non-stationarity is by using treed based model such as in \cite{Konomi2014JCGS}. Still, work needs to be done in developing new strategies for tuning the hyperparameters in more complex covariance functions with multiple parameters within a fidelity level.
\appendix
\section*{Appendix}
\section{NNGP specifications}\label{App1}
The posterior distribution of {\small{}{}{} \begin{align}
\tilde{p}(\mathbf w_{t}|\cdot) & \propto\text{exp}\left[-\frac{1}{2}\sum_{i=1}^{n_{t}}\left\{ w_{t}(s_{t,i})-\mathbf{B}_{t,s_{t,i}}\mathbf w_{t,N_{t}(s_{t,i})}\right\} ^{T}F_{t,s_{t,i}}^{-1}\left\{ w_{t}(s_{t,i})-\mathbf{B}_{t,s_{t,i}}\mathbf w_{t,N_{t}(s_{t,i})}\right\} \right]\nonumber \\
& =\text{exp}\left(-\frac{1}{2}\mathbf w_{t}^{T}\mathbf{B}_{t}^{T}\mathbf{F}_{t}^{-1}\mathbf{B}_{t}\mathbf w_{t}\right), \end{align} }{\small\par}
where $\mathbf{F}_{t}=\text{diag}(F_{t,s_{t,1}},F_{t,s_{t,2}},\ldots,F_{t,s_{t,n_{t}}})$, $\mathbf{B}_{t}=\Big{(}\mathbf{B}_{t,1}^{T},\mathbf{B}_{t,2}^{T},\ldots,\mathbf{B}_{t,n_{t}}^{T}\Big{)}^{T}$, and for each element in $\mathbf{B}_{t}$, we have $\mathbf{B}_{t,i}=\Big{(}\mathbf{B}_{t,s_{t,i},1}^{T},\mathbf{B}_{t,s_{t,i},2}^{T},\ldots,\mathbf{B}_{t,s_{t,i},n_{t}}^{T}\Big{)}^{T}$ and {\small{}{}{} \begin{align} \mathbf{B}_{t,s_{t,i},j}=\begin{cases} 1,\ \text{if}\ i=j,\\ -\mathbf{B}_{t,s_{t,i}}[,k],\ \text{if}\ s_{t,j}\ \text{is the}\ k^{th}\ \text{element in}\ N_{t}(s_{t,i}),\\ 0,\ \text{Others}. \end{cases} \end{align} }
\section{Mean and Variance Specifications}
The mean vector $\boldsymbol{\mu} = \left(\mu_1(s_{1,1}),\ldots,\mu_1(s_{1,n_1}),\ldots,\mu_T(s_{T,n_T})\right)$ is \allowdisplaybreaks \begin{align} \mu_t(s_{t,k})=&\mathbf{1}_{\{t>1\}}(t)\sum_{i=1}^{t-1}\left\{\prod_{j=i}^{t-1}\zeta_{j}(s_{t,k})\right\} \left\{\mathbf{h}_{i}^{T}(s_{t,k})\boldsymbol{\beta}_{i} + \mathbf{1}_{\{s_{t,k}\in \mathbf S_i\}}(s_{t,k})w_i(s_{t,k}) \right\} \nonumber \\ &+\mathbf{h}_{t}^{T}(s_{t,k})\boldsymbol{\beta}_{t} + w_t(s_{t,k}), \label{mean_function_original} \end{align} for $t=1,\ldots T$, $ i=1,\ldots, n_t$. $\mathbf{1}_{\{\cdot\}}(\cdot)$ is the indicator function, and covariance matrix $\boldsymbol{\Lambda}$ is a block matrix with blocks $\Lambda^{(1,1)},\ldots,\Lambda^{(1,T)},\ldots,\Lambda^{(T,T)}$, and the size of $\boldsymbol{\Lambda}$ is $\sum_{t=1}^{T}n_t \times \sum_{t=1}^{T}n_t$. The $\Lambda^{(t,t)}$ components are calculated as:
\begin{align}
& \Lambda^{(t,t)}_{k,l} = \text{cov}(z_{t}(s_{t,k}),z_{t}(s_{t,l})|\cdot)=\sum_{i=1}^{t-1} \mathbf{1}_{\{s_{t,k},s_{t,l}\notin \mathbf S_i\}}(s_{t,k},s_{t,l}) \left\{ \prod_{j=i}^{t-1}\zeta_{j}(s_{t,k})^{T}\zeta_{j}(s_{t,l})\right\} C_{i}(s_{t,k},s_{t,l}|\boldsymbol{\theta}_{i}) \nonumber \\
& \qquad \qquad \qquad +\mathbf{1}_{s_{t,k}=s_{t,l}}(s_{t,k},s_{t,l})\tau_{t}^{2},\nonumber
\end{align} for $t \ \text{and}\ t' = 1,\ldots,T$; $k=1,\ldots,n_t$; $l = 1,\ldots,n_{t'}$, and
\begin{align}
&\Lambda^{(t,t')}_{k,l} = \text{cov}(z_{t}(s_{t,k}),z_{t'}(s_{t',l})|\cdot)=\sum_{i=1}^{\text{min}(t,t')-1} \mathbf{1}_{\{s_{t,k},s_{t',l}\notin \mathbf S_i\}}(s_{t,k},s_{t',l}) \left\{ \prod_{j=i}^{\text{min}(t,t')-1}\zeta_{j}(s_{t,k})^{T}\zeta_{j}(s_{t',l})\right\} \nonumber \\
& \qquad \qquad \qquad \times C_{i}(s_{t,k},s_{t',l}|\boldsymbol{\theta}_{i}) + \mathbf{1}_{\{s_{t,k},s_{t',l}\notin \mathbf S_{\text{min}(t,t')}\}}(s_{t,k},s_{t',l}) C_{\text{min}(t,t')}(s_{t,k},s_{t',l}|\boldsymbol{\theta}_{\text{min}(t,t')}),\label{covariance_function_original} \end{align}
for $t\neq t'$, $\Lambda^{(t,t')}$.
\section{Gibbs Sampler } \begin{align} \mathbf{V}_{\beta_t}^* &= (\mathbf{h}_t(\mathbf S_t)\tilde{\Lambda}_t(\mathbf S_t,\boldsymbol \theta_{t})^{-1}\mathbf{h}^T_t(\mathbf S_t) + \mathbf{V}_{\beta_t}^{-1})^{-1},\nonumber \\ \boldsymbol \mu_{\beta_t}^* & = \mathbf{V}_{\beta_t}^{-1}\boldsymbol \mu_{\beta_t}+\mathbf{h}_t(\mathbf S_t)\tilde{\Lambda}_t(\mathbf S_t,\boldsymbol \theta_{t})^{-1}(z_t(\mathbf S_t)-\zeta_{t-1}(\mathbf S_t)\hat{y}_{t-1}(\mathbf S_t)). \\ \mathbf{V}_{\gamma_{t}}^* &= \left[(\mathbf{g}_t^T(\mathbf S_{t+1})\hat{y}_{t}(\mathbf S_{t+1}))^T\tilde{\Lambda}_{t+1}(\mathbf S_{t+1},\boldsymbol \theta_{t+1},\tau_{t+1})^{-1}(\mathbf{g}_t^T(\mathbf S_{t+1})\hat{y}_{t}(\mathbf S_{t+1})) + \mathbf{V}_{\gamma_{t}}^{-1}\right]^{-1},\nonumber \\ \boldsymbol \mu_{\gamma_{t}}^* & = \mathbf{V}_{\gamma_{t}}^{-1}\boldsymbol \mu_{\gamma_{t}}+(\mathbf{g}_t^T(\mathbf S_{t+1})\hat{y}_{t}(\mathbf S_{t+1}))^T\tilde{\Lambda}_{t+1}(\mathbf S_{t+1},\boldsymbol \theta_{t+1},\tau_{t+1})^{-1}(\mathbf Z_{t+1}-\mathbf{h}_t^T(\mathbf S_{t+1})\boldsymbol \beta_{t+1}).\label{AppendixC1} \end{align}
\section{Gibbs Sampler Conjugate} we derive the posterior distribution as \begin{align*}
p(&\boldsymbol \beta_t,\boldsymbol \gamma_{t-1},\sigma_t^2 |\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t)) \propto IG(\sigma_t^2|a_t,b_t)N(\boldsymbol \beta_t|\boldsymbol \mu_{\boldsymbol \beta_t},\sigma_t^2\mathbf{V}_{\boldsymbol \beta_t})N(\boldsymbol \gamma_{t-1}|\boldsymbol \mu_{\boldsymbol \gamma_{t-1}},\sigma_t^2\mathbf{V}_{\boldsymbol \gamma_{t-1}}) \nonumber \\
& \quad \ \times N(\mathbf Z_t|\zeta_{t-1}(\mathbf S_t)\circ \hat{y}_{t-1}(\mathbf S_t)+\mathbf{h}^T_t\boldsymbol \beta_t,\sigma_t^2\tilde{\boldsymbol \Sigma}_t) \\
& \propto p(\sigma_t^2|\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t))p(\boldsymbol \beta_t|\sigma_t^2,\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t))p(\boldsymbol \gamma_{t-1}|\boldsymbol \beta_t,\sigma_t^2,\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t)) \nonumber \\
& \propto (\sigma_t^2)^{a_t+0.5n_t}\text{exp}\left(-\frac{1}{2\sigma_t^2}(\boldsymbol \beta_t - \boldsymbol \mu_{\beta_t})^T\mathbf{V}_{\beta_t}^{-1}(\boldsymbol \beta_t - \boldsymbol \mu_{\beta_t})\right) \\
& \quad \ \times \text{exp}\left(-\frac{1}{2\sigma_t^2}(\boldsymbol \gamma_{t-1} - \boldsymbol \mu_{\gamma_{t-1}})^T\mathbf{V}_{\gamma_{t-1}}^{-1}(\boldsymbol \gamma_{t-1} - \boldsymbol \mu_{\gamma_{t-1}})\right) \\
& \quad \ \times \text{exp}\left(-\frac{1}{2\sigma_t^2}(\mathbf Z_t - \mathbf{g}^T(\mathbf S_t)\boldsymbol \gamma_{t-1} \hat{y}_{t-1}(\mathbf S_t)-\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t)^T \tilde{\boldsymbol \Sigma}_t^{-1}(\mathbf Z_t - \mathbf{g}^T(\mathbf S_t)\boldsymbol \gamma_{t-1} \hat{y}_{t-1}(\mathbf S_t)-\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t) \right).
\end{align*}
The full conditional density function of $\boldsymbol \gamma_{t-1}$ is \begin{align}
p(\boldsymbol \gamma_{t-1}&|\boldsymbol \beta_t,\sigma_t^2,\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t)) \propto \text{exp}\Bigg(-\frac{1}{2\sigma_t^2}[\mathbf{g}(\mathbf S_t)\boldsymbol \gamma_{t-1}^T\hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf{g}^T(\mathbf S_t)\boldsymbol \gamma_{t-1} \hat{y}_{t-1}(\mathbf S_t) \nonumber \\ & - 2(\mathbf Z_t-\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t)^T\tilde{\Sigma}_t^{-1}\mathbf{g}^T(\mathbf S_t)\boldsymbol \gamma_{t-1} \hat{y}_{t-1}(\mathbf S_t) ]\Bigg) \nonumber \\
&\propto N(\boldsymbol \gamma_{t-1}|\tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}}\tilde{\boldsymbol \mu}_{\boldsymbol \gamma_{t-1}},\sigma^2\tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}}), \nonumber \\ \tilde{\boldsymbol \mu}_{\boldsymbol \gamma_{t-1}}& = \mathbf{V}_{\boldsymbol \gamma_{t-1}}^{-1}\boldsymbol \mu_{\boldsymbol \gamma_{t-1}} +\mathbf{g}(\mathbf S_t) \hat{y}_{t-1}(\mathbf S_t)^T\tilde{\Sigma}_t^{-1}(\mathbf Z_t-\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t), \nonumber \\ \tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}} & = \left( \mathbf{V}_{\boldsymbol \gamma_{t-1}}^{-1} + \mathbf{g}(\mathbf S_t)\hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\hat{y}_{t-1}(\mathbf S_t)\mathbf{g}^T(\mathbf S_t) \right)^{-1}. \label{conjugate_gamma_b} \end{align}
After integrate $\boldsymbol \gamma_{t-1}$ out, the conditional posterior density function of $\boldsymbol \beta_t$ is \begin{align}
p(\boldsymbol \beta_{t}&|\sigma_t^2,\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t))\propto \text{exp}\Bigg(-\frac{1}{2\sigma_t^2}[(\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}(\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t)-2\mathbf Z_t^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf{h}^T_t(\mathbf S_t)\boldsymbol \beta_t]\Bigg) \nonumber \\
& \quad \ \times \text{exp}\Bigg( -\frac{1}{2\sigma_t^2}(\boldsymbol \beta_t-\boldsymbol \mu_{\beta_t})^T\mathbf{V}_{\beta_t}^{-1}(\boldsymbol \beta_t-\boldsymbol \mu_{\beta_t}) \Bigg) \text{exp}\Bigg(\frac{1}{2\sigma_t^2}\tilde{\boldsymbol \mu}_{\boldsymbol \gamma_{t-1}}^T\tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}}\tilde{\boldsymbol \mu}_{\boldsymbol \gamma_{t-1}} \Bigg), \nonumber \\
& \propto N(\boldsymbol \beta_t|\tilde{\mathbf{V}}_{\beta_t}\tilde{\boldsymbol \mu}_{\beta_t}, \sigma_t^2\tilde{\mathbf{V}}_{\beta_t}), \nonumber \\
\tilde{\boldsymbol \mu}_{\beta_t} & = \mathbf{V}_{\beta_t}^{-1}\boldsymbol \mu_{\beta_t} +\mathbf{h}_t(\mathbf S_t)\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t - (\mathbf{g}(\mathbf S_t)y_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf{h}^T_t(\mathbf S_t))^T \tilde{\mathbf{V}}_{\gamma_{t-1}}(\mathbf{V}_{\gamma_{t-1}}^{-1}\boldsymbol \mu_{\gamma_{t-1}}\nonumber \\
& \quad \ +\mathbf{g}(\mathbf S_t)\hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t), \nonumber \\
\tilde{\mathbf{V}}_{\beta_t} & = \Bigg( \mathbf{V}_{\beta_t}^{-1} + \mathbf{h}(\mathbf S_t)\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf{h}(\mathbf S_t)^T - (\mathbf{g}(\mathbf S_t)\hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf{h}^T_t(\mathbf S_t))^T \tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}} (\mathbf{g}(\mathbf S_t)y_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf{h}^T_t(\mathbf S_t)) \Bigg)^{-1}. \label{conjugate_beta5} \end{align}
The marginalized posterior density function of $\sigma_t$ is \begin{align}
&p(\sigma_t^2|\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t)) \propto \sigma_t^{-a_t-0.5n_t}\text{exp}\Bigg( -\frac{1}{2\sigma_t^2}\Bigg[2b_t + \mathbf Z_t^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t+ \boldsymbol \mu_{\beta_t}^T\mathbf{V}_{\beta_t}^{-1}\boldsymbol \mu_{\beta_t} +\boldsymbol \mu_{\gamma_{t-1}}^T\mathbf{V}_{\gamma_{t-1}}^{-1}\boldsymbol \mu_{\gamma_{t-1}} - \tilde{\boldsymbol \mu}_{\beta_t}^T\tilde{\mathbf{V}}_{\beta_t}\tilde{\boldsymbol \mu}_{\beta_t} \nonumber \\
& \qquad - (\mathbf{V}_{\boldsymbol \gamma_{t-1}}^{-1}\boldsymbol \mu_{\boldsymbol \gamma_{t-1}} +\mathbf{g}(\mathbf S_t) \hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t)^T \tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}} (\mathbf{V}_{\boldsymbol \gamma_{t-1}}^{-1}\boldsymbol \mu_{\boldsymbol \gamma_{t-1}} +\mathbf{g}(\mathbf S_t) \hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t)\Bigg] \Bigg), \nonumber \\
& \sigma_t^2|\mathbf Z_t,\hat{y}_{t-1}(\mathbf S_t) \sim IG(\sigma_t^2|a_t^*, b_t^*), \nonumber \\
& a_t^* = a_t + n_t/2, \nonumber \\
& b_t^* = b_t + 0.5\Bigg(\mathbf Z_t^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t+\boldsymbol \mu_{\beta_t}^T\mathbf{V}_{\beta_t}^{-1}\boldsymbol \mu_{\beta_t} + \boldsymbol \mu_{\gamma_{t-1}}^T\mathbf{V}_{\gamma_{t-1}}^{-1}\boldsymbol \mu_{\gamma_{t-1}} - \tilde{\boldsymbol \mu}_{\beta_t}^T\tilde{\mathbf{V}}_{\beta_t}\tilde{\boldsymbol \mu}_{\beta_t} \nonumber \\
& \qquad - (\mathbf{V}_{\boldsymbol \gamma_{t-1}}^{-1}\boldsymbol \mu_{\boldsymbol \gamma_{t-1}} +\mathbf{g}(\mathbf S_t) \hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t)^T \tilde{\mathbf{V}}_{\boldsymbol \gamma_{t-1}} (\mathbf{V}_{\boldsymbol \gamma_{t-1}}^{-1}\boldsymbol \mu_{\boldsymbol \gamma_{t-1}} +\mathbf{g}(\mathbf S_t) \hat{y}_{t-1}(\mathbf S_t)^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_t)\Bigg). \label{conjugate_sigma_3} \end{align}
the conditional posterior density function of $\boldsymbol \beta_1$ is \begin{align}
\boldsymbol \beta_{1}|\sigma_1^2,\mathbf Z_1 & \sim N(\boldsymbol \beta_1|\tilde{\mathbf{V}}_{\beta_1}\tilde{\boldsymbol \mu}_{\beta_1}, \sigma_1^2\tilde{\mathbf{V}}_{\beta_1}), \nonumber \\
\tilde{\boldsymbol \mu}_{\beta_1} & = \mathbf{V}_{\beta_1}^{-1}\boldsymbol \mu_{\beta_1} +\mathbf{h}_1(\mathbf S_1)\tilde{\boldsymbol \Sigma}_1^{-1}\mathbf Z_1, \nonumber \\
\tilde{\mathbf{V}}_{\beta_1} & = \Bigg( \mathbf{V}_{\beta_1}^{-1} + \mathbf{h}(\mathbf S_1)\tilde{\boldsymbol \Sigma}_1^{-1}\mathbf{h}(\mathbf S_1)^T \Bigg)^{-1}, \label{conjugate_beta1} \end{align} and the marginal posterior density function of $\sigma_1^2$ is \begin{align}
\sigma_1^2|\mathbf Z_1 & \sim IG(\sigma_1^2|a_1^*, b_1^*), \nonumber \\
a_1^* & = a_1 + n_1/2, \nonumber \\
b_1^* &= b_1 + 0.5\left(\mathbf Z_1^T\tilde{\boldsymbol \Sigma}_t^{-1}\mathbf Z_1 +\boldsymbol \mu_{\beta_1}^T\mathbf{V}_{\beta_1}^{-1}\boldsymbol \mu_{\beta_1} - \tilde{\boldsymbol \mu}_{\beta_1}^T\tilde{\mathbf{V}}_{\beta_1}\tilde{\boldsymbol \mu}_{\beta_1}\right). \label{conjugate_sigma_4} \end{align}
\end{document} | arXiv |
Let $x$ be a real number. Consider the following five statements:
$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$
What is the maximum number of these statements that can be true for any value of $x$?
One the first two statements, at most one of them can be true ($x^2$ cannot be both less than 1 and greater than 1). Of the next two statements, at most one of them can be true ($x$ cannot be both less than 0 and greater than 0). Hence, at most three statements can be true.
For $0 < x < 1,$ the first, fourth, and fifth statements are true, so the maximum number of statements that can be true is $\boxed{3}.$ | Math Dataset |
\begin{document}
\title{$\ \ $The structure of HCMU metric in a K-Surface} \author{ Qing Chen \ \ \, Xiuxiong Chen \ \ \ and \ \ Yingyi Wu }
\date{}
\maketitle
\begin{abstract}We study the basic structure of a HCMU metric in a K-Surface with prescribed singularities. When the underlying smooth surface is $S^2$, we prove the necessary condition given in [1] for the existence of HCMU metric is also sufficient.
\end{abstract} \maketitle
\section{Introduction}
Let M be any compact, oriented smooth Riemannian surface without boundary, and $M_{\{\alpha_1, \alpha_2, \cdots, \alpha_n\}}$ ( where $\alpha_i > 0, \forall i,\ 1 \le i \le n$ ) denotes a K-Surface associated with M. A Riemannian metric $g$ is said to be well defined or smooth in $M_{\{\alpha_1, \alpha_2, \cdots, \alpha_n\}}$ if it satisfies the following two conditions: \begin{description}\item[](1)\ \ $g$ is smooth everywhere on M except in a set of singular points $\{ p_1, p_2, \cdots, p_n \}$, \item[](2)\ For any $i (1 \le i \le n)$, the metric $g$ has a singular angle of $2\pi \alpha_i$ at the point $p_i$.\end{description} Here the condition (2) means that in a small neighborhood of $p_i$, there exists a local complex coordinate chart $ ( U,z)$ $ ( z(p_i)=0)),$ s.t.
$$ g |_U=h(z,\bar{z}){1 \over {{|z|}^{2-2{\alpha}_i}}}{|dz|}^2 ,$$
where $h:~U\to R$ is a continuous positive function and smooth on $U\setminus \{0\}$.
Two smooth Riemannian metrics on $M_{\{\alpha_1,\alpha_2,\cdots,\alpha_n\}}$ are pointwise conformal to each other if they are related by a multiple of a smooth positive function on $M$.
A natural question is whether or not there exists a ``best" metric in every conformal class of a K-Surface. This is an attempt to generalize the classical uniformization theorem to a K-Surface. Recalled that the classical uniformization theorem asserts that in every conformal class of $M$, there must exist a metric with constant scalar curvature. Many papers tried to generalize the uniformization theorem in K-Surfaces. For example, [5] and [3] independently found the sufficient condition under which in a K-Surface, there exists a constant scalar curvature metric. [6] found a necessary condition of the existence of a constant scalar curvature metric in a K-Surface, [9] proved that a uniqueness theorem on constant curvature metric in some K-surfaces. However, there does not exist a constant curvature metric in a K-Surface.
In a serial of papers [1] and [2], the second named author tried to find the ``best" metric
in a conformal class of a K-Surface,
through studying the critical point of the Calabi energy functional. He proposed that two types of metrics can be regarded as candidates of the ``best" metric: one is the extremal metric, another is the HCMU metric ( Definitions will be given later.).
Let $M_{\{\alpha_1,\alpha_2,\cdots,\alpha_n\}}$ be a K-Surface and $g_0$ be a smooth metric in it. Consider the conformal class of $g_0$ : $$\mathcal{S}(g_0)=\{ g= e^{2 \varphi}g_0,\varphi \in H^{2, 2}(M) \mid \int_{M\setminus \{p_1, p_2, \cdots, p_n \}} e^{2 \varphi}dg_0=\int_{M\setminus \{p_1, p_2, \cdots, p_n \}} dg_0\}.$$Define the Calabi energy functional:\begin{equation} E(g)=\int_{M \setminus \{p_1,p_2,\cdots,p_n\}} K^2 dg,\end{equation} here $K$ is the scalar curvature of the metric $g$. The Euler-Lagrange equation of $E(g)$ is (cf. [1] [8] ) \begin{equation} \triangle_g K+K^2=C,\end{equation}
or equivalently, in a local complex coordinate chart,
\begin{equation} {\partial \over {\partial \bar{z}}} K_{,zz}=0,\end{equation} where $K_{,zz}$ is the 2nd-order $(0,2)$ type covariant derivatives of $K$.
A metric which satisfies (2) or (3) is called an extremal metric. (3) has two special cases, one is \begin{equation} K\equiv Const, \end{equation} and the other is \begin{equation} K_{,zz}=0, \ K \neq Const. \end{equation} A metric which satisfies (5) is called a HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. Throughout this paper, we assume that a HCMU metric has finite area and finite Calabi energy.
Let us first quote an Obstruction Theorem from [1].\par
{\bf Theorem 1.} {\it Let $g$ be a HCMU metric in a K-Surface
$M_{\{\alpha_1, \alpha_2, \cdots, \alpha_n\}}$. Then the Euler
character of the underlying surface should be determined by
\begin{equation}\chi(M)=\sum_{i=1}^j(1-\alpha_i)+(n-j)+s\end{equation}
where $s$ is the number of critical points of the Curvature
$K$ ( excluding the singular points of $g$). Here we assume
that $\alpha_1, \alpha_2, \cdots, \alpha_k, ( 0\le k \le n)$ are
the only integers in the set of prescribed angles
$\{\alpha_1, \alpha_2, \cdots, \alpha_n\}$; and assume that
$\{p_{j+1}, p_{j+2}, \cdots, p_k\}$ are the only local extremal
points of $K$ in the set of singular points $\{p_j, 0\le j
\le k \}$.\par}
The formula (6) is an application of Poincar$\grave{\mbox{e} }$ -Hopf index theorem. When $g$ is a HCMU metric, the gradient vector field $\overrightarrow{V}$ of the scalar curvature $K$ is holomorphic. Hence, its real part is a Killing vector field. It was proved in [1] that the singularities of the Killing vector field is a finite set which is the union of the singularities of metric $g$ and the smooth critical points of function $K$. Consequently, any saddle point of $K$ must be the singularities of metric $g$. At these points the index of the vector field is $(1-\alpha_i)$. Other singularities of this gradient vector field must be local extremal points of $K$ with index $1$. Therefore, the Poincar$\grave{\mbox{e}}$-Hopf index theorem implies formula (6).
In this paper, we study the following question: whether or not the condition (6) is also sufficient to the existence of HCMU metrics in a K-Surface. Our main result in this paper is:
\noindent{\bf Theorem A.} For $S^2$, given $n$ points
$p_1,p_2,\cdots,p_n$ on $S^2$ and $n$ positive numbers $\alpha_1,
\alpha_2, \cdots, \alpha_n
$ with $ \alpha_1, \alpha_2, \cdots, \alpha_k$ being the only integers and $ \alpha_j \ge 2 (1 \le j \le k)$, suppose that $\alpha_1, \alpha_2, \cdots,
\alpha_n$
satisfy the following condition: $ \exists j_0 (1 \le j_0 \le k)$ and $\alpha_{\sigma{(1)}}, \alpha_{\sigma{(2)}},
\cdots, \alpha_{\sigma{(j_0)}}, \sigma(i)\in\{1, 2, \cdots
k\}(1 \le i \le j_0)$, s.t.\begin{equation}\sum_{i=1}^{j_0}\alpha_{\sigma(i)}+\chi(M)-n \ge 0.\end{equation}Then there exists a
HCMU metric whose scalar curvature $K$ is not a constant, s.t.
the angles of the metric at $p_1,p_2,\cdots,p_n$ are exactly
$\alpha_1, \alpha_2, \cdots, \alpha_n$ and $p_{\sigma{(1)}}, p_{\sigma{(2)}}, \cdots, p_{\sigma{(j_0)}}
$ are the only saddle points of the scalar curvature $K$.
In fact, we prove for $S^2$ the condition (7) is the necessary and sufficient
condition for the existence of a HCMU metric in it.
The simplest HCMU metric in $S^2$ is a football. It only has
two extremal points and it is a rotationally symmetric
metric. Fixing the area, a HCMU metric is uniquely determined by
the ratio of the two angles.
The proof of Theorem A is based on the following Theorem B, which says that any HCMU metric can be divided into a
finite number of
footballs. Under the condition (7), we can glue some suitable footballs together to
obtain a HCMU metric in $S^2$ as desired.
\begin{flushleft}{\bf Theorem B.} Let $g$ be a HCMU metric on a K-Surface
$M$, then there are a finite number of geodesics which connects extremal points and saddle points of the scalar
curvature $K$ together. In fact, $M$ can be divided into a finite number of pieces by cutting along these
geodesics where each piece is locally isometric to a HCMU
metric in some football.\end{flushleft}
We should point out that in [7], Lin and Zhu use ODE method and geometry of the scalar curvature of HCMU
metrics to construct a class of HCMU metrics with
finite conical singular angles $2 \pi\cdot integers$ on
$S^2$. This kind of HCMU metric is called
exceptional HCMU metric where all of its singularities are the saddle points of the scalar curvature $K$. A minimal exceptional
HCMU metric is an exceptional HCMU metric with only one minimum point of the scalar curvature $K$.
They give an explicit formula for minimal exceptional HCMU
metrics. Their theorem shows that a minimal exceptional HCMU metric is determined
by three parameters. In comparison, our existence theorem of HCMU metric is more general. Indeed
our construction in the proof of Theorem A is actually a minimal
exceptional HCMU metrics if all of the singularities are the
saddle points of the scalar curvature $K$.
The authors would like to thank the referee for many useful
suggestions. \section{Proof of Theorem B} \subsection{Preliminaries}
Let $M$ be a compact, oriented smooth Riemannian surface without boundary. $M_{\{\alpha_1,\alpha_2,\cdots,\alpha_n\}}$ denotes its K-Surface. $\{p_1,p_2,\cdots,p_n\}$ is the set of singular points. $\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ is the corresponding set of singular angles. $\forall p \in M \setminus \{p_1,p_2,\cdots,p_n\}$, assuming that $(U,z)$ is a complex coordinate chart around $p$, $g$ can be written as: $$
g=e^{2\varphi(z,\bar{z})}{|dz|}^2.$$ and $$K=-{\triangle \varphi \over {e^{2\varphi}}}.$$ Equation (5) can be written as:\begin{equation} K_{, zz}={\partial^2 K \over {\partial z} ^2}-2
{\partial K \over \partial z}{\partial \varphi \over \partial
z}=0,\end{equation} which means that the gradient vector field of the scalar curvature $K$ is holomorphic. The gradient vector field ${\nabla K}$ is: $$ {\nabla K}=\sqrt{-1}{K}^{\prime
\bar{z}}{\partial \over \partial z}=\sqrt{-1} e^{-2 \varphi}{\partial K \over \partial \bar{z}}{\partial
\over \partial
z},$$
and its real part is: $$\overrightarrow {V}={1 \over 2}(\sqrt{-1}{K}^{\prime z} {\partial \over \partial z}-\sqrt{-1}{K}^{\prime \bar{z}}{\partial \over \partial \bar {z}}).$$ Then $\overrightarrow {V}$ is a Killing vector field and its integral curve is the level set of the function $K$.
In fact, by studying the properties of the Killing vector field $\overrightarrow{V}$, the Obstruction Theorem is proved in [1]. We list here the main properties of $\overrightarrow {V}$.
\par\vskip0.3cm \noindent {\bf Proposition 1[1].} Let $Sing \overrightarrow{V} $ denote the set of all singular points of $\overrightarrow{V}$ and $\Omega_p$ denote the set of the integral curves of $\overrightarrow{V}$ which meet $p$ for $p \in Sing \overrightarrow {V}$. Then:\\
(1) $Sing \overrightarrow{V} = \{ \ smooth \ critical \ points \ of \ K \} \bigcup \{p_1,p_2,\cdots,p_n\}$ and is a finite set.\\
(2) $\Omega_p$ is empty or a finite set. Moreover, if $\Omega_p\neq \emptyset$, $\Omega_p$ has even number of points and
$g$ has angle $|\Omega_p|\pi$ at $p$. \\
(3) $K$ can be continuously extended to $M$.\\
(4) $Sing \overrightarrow{V}$ can be divided
into two parts, $ Sing \overrightarrow{V}=S_1 \bigcup S_2,\ $
such that\\
$~~~$ (a) $S_1=\{p \in Sing \overrightarrow{V}
|\Omega_p=\emptyset \}$,\ and if $p \in S_1$, then $p$ is an extremal point of
$K$;\\
$~~~$ (b) $S_2 =\{p \in Sing \overrightarrow{V}|\Omega_p \neq \emptyset\}$, if
$p \in S_2$, $p$ is a saddle point of $K$. and at $p$ the angle is
$\pi\cdot|\Omega_p|.$
\par\vskip0.3cm
\noindent{\bf Proposition 2[1].}\\
(1) Any integral curve of ${\overrightarrow
{V}}$ in the neighborhood of any local extremal of $K$ point is a topologically circle
which contains the point in its interior. \\
(2) If a closed
integral curve of $\overrightarrow{V}$ bounds a
topological disk in $M$, which contains only one extremal
point of $K$, then every integral curve of
$\overrightarrow{V}$ in this disk is also a topological
circle.
\par\vskip0.3cm
\noindent{\bf Proposition 3[1].} At a saddle point of $K$, the
included angle of two adjacent integral curves of $\nabla K$ is $\pi$.
\subsection{Proof of the Theorem B}
Let us begin with the study of a football, i.e. a HCMU metric $g$ in $S^2$, which is
rationally symmetric and has two extremal points.
According to Proposition 1, the scalar curvature $K$ is continuous. If
$K$ has only two extremal points $p$ and $q$, by Proposition 2.1, in a neighborhood of $p$, an integral curve $C$ of $\overrightarrow
{V}$ bounds a topological disk $D$ centered at $p$. By of Proposition 2.2, the
integral curves of $\overrightarrow
{V}$ in $D$ are all
topologically concentric circles containing $p$ in their interiors. Since
$\overrightarrow {V}$ is a Killing
vector field, $g$ is invariant along integral curves of $\overrightarrow {V}$, then
$g$ is rotationally symmetric in $D$.
On the other hand, $C$ is also a topological circle bounding
the disk $S^2 \setminus D$ which has only one extremal point $q$, by
Proposition 2.2 again, $g$ is also rotationally symmetric in $S^2 \setminus D$. Therefore, $g$ is globally rotationally symmetric.
It can be written as:
\begin{equation}g={du}^2+f^2(u) {d\theta}^2
\ \ (0 \le u \le l,\ 0 \le \theta \le 2 \pi),\end{equation}
with $p$ and $q$ corresponding to $u=0$ and $l$
respectively, and $\mbox{dist}_g(p,q)=l,$
see Figure 1.
\begin{center}
\includegraphics{111.eps}\\
Figure 1
\end{center}
If we assume the angle of the
metric at $p$ is $\alpha$, the angle at $q$ is $\beta$, $(\alpha
\ge \beta )$, then $f$ satisfies:
\parbox{9.75cm}{\[\left \{ \begin{array}{l}
f(0) = f(l)=0, \\ f^\prime(0)=\alpha, f^\prime(l)=-\beta, \\
f(u)>0 , u \in (0, l). \end{array}\right.\]}
\parbox{1cm}{\begin{eqnarray}\end{eqnarray}}
\noindent By (9), the scalar curvature $K$ is given by:\begin{equation}K=-{f^{\prime \prime} \over
f}.\end{equation}
\par \vskip0.3cm
\noindent {\bf Proposition 4.} There is a constant $c$ such that ${K}^\prime=cf
.$\par\vskip0.1cm
\noindent{\it Proof:} Since $g$ is a HCMU
metric, from (8) and (9) we have ${K}^{\prime \prime} f={K}^\prime f^\prime,$
that is
${K}^{\prime}=cf.$ $\Box$
\par\vskip0.3cm
\noindent {\bf Proposition 5.} ${K}^\prime \le 0, $ moreover, if
$K \neq Constant, $ then only when $u=0 \ \mbox{and}
\ u=l,\ {K}^\prime =0.$\par\vskip0.1cm
\noindent{\it Proof:} Define a function: $F={f^\prime}^2+K f^2.$ Then
$F(0)=\alpha^2,\ F(l)=\beta^2,\ F(l) \le F(0), $ so there is a $\xi \in (0, l)$ such that
$${F(l)-F(0) \over
l}=F^\prime (\xi)\le 0.$$
On the other
hand,
$$F^\prime=2f^\prime f^{\prime \prime}+{K}^\prime
f^2+2K f f^\prime,$$
by $K=-{f^{\prime \prime}\over
f},$ we have ${K}^\prime (\xi) \le 0.$ From {Proposition
4}, ${K}^\prime =cf$, and $f$ is positive on $(0,l)$, if $K$ is not a constant,
$K^\prime$ does not change its sign from $0$ to $l$. Hence, $K^\prime
(\xi) \le 0$ implies ${K}^\prime(u) < 0,\ (\forall u \in (0, l))$. Moreover,
$K^\prime=0$ only when $u=0$ and $l$. $\Box$
\par\vskip0.3cm
\noindent {\bf Remark 1:} From the proof of {Proposition 5}, we also get $K
\equiv Constant$ if and only if $\alpha=\beta$. \par\vskip0.3cm
In the following, we always assume $K \ne
Constant$. Therefore $K$ decreases monotonely from $p$ to
$q$. Substituting $f=\frac{{K}^\prime}c $
into $K=-{f^{\prime \prime} \over f},$ we
get $K'''+K'K=0$, that is
\begin{equation} {{{K}^\prime}^2 \over
2}=C_0K-{{K}^3 \over 6} +C_1,\end{equation}
here $C_0$ and $C_1$ are two constants. Assuming $K(0)=K_0, K(l)=K_1$ and
letting $u=0$ and $l$ in
(12), we know both $K_0$
and $K_1$ are roots of the equation $-{{K}^3
\over 6}+C_0K+C_1=0$. Then
$$-{{K}^3\over
6}+C_0K+C_1=-{1 \over 6}(K-K_0)(K-K_1)(K+K_0+K_1)$$
and \begin{equation} {{K}^\prime}^2=-{1 \over 3}
(K-K_0)(K-K_1)(K+K_0+K_1).\end{equation}
We take derivatives of (13) to
get:\begin{equation}{K}^{\prime \prime}=-{1 \over
6}[(K-K_0)(K+K_0+K_1)+(K-K_1)(K+K_0+K_1)+(K-K_0)(K-K_1)].\end{equation}
Using {Proposition 4}, we have:\begin{equation}
c f^\prime=-{1 \over
6}[(K-K_0)(K+K_0+K_1)+(K-K_1)(K+K_0+K_1)+(K-K_0)(K-K_1)].\end{equation}
Let $u=0 \ \mbox{and}\
l$ in (15), then by (10):\[ \left \{ \begin{array}{ll}
\alpha=\dfrac{(K_1-K_0)(2K_0+K_1)}{6c},
\\\beta=\dfrac{(K_1-K_0)(2K_1+K_0)}{6c}.\end{array}\right.
\]
Integrate the equation ${K}^\prime=cf$ from $0$ to $l$, we get:\begin{equation}
K_1-K_0=c\int_0^lf(u)du=c{A(g) \over 2\pi}, \end{equation}
here $A(g)$ denotes the area of the metric $g$. Then we have:
\begin{equation} \left\{
\begin{array}{l}\alpha=\dfrac{A(g)}{12 \pi} (2K_0+K_1), \\
\beta=\dfrac{A(g)}{12 \pi}(2K_1+K_0).
\end{array} \right.
\end{equation}
or:
\begin{equation} \left \{ \begin{array}{l} K_0=\dfrac{4\pi}{A(g)}(2\alpha-\beta), \\ K_1=\dfrac{4\pi}{A(g)}
(2\beta-\alpha).\end{array} \right.
\end{equation}
From (18) we see
if $\alpha,\ \beta,\ A(g)$ are fixed, then $K_0$ and $K_1$ are uniquely
determined, and by (13) $K$ is determined, again according to ${K}^\prime=cf,$ $f$ is
determined, i.e. the metric $g$ is determined. Therefore we get:
\begin{flushleft}{\bf Theorem C.} If area and angles at both extremal points are given, there exists a unique
rotationally symmetric HCMU metric in $S^2$, which is a football.\end{flushleft}
Meanwhile we get:
\begin{flushleft}{\bf Corollary 1.} In a football, assume that $\alpha$ is the angle of the HCMU
metric at the local maximum point of $K$, $\beta$ is the angle of the metric at the local minimum point of
$K$, $K_0=\max K$, $K_1=\min K$, then \\
(1) $K_0>0$, the sign of $K_1$ is the same as
$2\beta-\alpha.$\\
(2) $K_0>K_1>-(K_0+K_1).$\end{flushleft}
Next we consider a HCMU metric $g$ in a K-Surface $M_{\{\alpha_1,\alpha_2,\cdots,\alpha_n\}}$. Since the integral curves of $\overrightarrow {V}$ are the
level sets of $K,\ \nabla K \bot \overrightarrow
{V}$, integral curves of $\nabla K$ are geodesics. If $p$ is
a local minimum point of $K$, in a small neighborhood of $p$, the integral curves
of $\overrightarrow {V}$ are topologically concentric circles and the integral curves of $\nabla K$ are perpendicular to
them. Choose an integral curve $c(t)\ ~(t\in [0,T])$ of $\overrightarrow {V}$. For each $t$, there
exists a unique integral curve $C_t$ of $\nabla K$ starting from $p$ and
passing through the point $c(t)$. See Figure 2.
\begin{center}
\includegraphics{112.eps}\\
Figure 2
\end{center}
Obviously, $C_t$ must reach some saddle point of $K$ or some local
maximum point of $K$. We have the
following:\par\vskip0.3cm
\noindent{\bf Lemma 1.} For $t_1,t_2 \in (0,t_0)$, If both $C_{t_1}$ and $C_{t_2}$
reach local
maximum points $q_1$ and $q_2$ directly without passing through any
saddle point of $K$, then
$\mbox{dist}_g(p,q_1)=\mbox{dist}_g(p,q_2)=l$.\par\vskip0.2cm
\noindent{\it Proof:} Notice that in a small neighborhood of $p$ (we say
$D$), $g$ is rotationally symmetric. Therefore, in $D$, $g$ can be
written as $g={du}^2+f^2(u){dt}^2,$ where $t$ is the parameter of $c(t)$. Hence
$K=-{f^{\prime \prime} \over f},\ {K}^\prime=\frac{dK}{du}=cf.$
Then by (13)
\begin{equation} {K}^\prime=-\sqrt{-{1 \over 3} (K-K_0)({K}^2+a_0
K+a_1)}, \end{equation}\\ here $K_0=K(p),\ a_0 \ \mbox{and} \ a_1 \ \mbox{are
constants}$. On the other hand, if we restrict
$K$ at $C_{t_1}$, $K$ is a smooth function of the arc
length parameter $s\ (s=u)$ of $C_{t_1}$. Moreover, at $C_{t_1} \bigcap D$, we have
(19). According to ODE theory, we know that (19) holds true at the whole of $C_{t_1}$.
Since ${K}^\prime
(q_1)=0$ (It is because in a neighborhood of $q_1$, $g$ is also rationally
symmetric, we have $K'=\tilde{c}\tilde{f}$ and
$\tilde{f}(0)=0$),
we get:\begin{equation} {K}^\prime=-\sqrt{-{1 \over
3}(K-K_0)(K-K_1)(K+K_0+K_1)}, \end{equation}here $K_1=K(q_1)$. At $C_{t_2}\bigcap
D,$ we also have
$$\ {K}^\prime =-\sqrt{-{1 \over
3}(K-K_0)(K-K_1)(K+K_0+K_1)}.$$ Because along
$C_{t_2}$ except at $p$ and $q_2$, there is no point at which $\nabla K=0$, we get
$K(q_2)=K_1$. Furthermore, since
$${dK \over ds}=-\sqrt{-{1 \over
3}(K-K_0)(K-K_1)(K+K_0+K_1)},$$ \begin{equation}{ds \over dK}=-\dfrac{1}{ \sqrt{-{1 \over
3}(K-K_0)(K-K_1)(K+K_0+K_1)}}. \end{equation} Hence, the length of
a geodesic from $p$ to $q_1$ is:\begin{equation}
l=\int_{K_1}^{K_2}-{dK \over \sqrt{-{1 \over 3}
(K-K_1)(K-K_2)(K+K_1+K_2)}}. \end{equation}The same
as above, the length of a geodesic from $p$ to $q_2$ is
also $l$.$\Box$
\par\vskip0.3cm
\noindent{\bf Lemma 2.} Fix $t_0 \in [0,T]$ and suppose
$C_{t_0}$ reaches a maximum point $q$ of $K$ directly,
then $\exists \varepsilon>0$, s.t. $\forall t \in
(t_0-\varepsilon,t_0+\varepsilon)$, $C_t$ reaches the same
maximum point $q$ without passing through any saddle point of
$K$.\par\vskip0.2cm
\noindent{\it Proof:} Since the saddle points of $K$ are finite,
there exists a small neighborhood of $t_0 \ (t_0-\varepsilon
,t_0+\varepsilon)$ s.t. each $C_t (t \in (t_0-\varepsilon
,t_0+\varepsilon))$ does not reach any saddle point. The end points of $C_t$
are continuously dependent on $t$
and the maximum points of $K$ are finite. Therefore, $\exists \varepsilon>0$ s.t.
$\forall t \in (t_0-\varepsilon,
t_0+\varepsilon)$, $C_t$ reaches the same maximum point.
$\ \Box$ \par\vskip0.3cm
\noindent{\it Proof of Theorem B:} By virtue of Lemma 2,
$\bigcup\limits_{t \in (t_0-\varepsilon,t_0+\varepsilon)}C_t$ is a simply
connected domain in $M$. Suppose $F$ is the largest simply connected domain in $M$ which contains $\bigcup\limits_{t \in (t_0-\varepsilon,t_0+\varepsilon)}C_t$
and satisfies the following three properties: \begin{enumerate}\item[1] Any integral curve of $\nabla K$ in $F$
is from $p$ to $q$. \item[2] Any
integral curve of $\nabla K$ in $F$ does not pass through any
saddle point of $K$. \item[3] $\partial F$ are also integral curves of $\nabla K$, but they pass through some
saddle points of $K$.\end{enumerate}
$\partial F$ can be divided into two curves connecting
$p$ and $q$, we say $\gamma_1$ and $\gamma_2$. Along
$\gamma_1$, there are some saddle points which are connected by
geodesic segments (integral curves of $\nabla K$). At
each saddle point, by Proposition 3, the included angle
of two adjacent geodesics is $\pi$. Hence, $\gamma_1$ is
smooth at each saddle point. Therefore, $\gamma_1$ is a
smooth geodesic, either is $\gamma_2$.\par
On the other hand, in $F$, $g$ is invariant along integral curves of $\overrightarrow
{V}$, i.e. $g$ is rotationally symmetric in $F$.
Therefore, we can parameterize $g$ as: $
g={du}^2+f^2(u){d\theta}^2 $ as before. Hence, $(F,g)$ is
isometric to a football.\par
At a minimum point $p$ of $K$, since there are finite integral curves of $\nabla K$
starting from $p$ and reaching saddle points, we can
repeat the above proof to obtain finite pieces of the
largest simply connected domains in $M$, which are all
isometric to footballs and contain $p$ as a vertex. We
can also repeat this operation at each minimum point of
$K$ to obtain finite footballs. We claim that every point
of $M$ is contained in the union of these footballs. For
any point of $M$, there must be an integral curve of
$\nabla K$ starting from it or ending at it or passing
through it. If there are some saddle points at this integral curve, the point must
be on the boundary of some largest domain. If not, this
point must be in some largest domain. Therefore, every
point of $M$ is contained in the union of these
footballs. This completes the proof of Theorem B.
$\Box$\par
We also have following corollaries.
\begin{flushleft}{\bf Corollary 2. }On HCMU surface, the values of local minimum of scalar
curvature $K$ are the same to each other, and the same narration is true
for local maximum points of $K$.\end{flushleft}
{\bf Corollary 3.} Let $g$ be a HCMU metric in a K-surface, If scalar curvature $K$
has only extremal points in the Surface, then the
metric is a football.
\section{Proof of Theorem A}
We prove following lemma at first.\par\vskip0.2cm
\noindent {\bf Lemma 3.} Two footballs $S^2_{\{\alpha,\beta\}}$ and $S^2_{\{\alpha_1,\beta_1\}}$ can be smoothly glued along their meridians or some
segments of the two meridians, iff ${\alpha \over \beta}={\alpha_1
\over \beta_1}$ and $\frac{A(g)}{A(g_1)}=\frac{\alpha}{\alpha_1}$, here
$A(g)$ and $A(g_1)$ denote the areas of the two
metrics.\\ \vskip0.05cm
{\it Proof:} Suppose $K_0~(\tilde{K}_0)$ and $K_1~(\tilde{K}_1)$ are the maximum and minimum of
$K~(\tilde{K})$ of football $S^2_{\{\alpha,\beta\}}~(
S^2_{\{\alpha_1,\beta_1\}})$ respectively.\par
$(\Longrightarrow)$ If two footballs can be smoothly glued along their meridians or some
segments of the two meridians, we see at each segment being smoothly glued, the arc length is
the
same, $K$ is the same, so the derivative of $K$, ${K}^\prime$ is the same. Therefore
we get from the equation (13) and {Corollary 1}, $K_0=\tilde{K_0},\ K_1=\tilde{K_1}.$
Then from
the equation (17), we get ${\alpha \over \beta}={\alpha_1 \over
\beta_1}$ and $\frac{A(g)}{A(g_1)}=\frac{\alpha}{\alpha_1}.$\par
\vskip0.1cm
$(\Longleftarrow)$ Assume the metric of two football are given by
\begin{eqnarray*}
g &=& {du}^2+{\alpha_1}^2 f^2
{d\theta}^2 \ (0 \le u \le l, 0 \le \theta \le \frac{2 \pi}{\alpha_1}),\\
\tilde{g} &=& {du_1}^2+\alpha^2 {f_1}^2{d\theta_1}^2 \ (0 \le u_1 \le l_1,0 \le \theta_1 \le \frac{2 \pi}{\alpha}).
\end{eqnarray*}
If $\frac{\alpha}{\beta}=\frac{\alpha_1}{\beta_1}$ and
$\frac{A(g)}{A(g_1)}=\frac{\alpha}{\alpha_1}$, by the equation
(18), we get $K_0=\tilde{K_0},K_1=\tilde{K_1}$. Then by the
equation (21) and (20), $u_1=u,K'=\tilde{K}'$ and by the
equation (22), $l=l_1$. Meanwhile, we have
$\frac{f}{f_1}=\frac{\alpha}{\alpha_1}.$ Hence, if we let
$u=u_1$ and $\theta=\theta_1$, then $\alpha_1 f=\alpha f_1$, i.e. $g$ and $\tilde{g}$ are locally
isometric. Since the meridians are geodesics, two footballs
$S^2_{\{\alpha,\beta\}}$ and $S^2_{\{\alpha_1,\beta_1\}}$ can
be glued together along their meridians or some segments of
meridians.\ $\Box$\par\vskip0.2cm
\noindent{\it Proof of Theorem A.} ~~\\
{\it Claim: there exists a HCMU metric on $S^2$ whose singular angles
are exactly $\alpha_1,\alpha_2,\cdots,\alpha_n$ and the
angles at the $j_0$ saddle points of the scalar curvature $K$ are exactly $\alpha_{\sigma(1)},\alpha_{\sigma(2)},\cdots,
\alpha_{\sigma(j_0)}$.}\\
We will use some suitable footballs to construct $S^2_{\{\alpha_1, \alpha_2, \cdots, \alpha_n
\}}$ which satisfies the condition of the claim. Without loss of generality, we assume $\alpha_{\sigma(i)}=\alpha_i,\
i=1,2,\cdots,j_0.$ \par\vskip0.05cm
\noindent{\it Step 1: construction of a HCMU metric with one saddle point of
angle $2 \pi \alpha_i \ (i=1,2,\cdots,j_0).$}\par
Choose $\alpha_i$ footballs, say
$S^2_{\{x_1,y_1\}},S^2_{\{x_2,y_2\}},\cdots,S^2_{\{x_{\alpha_i},y_{\alpha_i}\}},$
they satisfy $$\frac{x_m}{y_m}=\frac{x_n}{y_n}>1,~~~
\frac{A(g_m)}{A(g_n)}=\frac{x_m}{x_n},$$
for any $m,n \in
\{1,2,\cdots,\alpha_i\}$. Then by Lemma 3, two
footballs can be glued together. We are going to glue these
footballs together. For example, when $\alpha_i=3,$ take 3
footballs
$S^2_{\{x_1,y_1\}},~~~~S^2_{\{x_2,y_2\}}~$, $S^2_{\{x_3,y_3\}}$, see
Figure 3.
\begin{center}
\includegraphics{116.eps}\\
Figure 3
\end{center}
\noindent Along the meridian, cut $S^2_{\{x_1,y_1\}}$ from $A$ to $Q$.
$\widehat{AQ}$ becomes two identical arcs: $B$ and $B'$. The
same as above, we cut $S^2_{\{x_2,y_2\}}$ along the meridian
and get $B,B''$; we cut $S^2_{\{x_3,y_3\}}$ along the
meridian and get $B',B''$. Then we glue $B$ in
$S^2_{\{x_1,y_1\}}$ and $B$ in $S^2_{\{x_2,y_2\}}$ together,
$B'$ in $S^2_{\{x_1,y_1\}}$ and $B'$ in $S^2_{\{x_3,y_3\}}$
together, $B''$ in $S^2_{\{x_2,y_2\}}$ and $B''$ in
$S^2_{\{x_1,y_1\}}$ together, to obtain a HCMU metric with
one saddle point $A\ (=A'=A'')$ of angle $6 \pi$. Meanwhile,
$Q=Q'=Q''$, at which the angle is $2 \pi(y_1+y_2+y_3)$.\par
Obviously we can glue $\alpha_i$ footballs together in the same way, to obtain a HCMU metric on $S^2$, which has $\alpha_i$ local maximum points of angles $2\pi x_1,2\pi x_2,\cdots,2\pi x_{a_{i}}$, one saddle point of angle $2\pi \alpha_i$, and one minimum point of angle $2\pi(y_1+y_2+\cdots+y_{\alpha_i})$.
\vskip0.1cm
\noindent{\it Step 2: construction of a HCMU metric with $j_0$ saddle
points.} \par
We choose $\alpha_1$ footballs $S^2_{\{x_k,y_k\}}~(k=1,2,\cdots,\alpha_1)$
to construct the first saddle point of angle $2 \pi
\alpha_1$ like Step 1. Then we choose another meridian on $S^2_{\{x_{\alpha_1},y_{\alpha_1}\}}$ which is different
from the one passing the previous saddle point, cut this meridian like Step 1, choose $\alpha_2-1$ footballs
$S^2_{\{x_k,y_k\}}~(k=\alpha_1+1,\cdots,\alpha_1+\alpha_2-1)$,
cut them and glue them together with the football $S^2_{\{x_{\alpha_1},y_{\alpha_1}\}}$ like Step 1, we
get the
second saddle point of angle $2 \pi \alpha_2$. And then we
choose $\alpha_3-1$ footballs $S^2_{\{x_k,y_k\}}~(k=\alpha_1+\alpha_2,\cdots,
\alpha_1+\alpha_2+\alpha_3-2)$
with
$S^2_{\{x_{\alpha_1+\alpha_2-1},y_{\alpha_1+\alpha_2-1}\}}$
to construct the third saddle point of angle $ 2\pi
\alpha_3$, and so on, finally we have chosen $\displaystyle\sum_{i=1}^{j_0}\alpha_i-(j_0-1)$ footballs to construct a HCMU metric on $S^2$ with $j_0$ saddle points, as we desire. \par\vskip0.1cm
\noindent {\it Step 3: construction of
$S^2_{\{\alpha_1,\alpha_2,\cdots,\alpha_n\}}$.}\par\vskip0.1cm
Denote $N=\displaystyle\sum_{i=1}^{j_0} \alpha_i-(j_0-1)$.
By Step 2, we have constructed a HCMU metric of $j_0$ saddle
points with the angles $2 \pi \alpha_1,2 \pi \alpha_2, \cdots,
2 \pi \alpha_{j_0}$, $N$ maximum points with the angles
$2 \pi x_k \ (k=1,2,\cdots,N)$, and one minimum point with the angle $2
\pi \displaystyle\sum_{k=1}^N y_k$. These angles satisfy:
\begin{equation}\label{23}\frac{x_k}{y_k}=\frac{x_l}{y_l}>1,~ k,l \in
\{1,2,\cdots,N\}.\end{equation}
In the following we adjust $x_k,\ y_k$ to make the metric
coincide with what we desire.\par
Without loss of generality, we assume
$\displaystyle{\alpha_n=\min_{j_0+1 \le k \le
n}\{\alpha_k\}}$
and denote $s=\displaystyle\sum_{i=1}^{j_0}\alpha_i-(j_0-1)-(n-j_0-1)$.
By the condition of Theorem A,
$$s=\sum_{i=1}^{j_0}\alpha_i-n+\chi(S^2)\geq
0.$$
{\it Case 1.} If $s=0$, we let $$ \begin{array}{ll} x_k=\alpha_{j_0+k} \ \ (1 \le k \le N),\\ \alpha_n=\displaystyle\sum_{k=1}^N y_k.\end{array}$$
By (23) the angles $y_1,y_2\cdots,y_N$ must satisfy the following equations: \[ \left \{ \begin{array}{l} \displaystyle\sum_{k=1}^N y_k =\alpha_n,\\ \dfrac{\alpha_{j_0+1}}{y_1}=\dfrac{\alpha_{j_0+2}}{y_2}=\cdots=\dfrac{\alpha_{n-1} }{y_{n-j_0-1}}. \end{array} \right. \] The equations have a unique solution $$y_k=\alpha_{j_0+k}{\displaystyle \frac{\alpha_n}{\sum \limits_{i=j_0+1}^{n-1}\alpha_i}}\ \ ~~(1 \le k \le N=n-j_0-1). $$ Thus we have proved the claim in this case.\par\vskip0.1cm
{\it Case 2.}
If $s>0,$ we let
\begin{eqnarray*} x_k &=& \alpha_{j_0+k},\ \ (1 \le k \le n-j_0-1),\\
x_k &=& 1\ \ \ \ \ \ (n-j_0 \le k \le N),\\
\sum \limits_{k=1}^{N}y_k&=&\alpha_n. \end{eqnarray*} This means that there are $s~(=N-(j_0-1))$ smooth maximum points in the surface, correspondent to angles $x_k ~(k=n-j_0,\cdots, N)$ .
The undetermined angles $y_k~(k=1,\cdots,N)$ satisfy
\[ \left \{ \begin{array}{l} \sum \limits_{k=1}^{N}y_k=\alpha_n,\\ \dfrac{\alpha_{j_0+1}}{y_1}=\dfrac{\alpha_{j_0+2}}{y_2}=\cdots=\dfrac{\alpha_{n-1} }{y_{n-j_0-1}}=\dfrac{1}{y_{n-j_0}}=\cdots=\dfrac{1}{y_{N}}. \end{array} \right. \] The equations have a unique solution \begin{eqnarray*} y_k &=& \alpha_{j_0+k}{\displaystyle \frac{\alpha_n}{s+ \sum \limits_{i=j_0+1}^{n-1}\alpha_i}}\ \ (1 \le k \le n-j_0-1),\\y_k &=& {\displaystyle \frac{\alpha_n}{s +\sum \limits_{i=j_0+1}^{n-1}\alpha_i}}\ \ (n-j_0 \le k \le N).\end{eqnarray*} We also prove the claim in this case.
Till now we have constructed a HCMU metric on $S^2$ with the singular angles $\alpha_1, \alpha_2, \cdots, \alpha_n$ and the correspondent singular points $q_1,q_2,\cdots,q_n$, which may be different from the prescribed points $p_1,p_2,\cdots,p_n$. However, there exists a diffeomorphism $h: S^2 \longrightarrow S^2_{\{q_1,q_2,\cdots,q_n\}}$, s.t. $h(p_i)=q_i,\ i=1,2,\cdots,n$. Therefore, we can use $h$ to pull back the HCMU metric $g$ constructed by the claim on $S^2$ to obtain the HCMU metric $h^*g$ as Theorem A desires.$\Box$
\par\vskip0.2cm
{\bf Remark 2:} We must point out that the HCMU metric satisfying the condition
in Theorem A is not unique, the metric
we construct in the proof has exactly one minimum point of the
scalar curvature $K$. Indeed, by Theorem B we know that the
construction of a HCMU metric form footballs is a combination problem. If we can
find some suitable footballs and glue them together to satisfy the condition of
Theorem A, we have a HCMU metric as desired, and the resulting metric
may have more than one minimum point of the scalar curvature. For example, we can
choose two footballs, either $S^2_{\{\frac12,\frac19\}}$ and $S^2_{\{1,\frac29\}}$, or
$S^2_{\{\frac35,\frac12\}}$ and $S^2_{\{\frac25,\frac13\}}$, to construct
$S^2_{\{2,\frac12,\frac13\}} $, see following figures for
illustration.
\begin{center}
\includegraphics{e11.eps} \qquad \qquad
\includegraphics{e12.eps}\\
Figure 4 \qquad \qquad \qquad \qquad \qquad Figure 5
\end{center}
In fact, Figure 4 is the same as what we constructed in the proof of Theorem A. However, Figure 5 has two minimum point of the scalar curvature.
\par\vskip0.2cm
{\bf Remark 3:} We should point out that if $\chi(M) \le 0$, the condition (7) is
not sufficient. For example, if $\chi(M)=-2$, we have the following counter
example.\par {\bf Counter example:} Let $\chi(M)=-2,\ ~n=3, ~\alpha_i=2,\ i=1,\ 2,\ 3\ \mbox{and} \ j_0=3.$ Then $\sum_{i=1}^{j_0} \alpha_{\sigma(i)}+\chi(M)-n=6-2-3=1>0,$ but there is no HCMU metric whose scalar curvature is not a constant s.t. three angles of saddle points of $K$ are all $4\pi$. If the metric exists, according to {Proposition 1}, $K$ is continuous in a K-Surface. It must take its maximum and
minimum, but the point at which $K$ takes maximum or minimum
can not be a saddle point, so the point is a smooth critical
point of $K$. Hence the number of smooth critical
points of $K$ is more than 1. However, from the formula $$\chi
(M)=\sum_{i=1}^j(1-\alpha_i)+n-j+s,\ $$we see $s=1$. That means
the number of smooth critical points of $K$ is 1, a
contradiction. $\Box$
\par\vskip0.2cm
{\bf Remark 4:} At last we list some problems that might be
interesting for future study:\par \begin{enumerate} \item For other compact Riemannian surfaces, what is the sufficient and necessary condition for the existence of a HCMU metric? \par \item Is the extremal Hermitian metric unique when none of the
prescribed angles is an integer multiple of $2\pi$? \par \item Given any surface configuration, is the Calabi energy the only factor determining the connected components
in the moduli space of HCMU metrics? \item If we deform the complex structure as well, what is the structure of the moduli space of HCMU metrics? It will be interesting to compare this to the classical Teichm\"{u}ller space in Riemann surfaces. \end{enumerate}
\par\vskip1cm
\noindent Qing Chen and Yingyi Wu \\ Department of mathematics\\ University of Science and Technology of China\\ Hefei, Anhui, 230026\\ P. R. China\\ [email protected] \\[email protected] \\ \vskip0.5cm
\noindent Xiuxiong Chen\\
Department of Mathematics\\
University of Wisconsin-Madison\\
Madison WI 53706\\
USA\\
[email protected]
\end{document} | arXiv |
Find the largest value of $x$ that satisfies the equation $|x-5|=12$.
We can split the expression $|x-5|=12$ into two separate cases: $x-5=12$ and $x-5=-12$. For the first case, solving for $x$ would give us $x=12+5=17$. For the second case, we would get $x=-12+5=-7$. Therefore, $x=17$ and $x=-7$ both satisfy the equation. Since the problem asks for the largest value of $x$, our solution is $\boxed{17}$. | Math Dataset |
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According to the Common Core State Standards Initiative, Euclidean geometry is characterized most importantly by 2. Each Non-Euclidean geometry is a consistent View Geometry 03. A fitting analogy is the globe. ) A straight line segment can be drawn joining any two points. 2. A paradigm shift occurs when one or more fundamental axioms get rejected. In this paper, I aim to expose a completely di erent notion of geometry - fractal geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. This takes into consideration, the Universe may be a closed surface. 'A Euclidean geometry is based on false assumptions, which are called definitions, axioms, and postulates. Definition. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms. A necessary assumption These are general statements, not specific to geometry, whose truth is obvious or self-evident. Nov 07, 2016 · A In Euclidean geometry, how many lines are parallel to a line ℓ through a point P that does not lie on the line? Draw a sketch to support your answer. Jan 01, 2021 · sphere S2, which can be seen most easily by noting that the corresponding Euclidean space-time R S2 is conformally equivalent to Euclidean at space R3 on which the vacuum stress tensor vanishes. Euclidean/Non-Euclidean Geometry Date: 02/21/2003 at 08:01:03 From: Sue Subject: Euclidean/Non Euclidean Geometry Consider the following geometry called S: Undefined terms: point, line, incidence Use Logic Rules 0-11 Axioms: I) Each pair of lines in S has precisely one point in common. math. So it needs a lot of work, and some of it should be moved to Non-euclidean geometry , but it should remain a separate article. For example, in IBC Jun 17, 2015 · Understanding these Euclidean symmetry arguments from a conceptual standpoint showed us that Euclidean geometry at the cortical level is a way to enforce conditions that are not specific to Euclidean geometry but have a meaning on every "symmetric enough" space, and we thus saw how a unique Gaussian random field providing V1-like maps can Deductive reasoning has long been an integral part of geometry, but the introduction in recent years of inexpensive dynamic geometry software programs has added visualization and individual exploration to the study of geometry. " Because mass and energy distort the shape of spacetime, the Euclidean geometry of standard textbooks can't accurately describe it. Dec 25, 2019 · However, whether there are situations in Nature where these statements are (exactly) true is unclear, because the geometry of the physical world is not known to be Euclidean. E lies on AB so that AE = BC. We develop the concepts of congruence and similarity of triangles, and, in particular, prove that corresponding sides of similar triangles are in proportion. Full curriculum of exercises and videos. As to 3), you would have to associate some quantifiable Natural phenomena with the sequence ##\{1/n\}##, such as volumes of water and imagine a sequence of events taking On traditional maps, earth is represented in a flat plane, or by euclidean geometry. In Euclidean plane geometry, there are three different relations indicated by the same words "is congruent to", one for segments, one for angles and one for triangles. . As far as I can tell the author just draws an analogy and wants to say that LISP is constructed from its ten atoms, just like Euclid's plane geometry is constructed from its five axioms. Euclidean geometry dates back to the Greek mathematician, Euclid, who lived around 300 BCE. Euclid definition, Greek geometrician and educator at Alexandria. 1, 1. The second type of non-Euclidean geometry in this text is called elliptic geometry, which models geometry on the sphere. Undefined terms are used to define other concepts. create formal constructions using a straight edge and compass. In constructions explained in the chapter, aids that have been used are not supplied by the geometry under investigation. The sum is greater than 180o. seen in a series of activities in solving the Euclidean Geometry problems (In'am, 2003). There are 12. Here's how Andrew Wiles, who proved Fermat's Last Theorem, described the process: Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. Midterm 1 February 10 Spring 2009, questions and answers Psychology Notes for Final - Evolution and Emotion fall 2018 quiz 7 questions and solutions Midterm 2016, questions and answers Midterm 2015 questions Midterm 2015, questions and answers Nov 29, 2011 · Euclid's Elements contained five postulates which form the basis for Euclidean geometry. Euclid was Any claims that all Euclidean geometry problems are decidable, as given in the comments to the question, will depend on some restricted definition regarding the form that a "problem" can take. We also present a proof of the Pythagorean Theorem. when s = 0 s = 0 you're doing Euclidean geometry. There are many different possibilities, but the shortest line lies on an imaginary "equator" through the two points. Greek geometry eventually passed into the hands of the great Islamic scholars, who translated it and added to it. Parallelograms In Euclidean geometry the following statements can all be used to define a parallelogram. co ordinate systems are a way of translating geometry to algebra or vice versa. Postulate 4. The generalization to non-Euclidean geometry is the following step to develop the language of Special and General Relativity. This survey highlights some Euclidean Hyperbolic Elliptic Which of the following best describes the sum of the angle measures of a triangle in hyperbolic geometry? F. The legacy of Euclidean geometry . Euclidean geometry is the starting point to understand all other geometries and it is the cornerstone for our basic intuition of vector spaces. Independ ent observati on c. Im realy having trouble on this quesiton & I need some help:) thankyou! The Axioms of Euclidean Plane Geometry. The modern fields of differential geometry and algebraic geometry 39 generalize geometry in different directions. " Note that the G-B-L geometry cannot be fully embedded in a three-dimensional Euclidean space, although nite patches of it can be so embedded. For more on non-Euclidean geometries, see the notes on hyperbolic geometry after I. In proof and congruence, students will use deductive reasoning to justify, prove and apply theorems about geometric figures. The independence of the parallel postulate is supremely important for the axiomatic logic of traditional geometry, but it says nothing to disturb the validity and completeness of linear algebra. Both Euclidean and non-Euclidean geometry are models. The sum is equal to 360o. Sumerians invent "placeholder" (zero) but don't consider it a number. In 1871, Klein completed the ideas of non-euclidean geometry and gave the solid underpinnings to the subject. ' 'It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry. Next Lesson: Geometric Figures Euclidean Geometry: An Introduction to Mathematical Work Math 3600 Spring 2017 The Geometry of Rectangles Rectangles are probably familiar to you, but to be clear we give a precise definition. Models are deduced from axioms. We give an overview of a piece of this structure below. Aug 01, 2016 · Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry the Elements. 1 Hyperbolic Geometry Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land. Euclid is best known for his five postulates, or axioms. (C) it will show that Euclidean geometry is inconsistent. On the other hand, in taxicab geometry, there are usually many different geodesics between two given points. Euclidean geometry, in the guise of plane geometry, is used to this day at the junior high level as an introduction to more advanced and more accurate forms of geometry. Riemannian Geometry deals with a spherical world that says parallel lines will intersect. Taxicab geometry is a form of geometry, where the distance between two points A and B is not the length of the line segment AB as in the Euclidean geometry, but the sum of the absolute differences of their coordinates. 06. There is exactly one line through point P that is parallel to line ℓ. Thanks!!! In Euclidean and hyperbolic geometry, the two lines are then parallel. It was not until the 1800s that Euclid's view of the world was shown to be inadequate as a model of the real world. Learn high school geometry for free—transformations, congruence, similarity, trigonometry, analytic geometry, and more. Euclides built his model of planar geometry on five axioms. "In my original work, I was making Euclidean wallpaper," says Farris. 23. Based on these axioms, he proved theorems - some of the earliest uses of proof in the history of mathematics. A rectangle is a quadrilateral which has all four interior angles that are right angles. Extended Euclidean Plane. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. He is the Father of Geometry for formulating these five axioms that, together, form an axiomatic system of geometry: (h) Which of these statements is TRUE? If someone produces a proof of the Euclidean parallel postulate within neutral geometry, (A) the author will be instantly acclaimed as a genius. P1: A quadrilateral with opposite sides parallel and equal in length, and opposite angles equal. ca Euclidean Geometry is considered as an axiomatic system, where all the theorems are derived from the small number of simple axioms. Examples Jun 14, 2011 · Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. On the contrary, fractal geometry and its exten- The study of space38 originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space, but later also generalized to non-Euclidean geometries which play a central role in general relativity. Now Euclidean geometry alone is similarity, the preservation of shape across variations of size, possible —and similarity is the sine qua nonof imaging. Euclidean geometry, codified around 300 BCE by Euclid of Alexandria in one of the most influential textbooks in history, is based on 23 definitions, 5 postulates, and 5 axioms, or "common notions. Which doesn't express equivalence. One of these, the parallel postulate has been the subject of debate among mathematicians for many centuries. C) Two perpendicular lines create four right angles. The modern version of Euclidean geometry is the theory of Euclidean (coordinate) spaces of multiple dimensions, where distance is measured by a suitable generalization of the Pythagorean theorem. the projective geometry a 3d engine uses). So, most of geometry which is done in this level is based on abstract and proof-oriented. Euclidean geometry, which had been for a long time one of the cornerstones of classical education, is not taught properly nowadays, and the role of geometry in education is obviously underestimated. The sum is less than 180o. However, a globe is a more accurate model that comes from elliptical geometry. The short first chapter consists of two sections: "The Origins of Geometry" and "A Few Words About Euclid's Elements", each two-pages long; the former including the motivation and statement of Fermat's Last Theorem. " A theorem is any statement that can be proven using logical deduction from the axioms. Start studying Euclidean Geometry. The Five Axioms of Euclidean Geometry Below are the ve postulates (axioms) of Euclidean Geometry. Geometry has always been an important topic in the eld of mathematics. (E) 5. However, mathematicians of the modern era developed new theorems and ideas pertaining to geometry and divided the subject to 'Euclidean Geometry' and 'Non-Euclidean Dec 03, 2020 · Euclidean and Non-Euclidean Geometry Euclidean Geometry Euclidean Geometry is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (330 B. These are from Hilbert's The Foundations of Geometry. edu 1. We will see in this handout and in Venema's Chapter 7 that many familiar properties of Euclidean geometry follow from this postulate. Jul 03, 2012 · Here, we find that the underlying non-Euclidean geometry of twisted fiber packing disrupts the regular lattice packing of filaments above a critical radius, proportional to the helical pitch. Euclid was Jan 06, 2021 · Let's define a new coord system in which the origin is moving to the right with velocity $(v_x, 0)$ in the old coordinate system. Examples The study of hyperbolic geometry—and non-euclidean geometries in general— dates to the 19 th century's failed attempts to prove that Euclid's fifth postulate (the parallel postulate) could be derived from the other four postulates. In Euclidean geometry, a line segment is that portion of a line which falls Which statements about the sum of the interior angle measures of a triangle in Euclidean and non-Euclidean geometries are true? A) In Euclidian geometry the sum of the interior angle measures of a triangle is 180 degrees, but in elliptical or spherical geometry the sum is less than 180 degrees. It was also the earliest known systematic discussion of geometry. The five postulates of Euclidean Geometry define the basic rules governing the creation and extension of geometric figures with ruler and compass. 16. Since the term "Geometry" deals with things like points, line, angles, square, triangle, and other shapes, the Euclidean Geometry is also known as the "plane geometry". Thus we see that different systems of geometry can describe the same physical situation, provided that the physical objects (in this case, light rays) are correlated with A postulate is a statement that is assumed to be true without proof. The customary tools of Euclidean geometry are compass and ruler. Readings: Stephen Toulmin: The Claims of Logic. A straight line along the globe describes a great circle 8, which goes once around the globe and comes back to its starting point. The lack of any conformal anomaly means that the same must be true on the original geometry R S2, and hence E[S2] = 0. To describe the whole space, it is necessary to describe it in terms of its inner properties. In Euclidean geometry, however, the lines remain at a constant distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular. stanford. A Euclidean geometric plane (that is, the Cartesian plane) is a sub-type of neutral plane geometry, with the added Euclidean parallel postulate. Non-Euclidean Geometry Asked by Brent Potteiger on April 5, 1997: I have recently been studying Euclid (the "father" of geometry), and was amazed to find out about the existence of a non-Euclidean geometry. Euclid (Greek) publishes Elements; basis for Euclidean Geometry. Yet the followers of Kant did not object when formulas in algebra no longer seemed to describe reality. Is it a true statement? In Euclidean Geometry, any Regular Polygon can be inscribed in a circle. The second midterm exam will cover units 3, 4, 5, and 6 of the course: Unit 3. Moreover, the shape of a geometric object will not change. Though this course is primarily Euclidean geometry, students should complete the course with an understanding that non-Euclidean geometries exist. That geometry is determined by the "Hyperbolic metric" for , as opposed to the Euclidean metric. Jan 03, 2020 · As another example, so-called "non-Euclidean geometries," which defy the parallel postulate of traditional Euclidean geometry (more below), came as a shock to the mathematical community in the Because mass and energy distort the shape of spacetime, the Euclidean geometry of standard textbooks can't accurately describe it. Spherical geometry states that there are no parralels to a given line through an external point and the sum of angles and triangles is greater than 180 degrees. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. Another copy is available here. ' Jun 17, 2005 · The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. To draw a straight line from any point to any point. Postulates These are the basic suppositions of geometry. Prove theorems about symmetries and transformations. Certainly a great many of thc most familiar algebraic relationships originated from real problems, some of them geometric and some from economics it can be seen that non-euclidean geometry is just as consistent as euclidean geometry. 7701 Quiz 2 _____ enhances the understanding of natural phenomena by enabling scientists to describe behavior accurately. Euclidean geometry In several ancient cultures there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. 2-dimensional neutral geometry without continuity axiom. Even drawing a "straight" line between two points on the surface of a sphere is problematic. Students in this stage are capable to compare axioms systems such as Euclidean and Non-Euclidean. Learn vocabulary, terms, and more with flashcards, games, and other study tools. In, Hyperbolic Geometry, this is not an obvious statement. Looking back at Gauss's work one gets This site includes statements of definitions, common notions, postulates, and propositions, but not proofs. In this geometry, Euclid's fifth postulate is replaced by this: 5E. In this study of Greek geometry, there were many more Greek mathematicians and geometers who contributed to the history of geometry, but these names are the true giants, the ones that developed geometry as we know it today. Systemati c observati on b. R. You can also identify and describe the undefined term, set, used in geometry and set theory. At non-Euclidean geometry constructing visual models for recognition is not easy and useful, so the focus is more on abstract concepts. edu In Euclidean Geometry, any Polygon can be completely enclosed in some sufficiently large triangle. In fact, these two kinds of geometry, together with Euclidean geometry, fit into a unified framework with a parameter s ∈ ℝ s \in \mathbb{R} that tells you the curvature of space: when s > 0 s \gt 0 you're doing elliptic geometry. Spherical geometry contradicts Euclidean geometry in two ways. Used in science to observe and describe throughout the Universe. Euclidean geometry Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. of the fundamental differences that separates euclidean geometry from non-euclidean geometry. This is so obvious a statement that I have never even seen it written as a theorem. ) During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. To describe a circle with any centre and distance (radius). Subject Focus: Shape, Space & Measures – Euclidean Geometry Lines & Line Segments Year 3 Year 4 A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. This procedure enables in constructing the midpoint of a segment. Feb 26, 2014 · None of them are axioms of Euclidean geometry. Let the following be postulated: Postulate 1. Aug 23, 2020 · It is non-Euclidean, as we can demonstrate by exhibiting at least one proposition that is false in Euclidean geometry. Non-Euclidean Geometry - Special Topics - This Second Edition is organized by subject matter: a general survey of mathematics in many cultures, arithmetic, geometry, algebra, analysis, and mathematical inference. See more. It is a very basic concept which cannot be defined. This chapter discusses the projective geometry as an abstract theory, self-contained and independent. See full list on cs. Please use CIM for all new proposals. Geometric content The geometry studied in this book is Euclidean geometry. edu 'A Euclidean geometry is based on false assumptions, which are called definitions, axioms, and postulates. Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described (although non-rigorously by modern standards) in his textbook on geometry: the Elements. To describe a circle with any center and radius. Poincaré: — Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity. 1. Given: A triangle ABC with A = 20 and B = C = 80. Then bobbym gave me a new puzzle to try. Find angle AEC. Euclidean Geometry requires the earners to have this knowledge as a base to work from. All three can be combined under a generalized defination that applies to arbitrary geometrical objects in the plane in transformation geometry. · how to construct and justify statements, determine truthfulness of a converse, an inverse and a contrapositive statement and demonstrate what it means to prove a statement is true. See full list on quickanddirtytips. AB There are three ways to prove that a quadrilateral is a rectangle. Euclid saw only part of the picture, however. For well over two thousand years, people had believed that only one geometry was possible, and they had accepted the idea that this geometry described reality. Start studying Euclidean Geometry. 8. Which of the following statements from Euclidean geometry is also true of spherical geometry? A) A line has infinite length. The problem solving in the Euclidean Geometry is done by giving some statements in line with the logical orders and on the basis of reasons by basing oneself on the concerned theorems, postulates or definitions. Any location on the earth can be found with its latitudes & longitudes. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. One can look for an alternate development of the theory by starting with different primitive concepts or different axioms. It presents a collection of concepts and a body of statements in the Euclidean theory. See analytic geometry and algebraic geometry. mcgill. The system that Euclid went on to describe in the 'Elements' was commonly known as the only form of geometry the world had witnessed and seen up until the 19th century. When we eventu-ally turn our attention to non-euclidean geometry, i want 2-dimensional neutral geometry without continuity axiom. There are plenty of unsolved geometry problems. This new organization enables students to focus on one complete topic and, at the same time, compare how different cultures approached each topic. 4. As of November 15, 2019 new proposals can no longer be created in the CPS. Feb 23, 2015 · Euclidean geometry cannot be used to Model the earth because it is a sphere. Fractal geometry and multifractals in analyzing and processing medical data and images T A traditional way for describing objects, based on the well-known Euclidean geometry, is not capable to describe different natural objects and phenomena such as clouds, relief shapes, trends in economy, etc. Midterm Exam #2 Review Sheet. Prove statements about parallelograms, circles, and the coordinate plane (A, B, E, F & G) 4. Computer Aided Design- CAD. Since the first 28 postulates of Euclid's Elements do not use the Parallel Postulate, then these results will also be valid in our first example of non-Euclidean geometry called hyperbolic geometry. In it Euclid laid down the rules of geometry. At the moment it doesn't actually say much about Euclidean geometry, and instead spends too much time discussing non-euclidean geometry, which is already discussed in Non-euclidean geometry. Part 1 In ΔABC shown below: The following flowchart proof with missing statements and reasons proves that if a line Describe the historic role of Euclid's Elements in the development of modern geometry State the axioms ("postulates" and "common notions") that are the foundation of Euclid's geometry Upon reviewing familiar results of "high school" Euclidean geometry, write basic proofs using the Elements as the source (focus on definitions and theorems of See full list on plato. Is the Parallel Postulate true in spherical like Euclidean geometry, there is exactly one geodesic through any two different points. To produce a finite straight line continuously in a straight line. For example: 1. While many of Euclid's findings had been previously stated by earlier Greek mathematicians, Euclid Euclidean Geometry Euclid's Axioms Momento della lettura: ~25 min Rivela tutti i passaggi Before we can write any proofs, we need some common terminology that will make it easier to talk about geometric objects. For each property listed from plane Euclidean geometry, choose a corresponding statement for non-Euclidean spherical geometry 1. Euclidean transformations preserve length and angle measure. An axiomatic description of it is in Sections 1. The sum is equal to 180o. Things which are equal to the same thing are equal to one another. Dec 14, 2020 · The duality principle is applicable in the elliptic geometry of space: The terms "point" and "plane" can be interchanged in every true statement, and a true statement is obtained. In addition to The primitives are analogous to the 5 axioms of Euclidean plane geometry. The geometry most people know of is called the Euclidean geometry, which is widely favored by mathematicians. the real content of the Euclidean Parallel Postulate is the statement that there is only one such line. The plane may be given a spherical geometry by using the stereographic projection. I am sure such a course is needed for all young students, not just for those who are going to pursue careers in science or engineering. when s < 0 s \lt 0 you're doing hyperbolic geometry. We use a TPTP inspired language to write a semi-formal proof of a theorem, that fairly accurately depicts a proof that can be found in mathemati-cal textbooks. Many new photographs and diagrams Dec 31, 2020 · If we deny the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, [ 34] H. To unlock this lesson you Another weakening of Euclidean geometry is affine geometry, first identified by Euler, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become Euclidean Geometry, has three videos and revises the properties of parallel lines and their transversals. We aim at giving a general intuitive grasp of the subject, and refer the reader to more orthodox literature for a rigorous treatement of the subject. Above this critical radius, the ground-state packing includes the presence of between one and six excess fivefold disclinations in the cross-sectional order. Continue this thread level 2 Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By elementary plane geometry I mean the geometry of lines and circles straight-edge and compass constructions in both Euclidean and non-Euclidean planes. All the constructions underlying Euclidean plane geometry can now be made accurately and conveniently. The text consists almost entirely of exercises that guide studentsas they discover the mathematics and then come to understandit for themselves. An axiomatic system has four parts: undefined terms axioms (also called postulates) definitions theorems A postulate or axiom is a mathematical statement which is so obviously true that it does not need to be proved and can be relied upon to be used to prove other statements. We start with the idea of an axiomatic system. (e) In 3-dimensional Euclidean geometry, the set of all points that are a fixed distance away from a specific point is called a _____. C. Learning Area Outcome: I can recognise and describe the properties of shapes. Dec 01, 2001 · Jan 2002 Euclidean Geometry The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in - that is, to describe the geometric rules which govern it. ) Euclid's text Elements was the first systematic discussion of geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely Jan 08, 2013 · On a sphere, however, if two parallel lines – great circles – are extended, they will end up intersecting. One obtains a mathematical theory by proving new statements, called theorems, using only the axioms (postulates), logic system, and previous theorems The moment is a reproduction of the coastal surveillance projects of the mid-18th century: just as surveyors worked to accurately remap the British coastline by measuring the curves of the shore, the character remapped his history by observing the geography. ANALYSIS: The argument didn't tell us whether or not the non-Euclidean system can have parallel lines. Notice that the definition only speaks about angles. Practical geometry or Euclidean geometry is the most pragmatic branch of geometry that deals with the construction of different geometrical figures using geometric instruments such as rulers, compasses and protractors. 4 Non-Euclidean Geometry There are other geometries in addition to Euclidean geometry. Recall that one Put simply, Euclidean geometry is a system in which all the theorems are derived and based on a set of 5 postulates, or proved things in mathematics. E. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. 2 Compare and contrast inductive reasoning and deductive reasoning for making predictions and valid conclusions based on contextual situations. Jul 26, 1999 · Riemann extends to n dimensions the methods employed by Gauss (1828) in his study of the intrinsic geometry of curved surfaces embedded in Euclidean space (called 'intrinsic' because it describes the metric properties that the surfaces display by themselves, independently of the way they lie in space). Which is the best description of the baseline in graph number 5? From that basic foundation we derive most of our geometry (and all Euclidean geometry). create a model in which some statement is true, and another model in which the statement is false. explored by geometers. ) Any straight line segment can (h) Which of these statements is TRUE? If someone produces a proof of the Euclidean parallel postulate within neutral geometry, (A) the author will be instantly acclaimed as a genius. More precisely, our formal proofs are based on the rst eleven chapters and some results from the twelfth of [SST83] which are valid in neutral geometry. • This is easy in geometry, but it is much harder when your axiom set describes most of mathematics – like the axioms of set theory. Geometry, one of the principle concepts of mathematics, entails lines, curves, shapes, and angles. For example, the meridians are such great circles. Writers were just as sensitive to the changes taking place during the Late-Euclidean era. Nov 10, 2011 · He found it in the non-Euclidean geometry of 19th-century mathematician Georg F. By contrast, non-Euclidean geometries are curved. I can use these properties to construct shapes using appropriate mathematical instruments and to prove given geometric statements. in non-euclidean geometry, the fourth angle cannot be a right angle, so there are no rectangles. Note that the second and third methods require that you first show (or be given) that the quadrilateral in question is a parallelogram: If all angles in a quadrilateral are right angles, then it's a rectangle (reverse of the rectangle definition). In this chapter we are going to describe the basic principles of plane projective geometry and euclidean 3-D geometry in a fairly unformal manner. For her map project, Clarke chose to study the Hobo-Dyer projection after learning that it better preserves the true proportions of the areas represented. They all describe the exact same family of polygons. (line from centre ⊥ to chord) If OM AB⊥ then AM MB= Proof Join OA and OB. In hyperbolic geometry, the sum of the angles of any triangle is less than 180\(^\circ\text{,}\) a fact we prove in Chapter 5. Euclidean geometry can be this "good stuff" if it strikes you in the right way at the right moment. As soon as I've figured how to hide it, I'll post it. See full list on pi. in euclidean geometry, the fourth angle is a right angle, so there are rectangles. D)The intersection of two lines creates four angles. It is used to describe points in space or on a plane to express geometric relations. II) Each point in S is incident with precisely 2 lines. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such. 1 Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true and discuss unique properties of each. Being as curious as I am, I would like to know about non-Euclidean geometry. Circle the most appropriate word or phrase to make each statement true. So Neutral Geometry gives the theorems that are common to both of these important geometries. Eucliean and Non-Euclidean Geometry – Fall 2007 Dr. Identify and describe the main properties of hyperbolic and spherical geometry. ) Euclid's text Elements was the first systematic discussion of geometry. Used in science to observe and describe things on Earth. Gauss's second central idea had to do with the form of the distance function d(1;2). Please refer to the graph image. In this article, we list the basic tools of geometry, their description and uses. But if you add the negation instead, you get Hyperbolic Geometry. spherical & cylindrical co ordinate systems can describe 3-D Euclidean geometry in terms of sets of numbers. It is Euclid's of the fundamental differences that separates euclidean geometry from non-euclidean geometry. So spherical geometry, and basic facts about navigation on a sphere such as the Earth, is fundamentally different from Euclidean geometry, or geometry on a flat surface. e. In order to understand elliptic geometry, we must first distinguish the defining characteristics of neutral geometry and then establish how elliptic geometry differs. Einstein's general relativity uses more complicated math Euclidean geometry, which had been for a long time one of the cornerstones of classical education, is not taught properly nowadays, and the role of geometry in education is obviously underestimated. Thus, we know now that we must include the parallel postulate to derive Euclidean geometry. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles shapes that can be drawn on a piece of paper Learning Area Outcome: I can recognise and describe the properties of shapes. If equals be added to equals, the wholes are equal. He shows that there are essentially three types of geometry: • that proposed by Bolyai and Lobachevsky, where straight lines have . In Euclidean geometry, for example, point, line and plane are not defined. According to Geometry from Multiple Perspectives (1991) from the National Council of Teachers of Mathematics, many manufactured items are made of parts that are linear or circular in shape and are based on the geometry of Euclid, which is the geometry of the point set, of the straight line, and of the Euclidean tools of construction. 300 BC. Euclidean geometry is flat—it is the geometry of a tabletop, infinitely extended. however, a globe is a more accurate model that comes from elliptical geometry. 1 Hyperbolic Geometry Euclidean Geometry (the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (330 B. May 19, 2006 · Thus A is the only statement that can be true. In this axiomatic approach, projective geometry means any collection of things called "points" and things called "lines" that obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of The suggestion that some new system of statements deserved to be called geometry was a threat. Dec 31, 2020 · the thread was moved here by a mod. (B) it will show that Euclidean geometry is consistent. Which of the following best represents a segment in Euclidean geometry? A. B) Two intersecting lines divide the plane into four regions. In this theory, neither of these statements is a theorem nor contradicts the axioms. Riemann, which provided just the tool he required: a geometry of curved spaces in any number of dimensions. In order to compare paths, we would like a way to measure curvature. " [1] These are used as foundation for geometry [2] and occasionally applied to other sciences, such as special relativity . 29 and elliptic geometry after I. Euclidean geometry was first conceived as a realistic theory of applied mathematics (for its role of first theory of physics), then became understood as an axiomatic theory of pure mathematics among diverse other, equally legitimate geometries in a mathematical sense; while the real physical space is more accurately described by the non Euclidean geometry is our way of measuring on Earth. In Euclidean Geometry, any Polygon can be completely enclosed in some sufficiently large triangle. spherical coordinates without the radius can describe 2-D elliptic geometry in terms of sets of numbers. Riemann in his lecture [1] (1854, published in 1867). Jan 27, 2016 · Galileo Galilei and Isaac Newton founded modern physics on the assumption that space is Euclidean, but Albert Einstein's equations of general relativity describe a universe that can have complex Learning Area Outcome: I can recognise and describe the properties of shapes. how does a globe represent the fact that there are no parallel lines in elliptical geometry? So by 1870, it was absolutely clear that this open geometry, this non-Euclidean geometry, was a perfectly consistent, is a perfectly consistent, formulation of geometry. Once you have learned the basic postulates and the properties of all the shapes and lines, you can begin to use this information to solve geometry problems. Until the late nineteenth century, all geometry was Euclidean. 3108. Another weakening of Euclidean geometry is affine geometry, first identified by Euler, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become Euclid definition, Greek geometrician and educator at Alexandria. The paper is organized into ve parts. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. Theorems - proved statements An axiomatic system consists of some undefined terms (primitive terms) and a list of statements, called axioms or postulates, concerning the undefined terms. For example, construct a triangle on the earth's surface with one corner at the north pole, and the other two at the equator, separated by 90 degrees of longitude. There is a lot of work that must be done in the beginning to learn the language of geometry. If one has a prior background in Euclidean geometry, it takes a little while to be comfortable with the idea that space does not have to be Euclidean and that other geometries are quite possible. Variable presentati on d. Jun 21, 2001 · Geometry down the centuries. a. G. Of course, we have to understand what a segment is in spherical geometry. Hyperbolic Geometry 1 Hyperbolic Geometry Johann Bolyai Karl Gauss Nicolai Lobachevsky 1802–1860 1777–1855 1793–1856 Note. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Axioms are statements which are assumed to be true (but are not necessarily proven). GeoGebra, to explore the statements and proofs of many of the most interestingtheorems in advanced Euclidean geometry. Until the advent of non-Euclidean geometry, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. 2, and 1. As a claim about human imaging capabilities, this might well be accepted even by life-long specialists in non-Euclidean geometry and Einsteinean physics. Postulate 2. ' Euclidean geometry is of great practical value. (Actually, you […] MTH 338: Non-Euclidean Geometry The world we live in is not Euclidean! Euclid tried (unsuccessfully!) to formulate a series of postulates for the geometry of a (flat, infinite) piece of paper. One of the greatest Greek achievements was setting up rules for plane geometry. Proposals that were submitted to liaisons by November 15 must be submitted into workflow by December 2, 2019 to continue in the CPS. His abstract model was supposed to accurately reflect the world around us in that his postulates were to be "self-evident". Hilberts Axioms of Euclidean geometry. So far the best math solution i had was to model everything in differential geometry of a Riemann manifold that uses a special custom metric - and then finding an Euclidean geometry approximation that looks mostly the same when projected down onto the sphere (i. Goals: To explore ways in which Euclidean Geometry served as a model for other thinkers, both in its content (which seemed to be certain knowledge about the universe) and in its form (a template for a worthy body of knowledge). Subject Focus: Shape, Space & Measures – Euclidean Geometry Lines & Line Segments Year 5 Year 6 In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. 10. cornell. See full list on mathstat. Why is the equivalence of the practically-rigid body and the body of geometry—which suggests This article describes how the idea of comparing Euclidean geometry with two non- Euclidean geometries (taxicab and spherical) provides participants with engaging mathematical tasks in a professional development geometry course for K-12 teachers. Like the Euclidean metric, it is defined by a non-degenerate inner product, but unlike that metric, the inner product is not positive definite, as I shall explain below. Now that you have navigated your way through this lesson, you are able to identify and describe three undefined terms (point, line, and plane) that form the foundation of Euclidean geometry. (a) In Euclidean geometry, the sum of angles inside a triangle is ( less than / equal to / more than ) 180 degrees. Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of simple axioms. 1 Neutral Geometry remedies some of the weaknesses of IBC Geometry. Subject Focus: Shape, Space & Measures – Euclidean Geometry Lines & Line Segments Year 5 Year 6 According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often. " Euclid's version: "Suppose that a line l meets two other lines m and n so that the sum of the interior angles on one side of l is less than 180°. geometry") withstood centuries of scrutiny by the best minds of the day. Euclidean Geometry is the system of postulates and proofs that we study in Course II. Unit 3. As its subtitle suggests, Euclid's Window aims to take the general reader from elementary ideas of euclidean geometry to Einstein's theory of general relativity and Sep 12, 2019 · On traditional maps, Earth is represented in a flat plane, or by Euclidean geometry. Goals: To explore ways in which Euclidean Geometry served as a model for other thinkers, both in its Nov 03, 2017 · Euclidean geometry: Playfair's version: "Given a line l and a point P not on l, there exists a unique line m through P that is parallel to l. Geometry. In designing, geometry has a symbolic role to play; as is evident from the carvings on the walls, roofs, and doors of various architectural marvels. Geometry is founded by a set of basic elements and basic relations between them (which are not defined, but are intuitively clear) and a system of axioms (statements which are not proved, but are considered to be true and are intuitively obvious) from which other figures are defined and all corresponding consequences (theorems) are deduced. CHAPTER 8 EUCLIDEAN GEOMETRY BASIC CIRCLE TERMINOLOGY THEOREMS INVOLVING THE CENTRE OF A CIRCLE THEOREM 1 A The line drawn from the centre of a circle perpendicular to a chord bisects the chord. axiomatic Euclidean geometry. The axioms are as follows Postulates. Sep 26, 2017 · Euclidean Geometry. They can do their work abstractly, This is only one of the facts which distinguish geometry on flat surfaces (Euclidean geometry) from spherical geometry. In this chapter , we will give an illustration of what it is like to do geometry in a space governed by an alternative to Euclid's fifth postulate. When we eventu-ally turn our attention to non-euclidean geometry, i want But Euclidean geometry will also apply in this world; instead of being non-Euclidean "straight lines," the light rays would be Euclidean circles perpendicular to C. (B, E, F & G) The axioms of Euclidean geometry were chosen by observation; that is, Euclidean geometry accurately describes the Universe locally because it was constructed for that purpose. ' 'Today we call these three geometries Euclidean, hyperbolic, and absolute. The first of these properties is a converse to the Alternate Interior Angles Theorem. com To find (by Euclidean geometry) x = EDB . Instead of the Cartesian coordinates used In Euclidean geometry, longitudes & latitudes are used as to define the points on the earth. and so on. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. The examples of non-Euclidean geometries considered in this paper are hyperbolic and elliptic geometries. Prove the Euclidean Geometry Theorems for similar, congruent and right triangles (A, E & F) 3. However, some authors allow a line to be parallel to itself, so that "is parallel to" forms an equivalence relation. 4. Jun 18, 2008 · In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere It describes a procedure for constructing the perpendicular bisector of a segment. In ΔΔOAM and OBM: (a) OA OB= radii REASONING: The non-Euclidean system of geometry with the most empirical verification (real world proof) is seen by some physicists think it accurately describes our universe. Conceptual framework What is crucial when measuring pupils' attitudes towards Euclidean geometry is that one should confidently accept the resulting measure produced by the attitudinal scale used. In this NEW system, the question is the same as the original, but the initial velocity is zero. H. They may have the positive curvature of a sphere, or they may have negative curvature, which is harder to visualize but may be compared to the frilly surface of some leafy vegetables. They are called non-Euclidean geometries and consider ideas that are not necessarily true in Euclidean geometry. Learners should know this from previous grades but it is worth spending some time in class revising this. The most direct route between two points is the one with the least curvature. The concept of elliptic geometry was apparently introduced by B. Still, his geometry (which, throughout the remainder of this discussion, will be referred to as "Euclidean . Maths is a very odd activity. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. It's taken me a while but I've finally got a solution that I think stands up OK. Euclid's book The Elements is one of the most successful books ever — some say that only the bible went through more editions. Dec 22, 2020 · This semester, Clarke and her classmates looked at three different types of geometry—Euclidean, spherical, and hyperbolic geometry—which each have a different set of guiding principles. If you E5P (or something equivalent) to Neutral Geometry, then you get Euclidean Ge-ometry. B. 3. B The statement in part (A) is the Parallel Postulate in Euclidean geometry. A list of axioms to develope Euclidean geometry in a modern way. which statements accurately describe euclidean geometry
zwbr, hog3, ns, xop, cr7, mf, cwnb, m2, 0iu4, mh, et1x, ogn, vhb, ju, cn, | CommonCrawl |
I want to compute the estimate of $\beta$ for a linear model $Y = X\beta + \varepsilon $ with $$\varepsilon \sim N_d(0, \sigma^2V),$$ where $V$ is a $d\times d$ definitive posive, symmetric matrix.
It is straightforward to generalize the theory of ordinary linear models (where $\varepsilon \sim N_d(0, \sigma^2I_d))$, to this general case. In fact, it is sufficient to consider the Cholesky decomposition of the matrix $V$ and then transform the variables (details, for instance, here).
Now that the theory is clear, how can I specify and solve this problem in R?
Browse other questions tagged r regression generalized-linear-model cholesky or ask your own question.
Distribution of sum of squares error for linear regression? | CommonCrawl |
HH-suite3 for fast remote homology detection and deep protein annotation
Martin Steinegger1,2,
Markus Meier1,
Milot Mirdita1,
Harald Vöhringer1,3,
Stephan J. Haunsberger4 &
Johannes Söding ORCID: orcid.org/0000-0001-9642-82441
HH-suite is a widely used open source software suite for sensitive sequence similarity searches and protein fold recognition. It is based on pairwise alignment of profile Hidden Markov models (HMMs), which represent multiple sequence alignments of homologous proteins.
We developed a single-instruction multiple-data (SIMD) vectorized implementation of the Viterbi algorithm for profile HMM alignment and introduced various other speed-ups. These accelerated the search methods HHsearch by a factor 4 and HHblits by a factor 2 over the previous version 2.0.16. HHblits3 is ∼10× faster than PSI-BLAST and ∼20× faster than HMMER3. Jobs to perform HHsearch and HHblits searches with many query profile HMMs can be parallelized over cores and over cluster servers using OpenMP and message passing interface (MPI). The free, open-source, GPLv3-licensed software is available at https://github.com/soedinglab/hh-suite.
The added functionalities and increased speed of HHsearch and HHblits should facilitate their use in large-scale protein structure and function prediction, e.g. in metagenomics and genomics projects.
A sizeable fraction of proteins in genomics and metagenomics projects remain without annotation due to the lack of an identifiable, annotated homologous protein [1]. A high sensitivity in sequence similarity searches increases the chance of finding a homologous protein with an annotated function or a known structure from which the function or structure of the query protein can be inferred [2]. Therefore, to find template proteins for comparative protein structure modeling and for deep functional annotation, the most sensitive search tools such as HMMER [3, 4] and HHblits [5] are often used [6–9]. These tools can improve homology detection by aligning not only single sequences against other sequences, but using more information in form of multiple sequence alignments (MSAs) containing many homologous sequences. From the frequencies of amino acids in each column of the MSA, they calculate a 20×length matrix of position-specific amino acid substitution scores, termed "sequence profile".
A profile Hidden Markov Model (HMM) extends sequence profiles by augmenting the position-specific amino acid substitution scores with position-specific penalties for insertions and deletions. These can be estimated from the frequencies of insertions and deletions in the MSA. The added information improves the sensitivity of profile HMM-based methods like HHblits or HMMER3 over ones based on sequence profiles, such as PSI-BLAST [10].
Only few search tools represent both the query and the target proteins as sequence profiles built from MSAs of homologous proteins [11–14]. In contrast, HHblits / HHsearch represent both the query and the target proteins as profile HMMs. This makes them among the most sensitive tools for sequence similarity search and remote homology detection [5, 15].
In recent years, various sequence search tools have been developed that are up to four orders of magnitude faster than BLAST [16–19]. This speed-up addresses the need to search massive amounts of environmental next-generation sequencing data against the ever-growing databases of annotated sequences. However, no homology can be found for many of these sequences even with sensitive methods, such as BLAST or MMseqs2 [19].
Genomics and metagenomics projects could annotate more sequence by adding HHblits searches through the PDB, Pfam and other profile databases to their pipelines [8]. Additional computation costs would be marginal, since the version of HHblits presented in this work runs 20 times faster than HMMER, the standard tool for Pfam [20] and InterPro [21] annotations.
In this work, our goal was to accelerate and parallelize various HH-suite algorithms with a focus on the most time-critical tools, HHblits and HHsearch. We applied data level parallelization using Advanced Vector Extension 2 (AVX2) or Streaming SIMD Extension 2 (SSE2) instructions, thread level parallelization using OpenMP, and parallelization across computers using MPI. Most important was the ample use of parallelization through SIMD arithmetic units present in all modern Intel, AMD and IBM CPUs, with which we achieved speed-ups per CPU core of a factor 2 to 4.
Overview of HH-suite
The software HH-suite contains the search tools HHsearch [15] and HHblits [5], and various utilities to build databases of MSAs or profile HMMs, to convert MSA formats, etc.
HHsearch aligns a profile HMM against a database of target profile HMMs. The search first aligns the query HMM with each of the target HMMs using the Viterbi dynamic programming algorithm, which finds the alignment with the maximum score. The E-value for the target HMM is calculated from the Viterbi score [5]. Target HMMs that reach sufficient significance to be reported are realigned using the Maximum Accuracy algorithm (MAC) [22]. This algorithm maximizes the expected number of correctly aligned pairs of residues minus a penalty between 0 and 1 (parameter -mact). Values near 0 produce greedy, long, nearly global alignments, values above 0.3 result in shorter, local alignments.
HHblits is an accelerated version of HHsearch that is fast enough to perform iterative searches through millions of profile HMMs, e.g. through the Uniclust profile HMM databases, generated by clustering the UniProt database into clusters of globally alignable sequences [23]. Analogously to PSI-BLAST and HMMER3, such iterative searches can be used to build MSAs by starting from a single query sequence. Sequences from matches to profile HMMs below some E-value threshold (e.g. 10−3) are added to the query MSA for the next search iteration.
HHblits has a two-stage prefilter that reduces the number of database HMMs to be aligned with the slow Viterbi HMM-HMM alignment and MAC algorithms. For maximum speed, the target HMMs are represented in the prefilter as discretized sequences over a 219-letter alphabet in which each letter represents one of 219 archetypical profile columns. The two prefilter stages thus perform a profile-to-sequence alignment, first ungapped then gapped, using dynamic programming. Each stage filters away 95 to 99% of target HMMs.
Overview of changes from HH-suite version 2.0.16 to 3
Vectorized viterbi HMM-HMM alignment
Most of the speed-up was achieved by developing efficient SIMD code and removing branches in the pairwise Viterbi HMM alignment algorithm. The new implementation aligns 4 (using SSE2) or 8 (using AVX2) target HMMs in parallel to one query HMM.
Fast MAC HMM-HMM alignment
We accelerated the Forward-Backward algorithm that computes posterior probabilities for all residue pairs (i,j) to be aligned with each other. These probabilities are needed by the MAC alignment algorithm. We improved the speed of the Forward-Backward and MAC algorithms by removing branches at the innermost loops and optimizing the order of indices, which reduced the frequency of cache misses.
Memory reduction
We reduced the memory required during Viterbi HMM-HMM alignment by a factor of 1.5 for SSE2 and implemented AVX2 with only a 1.3 times increase, despite the need to keep scores for 4 (SSE2) or 8 (AVX2) target profile HMMs in memory instead of just one. This was done by keeping only the current row of the 5 scoring matrices in memory during the dynamic programming ("Memory reduction for backtracing and cell-off matrices" section), and by storing the 5 backtrace matrices, which previously required one byte per matrix cell, in a single backtrace matrix with one byte per cell ("From quadratic to linear memory for scoring matrices" section). We also reduced the memory consumption of the Forward-Backward and MAC alignment algorithms by a factor of two, by moving from storing posterior probabilities with type double to storing their logarithms using type float. In total, we reduced the required memory by roughly a factor 1.75 (when using SSE2) or 1.16 (when using AVX2).
Accelerating sequence filtering and profile computation
For maximum sensitivity, HHblits and HHsearch need to reduce the redundancy within the input MSA by removing sequences that have a sequence identity to another sequence in the MSA larger than a specified cutoff (90% by default) [15]. The redundancy filtering takes time O(NL2), where N is the number of MSA sequences and L the number of columns. It can be a runtime bottleneck for large MSAs, for example during iterative searches with HHblits. A more detailed explanation is given in "SIMD-based MSA redundancy filter" section.
Additionally, the calculation of the amino acid probabilities in the profile HMM columns from an MSA can become time-limiting. Its run time scales as O(NL2) because for each column it takes a time ∼O(NL) to compute column-specific sequence weights based on the subalignment containing only the sequences that have no gap in that column.
We redesigned these two algorithms to use SIMD instructions and optimized memory access through reordering of nested loops and array indices.
Secondary structure scoring
Search sensitivity could be slightly improved for remote homologs by modifying the weighting of the secondary structure alignment score with respect to profile column similarity score. In HH-suite3, the secondary structure score can contribute more than 20% of the total score. This increased the sensitivity to detect remote homologs slightly without negative impact on the high-precision.
New features, code refactoring, and bug fixes
HH-suite3 allows users to search a large number of query sequences by parallelizing HHblits/HHsearch searches over queries using OpenMP and MPI (hhblits_omp, hhblits_mpi, hhsearch_omp, hhsearch_mpi). We removed the limit on the maximum number of sequences in the MSAs (parameter -maxseqs < max>). We ported scripts in HH-suite from Perl to Python and added support for the new PDB format mmCIF, which we use to provide precomputed profile HMM and MSA databases for the protein data bank (PDB) [24], Pfam [20], SCOP [25], and clustered UniProt databases (Uniclust) [23].
We adopted a new format for HHblits databases in which the column state sequences used for prefiltering (former *.cs219 files) are stored in the FFindex format. The FFindex format was already used in version 2.0.16 for the a3m MSA files and the hhm profile HMM files. This resulted in a ∼4 s saving for reading the prefilter database and improved scaling of HHblits with the number of cores. We also integrated our discriminative, sequence context-sensitive method to calculate pseudocounts for the profile HMMs, which slightly improves sensitivities for fold-level homologies [26].
To keep HH-suite sustainable and expandable in the longer term, we extensively refactored code by improving code reuse with the help of new classes with inheritance, replacing POSIX threads (pthreads) with OpenMP parallelization, removing global variables, moving from make to cmake, and moving the HH-suite project to GitHub (https://github.com/soedinglab/hh-suite). We fixed various bugs such as memory leaks and segmentation faults occurring with newer compilers.
Supported platforms and hardware
HHblits is developed under Linux, tested under Linux and macOS, and should run under any Unix-like operating systems. Intel and AMD CPUs that offer AVX2 or at least SSE2 instruction sets are supported (Intel CPUs: since 2006, AMD: since 2011). PowerPC CPUs with AltiVec vector extensions are also supported.
Because we were unable to obtain funding for continued support of HH-suite, user support is unfortunately limited to bug fixes for the time being.
Parallelization by vectorization using SIMD instructions
All modern CPUs possess SIMD units, usually one per core, for performing arithmetic, logical and other operations on several data elements in parallel. In SSE2, four floating point operations are processed in a single clock cycle in dedicated 128-bit wide registers. Since 2012, the AVX standard allows to process eight floating point operations per clock cycle in parallel, held in 256 bit AVX registers. With the AVX2 extension came support for byte-, word- and integer-level operations, e.g. 32 single-byte numbers can be added or multiplied in parallel (32×1 byte=256 bits). Intel has supported AVX2 since 2013, AMD since 2015.
HHblits 2.0.16 already used SSE2 in its prefilter for gapless and gapped profile-to-sequence alignment processing 16 dynamic programming cells in parallel, but it did not support HMM-HMM alignment using vectorized code.
Abstraction layer for SIMD-based vector programming
Intrinsic functions allow to write SIMD parallelized algorithms without using assembly instructions. However, they are tied to one specific variant of SIMD instruction set (such as AVX2), which makes them neither downwards compatible nor future-proof. To be able to compile our algorithms with different SIMD instruction set variants, we implemented an abstraction layer, simd.h. In this layer, the intrinsic functions are wrapped by preprocessor macros. Porting our code to a new SIMD standard therefore merely requires us to extend the abstraction layer to that new standard, whereas the algorithm remains unchanged.
The simd.h header supports SSE2, AVX2 and AVX-512 instruction sets. David Miller has graciously extended the simd.h abstraction layer to support the AltiVec vector extension of PowerPC CPUs. Algorithm 1 shows a function that computes the scalar product of two vectors.
Vectorized viterbi HMM-HMM alignments
The viterbi algorithm for aligning profile hMMs
The Viterbi algorithm, when applied to profile HMMs, is formally equivalent to global sequence alignment with position-specific gap penalties [27]. We had previously introduced a modification of the Viterbi algorithm that is formally equivalent to Smith-Waterman local sequence alignment [15]. In HH-suite we use it to compute the best-scoring local alignment between two profile HMMs.
HH-suite models MSA columns with <50% gaps (default value) by match states and all other columns as insertion states. By traversing through the states of a profile HMM, the HMM can "emit" sequences. A match state (M) emits amino acids according to the 20 probabilities of amino acids estimated from their fraction in the MSA column, plus some pseudocounts. Insert states (I) emit amino acids according to a standard amino acid background distribution, while delete states (D) do not emit any amino acids.
The alignment score between two HMMs in HH-suite is the sum over all co-emitted sequences of the log odds scores for the probability for the two aligned HMMs to co-emit this sequence divided by the probability of the sequence under the background model. Since M and I states emit amino acids and D states do not, M and I in one HMM can only be aligned with M or I states in the other HMM. Conversely, a D state can only be aligned with a D state or with a Gap G (Fig. 1). The co-emission score can be written as the sum of the similarity scores of the aligned profile columns, in other words the match-match (MM) pair states, minus the position-specific penalties for indels: delete-open, delete-extend, insert-open and insert-extend.
HMM-HMM alignment of query and target. The alignment is represented as red path through both HMMs. The corresponding pair state sequence is MM, MM, MI, MM, MM, DG, MM
We denote the alignment pair states as MM, MI, IM, II, DD, DG, and GD. Figure 1 shows an example of two aligned profile HMMs. In the third column HMM q emits a residue from its M state and HMM p emits a residue from the I state. The pair state for this alignment column is MI. In column six of the alignment HMM q does not emit anything since it passes through the D state. HMM p does not emit anything either since it has a gap in the alignment. The corresponding pair state is DG. To speed up the alignment, we exclude pair states II and DD, and we only allow transitions between a pair state and itself and between pair state MM and pair states MI, IM, DG, or GD.
To calculate the local alignment score, we need five dynamic programming matrices SXY, one for each pair state XY ∈{MM, MI, IM, DG, GD }. They contain the score of the best partial alignment which ends in column i of q and column j of p in pair state XY. These five matrices are calculated recursively.
$$\begin{array}{*{20}l} &S_{\text{MM}}\left(i,j\right) = S_{\text{aa}}\left(q^{p}_{i},t^{p}_{j}\right)\ + S_{\text{ss}}\left(q^{ss}_{i},t^{ss}_{j}\right)\ + \\ &\max \left\{ \!\! \begin{array}{c} \begin{aligned} &0 \text{ (for {local} alignment)} \\ &S_{\text{MM}}(i\,-\,1,j\,-\,1) + \log \left(q_{i\,-\,1}(\text{M,M}) \: t_{j\,-\,1}(\text{M,M}\right)) \\ &S_{\text{MI}}(i\,-\,1,j\,-\,1) \;\,+ \log \left(q_{i\,-\,1}(\text{M,M}) \: t_{j\,-\,1}(\text{I,M}) \right) \\ &S_{\text{II}}(i\,-\,1,j\,-\,1) \;\;\: + \log \left(q_{i\,-\,1}(\text{I,M}) \: t_{j\,-\,1}(\text{M,M}) \right) \\ &S_{\text{DG}}(i\,-\,1,j\,-\,1) \: + \log \left(q_{i\,-\,1}(\text{D,M}) \: t_{j\,-\,1}(\text{M,M}) \right) \\ &S_{\text{GD}}(i\,-\,1,j\,-\,1) \: + \log \left(q_{i\,-\,1}\left(\text{M,M}\right) \: t_{j\,-\,1}(\text{D,M}) \right) \end{aligned} \end{array} \right. \end{array} $$
$$ {}{\begin{aligned} &S_{\text{MI}}\left(i,j\right) = \max \left\{ \!\! \begin{array}{c} S_{\text{MM}}(i\,-\,1,j) + \log \left(q_{i\,-\,1}(\text{M,M}) \: t_{j}(\text{D,D}) \right) \\ S_{\text{MI}}(i\,-\,1,j) + \log \left(q_{i\,-\,1}(\text{M,M}) \: t_{j}(\text{I,I}) \right) \end{array} \right. \end{aligned}} $$
$$\begin{array}{*{20}l} &S_{\text{DG}}\left(i,j\right) = \max \left\{ \!\! \begin{array}{c} S_{\text{MM}}(i\,-\,1,j) + \log \left(q_{i\,-\,1}(\text{D,M}) \right) \\ S_{\text{DG}}(i\,-\,1,j) + \log \left(q_{i\,-\,1}(\text{D,D}) \right) \end{array} \right. \!\!\! \end{array} $$
$$\begin{array}{*{20}l} &S_{aa}\left(q^{p}_{i}, t^{p}_{j}\right) = \log \sum_{a=1}^{20} \frac{q^{p}_{i}(a)\,t^{p}_{j}(a)}{f_{a}} \end{array} $$
Vector \(q^{p}_{i}\) contains the 20 amino acid probabilities of q at position i, \(t^{p}_{j}\) are the amino acid probabilities t at j, and fa denotes the background frequency of amino acid a. The score Saa measures the similarity of amino acid distributions in the two columns i and j. Sss can optionally be added to Saa. It measures the similarity of the secondary structure states of query and target HMM at i and j [15].
Vectorizations of smith-Waterman sequence alignment
Much effort has gone into accelerating the dynamic programming based Smith-Waterman algorithm (at an unchanged time complexity of O(LqLt)). While substantial accelerations using general purpose graphics processing units (GPGPUs) and field programmable gated arrays (FPGAs) were demonstrated [28–31], the need for a powerful GPGPU and the lack of of a single standard (e.g. Nvidia's proprietary CUDA versus the OpenCL standard) have been impediments. SIMD implementations using the SSE2 and AVX2 standards with on-CPU SIMD vector units have demonstrated similar speed-ups as GPGPU implementations and have become widely used [3, 4, 32–35].
To speed up the dynamic programming (DP) using SIMD, multiple cells in the DP matrix are processed jointly. However the value in cell (i,j) depends on those in the preceding cells (i−1,j−1), (i−1,j), and (i,j−1). This data dependency makes acceleration of the algorithm challenging.
Four main approaches have been developed to address this challenge: (1) parallelizing over anti-diagonal stretches of cells in the DP matrices ((i,j),(i+1,j−1),…(i+15,j−15), assuming 16 cells fit into one SIMD register) [32], (2) parallelizing over vertical or horizontal segments of the DP matrices (e.g. (i,j),(i + 1,j),…(i + 15,j)) [33], (3) parallelizing over stripes of the DP matrices ((i,j),(i+1×D,j),…(i+15×D,j) where D:=ceil(query_length/16)) [34] and (4) where 16 cells (i,j) of 16 target sequences are processed in parallel [35].
The last option is the fastest method for sequence-sequence alignments, because it avoids data dependencies. Here we present an implementation of this option that can align one query profile HMM to 4 (SSE2) or 8 (AVX2) target profile HMMs in parallel.
Vectorized viterbi algorithm for aligning profile HMMs
Algorithm 2 shows the scalar version of the Viterbi algorithm for pairwise profile HMM alignment based on the iterative update Eqs. (1)–(3). Algorithm 3 presents our vectorized and branch-less version (Fig. 2). It aligns batches of 4 or 8 target HMMs together, depending on how many scores of type float fit into one SIMD register (4 for SSE2, 8 for AVX).
SIMD parallelization over target profile HMMs. Batches of 4 or 8 database profile HMMs are aligned together by the vectorized Viterbi algorithm. Each cell (i,j) in the dynamic programming matrix is processed in parallel for 4 or 8 target HMMs
The vectorized algorithm needs to access the state transition and amino acid emission probabilities for these 4 or 8 targets at the same time. The memory is laid out (Fig. 3), such that the emission and transition probabilities of 4 or 8 targets are stored consecutively in memory. In this way, one set of 4 or 8 transition probabilities (for example MM) of the 4 or 8 target HMMs being aligned can be loaded jointly into one SIMD register.
The layout of the log transition probabilities (top) and emission probabilities (bottom) in memory for single-instruction single data (SISD) and SIMD algorithms. For the SIMD algorithm, 4 (using SSE2) or 8 (using AVX 2) target profile HMMs (t1 – t4) are stored together in interleaved fashion: the 4 or 8 transition or emission values at position i in these HMMs are stored consecutively (indicated by the same color). In this way, a single cache line read of 64 bytes can fill four SSE2 or two AVX2 SIMD registers with 4 or 8 values each
The scalar versions of the functions MAX6, MAX2 contain branches. Branched code can considerably slow down code execution due to the high cost of branch mispredictions, when the partially executed instruction pipeline has to be discarded to resume execution of the correct branch.
The functions MAX6 and MAX2 find the maximum score out of two or six input scores and also return the pair transition state that contributed the highest score. This state is stored in the backtrace matrix, which is needed to reconstruct the best-scoring alignment once all five DP matrices have been computed.
To remove the five if-statement branches in MAX6, we implemented a macro VMAX6 that implements one if-statement at a time. VMAX6 needs to be called 5 times, instead of just once as MAX6, and each call compares the current best score with the next of the 6 scores and updates the state of the best score so far by maximization. At each VMAX6 call, the current best state is overwritten by the new state if it has a better score.
We call the function VMAX2 four times to update the four states GD, IM, DG and MI. The first line in VMAX2 compares the 4 or 8 values in SIMD register sMM with the corresponding values in register sXY and sets all bits of the four values in SIMD register res_gt_vec to 1 if the value in sMM is greater than the one in sXY and to 0 otherwise. The second line computes a bit-wise AND between the four values in res_gt_vec (either 0x00000000 or 0xFFFFFFFF) and the value for state MM. For those of the 4 or 8 sMM values that were greater than the corresponding sXY value, we obtain state MM in index_vec, for the others we get zero, which represents staying in the same state. The backtrace vector can then be combined using an XOR instruction.
In order to calculate suboptimal, alternative alignments, we forbid the suboptimal alignment to pass through any cell (i,j) that is within 40 cells from any of the cells of the better-scoring alignments. These forbidden cells are stored in a matrix cell_off[i][j] in the scalar version of the Viterbi algorithm. The first if-statement in Algorithm 2 ensures that these cells obtain a score of −∞.
To reduce memory requirements in the vectorized version, the cell-off flag is stored in the most significant bit of the backtracing matrix (Fig. 5) (see "Memory reduction for backtracing and cell-off matrices" section). In the SIMD Viterbi algorithm, we shift the backtracing matrix cell-off bit to the right by one and load four 32bit (SSE2) or eight 64bit (AVX2) values into a SIMD register (line 23). We extract only the cell-off bits (line 24) by computing an AND between the co_mask and the cell_off register. We set elements in the register with cell_off bit to 0 and without to 0xFFFFFFFF by comparing if cell_mask is greater than cell_off (line 25). On line 26, we set the 4 or 8 values in the SIMD register cell_off to −∞ if their cell-off bit was set and otherwise to 0. After this we add the generated vector to all five scores (MM, MI, IM, DG and GD).
Two approaches to reduce the memory requirement for the DP score matrices from O(LqLt) to O(Lt), where Lq and Lt are lengths of the query and target profile, respectively. (Top) One vector holds the scores of the previous row, SXY(i−1,·), for pair state XY ∈{MM, MI, IM, GD and DG}, and the other holds the scores of the current row, SXY(i,·) for pair state XY ∈{MM, MI, IM, GD and DG}. Vector pointers are swapped after each row has been processed. (Bottom) A single vector per pair state XY holds the scores of the current row up to j−1 and of the previous row for j to Lt. The second approach is somewhat faster and was chosen for HH-suite3
A small improvement in runtime was achieved by compiling both versions of the Viterbi method, one with and one without cell-off logic. For the first, optimal alignment, we call the version compiled without the cell off logic and for the alternative alignments the version with cell-off logic enabled. In C/C++, this can be done with preprocessor macros.
Shorter profile HMMs are padded with probabilities of zero up to the length of the longest profile HMM in the batch (Fig. 2). Therefore, the database needs to be sorted by decreasing profile HMM length. Sorting also improves IO performance due to linear access to the target HMMs for the Viterbi alignment, since the list of target HMMs that passed the prefilter is automatically sorted by length.
Vectorized column similarity score
The sum in the profile column similarity score Saa in the first line in Algorithm 4 is is computed as the scalar product between the precomputed 20-dimensional vector \(q^{p}_{i}(a)/f_{a}\) and \(t^{p}_{j}(a)\). The SIMD code takes 39 instructions to compute the scores for 4 or 8 target columns, whereas the scalar version needed 39 instructions for a single target column.
From quadratic to linear memory for scoring matrices
Most of the memory in Algorithm 2 is needed for the five score matrices for pair states MM, MI, IM, GD and DG. For a protein of 15 000 residues, the five matrices need 15 000×15 000×4byte×5 matrices=4.5GB of memory per thread.
In a naive implementation, the vectorized algorithm would need a factor of 4 or 8 more memory than that, since it would need to store the scores of 4 or 8 target profile HMMs in the score matrices. This would require 36GB of memory per thread, or 576GB for commonly used 16 core servers.
However, we do not require the entire scoring matrices to reside in memory. We only need the backtracing matrices and the position (ibest,jbest) of the highest scoring cell to reconstruct the alignment.
We implemented two approaches. The first uses two vectors per pair state (Fig. 4 top). One holds the scores of the current row i, where (i,j) are the positions of the cell whose scores are to be computed, and the other vector holds the scores of the previous row i−1. After all the scores of a row i have been calculated, the pointers to the vectors are swapped and the former row becomes the current one.
Predecessor pair states for backtracing the Viterbi alignments are stored in a single byte of the backtrace matrix in HH-suite3 to reduce memory requirements. The bits 0 to 2 (blue) are used to store the predecessor state to the MM state, bits 3 to 6 store the predecessor of GD, IM, DG and MI pair states. The last bit denotes cells that are not allowed to be part of the suboptimal alignment because they are near to a cell that was part of a better-scoring alignment
The second approach uses only a single vector (Fig. 4 bottom). Its elements from 1 to j−1 hold the scores of the current row that have already been computed. Its elements from j to the last position Lt hold the scores from the previous row i−1.
The second variant turned out to be faster, even though it executes more instructions in each iteration. However, profiling showed that this is more than compensated by fewer cache misses, probably owed to the factor two lower memory required.
We save a lot of memory by storing the currently needed scores of the target in a linear ring buffer of size O(Lt). However, we still need to keep the backtracing matrix (see next subsection), of quadratic size O(LqLt) in memory. Therefore the memory complexity remains unaffected.
Memory reduction for backtracing and cell-off matrices
To compute an alignment by backtracing from the cell (ibest,jbest) with maximum score, we need to store for each cell (i,j) and every pair state (MM,GD,MI,DG,IM) the previous cell and pair state the alignment would pass through, that is, which cell contributed the maximum score in (i,j). For that purpose it obviously suffices to only store the previous pair state.
HHblits 2.0.16 uses five different matrices of type char, one for each pair state, and one char matrix to hold the cell-off values (in total 6 bytes). The longest known protein Titin has about 33 000 amino acids. To keep a 33 000×33 000×6byte matrix in memory, we would need 6GB of memory. Since only a fraction of ∼10−5 sequences are sequences longer than 15 000 residues in the UniProt database, we restrict the default maximum sequence length to 15 000. This limit can be increased with the parameter -maxres.
But we would still need about 1.35GB to hold the backtrace and cell-off matrices. A naive SSE2 implementation would therefore need 5.4GB, and 10.8GB with AVX2. Because every thread needs its own backtracing and cell-off matrices, this can be a severe restriction.
We reduce the memory requirements by storing all backtracing information and the cell-off flag in a single byte per cell (i,j). The preceding state for the IM, MI, GD, DG states can be held as single bit, with a 1 signifying that the preceding pair state was the same as the current one and 0 signifying it was MM. The preceding state for MM can be any of STOP, MM, IM, MI, GD, and DG. STOP represents the start of the alignment, which corresponds to the 0 in (eq. 1) contributing the largest of the 6 scores. We need three bits to store these six possible predecessor pair states. The backtracing information can, thus, be held in '4 + 3' bits, which leaves one bit for the cell-off flag (Fig. 5). Due to the reduction to one byte per cell we need only 0.9GB (with SSE2) or 1.8GB (with AVX2) per thread to hold the backtracing and cell-off information.
Viterbi early termination criterion
For some query HMMs, a lot of non-homologous target HMMs pass the prefiltering stage, for example when they contain one of the very frequent coiled coil regions. To avoid having to align thousands of non-homologous target HMMs with the costly Viterbi algorithm, we introduced an early termination criterion in HHblits 2.0.16. We averaged 1/(1+E-value) over the last 200 processed Viterbi alignments and skipped all further database HMMs when this average dropped below 0.01, indicating that the last 200 target HMMs produced very few Viterbi E-values below 1.
This criterion requires the targets to be processed by decreasing prefilter score, while our vectorized version of the Viterbi algorithm requires the database profile HMMs to be ordered by decreasing length. We solved this dilemma by sorting the list of target HMMs by decreasing prefilter score, splitting it into equal chunks (default size 2 000) with decreasing scores, and sorting target HMMs within each chunk by their lengths. After each chunk has been processed by the Viterbi algorithm, we compute the average of 1/(1+E-value) for the chunk and terminate early when this number drops below 0.01.
SIMD-based MSA redundancy filter
To build a profile HMM from an MSA, HH-suite reduces the redundancy by filtering out sequences that have more than a fraction seqid_max of identical residues with another sequence in the MSA. The scalar version of the function (Algorithm 5) returns 1 if two sequences x and y have a sequence identity above seqid_min and 0 otherwise. The SIMD version (Algorithm 6) has no branches and processes the amino acids in chunks of 16 (SSE2) or 32 (AVX2). It is about ∼11 times faster than the scalar version.
Speed benchmarks
Speed of HHsearch 2.0.16 versus HHsearch 3
Typically more than 90% of the run time of HHsearch is spent in the Viterbi algorithm, while only a fraction of the time is spent in the maximum accuracy alignment. Only a small number of alignments reach an E-value low enough in the Viterbi algorithm to be processed further. HHsearch therefore profits considerably from the SIMD vectorization of the Viterbi algorithm.
To compare the speed of the HHsearch versions, we randomly selected 1 644 sequences from Uniprot (release 2015_06), built profile HMMs, and measured the total run time for searching with the 1644 query HMMs through the PDB70 database (version 05Sep15). The PDB70 contains profile HMMs for a representative set of sequences from the PDB [24], filtered with a maximum pairwise sequence identity of 70%. It contained 35 000 profile HMMs with an average length of 234 match states.
HHsearch with SSE2 is 3.2 times faster and HHsearch with AVX2 vectorization is 4.2 times faster than HHsearch 2.0.16, averaged over all 1644 searches (Fig. 6a). For proteins longer than 1000, the speed-up factors are 5.0 and 7.4, respectively. Due to a runtime overhead of ∼20 s that is independent of the query HMM length (e.g. for reading in the profile HMMs), the speed-up shrinks for shorter queries. Most of this speed-up is owed to the vectorization of the Viterbi algorithm: The SSE2-vectorized Viterbi code ran 4.2 times faster than the scalar version.
Speed comparisons. a runtime versus query profile length for 1644 searches with profile HMMs randomly sampled from UniProt. These queries were searched against the PDB70 database containing 35 000 profile HMMs of average length 234. The average speedup over HHsearch 2.0.16 is 3.2-fold for SSE2- vectorized HHsearch and 4.2-fold for AVX2-vectorized HHsearch. b Box plot for the distribution of total runtimes (in logarithmic scale) for one, two, or three search iterations using the 1644 profile HMMs as queries. PSI-BLAST and HHMER3 searches were done against the UniProt database (version 2015_06) containing 49 293 307 sequences. HHblits searches against the uniprot20 database, a clustered version of UniProt containing profile HMMs for each of its 7 313 957 sequence clusters. Colored numbers: speed-up factors relative to HMMER3
In HHblits, only part of the runtime is spent in the Viterbi algorithm, while the larger fraction is used by the prefilter, which was already SSE2-vectorized in HHblits 2.0.16. Hence we expected only a modest speed-up between HHblits 2.0.16 and SSE2-vectorized HHblits 3. Indeed, we observed an average speed-up of 1.2, 1.3, and 1.4 for 1, 2 and 3 search iterations, respectively (Fig. 6b), whereas AVX2-vectorized version is 1.9, 2.1, and 2.3 times faster than HHblits 2.0.16, respectively. AVX2-vectorized HHblits is 14, 20, and 29 times faster than HMMER3 [4] (version 3.1b2) and 9, 10, and 11 times faster than PSI-BLAST [10] (blastpgp 2.2.31) for 1, 2, and 3 search iterations.
All runtime measurements were performed using the Unix tool time on a single core of a computer with two Intel Xeon E5-2640v3 CPUs with 128GB RAM.
Sensitivity benchmark
To measure the sensitivity of search tools to detect remotely homologous protein sequences, we used a benchmarking procedure very similar to the one described in [5]. To annotate the uniprot20 (version 2015_06) with SCOP domains, we first generated a SCOP20 sequence set by redundancy-filtering the sequences in SCOP 1.75 [25] to 20% maximum pairwise sequence identity using pdbfilter.pl with minimum coverage of 90% from HH-suite, resulting in 6616 SCOP domain sequences. We annotated a subset of uniprot20 sequences by the presence of SCOP domains by searching with each sequence in the SCOP20 set with blastpgp through the consensus sequences of the uniprot20 database and annotated the best matching sequence that covered ≥90% of the SCOP sequence and that had a minimum sequence identity of at least 30%.
We searched with PSI-BLAST (2.2.31) and HMMER3 (v3.1b2) with three iterations, using the 6616 sequences in the SCOP20 set as queries, against a database made up of the UniProt plus the SCOP20 sequence set. We searched with HHblits versions 2.0.16 and 3 with three iterations through a database consisting of the uniprot20 HMMs plus the 6616 UniProt profile HMMs annotated by SCOP domains.
We defined a sequence match as true positive if query and matched sequence were from the same SCOP superfamily and as false positive if they were from different SCOP folds and ignore all others. We excluded the self-matches as well as matches between Rossman-like folds (c.2-c.5, c.27 and 28, c.30 and 31) and between the four- to eight-bladed β-propellers (b.66-b.70), because they are probably true homologs [2]. HMMER3 reported more than one false positive hit just in one out of three queries, despite setting the maximum E-value to 100 000, and we therefore measured the sensitivity up to the first false positive (AUC1) instead of the AUC5 we had used in earlier publications.
We ran HHblits using hhblits -min_prefilter_hits 100 -n 1 -cpu $NCORES -ssm 0 -v 0 -wg and wrote checkpoint files after each iteration to restart the next iteration. We ran HMMER3 (v3.1b2) using hmmsearch –chkhmm -E 100000 and PSI-BLAST (2.2.31) using -evalue 10000 -num_descriptions 250000.
The cumulative distribution over the 6616 queries of the sensitivity at the first false positive (AUC1) in Fig. 7a shows that HHblits 3 is as sensitive as HHblits 2.0.16 for 1, 2, and 3 search iterations. Consistent with earlier results [5, 26], HHblits is considerably more sensitive than HMMER3 and PSI-BLAST.
Sensitivity of sequence search tools. a We searched with 6616 SCOP20 domain sequences through the UniProt plus SCOP20 database using one to three search iterations. The sensitivity to detect homologous sequences is measured by cumulative distribution of the Area Under the Curve 1 (AUC1), the fraction of true positives ranked better than the first false positive match. True positive matches are defined as being from the same SCOP superfamily [25], false positives have different SCOP folds, excepting known cases of inter-fold homologies. b Sensitivity of HHsearch with and without scoring secondary structure similarity, measured by the cumulative distribution of AUC1 for a comparison of 6616 profile HMMs built from SCOP20 domain sequences. Query HMMs include predicted secondary structure, target HMMs include actual secondary structure annotated by DSSP. True and false positives are defined as in A
We also compared the sensitivity of HHsearch 3 with and without scoring secondary structure similarity, because we slightly changed the weighting of the secondary structure score (Methods). We generated a profile HMM for each SCOP20 sequence using three search iterations with HHblits searches against the uniprot20 database of HMMs. We created the query set of profile HMMs by adding PSIPRED-based secondary structure predictions using the HH-suite script addss.pl, and we added structurally defined secondary structure states from DSSP [36] using addss.pl to the target profile HMMs. We then searched with all 6616 query HMMs through the database of 6616 target HMMs. True positive and false positive matches were defined as before.
Figure 7b shows that HHsearch 2.0.16 and 3 have the same sensitivity when secondary structure scoring is turned off. When turned on, HHsearch 3 has a slightly higher sensitivity due to the better weighting.
We have accelerated the algorithms most critical for runtime used in the HH-suite, most importantly the Viterbi algorithm for local and global alignments, using SIMD vector instructions. We have also added thread parallelization with OpenMP and parallelization across servers with Message Passing Interface (MPI). These extensions make the HH-suite well suited for large-scale deep protein annotation of metagenomics and genomics datasets.
Project name: HH-suite
Project page: https://github.com/soedinglab/hh-suite
Operating systems: Linux, macOS
Programming languages: C++, Python utilities
Other requirements: support for SSE2 or higher
The datasets used and/or analysed during the current study are available from the corresponding author on request.
AVX2:
advanced vector extension (SIMD instruction set standards)
HMM:
hidden Markov model
MSA:
multiple sequence alignment
SIMD:
single-instruction multiple-data
SSE2:
streaming SIMD extensions 2
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We thank the HH-suite community for their contributions and bug reports. We want to especially thank Lim Heo (Michigan State University) for fixing a bug in the Viterbi global alignment mode and David Miller for adding PowerPC support to the HH-suite.
This work was supported by the European Research Council's Horizon 2020 Framework Programme for Research and Innovation ("Virus-X", project no. 685778).
Quantitative and Computational Biology Group, Max-Planck Institute for Biophysical Chemistry, Am Fassberg 11, Munich, 81379, Germany
Martin Steinegger, Markus Meier, Milot Mirdita, Harald Vöhringer & Johannes Söding
Center for Computational Biology, McKusick-Nathans Institute of Genetic Medicine, Johns Hopkins School of Medicine, Baltimore, MD, USA
Martin Steinegger
European Bioinformatics Institute, Cambridge, CB10 1SD, United Kingdom
Harald Vöhringer
Royal College of Surgeons, Dublin, D02 YN77, Ireland
Stephan J. Haunsberger
Markus Meier
Milot Mirdita
Johannes Söding
MS & JS designed research, MS developed vectorized code and performed analyses, M. Meier refactored code, added features, fixed bugs and performed benchmarks, M. Mirdita added features, fixed bugs and maintains databases, HV implemented mmCIF support, SH optimized the MAC algorithm memory usage, MS and JS wrote the manuscript. All authors read and approved the final manuscript.
Correspondence to Johannes Söding.
Steinegger, M., Meier, M., Mirdita, M. et al. HH-suite3 for fast remote homology detection and deep protein annotation. BMC Bioinformatics 20, 473 (2019). https://doi.org/10.1186/s12859-019-3019-7
Homology detection
Protein alignment
Profile HMM
SIMD
Functional annotation | CommonCrawl |
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Molecular dynamics investigation of charging process in polyelectrolyte-based supercapacitors
Nasrin Eyvazi, Morad Biagooi & SeyedEhsan Nedaaee Oskoee
Unidirectional ion transport in nanoporous carbon membranes with a hierarchical pore architecture
Lu Chen, Bin Tu, … Kai Xiao
C. Prehal1 nAff5,
C. Koczwara ORCID: orcid.org/0000-0002-8905-05861,
H. Amenitsch ORCID: orcid.org/0000-0002-0788-13362,
V. Presser ORCID: orcid.org/0000-0003-2181-05903,4 &
O. Paris1
Nature Communications volume 9, Article number: 4145 (2018) Cite this article
An Author Correction to this article was published on 11 January 2019
This article has been updated
A fundamental understanding of ion charge storage in nanoporous electrodes is essential to improve the performance of supercapacitors or devices for capacitive desalination. Here, we employ in situ X-ray transmission measurements on activated carbon supercapacitors to study ion concentration changes during electrochemical operation. Whereas counter-ion adsorption was found to dominate at small electrolyte salt concentrations and slow cycling speed, ion replacement prevails for high molar concentrations and/or fast cycling. Chronoamperometry measurements reveal two distinct time regimes of ion concentration changes. In the first regime the supercapacitor is charged, and counter- and co-ion concentration changes align with ion replacement and partially co-ion expulsion. In the second regime, the electrode charge remains constant, but the total ion concentration increases. We conclude that the initial fast charge neutralization in nanoporous supercapacitor electrodes leads to a non-equilibrium ion configuration. The subsequent, charge-neutral equilibration slowly increases the total ion concentration towards counter-ion adsorption.
The interactions between ions, solvent molecules, and the internal surface of an electrically conductive, nanoporous electrode material determine ion electrosorption mechanisms and their related phenomena1,2,3,4. The request for further increasing the performance of supercapacitors and devices for capacitive deionization (CDI) demands a fundamental, microscopic understanding of both equilibrium and dynamic behavior of ion charge storage1,5.
When carbon-based supercapacitors are charged, the (non-Faradaic) electrode charge is counter-balanced by the ionic charge within the pore space. At the potential of zero charge (PZC), the number of cations and anions within the pores is balanced. Upon charging, there are three modes for charge-balancing: the adsorption of additional counter-ions (counter-ion adsorption), the desorption of co-ions (co-ion expulsion), or the concurrence of counter-ion adsorption and co-ion desorption (ion replacement or ion swapping)3,5. The charging mechanism is typically characterized by either identifying the difference between counter-ion and co-ion concentration at a certain electrode charge3,6 or the derivative of the latter, that is, the change of counter- and co-ion concentrations with increasing electrode charge7.
Cation and anion concentration changes during charging can be measured by different experimental methods like in situ nuclear magnetic resonance (NMR)6, electrochemical quartz crystal microbalance (eQCM)8, or in situ X-ray transmission (XRT) measurements9. In situ small-angle X-ray scattering (SAXS) and atomistic modeling10,11 have shown that in addition to concentration changes, there is local ion rearrangement across the nanopores combined with partial desolvation. Ions rearrange to optimally screen repulsive interactions between counter-ions by preferentially occupying sites with highest possible degree of confinement12. This mechanism naturally explains the often reported increase of surface-normalized capacitance with decreasing micropore size13,14. Spectroscopic techniques6,15 allow the effective measurement of concentration changes of specific chemical species within the system. By use of XRT, both cation and anion concentration changes can be quantified at the same time and correlated to the electrode charge16. Key advantages of in situ XRT are the simple experimental setup, the high time resolutions, and the flexibility of cell designs. So far, ion replacement6,9, counter-ion adsorption7,17,18, and to some extend co-ion expulsion6 have been observed during ion electrosorption in organic and aqueous electrolytes. While eQCM experiments7,8,18,19 preferentially obtained counter-ion adsorption for a number of different systems, in situ NMR6,20,21 and in situ XRT9,10 studies typically indicate the dominance of ion replacement. However, experimental conditions and key-parameters determining the dominating ion charge storage mechanism still remain to be identified. Both atomistic/molecular parameters, such as carbon/ion interactions, ion mobilities or hydration enthalpies, and macroscopic properties of the entire system, like cell design or cycling rates, might ultimately influence the charge storage mechanism in a yet unknown way.
Here we present a systematic investigation of ion electrosorption mechanisms in a microporous activated carbon-based electrical double-layer capacitor (EDLC) using aqueous electrolytes with different salt concentrations (details of all materials used, see Methods section). In situ XRT and small-angle X-ray scattering experiments during charging and discharging in a custom-built supercapacitor cell16 reveal distinct dependencies of the ion charge storage mechanism on the electrolyte salt concentration, the charging and discharging rates, the specific cell design and partially the nature of the used ions. Cation and anion concentration changes are discussed based on cyclic voltammetry (CV) data at four different scan rates. Varying the type of ions, and thus the sensitivity of the X-ray transmission of cations and anions, provides compelling evidence for the strong dependence of the storage mechanism on ion concentration, cycling speed, and cell design. Moreover, changes of cation and anion concentrations on time scales much larger than the time of the actual charging were detected during chronoamperometry (CA) measurements, suggesting that the first fast time regime does not lead to the final equilibrium configuration of the system.
Electrochemical characteristics
Cyclic voltammograms (corrected for leakage currents, see Supplementary Fig. 1, Supplementary Note 1) of in situ cells using aqueous 1, 0.1, and 0.01 M RbBr electrolyte (Fig. 1a–c) reveal differences in the capacitance and its voltage dependence. CV curves of cells with the lowest salt concentration tend to show a distinct minimum around the potential of zero charge (PZC) at low scan rates. For high molar electrolytes, such butterfly-shape is often referred to the capacitance contribution of the carbon electrode, which depends on the voltage-dependent electronic charge carrier density at the Fermi level22,23,24. In most nanoporous carbon materials, this "space charge" contribution Csc from the carbon material is not negligible compared to the contribution from the Helmholtz layer CH and the diffuse layer \(C_{\mathrm {diff}}\left( {\frac{1}{{C_{\mathrm {total}}}} = \frac{1}{{C_{\mathrm {SC}}}} + \frac{1}{{C_{\mathrm {H}}}} + \frac{1}{{C_{\mathrm {diff}}}}} \right)\). Within those models, the voltage dependence reflects the electronic density of states of the carbon material and is referred to as quantum capacitance. However, in the present work a distinct minimum around the PZC was not only visible at high concentrations but was even more pronounced for low molar electrolytes (Fig. 1c). It is therefore most probably caused by the capacitance contribution of the diffuse layer Cdiff, as predicted by the Gouy–Chapman–Stern (GCS) theory in low-concentration electrolytes25,26. The kinetic offset between the capacitance minimum in Fig. 1c during charge and discharge is largely induced by the limited ionic conductivity in low molar electrolytes. Moreover, the particular design of the in situ cell16 enlarges the diffusion pathways for ions diffusing from one electrode to the other (compared to, for example, a standard cell assembly in a Swagelok cell), also being detrimental for a good rate performance.
Cyclic voltammograms of all investigated in situ cells. Specific capacitance versus cell voltage for the in situ cells using the same activated carbon as working electrode (WE) material and different aqueous electrolytes. 1 M RbBr (a), 0.1 M RbBr (b), 0.01 M RbBr (c), 1 M CsCl (d), and 1 M NaCl (e) tested at four different scan rates (0.1–0.8 mV s−1) with ± 0.5 V applied cell voltage
Apart from the minimum around the PZC, the decrease in capacitance with lower molarities is well seen at higher scan rates. This effect is caused by the lowered ionic conductivity, and reflects the limited power handling of the system under these conditions. 1 M CsCl and 1 M NaCl cells show, compared to 1 M RbBr, neither a clear difference in capacitance nor in the shape of the CV curves (Fig. 1d, e). In addition, gravimetric capacitance, cyclic stability, and rate handling of 1 M RbBr, 1 M CsCl, and 1 M NaCl electrolytes were determined in a symmetric custom-built cell, optimized for supercapacitor performance testing27 (Supplementary Fig. 2, Supplementary Note 2). The specific capacitance at lower charging/discharging currents is practically equal for all three electrolytes and remains stable for at least 1000 cycles. A detailed discussion regarding the specific selection of electrolytes is given in Supplementary Note 3.
In situ X-ray transmission
In situ XRT allows quantifying ion concentration and corresponding changes during charging and discharging of nanoporous carbon supercapacitor electrodes9,16. The X-ray transmission τ is defined as the ratio between transmitted intensity of the (primary) X-ray beam and the incident beam intensity. According to Lambert–Beers law, the X-ray intensity decays exponentially with sample thickness d when penetrating a material (τ=exp(−μd)). The linear attenuation coefficient μ is a material-specific parameter and depends also on the energy of the incident X-ray photons. The negative logarithm of the transmission τ for the in situ supercapacitor cell corresponds to the sum of cation and anion concentrations (ccat, can) weighted by their respective mass attenuation coefficient \(\left( {\frac{\mu }{\rho }} \right)\), their molar mass M and the electrolyte "thickness" del in beam direction. Additionally, it includes a term accounting for the solvent (water) absorption (with ρH2O being the water mass density) and a constant term considering the solid phases in the beam (carbon, current collector, separator, and tape windows)16.
$$- \ln \left( \tau \right) = \left[ {c_{\mathrm{cat}}{M}_{\mathrm {cat}}\left( {\frac{\mu }{\rho }} \right)_{\mathrm{cat}} + {c}_{\mathrm{an}}{M}_{\mathrm{an}}\left( {\frac{\mu }{\rho }} \right)_{\mathrm{an}} + \rho _{\mathrm {H}_2{\mathrm{O}}}\left( {\frac{\mu }{\rho }} \right)_{\mathrm {H}_2{\mathrm{O}}}} \right]d_{\mathrm {el}} + \rho _C\left( {\frac{\mu }{\rho }} \right)_Cd_C.$$
Raw transmission data for scan rates of 0.1 mV s−1 using cells with 1 M CsCl, 1 M RbBr, 1 M NaCl, 0.1 M RbBr and 0.01 M RbBr are shown in Supplementary Fig. 3. To obtain cation and anion concentration changes independently from each other, the electrode charge needs to be calculated by integrating the measured current over time9,16. In addition, the initial cation and anion concentration within the working electrode (WE) at the PZC must be estimated16. In electrolytes with a 1 M salt concentration, the initial concentration within the WE pores should correspond in a good approximation to the bulk concentration of 1 M. For smaller salt concentrations, however, image forces of the conducting electrode attract both sorts of ions, resulting in an increased concentration within the nanopores, even at zero applied voltage28. Since the absolute value of the transmission signal (Eq. 1) contains contributions from all species in the irradiated volume, like carbons atoms, water molecules as well as cations and anions both in micropores and the bulk electrolyte in the macropores, the initial micropore ion concentration at the PZC is experimentally difficult to access.
Therefore, the ion concentration change is visualized here by plotting the negative logarithm of the transmission signal subtracted by its value at zero electrode charge In (τ0) as a function of the electrode charge (Eq. 2).
$$A = - \ln \left( \tau \right) + \ln \left( {\tau _0} \right) = \left[ {\Delta c_{\mathrm {cat}}{M}_{\mathrm {cat}}\left( {\frac{\mu }{\rho }} \right)_{\mathrm {cat}} + \Delta c_{\mathrm {an}}{M}_{\mathrm{an}}\left( {\frac{\mu }{\rho }} \right)_{\mathrm {an}} + \Delta \rho _{\mathrm {H}_2{\mathrm{O}}}\left( {\frac{\mu }{\rho }} \right)_{\mathrm {H}_2{\mathrm{O}}}} \right]d_{\mathrm {el}}.$$
Using this approach, the curves in Fig. 2 can be understood as the sum of cation and anion concentration change weighted by their corresponding effective mass attenuation coefficients (and del). We refer to the parameter A=−ln(τ)+ln(τ0) as (relative) X-ray attenuation. The representation of the parameter A in Fig. 2 is therefore in full analogy with the usual visualization of cation and anion fluxes from eQCM experiments7, and can be interpreted in the same manner.
Quantification of parameters controlling ion charge storage mechanisms. a–f Relative attenuation A (Eq. 2) vs. electrode charge for in situ cells using different aqueous electrolytes at four different scan rates with ±0.5 V maximum applied cell voltage. In a–c, the attenuation coefficient of cations (Table 1) becomes smaller from left to right, whereas in d–f, the salt concentration was decreased from left to right. The black lines indicate the theoretical attenuation curves for pure ion swapping (dashed) and counter-ion adsorption (dashed dotted) calculated from Eq. 2. Note the different scales in a–c. The charge storage mechanism is quantified in g plotting the charge storage parameter X (see Eq. 3) vs. the scan rate for RbBr. With decreasing salt concentration counter-ion adsorption becomes the dominant charge storage mechanism, as visualized in g on the right
The average density of water within the pores (i.e., the third term in Eq. 2) might also change during charging and discharging. The ions are covered by hydration shells with different water density. If the concentration of the specific ions is changing during charging, the average water density is also changing. Another reason for water densification might be an increased osmotic pressure if ion concentrations increase during charging. However, ion concentrations do not exceed 1 M and the ions are of similar size. Consequently, the influence of the water term in Eq. 2 is expected to be small and can be neglected on first approximation in this work.
To experimentally observe possible counter-ion adsorption for electrolytes with lower salt concentrations, the cell assembly had to be re-designed as compared to previous experiments9,16. The electrolyte reservoir was strongly enlarged (partially to avoid ion starvation29 in low molar electrolytes), and the WE mass was kept as small as possible. Only such large bulk electrolyte volume (about a factor of 100 larger than the WE micropore volume) ensures that the mean bulk electrolyte concentration change will be negligible during possible counter-ion adsorption when the total ion concentration increases in both electrodes.
In situ X-ray transmission during cyclic voltammetry
Cells with 1 M CsCl, 1 M RbBr, and 1 M NaCl show very similar electrochemical response (Fig. 1a, d, e), but a notably different X-ray attenuation behavior (Fig. 2a–c). These salts with their strongly different ratios of cation and anion attenuation coefficients (Table 1) enable a consistent interpretation of the experimental data using Eq. 2 (detailed discussion see Supplementary Note 3). Thus differences in the experimental data of the different salts are mainly induced by the different X-ray attenuation coefficients and not by differences regarding the ion charge storage mechanism or the electrochemical performance9. This "contrast variation" is an additional source of information and proves the general validity of the experimental approach and the corresponding model in Eq. 2. The predictions for pure counter-ion adsorption (black, dashed dotted line in Fig. 2a–f) and pure ion swapping (black dashed line) are based on Eq. 2 using the charge as input. Pure ion swapping corresponds to a linear behavior for all (both positive and negative) values of electrode charge, while counter-ion adsorption is related to a deviation from this level towards more positive values. According to Fig. 2a–c, large salt concentrations of 1 M show almost pure ion swapping as the main charging mechanism in nanoporous carbon electrodes.
Table 1 Effective (molar) attenuation coefficients, bare (crystal) ion radii, hydration enthalpies and ion mobilities for Cs+, Cl− Rb+, Br−, and Na+ ions in aqueous solution
At smaller salt concentrations and low scan rates, the counter-ion adsorption becomes more dominant (Fig. 2d–f). Notably, at larger scan rates ion swapping was observed for 0.1 M and even for 0.01 M concentrations.
It has to be noted that the theoretical prediction for pure counter-ion adsorption may not be perfectly reached by any system, since an increase of the total ion concentration within the pores should always be accompanied by a release of some water molecules and thus by a decrease of the relative attenuation A. In addition, to some degree ion swapping (or permselectivity failure), is always present close to the PZC7, making the theoretical curves for counter-ion adsorption hard to reach.
Equivalently to the point of zero mass change in eQCM data7, the minima in Fig. 2e, f might be attributed to the point of zero attenuation change (PZAC). In the case that cation and anion attenuation coefficients are similar, the PZAC might be close to the PZC. However, considering the relatively large errorbars in Fig. 2 and the influence of the actual, prevalent ion charge storage mechanism the PZAC can be only seen as a rough estimate for the position of the PZC.
In order to parametrize the charging mechanism with a single number, Forse et al3. have introduced the charging mechanism parameter X, which is +1 for pure counter-ion adsorption, 0 for pure ion swapping and −1 for pure co-ion expulsion. We calculate X for an electrode charge Q of 0.02 C m−2 via Eq. 3:
$${X}\left( {Q} \right) = \frac{{A_{\mathrm {meas}}\left( Q \right) - A_{\mathrm {theo}}^{\mathrm {swapp}}(Q)}}{{A_{\mathrm {theo}}^{\mathrm {counter}}(Q) - A_{\mathrm {theo}}^{\mathrm {swapp}}(Q)}}.$$
As an alternative definition of X(Q) the derivative dA(Q)/dQ could be used instead of A(Q) in Eq. 3. This would reflect the mass transport of counter-ions and co-ions in and out of the electrode at a certain charge, where the definition of the charging mechanism in this work is based on the total counter-ion and co-ion concentration in the pores at a specific charge. The latter is equivalent to the definition in ref. 3.
The average of X(Q) at negative and positive polarization (at ±0.02 C m−2) as a function of the scan rate (Fig. 2g) shows a distinct dependence of the charging mechanism on the scan rate and the salt concentrations. For cell designs used in this work and for large enough scan rates (≥0.4 mV s−1), ion swapping dominates for all salt concentrations investigated. However, at very small scan rates and for low salt concentrations, counter-ion adsorption tends to become competitive. This finding strongly suggests that the detailed mechanism (ion swapping or counter-ion adsorption) does not only depend on salt concentration, but also on the scan rate. A transition from ion swapping towards counter-ion adsorption is clearly observed if the scan is performed slow enough. Thus, we conclude that a simple exchange of counter- and co-ions observed for large scan rates even for very low salt concentrations must correspond to a non-equilibrium or transient state of cation and anion concentrations within the pores.
Besides the scan rate dependence (i.e., the kinetic behavior of the system), the equilibrium ion concentration is influenced by the salt concentration of the bulk electrolyte. Consequently, we expect also a dependence of the ion charge storage mechanism on the specific assembly of the EDLC cell. Cell designs with relatively small bulk electrolyte volume should suppress counter-ion adsorption, as supported by XRT data using an alternative cell assembly (Supplementary Fig. 4 and related discussion28 in Supplementary Note 4). This finding is of high relevance for the experimental characterization of supercapacitors or related technologies like CDI. Our data indicate that not only the comparability between different in situ techniques (all using very specific cell designs), but also between sophisticated experiments and commercial devices is limited. While in many eQCM studies7,17,18, a tendency toward counter-ion adsorption can be observed even for high molar electrolytes, the finite electrolyte reservoir in in situ XRT/SAXS9 or NMR experiments6 seems to support ion swapping. Consequently, specific charge storage mechanisms deduced from experimental data on the smallest accessible length scale should be generalized only if cell design, scan rate, and salt concentration are properly considered.
In situ X-ray transmission during chronoamperometry
Since the ion charge storage mechanism shows a distinct dependence on the CV scan rate even at very low scan rates, it is worthwhile to look more closely at the time dependence of the transmission signal and the relative absorption A using chronoamperometry (CA) measurements. At positive polarization, the 1 M CsCl cell (Fig. 3a) shows a fast decline of the relative absorption before it slowly increases again. The gray dashed lines indicate the theoretical curves for pure ion swapping (X = 0) and pure counter-ion adsorption (X = 1). Similarly, at negative polarization (Fig. 3b) two different regimes of the relative absorption signal can be identified: a fast increase with the same magnitude as the fast decline at positive polarization, and a subsequent slower increase. Notably, global electrode charging had stopped already after very short times (black curve on the top of Fig. 3a, b). This implies that for both polarizations the electrode charge is first counter-balanced via ion swapping by a first, fast process, while on the longer term the ion concentrations change towards counter-ion adsorption.
Ion concentration change during potentiostatic charge/discharge. Cell voltage U (blue), charge signal Q (black), and relative attenuation A (red) versus time for 1 M CsCl, 1 M RbBr, and 1 M NaCl for a single chronoamperometry step at + 0.5 V (a, c, e) and -0.5 V (b, d, f). The thick gray lines indicate the theoretical attenuation curves calculated from Eq. 2 for pure ion swapping (X = 0) and counter-ion adsorption (X = 1). Note the different ratios between cation and anion attenuation coefficients μeff of the different salts (Table 1)
The 1 M RbBr cell shows for positive polarization again two processes within strongly different time regimes (Fig. 3c). For charging, at first the effective absorption decreases, before it increases again. At negative polarization (Fig. 3d) the effective absorption shows only a very slight increase upon charging and remains rather constant after a short period of time.
The 1 M NaCl (Fig. 3e, f) cell behaves essentially like the 1 M CsCl cell, but with reversed polarization dependence due to the reversed order of the cation and anion absorption strength (see Table 1). The transmission raw-data for all CA measurements, including 0.1 M and 0.01 M RbBr, are given in Supplementary Fig. 5.
To verify whether changes of the effective attenuation can be attributed to changes within the carbon micropores only, in situ SAXS measurements were carried out using the AC electrode and aqueous 0.1 M RbBr (Supplementary Fig. 6 and Supplementary Note 5). Equivalent features of a two-step process (like in Fig. 3) were observed in the time-dependent SAXS intensity in a scattering angle regime covering ion concentration changes from micropores only9,10,16.
The observed two-step process shall now be discussed in more detail for the case of the 1 M CsCl cell (Fig. 4a). On a timescale that corresponds to conventional supercapacitor charging times (i.e., seconds to several hundred seconds) the charging mechanism corresponds to pure ion swapping. According to the electrode charge signal the actual charging process has finished after a very short period. On larger time scales, the relative absorption A in Fig. 4 increases again, while the electrode charge remains perfectly constant on these time scales. Thus, the increase can only be interpreted as a concentration increase of both cations and anions at exactly the same amount. This means that the net current is zero, but the ratio between counter-ion and co-ion concentration has changed. Effectively, this leads to a transition from ion swapping to counter-ion adsorption. In equilibrium, aqueous 1 M CsCl tends toward counter-ion adsorption for both positive and negative polarization. This implies that the number of co-ions is initially decreasing before it slightly increases again.
Two distinct time regimes of ion concentration changes. a Relative attenuation A (red data points), applied cell voltage U (blue) and charge Q (black) vs. time are shown exemplarily for a single chronoamperometry charging step at +0.5 V (left) for 1 M CsCl. The gray and the black dashed line indicate the theoretical attenuation curves for pure ion swapping (X=0) and pure counter-ion adsorption (X=1), respectively. As visualized on the right (b), ion concentration changes can be separated in two processes occurring at significantly different time scales: a fast charging regime with ion swapping as the main charging mechanism (recorded by in situ X-ray transmission, XRT) in combination with local ion rearrangement (recorded by in situ small angle X-ray scattering, SAXS) and a subsequent neutral increase of the total ion concentration (recorded by in situ XRT) causing a slow transition towards counter-ion adsorption
At time scales typically used in supercapacitor research, the dominant charge storage mechanism is ion replacement, but ion concentrations in the pores represents only a transient state. We have shown previously10, that besides global ion concentration changes, counter-ions preferably move into pore-sites with highest possible geometrical confinement (local rearrangement) and, if necessary, do so by partially striping off their solvation shell (charging regime, Fig. 4b). Features in the in situ SAXS data (Supplementary Fig. 6d) suggest that the local rearrangement mainly takes place in the charging time regime. The slow equilibration process implies only a small gain in the free energy of the system when increasing the total ion concentration (i.e., also changing the ratio of cations and anions), and the net ionic charge remains constant (neutral equilibration regime, Fig. 4b).
While the CsCl and NaCl systems show a very similar behavior (Fig. 3a, b, e, f), RbBr behaves differently. At positive polarization, co-ion expulsion occurs during charging. In the neutral equilibration regime, the total ion concentration increases, causing a transition towards ion swapping and subsequently (partial) counter-ion adsorption (Fig. 3c). At negative polarization, we see ion swapping in the charging regime (Fig. 3d). Here, cation and anion concentrations remain constant after electrode charging has stopped. Since the in-pore ion concentration in the charging regime represents a transient state, diffusive properties of the different ion species might play a significant role. Co-ion expulsion of Br− ions at positive polarization might be attributed to their high ion mobilities and consequently their smaller hydration enthalpies compared to Rb+ ions (Table 1). A high Br− mobility would imply a fast expulsion of Br− ions and a slower adsorption of Rb+ ions at positive polarization, effectively leading to co-ion expulsion at small timescales.
In summary, the present study demonstrates a clear non-equilibrium behavior of ionic charging in nanoporous supercapacitors even at slow scan rates and for highly conductive aqueous electrolytes. This emphasizes an important issue in supercapacitor research. It is extremely difficult (and requires extremely long charging times) to obtain actual stable equilibrium ion configurations in the nanoporous electrodes of a supercapacitor30. In an experimental situation, the behavior of ions on the atomic scale usually must be drawn from non-equilibrium ion configurations that can change significantly by slightly changing, e.g., temperature30, the size of the electrolyte reservoir or the salt concentration5,28.
Ion charging mechanisms on short time scales, typical for charging carbon-based supercapacitor electrodes, were shown to depend on the ionic strength of the electrolyte, the charging/discharging rate and the specific design of the supercapacitor cell. These dependencies should always be considered when comparing different in situ methods with each other or with simulation studies. So far however, the measured ion charge storage mechanisms have in most studies only been discussed regarding local ion-carbon, ion–solvent and ion–ion interactions. In situ XRT during CA measurements clearly demonstrate that the total ion concentration further increases after the actual charging/discharging of the supercapacitor has stopped. This implies a transition towards higher charging parameter X (i.e., counter-ion adsorption) although the difference between adsorbed counter- and co-ions remains equal on these timescales.
Since equilibrium ion concentration within the micropores have typically not been reached after fully charging of the supercapacitor, the actual charge storage mechanism in this time regime may be strongly influenced by kinetic properties of the different ion species. This may imply preferable co-ion desorption, if the co-ion has a high mobility or preferable counter-ion adsorption if the mobility of counter-ions is high. Moreover ion-concentration dependent changes of the ion diffusion coefficient due to mutual ion blocking might play a role31.
Further systematic investigations of non-equilibrium properties of the supercapacitors are essential to improve the comparability between fundamental studies using atomistic simulations and the various in situ experiments, enabling further progress in optimizing the performance of commercial devices in the future.
Notably most important parameters influencing ion charge storage mechanisms are properties of the entire system, such as salt concentration, charging velocity or cell design, rather than properties on the molecular scale. Therefore, most relations found in this work should be generally valid and applicable to a wide range of electrode–electrolyte combinations (including organic solvents). Subtle differences regarding the time-dependent charge storage mechanism were induced by the nature of the used ions and should depend on molecular phenomena, such as partial dehydration and the enhancement of the surface normalized capacitance in strong confinement.
The in situ experiment
In situ X-ray transmission (XRT) measurements were carried out on a laboratory SAXS instrument (NanoStar, Bruker AXS) using Cu Kα radiation and a Vantec 2000 area detector32. The transmission signal was measured using glassy carbon (GC) as a quantitative standard33, where a GC sample is placed in the beam right behind the measurement cell. In a good approximation, the transmission signal corresponds to the total intensity of the in situ cell plus GC measured by the area detector, divided by the integrated intensity of GC alone. While recording such 2D patterns from the working electrode every 90–180 s, CV or CA was applied to the in situ supercapacitor cell via a Gamry Ref600 potentiostat.
Since the photon flux of the laboratory X-ray source was too low to perform in situ SAXS experiments at higher charging rates, we used the Austrian SAXS beamline at the synchrotron radiation source ELETTRA (Trieste, Italy) to collect the SAXS data shown here. Measurements and data analysis was performed following the experimental setup and protocols described previously9,10,16.
All in situ XRT and SAXS measurements were performed with a custom-built in situ supercapacitor cell16. Holes of 6 mm diameter in the titanium and polyether ether ketone (PEEK) housing are sealed with tape, which ensures the almost undistorted transmission of the X-ray beam. The cell assembly used thin (ca. 200 nm) platinum paper as current collector (CC), an activated carbon (AC) working electrode (WE), an AC counter electrode (CE), and a Whatman GF/A glass fiber separator in-between. To provide a sufficiently large electrolyte volume for cells with low salt concentrations, five separator layers were stacked on top of each other. The asymmetric cell design (CE oversized by a factor of 15 in volume) guaranteed that the current is limited by the WE and almost the entire applied cell voltage drops at the WE. A hole of 3 mm in diameter in the CE (and in 4 out of 5 separators) ensured large enough transmission and warranted changes of the transmission and scattering signal originating from the WE only. The WEs had a diameter of 6 mm, a thickness of 200 ± 15 µm and a mass of 3.2 mg.
WEs were prepared by mixing the AC powder (MSP20, Kansai Coke and Chemicals) with ethanol and 10 mass% of dissolved polytetrafluoroethylene (PTFE, 60 mass% solution in water from Sigma Aldrich) in a mortar22. The material was rolled with a rolling machine (MSK-HRP-MR100A, MTI) to a 200 ± 15 µm thick free-standing film electrode and dried at 120 °C at 2 kPa for 24 h. Gas sorption analysis of the WE was performed using N2 and CO2 sorption of the AC electrode. Data analysis by quenched solid density functional theory34 revealed a specific surface area of 1707 m2 g−1 and an average pore size of 0.9 nm, as already reported in previous work16.
We used as electrolytes aqueous solutions of RbBr at concentrations of 1, 0.1, and 0.01 M; as well as CsCl and RbBr at concentrations of 1 M.
XRT raw data were generated at a laboratory SAXS instrument. SAXS raw data were generated at the large-scale synchrotron radiation facility ELETTRA. Derived data of this study are available from the corresponding author C.P. on request.
The original version of this Article contained an error in the Acknowledgements, which incorrectly omitted from the end the following: 'The research leading to these results has received funding from the European Community's Horizon 2020 Framework Programme under grant agreement nº 730872.' This has been corrected in both the PDF and HTML versions of the Article.
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O.P., C.P., and C.K. acknowledge financial support from the Austrian Klima- und Energiefonds via the FFG program 'Energieforschung' (Project: Hybrid Supercap). CP thanks Stefan Freunberger (ICTM, TU Graz) for discussions and support. V.P. thanks Prof. Eduard Arzt (INM) for his continuing support. The research leading to these results has received funding from the European Community's Horizon 2020 Framework Programme under grant agreement n° 730872.
C. Prehal
Present address: Institute for Chemistry and Technology of Materials, Graz University of Technology, Stremayrgasse 9/V, 8010, Graz, Austria
Institute of Physics, Montanuniversitaet Leoben, Franz-Josef Straße 18, 8700, Leoben, Austria
C. Prehal, C. Koczwara & O. Paris
Institute of Inorganic Chemistry, Graz University of Technology, Stremayrgasse 9/IV, 8010, Graz, Austria
H. Amenitsch
INM-Leibniz Institute for New Materials, Campus D2 2, 66123, Saarbrücken, Germany
V. Presser
Department of Materials Science and Engineering, Saarland University, Campus D2 2, 66123, Saarbrücken, Germany
C. Koczwara
O. Paris
C.P. carried out experiments and data analysis of the in situ XRT and SAXS measurements, and primarily developed the present work. C.K. and H.A. supported the in situ SAXS experiments. C.K. performed the electrochemical characterization of the standard supercapacitor cells. C.P., V.P., and O.P. conceptualized the work and wrote the paper.
Correspondence to C. Prehal or O. Paris.
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Prehal, C., Koczwara, C., Amenitsch, H. et al. Salt concentration and charging velocity determine ion charge storage mechanism in nanoporous supercapacitors. Nat Commun 9, 4145 (2018). https://doi.org/10.1038/s41467-018-06612-4
Received: 20 March 2018
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A trivial proof of Bertrand's postulate
Write the integers from any $n$ through $0$ descending in a column, where $n \geq 2$, and begin a second column with the value $2n$. For each entry after that, if the two numbers on that line share a factor, copy the the entry unchanged, but if they're coprime, subtract $1$.
We'll refer to the first column as $a$, where each value is the same as its index, and the second column as $b$, where the $a$th row's entry is $b_a$. The $0$-index refers to the bottom row. Equivalently,
$$ b_a = \begin{cases} 2n & \textrm{if } a = n \\ b_{a+1} - 1 & \textrm{if }\gcd(a+1,b_{a+1})=1 \\ b_{a+1} & \textrm{otherwise}\end{cases}$$
Consider the following example where $n=8$. I've also included a column showing $\gcd(a,b_a)$, and colored those $b_a$ that share a factor with $a$ and thus don't change.
$$ \begin{array}{|c|c|c|} \hline a & b_a & (a,b_a) \\ \hline 8 & \color{red}{16} & 8 \\ \hline 7 & 16 & 1 \\ \hline 6 & \color{red}{15} & 3 \\ \hline 5 & \color{red}{15} & 5 \\ \hline 4 & 15 & 1 \\ \hline 3 & 14 & 1 \\ \hline 2 & 13 & 1 \\ \hline 1 & 12 & 1 \\ \hline 0 & 11 & 11 \\ \hline \end{array} $$
Assertion: $b_0$ will always be prime.
Why? Well, suppose not, that some smaller prime $p<b_0$ divides it. In particular, let $p$ be the smallest prime factor that divides $b_0$. Since $b_0 \neq b_n$, and $p\geq 2$, we have $p<n$, so if a prime does divide $b_0$, it must be in our column of $a$ values.
$p \mid b_0 \implies p \mid b_p$. This is because $p$ can only divide $b_0$ if it has already been established by dividing $b_{kp}$ for some $k\geq 1$. A factor cannot have its first appearance at $b_0$ unless it is prime.
That said, $p \mid b_p \implies b_p = b_{p-1}$. However, that means $b_{p-1} \not\equiv b_0 \pmod {p}$, regardless of which $b_a$ decrement or not; there are one too few to make it back to our asserted divisibility, and we're left with $b_0 \not\equiv 0 \pmod {p}$, i.e. $p \nmid b_0$, a contradiction. (Recall that $b_1 - b_0 = 1$ always, preventing a constant $0 \pmod p$ all the way down.)
$$ \begin{array}{|l|l|} \hline n & 2n \\ \hline \dots & \dots \\ \hline p & b_p \equiv 0 \pmod{p} \\ \hline p-1 & b_{p-1} \equiv 0 \pmod {p} \\ \hline p-2 & b_{p-2} \equiv \{0\text{ or } p-1\} \pmod{p} \\ \hline p-3 & b_{p-3} \equiv \{0\text{ or } p-1 \text{ or }p-2\} \pmod{p} \\ \hline \dots & \dots \\ \hline 0 & b_0 \not\equiv 0 \pmod{p} \\ \hline \end{array} $$
Conclusion: As we've established there can be no smallest prime factor dividing $b_0$, it must be prime. Now that we have prime $b_0$, we can apply the same process arbitrarily with any $n$, and immediately we've shown there exists a prime in any $(n,2n)$ interval.
It's pretty clear I got the logic wrong for an important chunk of the proof, and I'm working on a clever way to solve that, but in the meantime, I have an idea for a less elegant fix.
If you look at the actual mechanism of what's going on, it's basically this. The subtracting one only when coprime essentially maintains a number (the difference $b_a - a$ for any $a$) which it's trying to rule out as a prime. This starts off as $n$, which is automatically bumped up to $n+1$ on the next line since $n \mid 2n$. Thereafter, whenever any factor in $a$ is shared by a factor in $b_a -a$, it's marking $b_a-a$ as composite and moving on. You can see that in this partial chart for $n=113$, where the right hand column is just the difference of the first two:
$$ \begin{array}{|l|l|l|} \hline 113 & 226 = 2 \cdot 113 & 113 \\ \hline 112 = 2^4\cdot 17 & 226 = 2 \cdot 113 & 114=2\cdot 3 \cdot 19 \\ \hline 111 = 3\cdot 37 & 226 = 2 \cdot 113 & 115 = 5 \cdot 23 \\ \hline 110 = 2\cdot 5\cdot 11 & 225 = 3^2 \cdot 5^2 & 115 = 5 \cdot 23 \\ \hline 109 & 225 = 3^2 \cdot 5^2 & 116 = 2^2 \cdot 29 \\ \hline 108 = 2^2 \cdot 3^3 & 224=2^5 \cdot 7 & 116 = 2^2 \cdot 29 \\ \hline 107 & 224=2^5 \cdot 7 & 117 = 3^2 \cdot 13 \\ \hline 106 = 2 \cdot 53 & 223 & 117 = 3^2 \cdot 13 \\ \hline 105 = 3 \cdot 5 \cdot 7 & 222 = 2\cdot 3 \cdot 37 & 117 = 3^2 \cdot 13 \\ \hline 104 = 2^3 \cdot 13 & 222 = 2\cdot 3 \cdot 37 & 118 = 2\cdot 59 \\ \hline 103 & 222 = 2\cdot 3 \cdot 37 & 119 = 7 \cdot 17 \\ \hline 102 = 2 \cdot 3 \cdot 17 & 221=13 \cdot 17 & 119 = 7 \cdot 17 \\ \hline 101 & 221=13 \cdot 17 & 120 = 2^3 \cdot 3 \cdot 5 \\ \hline 100 = 2^2 \cdot 5^2 & 220 = 2^2 \cdot 5 \cdot 11 & 120 = 2^3 \cdot 3 \cdot 5 \\ \hline \dots & \dots & \dots \\ \hline 88 = 2^3 \cdot 11 & 214 = 2 \cdot 107 & 126 = 2 \cdot 3^2 \cdot 7 \\ \hline 87 = 3 \cdot 29 & 214 = 2 \cdot 107 & 127 \\ \hline 86 = 2 \cdot 43 & 213 = 3 \cdot 71 & 127 \\ \hline \dots & \dots & \dots \\ \hline \end{array} $$
It takes $14$ non-decrements, which is exactly the amount needed to get from $113$ through the big gap there up to the next prime $127$, and thereafter there are no more shared factors and it stays $127$ the whole way down, and it does indeed always work like this.
So the size of the prime gap is one determiner of how long that "trial division" section lasts, and the other is the size of the factors themselves. As I said, any factor present will do, and I can't discern much rhyme or reason to it, so that leaves us with making a worst-case upper bound estimate of the sum of the least prime factors comprising every number in the prime gap. In this example, I think that adds up to $60$ or so, but it's one of if not the worst case around.
To make this rigorous, we can use the current best upper bound established on prime gap size for sufficiently large $x$ of $x^{0.525}$. If we consider some large $x$ as having a gap of that size, we can immediately mark half of those entries as being even, which means in the worst case, it would require two $a$-decrements to move past each of those entries within the gap. So that half of the gap is just
$$x^{0.525} / 2 \times 2 = x^{0.525},$$
and leaves us half left to deal with. Here, we could undoubtedly continue to whittle down our estimate by taking out other small factors, but I'm not sure that really helps anyway. Ignoring removing small factors, our bottom line is that we need
$$x^{0.525} x^k < x,$$
where $x^k$ represents an upper bound for the sum of the least prime factors in that gap, and it looks like we need $k<0.475$. I would expect that $x^k$ to work out to something more like $\log{x}$, but I'm not aware of any bounds that immediately say that.
So no, this isn't a completed proof either, but I thought I would share some of my thinking. I'm still hoping for a nice elegant solution to pop out. That said, if this approach can be made to work, that should instantly prove my approach valid for large $n$... but of course, using something more powerful than Bertrand's postulate to help do it sort of defeats the purpose. More updates later.
One other thing worth a mention. There's an easy workaround for scenarios where this fails. If $b_0=cd$, some composite, restart the process using $c, (c+1)d$, and repeat as necessary. This lets you do fun stuff like hit the prime values in $p(p+1)$.
For example, starting with $\{29, 29\cdot 30\}$ will yield $b_0=851=23\cdot 37$. Restart with $\{23, 23\cdot 37 + 23\}$, and you'll get a valid $b_0=853$. This seems to work fine empirically, but I have to doubt there's any way to justify it rigorously.
Update: Just a quicky. I got to thinking about Arnaud's note about reverse engineering, and I've got an idea to float. I tried doing some mapping of the origin possibilities for various $b_0$, and while the primes are nice and robust, the composites are not. The very best they have to offer in the first 500 or so is probably:
which makes sense, what with $209$ being a larger semiprime and $233$ up top being one half of a problem semiprime that shows up a bit.
I had hoped that that possibility graphs for the primes could be infinite, but if my code is right, it turns out they're merely far larger than the non-primes. Here's a sample:
\begin{array}{|l|l|l|l|} \hline \mathbf{b_0} & & \textbf{nodes} & \textbf{max length} \\ \hline 101 & 101 & 6206 & 818 \\ \hline 102 & 2\cdot 3\cdot 17 & 1 & 0 \\ \hline 103 & 103 & 9779 & 918 \\ \hline 104 & 2^3\cdot 13 & 1 & 0 \\ \hline 105 & 3\cdot 5\cdot 7 & 4 & 2 \\ \hline 106 & 2\cdot 53 & 1 & 0 \\ \hline 107 & 107 & 11059 & 1074 \\ \hline 108 & 2^2\cdot 3^3 & 1 & 0 \\ \hline 109 & 109 & 6293 & 1094 \\ \hline 110 & 2\cdot 5\cdot 11 & 1 & 0 \\ \hline 111 & 3\cdot 37 & 4 & 2 \\ \hline 112 & 2^4\cdot 7 & 1 & 0 \\ \hline 113 & 113 & 8886 & 1184 \\ \hline 114 & 2\cdot 3\cdot 19 & 1 & 0 \\ \hline 115 & 5\cdot 23 & 8 & 4 \\ \hline 116 & 2^2\cdot 29 & 1 & 0 \\ \hline 117 & 3^2\cdot 13 & 4 & 2 \\ \hline 118 & 2\cdot 59 & 1 & 0 \\ \hline 119 & 7\cdot 17 & 44 & 14 \\ \hline 120 & 2^3\cdot 3\cdot 5 & 1 & 0 \\ \hline 121 & 11^2 & 70 & 22 \\ \hline 122 & 2\cdot 61 & 1 & 0 \\ \hline 123 & 3\cdot 41 & 4 & 2 \\ \hline 124 & 2^2\cdot 31 & 1 & 0 \\ \hline 125 & 5^3 & 20 & 8 \\ \hline 126 & 2\cdot 3^2\cdot 7 & 1 & 0 \\ \hline 127 & 127 & 12230 & 1268 \\ \hline \end{array}
I also analyzed some parameters from the first $15000$ non-prime graphs. There are a few strong correlations, particularly that between large semiprimes and larger graphs, but the most promising find is what looks like a strong bound on the ratio of total nodes in the graph to $b_0$. It was $<1$ always, and looked to be decreasing, suggesting a strong bound might be possible. (This same ratio was $>1$ for all primes, and scaled very close to linearly.)
Since the maximum length (or height if you like) of the graph is the critical piece that determines whether or not this whole conjecture works, and since that length is a subset of the total graph, a hard limit on the number of nodes would effectively be a proof that the conjecture holds up.
To be clear, "nodes" correspond to starting pairs of numbers which would lead to a given $b_0$. The pair of numbers in question are the ones we previously called $n$ and $2n$, but we're being more flexible now. So, if it turned out there were some compelling reason why any given composite $m$ must have less than $m$ different starting pairs that led to its being $b_0$, that would be sufficient for a proof.
Latest attempt
All right. I'm going to try justifying the original $(n,2n)$ approach again.
First, however, I think it serves to look at $(n,n+2)$ as the seed pair. $n=16$ looks good for illustrative purposes. Here's a chart for it; as someone else pointed out, the $b$ column is unnecessary in this case. We could replace it with $c=b-a$, which is more clear and will share all of $b$'s relevant factors, since we're only interested in where $a$ and $b$ overlap. That said, we'll leave it in for this one.
$$ \begin{array}{|ll|ll|ll|} \hline \textbf{a} & & \textbf{b} & & \textbf{c} & \\ \hline 16 &= 2^4 & 18 &= 2 \cdot 3^2 & 2 & \\ \hline 15 &= 3\cdot 5 & 18 &= 2 \cdot 3^2 & 3 \\ \hline 14 &= 2 \cdot 7 & 18 &= 2 \cdot 3^2 & 4 &= 2^2 \\ \hline 13 & & 18 &= 2 \cdot 3^2 & 5 & \\ \hline 12 &= 2^2 \cdot 3 & 17 & & 5 & \\ \hline 11 & & 16 &= 2^4 & 5 & \\ \hline 10 &= 2 \cdot 5 & 15 &= 3\cdot 5 & 5 & \\ \hline 9 &= 3^2 & 15 &= 3\cdot 5 & 6 &= 2 \cdot 3 \\ \hline 8 &= 2^3 & 15 &= 3\cdot 5 & 7 \\ \hline 7 & & 14 &= 2 \cdot 7 & 7 & \\ \hline 6 &= 2 \cdot 3 & 14 &= 2 \cdot 7 & 8 &= 2^3 \\ \hline 5 & & 14 &= 2 \cdot 7 & 9 &=3^2 \\ \hline 4 &= 2^2 & 13 & & 9 &=3^2 \\ \hline 3 & & 12 &= 2^2 \cdot 3 & 9 &=3^2 \\ \hline 2 & & 12 &= 2^2 \cdot 3 & 10 &=2\cdot 5 \\ \hline 1 & & 12 &= 2^2 \cdot 3 & 11 &\\ \hline 0 & & 11 & & 11 &\\ \hline \end{array} $$
We're using the same system here for determining the successive values in $b$ as we did earlier: subtract $1$ when coprime with $a$, otherwise move it down unchanged.
I'd mostly like to use this table to point out there's nothing magical or inexplicable happening here. It's probably most clear in $c$: we're simply counting up from $2$, and keeping each value until it matches with a factor in $a$, and then we increment by one. Any factor at all will do, so long as it's shared with $a$.
A few things to notice. First, since $a$ is ascending with no pauses and $c$ is descending but waiting for a match before incrementing, it's natural that $c$ will grow more slowly, but given the large number of small factors available as $a$ flies by, it'll still grow a respectable amount.
Second thing, and this is really the important one, is to note the $11$ at the bottom of the column. Our whole system is predicated on the notion that this number will always be a prime, provided you plug in reasonable seed values. And this table shows why.
To state the obvious first off, we had to end on something. We didn't know it had to be prime, perhaps, but it's obvious $c$ was counting up and going to end somewhere. More to the point, note that we're not claiming that it's going to reach any specific prime yet, just that it's slowly growing. So the question is, why should we expect that bottom value to be prime necessarily?
Look at the penultimate prime, the $7$. It won't always be $7$, but there will always be a next-to-last prime, and after we hit it, there's often a spatter of small factor annihilation just like we see below. Whether this happened at $7$ or at $737$, the space and factors needed to bridge the gap to the next prime will always be available.
The upshot is that a prime will always be waiting there, since obviously no big factors are showing up between $1$ and $0$. In particular, only smaller factors come after the penultimate prime. Usually there's plenty of room; this example shows as close as it ever gets to having the prime superseded by small factors.
I realize this isn't proof-level justification that that can never happen. That said, I think I could explicitly point out a sufficiently covering bijective mapping of factors from one column to the other that always takes place, but at the moment I'm satisfied if that was persuasive.
And that's the bulk of it. I think taking $(n,n+2)$ better illustrates the underlying mechanism, but if you look closely, you'll notice that line $7$ with $14$ next to it. That means that from there down, this chart is identical to if we had used $(7,14)$ as our seed pair from the outset.
The same applies to any $(p,2p)$; there are arbitrarily many $(n,n+2)$ charts that can be cut off to get whatever pair you like. Presumably this is true for $(n,2n)$ as well, although we'll avoid that just to play it safe. And of course there is no need of actually finding such charts; if you subscribe to the validity of the example process, that should suffice to show the validity of using any $(p,2p)$ as a seed pair.
A couple of closing notes, then. When we do use $(p,2p)$, it has the additional handy feature of providing not only a prime in that range, but the very next prime larger than $p$. This should make sense after having seen our example.
And finally, do note that this gives us what we were after all along: a proof of primes in every $2n$ interval. We can of course apply this as much as we like using any arguments we want for $p$. Some of my additional data also suggest that after five or ten early exceptions have passed, we should be able to use $(4p,5p)$, and sometime after getting up to $1000$ or so, $(9p,10p)$ and even $(19p,20p)$, giving us far tighter bounds on those intervals.
I think that covers it. So what crucial element did I miss this time? Specifically, is the factor matchup stuff a critical tricky part which defeats the whole purpose if I omit it, or is it as straightforward to actually prove as I hope?
(...actually since writing that, I went and ran a battery of tests against that general principle of factor matching. It is ROBUST. This is the least of what it can do reliably. Still not a proof, but I'm much more convinced one would be easy to come up with now.)
elementary-number-theory alternative-proof solution-verification prime-gaps
TrevorTrevor
$\begingroup$ Why would it have to be $p+q$ in the $b$ column where $a=p$? This would be assuming that the number in the $b$ column keeps decreasing $1$ by $1$ (never stabilises even once) between this place and the bottom, I can't see why this would need to hold. $\endgroup$ – Arnaud Mortier Dec 13 '19 at 13:38
$\begingroup$ Seems clear to me there is a good idea here. I agree with the comments that suggest the phrasing could be improved, but it works far too well to be a simple blunder. $\endgroup$ – lulu Dec 13 '19 at 13:50
$\begingroup$ Is $p+q$ even in the table? I mean, does it have to be less than $2n$? $\endgroup$ – Arnaud Mortier Dec 13 '19 at 13:50
$\begingroup$ I don't understand why $p|b_0$ should imply $p|b_p$ - what does it mean to "establish" a factor and why should this occur exactly at $p$? The rest is clear now, but I'm unconvinced by this implication. (Also: it might help if you really fully illustrated a particular case, like $p=2$ to show that $b_0$ is odd. It's often helpful for others to include worked examples like that, since then you can avoid putting loads of indeterminates in tables and things like that) $\endgroup$ – Milo Brandt Dec 13 '19 at 22:11
$\begingroup$ @Trevor The justification that $p|b_0\implies p|b_p$ is still totally unclear to me. $\endgroup$ – Arnaud Mortier Dec 14 '19 at 10:11
Partial answer.
Conjecture 1: $b_0$ is the smallest prime larger than $n$.
Conjecture 2: $b_0$ is always a prime number as soon as $b_n$ is greater than $n+1$ and lower than some increasing bound. For a fixed $n$, all those prime values of $b_0$ make up a set of consecutive primes.
What is proved so far:
Regarding Conjecture 1
If the bottom-right value is a prime, then it's the smallest prime larger than $n$.
The conjecture is true when the gap between $n$ and the next prime is $|p-n|\leq 4$
The table below shows the range of $b_n$ values for which $b_0$ is a prime.
Proof of Conjecture 1 in the case where $n=p-1$ with $p$ prime.
The second row is $(p-2, p+(p-2))$, which are coprime numbers, and therefore by an immediate induction since $p$ is prime you can see that every subsequent row is of the form $$(a,p+a)$$ down to the last row $(0,p)$ as promised.$\,\,\square$
Proof in the case where $n=p-2$ with $p$ prime ($p>2$).
The second row is $(p-3, 2(p-2))$ and these two are not coprime: since $p>2$ is prime, $p-3$ is even. Therefore the third row is $(p-4, (p-4)+p)$ and from here we conclude the same way as before. $\,\,\square$
Proof in the case where $n=p-3$ with $p$ prime.
There you start to see some new arguments, where the proof is not constructive.
The second row is $(p-4, (p-4)+(p-2))$. They're coprime since $p$ is odd. You go down to $(p-5, (p-5)+(p-2))$. As long as you keep coprime pairs, you go down as $(p-k, (p-k)+(p-2))$. But the trick is that $p-2$ can't be prime, otherwise you wouldn't be in the case $n=p-3$, $p$ prime but rather $n=q-1$, $q$ prime (first case treated above) with $q=p-2$. So at the very least, when $a$ becomes a factor of $p-2$, you will get $(a,a+(p-2))$ and from there get down to $(a-1,(a-1)+(p-1))$.
From then on you can't stay at a difference $b-a=p-1$ for long, since $p-1$ is even. As soon as $a$ becomes even you will get up to $b-a=p$ and win.$\,\,\square$
Proof (sketch) in the case where $n=p-4$ with $p$ prime ($p>2$).
The proof for $n=p-3$ can be repeated: you're going to get rid of the difference $b-a=p-3$ very fast since $p$ is odd, you're getting rid of $b-a=p-2$ sooner or later since $p-2$ can't be a prime, and then you're getting rid of $b-a=p-1$ in at most two moves since $p$ is odd.$\,\,\square$
One problem in the general case is that you can't reverse-engineer the table, e.g. $(1,8)$ could come from $(2,8)$ or it could come from $(2,9)$.
If you add a column $b-a$, it starts at $n$, and goes non-decreasing. If it ever reaches a prime number, then it will stay at that prime number, since from then down you will have $(a=k, b=p+k)$ down to $(0,p)$ and the output will therefore be the smallest prime greater than $n$.
So all you've got to do is prove that you do reach a prime at some point. You could try to do that assuming Bertrand's postulate, it would already be some achievement.
Arnaud MortierArnaud Mortier
$\begingroup$ FYI - in general it seemed like it was possible to reverse engineer this sort of thing to the nearest prime if I remember right. $\endgroup$ – Trevor Dec 13 '19 at 14:25
The argument given makes no sense to me (and, judging from the comments, I'm not alone). To try to fix it, I suggest that you
Use some notation which lets you talk unambiguously about different rows in the table. It's fairly standard to use subscripts for states in a process, so define $$b_a = \begin{cases} 2n & \textrm{if } a = n \\ b_{a+1} - [a+1, b_{a+1} \textrm{ coprime}] & \textrm{otherwise} \end{cases}$$
Fix $2 \le p < q$ to be the smallest non-trivial factor of $q$ (assumed composite).
Work up from $a=0$ to $a=p$ rather than beginning the argument at $a=p$.
But it's not going to be an easy task, because there are unstated assumptions which don't seem to be justified. In particular, the line
And if $p \mid q$, then $p \mid q+p$. But if it did, then because the right side would be unchanged on the next line $p-1$
seems to assume that if $b_0$ is composite with prime factor $p$ then $b_p = b_0 + p$. It's easy to derive a contradiction from "$b_0$ is composite with prime factor $p$ and $b_p = b_0 + p$". It's easy to show that if $p$ is the smallest prime factor of $b_p$ then $b_0 = b_p - p$. But neither of those is anywhere near sufficient: the goal is to derive a contradiction from the much simpler statement that $b_0$ is composite.
Edit: it's now claimed explicitly that $p | b_0$ implies $p | b_p$, but to me it looks like a proof by assertion. This needs much more detail to show that there's a justified argument.
Another issue which I think should be addressed is the strength of the argument. In particular, why should the same argument not hold when we change the definition to $b_n = n^2$? It's still the case that if $b_0$ is composite then it has a prime factor $p$ which has appeared in the first column, but under these starting conditions e.g. $n=10$ yields $b_0 = 95$.
Peter TaylorPeter Taylor
$\begingroup$ Thanks for the notation suggestions, that's more or less what I intend to rewrite as when I get a chance. However, $b_a$ need not always be built around $2n$, and actually should only be when determining your $b_0$ value initially. And the fundamental contradiction is that $p \mid b_0 \implies b_p = b_{p-1}$ and then that $b_p \equiv b_{p-1} \not\equiv b_0 \pmod {p}$ implies $p \nmid b_0$ after all. $\endgroup$ – Trevor Dec 13 '19 at 19:04
$\begingroup$ I think your definition of $b_a$ needs some fixing, there are $3$ cases. $\endgroup$ – Arnaud Mortier Dec 14 '19 at 16:00
$\begingroup$ @ArnaudMortier, the square brackets in the second line are Iverson brackets. $\endgroup$ – Peter Taylor Dec 14 '19 at 19:48
$\begingroup$ @PeterTaylor Oh I see! Thanks. When I discovered the Kronecker delta as a student, we were making fun of the fact that such a trivial thing was named after someone - now I discover that it was actually named after two different people! $\endgroup$ – Arnaud Mortier Dec 14 '19 at 19:57
Let me start by saying, this is awesome!
Here is a partial answer.
Let me call the number next to $i$ on the table $a_i$. Also, I would rather work with $b_i=a_i-i$. Notice that $$ \operatorname{gcd}(i, a_i) = \operatorname{gcd}(i, a_i -i) = \operatorname{gcd}(i, b_i). $$ As we go down the table, we follow the rules:
$a_n = 2n$, so $b_n = n$.
$a_{n-1} = 2n$, so $b_{n-1} = n+1$.
If $(a_i, i) = 1$, then $a_{i-1} = a_i - 1$, so $b_{i - 1} = b_i$
If $(a_i, i) \neq 1$, then $a_{i-1} = a_i$, so $b_{i - 1} = b_i + 1$
At the end, $a_0 = b_0 = q$.
Now, if we look at the sequence $b_i$ as $i$ decreases, it will increase until it hits a prime and then it won't ever increase. I have no clue why it would reach this prime before $n$ steps.
I am like 85% confident on my coding skills and I think this works for all $n$'s up to $80000$. Also, if you look at the number of steps before you reach a prime, the numbers look half as long (as in it looks like the square root), so I am going to guess that the sequence reaches a prime pretty fast.
MoisésMoisés
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Relative sizes of prime gaps
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Convergence of the discrete to the continuous
1 Calculus vs. discrete calculus
2 Convergence under discretizations
3 Euler's method for ODEs
4 How do we measure the length of a curve in a digital image?
5 The arc-length as a limit of cell approximations of a curve
Calculus vs. discrete calculus
Calculus vs. discrete calculus? They have the same part and behave the same, so far. Indeed, this is how discrete calculus fits into the familiar calculus:
The diagram commutes. Indeed, given a function $f:{\bf R}\to {\bf R}$, we can proceed in two ways:
right then down: we acquire a $0$-form $g$ by sampling function $f$, and then we acquire $dg$ by taking the differences of the values of $g$; or, alternatively,
down then right: we acquire first the derivative $f'$ of $f$, and then we find the exterior derivative (a $1$-form) $dg$ by integrating $f'$ on the segments:
$$dg\Big([a,b] \Big):= \displaystyle\int_{[a,b]}f'dx.$$ The result is the same. Then we can say that calculus and discrete calculus develop in parallel.
But does the latter approximate the former? Will increasing the "resolution" of the discretization allow us to recover the original? What about convergence?
Example. Recall that the integrals of step-functions approximate the integral of a function, i.e.e, the area under its graph.
In contrast, an attempt to use the graphs of these functions to approximate the length of the graph fails, manifestly: $$\hspace{.29in}L:= \displaystyle\sum_{i=0}^{n-1} \Big( \tfrac{1}{n}+\tfrac{1}{n} \Big)=\tfrac{2}{n}\sum_{i=0}^{n-1} 1 =\tfrac{2}{n}n = 2.\hspace{.28in}$$
$\square$
Exercise. Use other step-functions to approximate the length of the diagonal segment.
Example. Let's use step-functions to approximate the area and the length of the circle:
Turns out that, in the digital world, $\pi =4$:
We can find other examples when a discrete geometry fails to mimic its continuous counterpart.
Exercise. Show that this is the case of the diagonal spread of liquid not being quite diagonal:
Exercise. What about the pattern of heat transfer below?
These mismatches are examples of the frequent lack of convergence of the discrete to the continuous.
Convergence under discretizations
Suppose we have a function $f$ and a sequence of real-valued $0$-forms $g_k$. Then, what do we mean when we say that, as $k\to \infty$, $$g_k:{\mathbb R} \to {\bf R} \text{ converges to } f: {\bf R} \to {\bf R}?$$
Let's start with the simplest case. Suppose an integer $k=0,1,2,...$ is given. Every function $f:{\bf R}\to {\bf R}$ is discretized by sampling points, $$g_k(s):=f\left( \frac{s}{2^k} \right), \ s\in {\bf Z}.$$ This creates a real-valued $0$-form $g_k$ but each time it is defined on a different copy of ${\mathbb R}$. Let's consider the whole setup.
Suppose we have a sequence of copies of the standard cubical complex ${\mathbb R}$, $$\{ {\mathbb R}_k:\ k=0,1,2,3...\}.$$ Then, we need to align them as if by inclusion: $${\mathbb R}_0 \subset {\mathbb R}_1 \subset {\mathbb R}_2 \subset {\mathbb R}_3 \subset ... \subset {\bf R}.$$ For that, we define these maps: $$i_k (s):=2s,\ s\in {\bf Z}.$$ They generate the chain maps $$i_k:C({\mathbb R}_k) \to C({\mathbb R}_{k+1}),\ k=0,1,2....$$
We also define the sampling maps: $$I_k:{\mathbb R}_k \to {\bf R},\ k=0,1,2...,$$ by $$I_k(s)= \frac{s}{2^k},\ s\in {\bf Z}.$$ Then, $$I_{k+1}i_k=I_k.$$
We can also endow each ${\mathbb R}_k$ with a geometry such that the edges are smaller and smaller: $$|AB|=\frac{1}{2^k}.$$ In that case, the maps above are isometries.
As $g_k$ are constructed from $f$ by sampling, the meaning of convergence is obvious:
The sequences are simply constant: for all $s\in {\bf Z}$ and for large enough $k$, we have $$g_k(s):=f(I_k(s))=f\left( \frac{s}{2^k} \right).$$
Next, what about convergence of the integrals?
We can use the outline in the last subsection. Then the approximations are, once again, "perfect". Indeed, suppose we are given an interval $[A,B]\subset {\bf R}$ and suppose $f$ is integrable on $[A,B]$. Then, for every complex $K$ representing $[A,B]$ we define $g$ as the $1$-form acquired by integrating $f$ on the edges of $K$. Then, for any $1$-chain $q$ in $K$ with $\partial q=B-A$, we have $$ \int_A^Bf(x)dx = \int_q g.$$
If, instead, we sample the original function, the approximations are non-trivial but they do converge.
We rephrase the familiar result from calculus about left-end approximations of the Riemann integral.
Theorem. Suppose we are given an interval $[A,B]\subset {\bf R}$ and suppose $f$ is integrable on $[A,B]$. Then, for every $\varepsilon >0$ there is a $\delta >0$ such that if $K$ is a geometric complex $K$ with a realization $r:K\to [A,B]$ and $\operatorname{mesh}(K)<\delta$ and if $q$ is a $1$-chain in $K$ with $\partial q=B-A$, then $$\left| \int_A^Bf(x)dx - \int_q g \right|<\varepsilon,$$ where $g$ is the $1$-form acquired by sampling $f$ at the vertices of $K$: $$g\left( PQ \right):=f(r(P)),\ PQ\in K.$$
Exercise. Prove the theorem.
Next, what about convergence of the derivatives?
The convergence is point-wise but the values are intervals rather than points. For a given $s\in {\bf Z},\ n\in {\bf Z},n\ge 0$, we have as $i\to +\infty$: $$\begin{array}{llll} \frac{dg_{n+i}\Big( [s2^i,s2^i+1] \Big)}{2^{n+i}}&=\frac{g_{n+i}(s2^i+1)-g_{n+i}(s2^i)}{2^{n+i}}\\ &=\frac{f\left( \frac{s2^i+1}{2^{n+i}} \right)- f\left( \frac{s2^i}{2^{n+i}} \right)}{2^{n+i}}\\ &=\frac{f\left( \frac{s}{2^{n}} +\frac{1}{2^{n+i}} \right)- f\left( \frac{s}{2^{n}} \right)}{2^{n+i}} \\ &\to f'\left( \frac{s}{2^n} \right) \ \text{ as } n\to \infty. \end{array}$$
Exercise. Show that, in this case, all the values of the derivative $f'$ of $f$ are the limits of sequences of values of $g_k'$, under the assumption that $f$ is continuously differentiable.
Exercise. Provide a similar analysis for ${\mathbb R}_k$ with $|AB|=1/k$.
We rephrase a familiar result from calculus.
Theorem. Suppose $f$ is continuously differentiable on $(A,B)$. Then, for every $\varepsilon >0$ there is a $\delta >0$ such that if $K$ is a geometric complex $K$ with a realization $r:K\to {\mathbb R}$ and $\operatorname{mesh}(K)<\delta$ and $x\in r(PQ)\cap (A,B),\ PQ\in K$, then $$\left| f'(x) - h'( PQ ) \right|<\varepsilon,$$ where $h$ is the $1$-form acquired by sampling $f$ on the edges of $K$: $$h\left( PQ \right):=f(r(Q))-f(r(P)).$$
Exercise. Prove the theorem. Hint: use the Mean Value Theorem.
Exercise. Provide a similar analysis for a convergent sequence of functions, $f_n\to f$.
Euler's method for ODEs
Given a function $$F:{\bf R}\times {\bf R} \to {\bf R},$$ an ordinary differential equation (ODE) of order $1$ with right-hand side function $P$ is: $$y'(t) = F(t,y(t)).$$ where we have
a differentiable function $y\in C^0({\bf R})$, and
a time instant $t\in {\bf R}$.
If we can't find $y$ explicitly, we think of the differential equation as a formula by which the slope of the tangent line to the graph of $y$ can be computed at any point on the curve, once the location of that point is known.
We use this information to approximate $y$. We do that one initial condition at a time. So, we are looking for a way to approximate $y$ that satisfies $$y'(t) = F(t,y(t)),\ \forall t\in [A,B], \qquad y(A)=X.$$
Each approximation will be a $0$-form $f$ on a complex that represents $[A,B]$ that satisfies the same initial condition, $f(A)=X$. It is built as follows.
The starting point $$(t_0,f(t_0)):=(A,X)$$ is known; then, from the differential equation, the slope at that point can be computed (it is exactly the slope of the tangent line of $y$). We take a step along that tangent line up to the next point. We choose a value for the horizontal component of this step: $$h:=(B-A)/N,$$ where $N$ is the number of steps. Then, for $n=0,1,2,...,N-1$, we have $$t_{n+1} := t_n + nh,$$ and $$f(t_{n+1}):= f(t_n) + hF(t_n,f(t_n)).$$
We recognize then that we have a geometric complex representing $[A,B]$: $$K:=\{t_{n}:n=0,1,2,...,N\}\cup \{t_{n}t_{n+1}:n=0,1,2,...,N-1\},$$ with $|t_{n}t_{n+1}|=h$.
Furthermore, we have an ODE of cochains on $K$ that approximates the original: $$f'(PQ)=G(P,f(P)),$$ where the right-hand side function $$G:C_0(K)\times {\bf R} \to {\bf R}$$ is given by the same formula as $F$. Here, we have
a cochain $f\in C^0(K)$,
a node $P\in K$, and
a edge $PQ\in K$.
On a finite interval, this approximate solution does not deviate too far from the original, unknown curve. Furthermore, the difference between the two curves can be made as small as we like by choosing the step size small enough. The general result is as follows.
Theorem. Suppose we are given an interval $[A,B]\subset {\bf R}$ and suppose $F$ has bounded partial derivatives on $[A,B]\times {\bf R}$. Suppose the initial position is given: $X\in {\bf R}$. Then, for every $\varepsilon >0$ there is a $\delta >0$ such that if $K$ is a geometric complex $K$ with a realization $r:K\to [A,B]$ and $\operatorname{mesh}(K)<\delta$, then $$\left| f(C) - y(r(C)) \right|<\varepsilon,\ \forall C\in K,$$ where
$y$ is the solution of the IVP $y'(t) = F(t,y(t)),\ y(A)=X$, and
$f$ is the solution of the IVP $f'(PQ)= F(r(P),f(P)),\ f(r^{-1}(A))=X$.
Exercise. Provide the analog of the above theorem for ODEs of chain maps $f:C(K)\to C({\mathbb R})$.
How do we measure the length of a curve in a digital image?
In order to understand the contents of an image, we need to be able to do some basic measurements of what is shown.
Example. Suppose we want to use computer vision to count nuts and bolts in an image:
Suppose we have detected and captured these objects. We can find their areas. However, since some of these objects have about the same size in terms of the area, we have to look at their shapes. We want to be able to evaluate shapes of objects and the most elementary way to do it is to compare their areas to their perimeters. The ability to compute lengths becomes crucial. $\square$
Specifically, we use roundness of objects, which is $$Roundness = 4\pi \frac{area}{perimeter^2}. $$ The roundness will tell circles from squares and squares from elongated rectangles. For example, a circle's roundness is $1$; while two rectangles may have the same area but a very different roundness:
In binary images (and cubical complexes, curves are represented as sequences of adjacent edges.
As we know, the length of the curve should be the total sum of distances between consecutive points. It is our concern however that the same "physical" curve will have different digital representations -- depending on the image's resolution and the curve's orientation with respect to the grid. For the length to be a meaningful characteristic of the "physical" curve, the representations should approximate the its length: as the resolution increases, the lengths of the digital representations should converge to the "true" length of the curve.
Exercise. Show that the area is independent of digital representation, in the above sense.
Example. Here is a simple example: the perimeters of a square and the inscribed circle are the same!
Indeed, it takes exactly as many vertical and horizontal steps to get from the left end of the circle to the top - no matter whether you make just one turn or many. The curve isn't really a curve... $\square$
Example. Let's consider a simpler example in more detail. If $r$ is the size of the pixel, a line segment of length $a$ will be represented by, roughly, $a/r$ pixels... but only if it is placed horizontally or vertically!
If it is placed diagonally, there will be $\sqrt{2}a/r$ pixels arranged in the staircase pattern. Thus we have the following:
Horizontal orientation: length $= a$;
Diagonal orientation: length $= \sqrt{2}a \approx 1.42a.$
Conclusion: the length depends on the orientation of the curve with respect to the grid of the image. $\square$
The problem described here is also known as "Weyl's tile paradox" in physics.
Thus, the digital length of a curve may vary by $40\%$!
We may try to deal with this problem the same way one would deal with other accuracy problems -- by increasing the resolution of the image. In fact, a curve can be approximated by a digital curve with any degree of accuracy:
Unfortunately, the error is independent of the resolution.
Exercise. Show why.
Example. Ignoring the error may be dangerous. The roundness is supposed to be lower for elongated objects, like bolts. Let's consider again what was shown above.
Diagonally oriented square with side a:
area $= a^2$,
perimeter $= 4\sqrt{2}a$.
Horizontally oriented rectangle with sides $b=(\sqrt{2}-1)a$ and $b'=(\sqrt{2}+1)a=5b$:
Thus, a horizontally oriented rectangle with proportions about $1$-to-$5$, just as in the image above, will have the same measurements as a diagonally oriented square. We can't tell a bolt from a nut... $\square$
Thus, approximations fail: even though computing the lengths of horizontal and vertical segments produces correct results, the lengths of diagonal segments are off by $40\%$.
A possible remedy may be to break the curve into longer pieces. Every time we have a triple of consecutive points arranged in a triangle we should replace $1+1=2$ in the computation with $\sqrt{2}$. The result is that now, at least, all $45$ degree segments have correct lengths!
But what about $30$ degree segments? Consider segments with $2$ horizontal steps followed by $1$ vertical. Then
"True" length $= \sqrt{2^2+1^2} = \sqrt{5} = 2.24. $
Old method: length $= 3$.
New method: length $= 1+\sqrt{2} = 2.41$.
The error is still almost $8\%$!
Exercise. If the relative error for the length is $n\%$, what is the relative error for the roundness?
Next we try an angle between $0$ and $30$ degrees, and so on:
To sum up:
one breaks a digital curve into a sequence of $n$-step curves,
each curve is assigned a length (such as the distance from the beginning to the end), then
the length of the original curve is the sum of those.
As the length of the edge approaches $0$, the length computed this way should converge to the "true" length. We prove that in general it does not:
For a given $n$, the relative error is constant for any resolution.
An informal "explanation" is that once $n$ is chosen, the number of possible angles that curves of $n$ steps can generate is finite. Then, there are slopes that are missed and they cannot be approximated...
We use this idea to provide a simple proof of something tangible. Suppose there are only two available slopes, $a_1,a_2>0$. Let $b$ be their average: $$b:=\frac{a_1+a_2}{2}.$$ Let's compare
the length the straight line from $(0,0)$ to $(1,b)$ with slope $b$, and
the combined lengths of two straight lines from $(0,0)$ to $(1/2,a_1/2)$ with slope $a_1$, and from $(1/2,a_1/2)$ to $(1,b)$ with slope $a_2$.
The former is $$l:=\sqrt{1+b^2}.$$ The latter is $$L_0:=\sqrt{(1/2)^2+(a_1/2)^2}+\sqrt{(1/2)^2+(a_2/2)^2}.$$ Of course, $L_0>l$.
Theorem. Given $a_1,a_2$, there is such an $\epsilon >0$ that, if $L$ is the length of any line from $(0,0)$ to $(1,\frac{a_1+a_2}{2})$ that consists of segments of slopes $a_1$ and $a_2$, then $L/l>1+\epsilon$, where $l$ is the distance between these two points.
A more general about the relative errors of these approximations is as follows.
Corollary. Given $a_1,a_2,...,a_n>0$, there is such an $\epsilon >0$ and a number $b$ with $$\min a_i \le b\le \max a_i$$ that, if $L$ is the length of any line from $(0,0)$ to $(1,b)$ that consists of segments of slopes $a_1,a_2,..., a_n$, then $L/l>1+\epsilon$, where $l$ is the distance between these two points.
Exercise. Prove the corollary. Hint: assume that $a_1>a_2>...> a_n$.
The arc-length as a limit of cell approximations of a curve
It might seem counter-intuitive that the approximations fail to converge to the correct length.
In fact, one might recall from calculus that the arc-length of a curve is defined as the limit of approximations of the curve by line segments. The difference is that here the end-points of the segments lie exactly on the curve while in our construction they may be -- within a pixel -- off the curve. As a result, in the latter case all slopes are permissible while in the former only finitely many, as we have seen. This is why these combinations of segments do converge as functions (with respect to the sup-norm) but their derivatives don't! Considering that the derivative appears in the arc-length formula itself, we shouldn't be surprised to see that there is no convergence of the lengths without convergence of the derivatives, i.e., the slopes.
A calculus-free explanation is that the geometry of ${\bf R}^2$ (and the universe) is independent of direction, i.e., isotropic. Limiting ourselves to grids with fixed directions makes our approximation of this universe anisotropic. And making the grid finer and finer won't make it isotropic.
Indeed, let's take a look at these are possible angles from a few grids:
We can always find an angle that isn't there! Then the difference of the slopes of this line and the nearest grid line determines the relative error of the approximation of the length.
Conclusion: the approximation scheme we have used is invalid.
Solution: make the set of available angles in the approximations denser and denser.
This is a way to do it: a sequence of square grids that alternates between
shrinking of the squares and
rotating the grids through smaller and smaller angles:
Specifically, we define $K_{nm}$ to be a copy of ${\mathbb R}^2$ that satisfies the following. For any $AB\in K_{nm}$ we have
$|AB|=1/n$, and
$AB$ is either parallel or perpendicular to the line $\pi/m$ from the $x$-axis.
Theorem. Suppose $\{K_{nm}\}$ is the sequence of cubical complexes defined above. Then, given a line segment $L$ between points $A$ and $B$ in ${\bf R}^2$, for any $\varepsilon >0$ there is an $N$ such that for any $n>N$ there is an $m$ and a $1$-chain $a_n$ in $K_{nm}$ such that $$\partial a_n=B_n-A_n,$$ where $A_n$ is in the star of $A$ and $B_n$ is in the star of $B$ in $K_{nm}$, and $$|a_n| < (1+ \varepsilon ) |L|.$$
Proof. For the sake of the proof we assume that it is the segment $L$ that is rotated:
For any $n$ there is an $m$ such that $$\sin \left(\alpha - \frac{m\pi}{n} \right)|L| < 1/n.$$ Now we simply choose $a_n$ to be the horizontal sequence of edges. Then, $$|a_n| < \cos \left(\alpha - \frac{m\pi}{n} \right)|L|.$$ $\blacksquare$
The problem we faced in the last subsection is resolved!
Now, what about actual curves?
Theorem. Suppose $\{K_{nm}\}$ is the sequence of cubical complexes defined above. Suppose also that $C$ is a rectifiable curve between points $A$ and $B$ in ${\bf R}^2$ with arc-length $|C|$. Then, for any $\varepsilon >0$ there is an $N$ such that for any $n>N$ there is an $m$ and a $1$-chain $a_n$ in $K_{nm}$ such that $$\partial a_n=B_n-A_n,$$ where $A_n$ is in the star of $A$ and $B_n$ is in the star of $B$ in $K_{nm}$, and $$|a_n| < (1+ \varepsilon ) |C|.$$
Proof. An outline... The curve is approximated by a sequence of segments just as in calculus. Then each segment is approximated by a $1$-chain according to the last theorem. $\blacksquare$
For the general setup, suppose $C$ is a rectifiable curve in ${\bf R}^n$ with arc-length $|C|$. To approximate it we follow the idea of the definition of the Riemann integral. Suppose we have
$\gamma:=\{K\}$ is the set of all metric cell complexes each representing ${\bf R}^n$.
Furthermore, in each $K\in\gamma$, we approximate $L$ with a sequence of edges, i.e., a $1$-chain, $$L_K=a_1+...+a_m.$$ Then the length of this curve is the sum of the lengths of the edges, $$|L_K|=|a_1|+...+|a_m|.$$ Thus we have
a collection of $1$-chains $L_K\in K\in\gamma$.
Exercise. When does this collection $\{ |L_K|: K\in\gamma \}$ "converge", to $|L|$? Hint: define two meshes.
Exercise. Consider also:
Delaunay triangulations;
barycentric subdivisions;
the averages of several rotated square grids;
random grids.
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\begin{document}
\parindent0pt \title{\bf Staircase algebras and graded nilpotent pairs}
\author{Magdalena Boos\\Ruhr-Universit\"at Bochum\\ Faculty of Mathematics\\ D - 44780 Bochum\\ [email protected]} \date{} \maketitle
\begin{abstract} We consider a class of finite-dimensional algebras, the so-called \textquotedblleft Staircase algebras\textquotedblright~ parametrized by Young diagrams. We develop a complete classification of representation types of these algebras and look into finite, tame (concealed) and wild cases in more detail. Our results are translated to the setup of graded nilpotent pairs for which we prove certain finiteness conditions.\\[1ex] Keywords: Graded nilpotent pairs, finite-dimensional algebras, representation type \end{abstract}
\section{Introduction}\label{intro} Graded nilpotent pairs naturally appear as a generalization of principal nilpotent pairs introduced by Ginzburg \cite{Gi} and studied by Panyushev \cite{Pan3}, Yu \cite{Yu} and others.\\[1ex]
Such pair is given as an element $(\varphi, \psi)$ of the nilpotent commuting variety of a fixed bigraded vector space $V$, such that both nilpotent operators are compatible with the bigrading in a natural commuting way: $\varphi$ respects the bigrading "horizontally" and $\psi$ respects the bigrading "vertically". The non-zero components of the bigrading of $V$ induce a partition $\lambda$ and the graded nilpotent pair is called $\lambda$-shaped, thus. \\[1ex] One of our main results gives an answer to a standard Lie-theoretic question: Are there only finitely many $\lambda$-shaped graded nilpotent pairs up to base change by a Levi respecting the grading? \begin{theorem*}[\ref{gradedLambda}] The number of $\lambda$-shaped graded nilpotent pairs is finite (modulo base change in the homogeneous components) if and only if \begin{enumerate}
\item $\lambda\in\{(n),~ (1^k,n-k),~ (2,n-2),~ (1^{n-4},2^2)\}$ for some $k\leq n$, \item $n\leq 8$ and $\lambda\notin\{(1,3,4),~ (2,3^2),~ (1,2^2,3),~ (1^2,2,4)\}$. \end{enumerate} \end{theorem*} Furthermore, we obtain (in)finiteness conditions for graded nilpotent pairs (modulo Levi-base change) on a fixed bigraded vector space $V$ in Lemma \ref{gradedTame} and Lemma \ref{gradedInfinite}.\\[1ex] In order to obtain our results on graded nilpotent pairs, we
introduce a class of finite-dimensional algebras, the so-called staircase algebras in Section \ref{SectStaircase}. We show that every graded nilpotent pair can be considered as a representation of such a staircase algebra and approach this class of algebras from the angle of representation theory of finite-dimensional algebras (for example with Auslander-Reiten techniques). Our results are translated to graded nilpotent pairs in Section \ref{SectGradedNilp}.\\[1ex] Staircase algebras, denoted by $\mathcal{A}(\lambda)$, are quite interesting themselves - they are parametrized by Young diagrams $Y(\lambda)$ (or, equivalently, partitions $\lambda$) and are defined by quivers with (exclusively) commutativity relations. Our main aim concerning staircase algebras is to classify their representation types completely in Section \ref{SectRepTypes}. \begin{theorem*}[\ref{reptype}] A staircase algebra $\mathcal{A}(\lambda)$ is \begin{itemize} \item representation-finite if and only if the conditions of Theorem \ref{gradedLambda} hold true. \item tame concealed if and only if $\lambda$ comes up in the following list:\\ $(3,6)$, $(1,2,6)$, $(1,3,4)$, $(2^2,5)$, $(1^2,2,4)$, $(1,2^2,3)$, $(1^3,3^2)$, $(1^3,2^3)$, $(1^4,2,3)$. \item tame, but not tame concealed if and only if $\lambda$ comes up in the following list:\\ $(4,5)$, $(5^2)$, $(1,4^2)$, $(2,3^2)$, $(3^3)$, $(2^3,3)$, $(1,2^4)$, $(2^5)$. \end{itemize}
Otherwise, $\mathcal{A}(\lambda)$ is of wild representation type. \end{theorem*} The orbit type of each staircase algebra is classified in Lemma \ref{orbittype} and we prove a correlation between representation types and orbit types for the class of staircase algebras. We obtain a complete hierarchy of algebras which makes clear the transitions between representation types and is visualized in Appendix \ref{hierarchy}.\\[1ex] In Section \ref{SectCasestudy}, the finite, tame (concealed) and minimal wild cases are examined in more detail. For example, all staircase algebras of finite type and most of the tame ones are tilted and (except for infinite families of representation-finite algebras) all Auslander-Reiten quivers of the former are attached in Appendix \ref{appendix}.\\[1ex] For each tame case, minimal nullroots which admit infinitely many isomorphism classes of representations are provided initiating the study of finiteness criteria for nilpotent graded pairs of a fixed bi-graded vector space. For each case of wild representation type, we construct a $2$-parameter family of pairwise non-isomorphic representations.~\\[2ex] {\bf Acknowledgments:} I am grateful to K. Bongartz and M. Reineke for various helpful discussions and ideas. I would like to thank M. Bulois and J. K\"ulshammer for helpful remarks and suggestions. Furthermore, I am thankful to O. Kerner and L. Unger for helping me with finding and getting to know literature and known results. \section{Theoretical background}\label{SectTheory} Let $K=\overline{K}$ be an algebraically closed field and let $\GL_n\coloneqq\GL_n(K)$ be the general linear group for a fixed integer $n\in\textbf{N}$ regarded as an affine variety. We begin by defining the notion of a graded nilpotent pair before including some facts about the representation theory of finite-dimensional algebras. We refer to \cite{ASS} for a thorough treatment of the latter. \subsection{Graded nilpotent pairs}\label{gradNilpTheory} Let $V= \bigoplus_{s,t\in\mathbf{Z}_{\geq 1}} V_{s,t}$ be an $N$-dimensional bigraded $K$-vector space; we formally set $V_{x,y}:=0$ for $(x,y)\notin \mathbf{Z}_{\geq 1}^2$.\\[1ex]
Denote by $\mathcal{N}(V)$ the \textit{nilpotent cone} of nilpotent operators on $V$. The \textit{nilpotent commuting variety} of $V$ is defined by \[\mathcal{C}(V):=\{(\varphi, \psi)\in \mathcal{N}(V)\times \mathcal{N}(V)\mid [\varphi, \psi]=0\},\] its elements are called \textit{commuting nilpotent pairs}.\\[1ex] Such pair is called \textit{graded nilpotent pair}, if $\varphi$ restrict to each $V_{s,t}$ "horizontally" via \[\varphi_{s,t}:=\varphi\mid_{V_{s,t}}: V_{s,t} \rightarrow V_{s-1,t}\] and $\psi$ restrict to each $V_{s,t}$ "vertically" via \[\psi_{s,t}:=\psi\mid_{V_{s,t}}: V_{s,t} \rightarrow V_{s,t-1}.\] Note that the study of this setup is quite natural regarding the context of principal nilpotent pairs \cite{Gi}.\\[1ex] We define the \textit{shape} of $V$ by \[\sh(V):=\{(s,t)\mid \exists p,q\in\mathbf{Z}_{\geq 1}, p\geq s, q\geq t: V_{p,q}\neq 0 \}\] which defines a Young diagram corresponding to a partition $\lambda(V)$. In more detail, the latter is given by $\lambda(V)_{i}= \sharp\{(s,h(V)-i)\in\sh(V) \mid s\in \mathbf{Z}_{\geq 1}\} $ if we define $h(V):= \max\{t\mid V_{1,t}\neq 0\} +1$.\\[1ex]
In this case, $(\varphi,\psi)$ is called a \textit{$\lambda$-graded nilpotent pair}. Note that these definitions depend on a fixed chosen grading of $V$, but not on the nilpotent pairs compatible with it. \begin{example}\label{gradedExample} Let $V:= \bigoplus_{s,t\in\mathbf{Z}_{\geq 1}} V_{s,t}$ be a bigraded $K$-vector space, such that $V_{3,1}= V_{1,3}=V_{4,1}=K$, $V_{2,1}= V_{1,2}= K^2$, $V_{2,2}=K^3$ and $V_{s,t}=0$, otherwise. Let $(\varphi, \psi)$ be a graded nilpotent pair on $V$. Then we can illustrate the latter by \begin{center}\small\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=1.81em, column sep=1.8em, text height=0.9ex, text depth=0.1ex] { V_{4,3} &V_{3,3} & V_{2,3} & V_{1,3} & & 0 & 0 & 0 & K \\ V_{4,2} & V_{3,2} & V_{2,2} & V_{1,2} & = & 0 & 0 & K^3 & K^2 \\ V_{4,1} & V_{3,1} & V_{2,1} & V_{1,1} & & K & K & K^2 & 0 \\
}; \path[->] (m-1-4) edge (m-2-4) (m-1-3) edge (m-2-3) (m-1-2) edge (m-2-2) (m-1-1) edge (m-2-1) (m-1-1) edge (m-1-2) (m-1-2) edge (m-1-3) (m-1-3) edge (m-1-4) (m-2-1) edge (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge (m-2-4) (m-2-1) edge (m-3-1) (m-2-2) edge (m-3-2) (m-2-3) edge (m-3-3) (m-2-4) edge (m-3-4) (m-3-1) edge (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge (m-3-4) (m-1-1) edge[-,dotted] (m-2-2) (m-1-2) edge[-,dotted] (m-2-3) (m-1-3) edge[-,dotted] (m-2-4) (m-2-1) edge[-,dotted] (m-3-2) (m-2-2) edge[-,dotted] (m-3-3) (m-2-3) edge[-,dotted] (m-3-4) (m-2-8) edge node[above]{$\varphi_{2,2}$} (m-2-9) (m-2-8) edge node[left]{$\psi_{2,2}$} (m-3-8) (m-1-9) edge node[right]{$\psi_{1,3}$} (m-2-9) (m-2-9) edge (m-3-9) (m-3-6) edge node[below]{$\varphi_{4,1}$} (m-3-7) (m-3-7) edge node[below]{$\varphi_{3,1}$} (m-3-8) (m-3-8) edge (m-3-9) (m-2-8) edge[-,dotted] (m-3-9) ;\end{tikzpicture}\end{center}
The Young diagram is accentuated on the right hand side, it is given by
\[Y(\lambda) = \begin{tabular}{|c|c|c}\cline{1-1} (4,1)&\multicolumn{2}{c}{}\\ \cline{1-1} (3,1)&\multicolumn{2}{c}{}\\ \cline{1-2} (2,1)&(2,2)&\\ \hline
(1,1)&\multicolumn{2}{c|}{(1,2) \hspace{0.12cm}\vline ~~(1,3)} \\ \hline \end{tabular}.\] Thus, $\lambda(V)=(1,1,2,3)$ and $(\varphi,\psi)$ is a $\lambda(V)$-graded nilpotent pair. \end{example}
\subsection{Basics of Representation Theory}\label{reptheory} A \textit{finite quiver} $\mathcal{Q}$ is a directed graph $\mathcal{Q}=(\mathcal{Q}_0,\mathcal{Q}_1,s,t)$, given by a finite set of \textit{vertices} $\mathcal{Q}_0$, a finite set of \textit{arrows} $\mathcal{Q}_1$ and two maps $s,t:\mathcal{Q}_1\rightarrow \mathcal{Q}_0$ defining the source $s(\alpha)$ and the target $t(\alpha)$ of an arrow $\alpha$. A \textit{path} is a sequence of arrows $\omega=\alpha_s\punkte\alpha_1$, such that $t(\alpha_{k})=s(\alpha_{k+1})$ for all $k\in\{1,\punkte,s-1\}$.\\[1ex]
We define the \textit{path algebra} $K\mathcal{Q}$ as the $K$-vector space with a basis consisting of all paths in $\mathcal{Q}$, formally included is a path $\varepsilon_i$ of length zero for each $i\in \mathcal{Q}_0$ starting and ending in $i$. The multiplication is defined by concatenation of paths, if possible, and equals $0$, otherwise.\\[1ex] Let us define the \textit{arrow ideal} $R_{\mathcal{Q}}$ of $K\mathcal{Q}$ as the (two-sided) ideal which is generated (as an ideal) by all arrows in $\mathcal{Q}$. In particular, the arrow ideal of $\mathcal{Q}$ equals the radical of $K\mathcal{Q}$ if $\mathcal{Q}$ does not contain oriented cycles. An arbitrary ideal $I\subseteq K\mathcal{Q}$ is called \textit{admissible} if there exists an integer $s$, such that $R_{\mathcal{Q}}^s\subset I\subset R_{\mathcal{Q}}^2$. Given such admissible ideal $I$, we denote by $I(i,j)$ the set of paths in $I$ starting in $i$ and ending in $j$.\\[1ex] Given such admissible ideal $I$, the path algebra of $\mathcal{Q}$, \textit{bound by} $I$ is defined as $\mathcal{A}:=K\mathcal{Q}/I$; it is a basic and finite-dimensional $K$-algebra \cite{ASS}; the elements of $I$ are the so-called \textit{relations} of $\mathcal{A}$. Then $\mathcal{A}$ is called \textit{triangular}, if $\mathcal{Q}$ does not contain an oriented cycle.\\[1ex] Let $\rep_K\mathcal{A}$ be the abelian $K$-linear category of all finite-dimensional $\mathcal{A}$-representations which is equivalent to the category of \textit{$K$-representations} of $\mathcal{Q}$, which are \textit{bound by $I$}, defined as follows: The objects are given by tuples $((M_i)_{i\in \mathcal{Q}_0},(M_\alpha)_{\alpha\in \mathcal{Q}_1})$, where the $M_i$ are $K$-vector spaces, and the $M_\alpha\colon M_{s(\alpha)}\rightarrow M_{t(\alpha)}$ are $K$-linear maps. For each path $\omega$ in $\mathcal{Q}$ as above, we denote $M_\omega=M_{\alpha_s}\cdot\punkte\cdot M_{\alpha_1}$ and ask a representation $M$ to fulfill $\sum_\omega\lambda_\omega M_\omega=0$ whenever $\sum_\omega\lambda_\omega\omega\in I$. A \textit{morphism of representations} $M=((M_i)_{i\in \mathcal{Q}_0},(M_\alpha)_{\alpha\in \mathcal{Q}_1})$ and
\mbox{$M'=((M'_i)_{i\in \mathcal{Q}_0},(M'_\alpha)_{\alpha\in \mathcal{Q}_1})$} consists of a tuple of $K$-linear maps $(f_i\colon M_i\rightarrow M'_i)_{i\in \mathcal{Q}_0}$, such that $f_jM_\alpha=M'_\alpha f_i$ for every arrow $\alpha\colon i\rightarrow j$ in $\mathcal{Q}_1$. \\[1ex] The \textit{dimension vector} of an $\mathcal{A}$-representation $M$ is defined by $\underline{\dim} M\in\mathbf{N}^{\mathcal{Q}_0}$; in more detail $(\underline{\dim} M)_{i}=\dim_K M_i$ for $i\in \mathcal{Q}_0$. Let us fix such a dimension vector $\underline{d}\in\mathbf{N}^{\mathcal{Q}_0}$ and denote by $\rep_K\mathcal{A}(\underline{d})$ the full subcategory of $\rep_K\mathcal{A}$ which consists of all representations of dimension vector $\underline{d}$.\\[1ex] By defining the affine space $R_{\underline{d}}K\mathcal{Q}:= \bigoplus_{\alpha\colon i\rightarrow j}\Hom_K(K^{d_i},K^{d_j})$, one realizes that its points $m$ naturally correspond to representations $M\in\rep_K K\mathcal{Q}(\underline{d})$ with $M_i=K^{d_i}$ for $i\in \mathcal{Q}_0$.
Via this correspondence, the set of such representations bound by $I$ corresponds to a closed subvariety $R_{\underline{d}}\mathcal{A}\subset R_{\underline{d}}K\mathcal{Q}$.\\[1ex] The algebraic group $\GL_{\underline{d}}=\prod_{i\in \mathcal{Q}_0}\GL_{d_i}$ acts on $R_{\underline{d}}K\mathcal{Q}$ and on $R_{\underline{d}}\mathcal{A}$ via base change, furthermore the $\GL_{\underline{d}}$-orbits $\mathcal{O}_M$ of this action are in bijection to the isomorphism classes of representations $M$ in $\rep_K\mathcal{A}(\underline{d})$.\\[1ex] Due to Krull, Remak and Schmidt, every finite-dimensional $\mathcal{A}$-representation decomposes into a direct sum of \textit{indecomposables} (which by definition do not decompose further). For certain classes of finite-dimensional algebras, a convenient tool for the classification of these indecomposable representations is the \textit{Auslander-Reiten quiver} $\Gamma_{\mathcal{A}}=\Gamma(\mathcal{Q},I)$ of $\rep_K(\mathcal{Q},I)$. Its vertices $[M]$ are given by the isomorphism classes of indecomposable representations of $\rep_K(\mathcal{Q},I)$; the arrows between two such vertices $[M]$ and $[M']$ are parametrized by a basis of the space of so-called \textit{irreducible maps} $f\colon M\rightarrow M'$. One standard technique to calculate $\Gamma_{\mathcal{A}}$ is the \textit{knitting process} (see, for example, \cite[IV.4]{ASS}).\\[1ex]
A component $\Pi_{\mathcal{A}}$ of $\Gamma_{\mathcal{A}}$ is called \textit{preprojective} if it does not contain oriented cycles and if every module in $\Pi_{\mathcal{A}}$ lies in the $\tau$-orbit of some projective. Its corresponding \textit{orbit quiver} $\Upsilon_{\mathcal{A}}$ is defined as the quiver whose vertices are given by the $\tau$-orbits $[X]$ of $\Pi_{\mathcal{A}}$. The number of arrows $[X] \rightarrow [X']$ in $\Upsilon_{\mathcal{A}}$ coincides with the maximal number of arrows $M\rightarrow M'$ in $\Pi_{\mathcal{A}}$, where $M\in [X], M'\in [X']$. The orbit type of $\mathcal{A}$ is defined to be the type of $\Upsilon_{\mathcal{A}}$. A representation $M$ is called \textit{sincere}, if every simple indecomposable of $\mathcal{A}$ comes up in a composition series of $M$.\\[1ex] We say that an algebra $\mathcal{B} = K\mathcal{Q}'/I'$ is a \textit{convex subcategory} of $\mathcal{A} = K\mathcal{Q}/I$, if $\mathcal{Q}'$ is a \textit{convex subquiver} of $\mathcal{Q}$ (that is, if two vertices $i,j$ are contained in $\mathcal{Q}'$, then every path of $\mathcal{Q}$ from $i$ to $j$ is completely contained in $\mathcal{Q}'$) and $I'\coloneqq \langle I(i,j)\mid i,j\in \mathcal{Q}'\rangle$. \\[1ex] An indecomposable projective $P$ has a so-called \textit{separated} radical, if for arbitrary two non-isomorphic direct summands of its radical, their supports are contained in different components of the subquiver $\mathcal{Q}$ obtained by deleting all starting points of paths ending in $i$. We say that $\mathcal{A}$ \textit{fulfills the separation condition}, if every projective indecomposable has a separated radical. If this condition is fulfilled, $\mathcal{A}$ admits a preprojective component, see \cite{Bo4}. In general, the definition of an algebra to be \textit{strongly simply connected} algebra is quite involved. In case of a triangular algebra $\mathcal{A}$, there is an equivalent description, though : $\mathcal{A}$ is \textit{strongly simply connected} if and only if every convex subcategory of $\mathcal{A}$ satisfies the separation condition \cite{Sko2}. \subsection{Representation types}\label{reptypetheory} Consider a finite-dimensional basic $K$-algebra $\mathcal{A}:=K\mathcal{Q}/I$. It is called of \textit{finite representation type}, if there are only finitely many isomorphism classes of indecomposable representations. If it is not of finite representation type, the algebra is of \textit{infinite representation type}. These infinite types split up into two disjoint cases; we say that the algebra $\mathcal{A}$ has
\begin{itemize}
\item \textit{tame representation type} (or \textit{is tame}) if for every integer $n$ there is an integer $m_n$ and there are finitely generated $K[x]$-$\mathcal{A}$-bimodules $M_1,\punkte,M_{m_n}$ which are free over $K[x]$, such that for all but finitely many isomorphism classes of indecomposable right $\mathcal{A}$-modules $M$ of dimension $n$, there are elements $i\in\{1,\punkte,m_n\}$ and $\lambda\in K$, such that $M\cong K[x]/(x-\lambda)\otimes_{K[x]}M_i$.
\item \textit{wild representation type} (or \textit{is wild}) if there is a finitely generated $K\langle X,Y\rangle$-$\mathcal{A}$-bimodule $M$ which is free over $K\langle X,Y\rangle$ and sends non-isomorphic finite-dimensional indecomposable $K\langle X,Y\rangle$-modules via the functor $\_\otimes_{K\langle X,Y\rangle}M$ to non-isomorphic indecomposable $\mathcal{A}$-modules. \end{itemize} In 1979, Drozd proved the following dichotomy statement \cite{Dr}. \begin{theorem}\label{dichotomy}
Every finite-dimensional algebra is either tame or wild. \end{theorem} The notion of a tame algebra $\mathcal{A}$ yields that there are at most one-parameter families of pairwise non-isomorphic indecomposable $\mathcal{A}$-modules; in the wild case there are parameter families of arbitrary many parameters.\\[1ex]
A finite-dimensional $K$-algebra is called of \textit{finite growth}, if there is a natural number $m$, such that the indecomposable finite-dimensional modules occur in each dimension $d\geq 2$ in a finite number of discrete and at most $m$ one-parameter families. \subsubsection{Criteria (via quadratic forms)}\label{quadforms} For a triangular algebra $\mathcal{A} = K\mathcal{Q}/I$, the \textit{Tits form} $q_{\mathcal{A}}:\mathbf{Z}^{\mathcal{Q}_0}\rightarrow \mathbf{Z}$ is the integral quadratic form defined by \[q_{\mathcal{A}}(v) = \sum_{i\in\mathcal{Q}_0} v_i^2 - \sum_{\alpha:i\rightarrow j\in\mathcal{Q}_1} v_iv_j + \sum_{i,j\in\mathcal{Q}_0} r(i,j)v_iv_j;\] here $r(i,j)$ equals the number of elements in $R\cap I(i,j)$ whenever $R$ is a minimal set of generators of $I$, such that $R\subseteq \bigcup_{i,j\in\mathcal{Q}_0} I(i,j)$. The corresponding symmetric bilinear form is denoted $b_{\mathcal{A}}(\_,\_)$ and fulfills the condition $q_{\mathcal{A}}(v+w)=q_{\mathcal{A}}(v)+b_{\mathcal{A}}(v,w) +q_{\mathcal{A}}(w)$.\\[1ex] Any non-zero vector $v\in \mathbf{N}^{\mathcal{Q}_0}$ is called \textit{positive}. The quadratic form $q_{\mathcal{A}}$ is called \textit{weakly positive}, if $q_{\mathcal{A}}(v) > 0$ for every $v\in\mathbf{N}^{\mathcal{Q}_0}$; and \textit{(weakly) non-negative}, if $q_{\mathcal{A}}(v) \geq 0$ for every $v\in\mathbf{Z}^{\mathcal{Q}_0}$ (or $v\in\mathbf{N}^{\mathcal{Q}_0}$, respectively).\\[1ex] For a non-negative form $q_{\mathcal{A}}$, the \textit{radical} of $q_{\mathcal{A}}$ is $\rad q_{\mathcal{A}}:=\{u\in\mathbf{Z}^{\mathcal{Q}_0} \mid q_{\mathcal{A}}(u)=0\}$, we call its elements \textit{nullroots}. In a similar manner, we define the \textit{set of isotropic roots} as $\rad^0 q_{\mathcal{A}}:=\{u\in\mathbf{N}^{\mathcal{Q}_0}\mid q_{\mathcal{A}}(u)=0\}$ and the \textit{set of rational isotropic roots} to be $\rad_{\mathbf{Q}}^0 q_{\mathcal{A}}:=\{u\in\mathbf{Q_+}^{\mathcal{Q}_0}\mid q_{\mathcal{A}}(u)=0\}$ (here, $\mathbf{Q_+}$ is the set of non-negative rational numbers). The maximal dimension of a connected halfspace in $\rad_{\mathbf{Q}}^0 q_{\mathcal{A}}$ is the \textit{isotropic corank} $\corank^0 q_{\mathcal{A}}$ of $q_{\mathcal{A}}$.\\[1ex] The definiteness of the Tits form is closely related to the representation type of $\mathcal{A}$, and there are connections between roots and certain dimension vectors of representations. Many results are, for example, summarized by De la Pe\~na and Skowro\'{n}ski in \cite{DlPS} where all definitions can be found, too. It is well known that $q_{\mathcal{A}}$ is weakly positive if $\mathcal{A}$ is representation-finite and $q_{\mathcal{A}}$ is weakly non-negative if $\mathcal{A}$ is tame. In certain cases, the opposite directions are true, as well. The following criterion for finite representation type is due to Bongartz \cite{Bo4}. \begin{theorem}\label{criterionFinite} Let $\mathcal{A} = K\mathcal{Q}/I$ be a triangular algebra, which admits a preprojective component. Then $\mathcal{A}$ is representation-finite if and only if the Tits form $q_{\mathcal{A}}$ is weakly positive. If the equivalent conditions hold true, then the dimension vector function $X\mapsto \dim X$ induces a bijection between the set of isomorphism classes of indecomposable $\mathcal{A}$-modules and the set of positive roots of $q_{\mathcal{A}}$. \end{theorem} Assume that $\Gamma_{\mathcal{A}}$ has a preprojective component. The algebra $\mathcal{A}$ is called \textit{critical} if $q_{\mathcal{A}}$ is not weakly positive, but every proper restriction of $q_{\mathcal{A}}$ is weakly positive. The term "critical" is actually intuitive, since the conditions of Theorem \ref{criterionFinite} are equivalent to $\mathcal{A}$ not having a convex subcategories which is critical; a classification of the critical algebras can be found in \cite{Bo5,HaVo}.\\[1ex]
The algebra $\mathcal{A}$ is called \textit{hypercritical} if $q_{\mathcal{A}}$ is not weakly non-negative while every proper restriction of $q_{\mathcal{A}}$ is weakly non-negative. The hypercritical algebras have been classified in Unger's list \cite{Un}. For strongly simply connected algebras, they turn out to be the minimal wild algebras as the following classification of tame types due to of Br\"ustle, De la Pe\~na and Skowro{\'n}ski yields \cite{BdlPS}.
\begin{theorem}\label{criterionTame}
Let $\mathcal{A}$ be strongly simply connnected. Then the following are equivalent: \begin{enumerate} \item $\mathcal{A}$ is tame; \item $q_{\mathcal{A}}$ is weakly non-negative; \item $\mathcal{A}$ does not contain a full convex subcategory which is hypercritical. \end{enumerate} \end{theorem}
Theorem \ref{criterionTame} and Theorem \ref{dichotomy} yield a sufficient criterion for wildness. \begin{corollary}\label{criterionWild}
Let $\mathcal{A}$ be strongly simply connnected. Whenever there exists $v\in \mathbf{N}^{\mathcal{Q}_0}$, such that $q_{\mathcal{A}}(v)\leq -1$, then $\mathcal{A}$ is of wild representation type. \end{corollary} By De la Pe\~na \cite{DlP2}, the following description of finite growth is available. \begin{proposition}\label{criterionFinGrowth} Let $\mathcal{A}$ be a strongly simply connected algebra. Then $\mathcal{A}$ is of finite growth if and only if $q_{\mathcal{A}}$ is weakly non-negative and $\corank^0 q_{\mathcal{A}}\leq 1$. \end{proposition}
\subsection{Tilted algebras}
Some of our algebras $\mathcal{A}$ do appear as tilted algebras, that is, there is a hereditary algebra $\mathcal{B}$ and a $\mathcal{B}$-tilting module $T$, such that $\End_{\mathcal{B}}(T) =\mathcal{A}$. In general, there is a sufficient criterion for an algebra to be tilted which reads as follows \cite{Ri}: \begin{lemma}\label{criterionTiltedSincere} If $\mathcal{A}$ has a preprojective component which contains an indecomposable sincere representation, then $\mathcal{A}$ is a tilted algebra. \end{lemma}
The following lemma provides a classification for certain cases.
\begin{lemma}\label{criterionTameTilted} If there is a preprojective component of $\mathcal{A}$ containing all indecomposable projective $\mathcal{A}$-modules, but no injective module, then the following are equivalent: \begin{enumerate} \item $\mathcal{A}$ is tilted of Euclidean type; \item $\mathcal{A}$ is tame; \item $q_{\mathcal{A}}$ is non-negative. \end{enumerate} \end{lemma} Especially the notion of a tame concealed algebra comes up in our setup. These are the Euclidean tilted algebras, such that the tilting module is preprojective; also shown to be the minimal representation-infinite, that is, critical algebras \cite{Ri}. The so-called Bongartz-Happel-Vossieck list (BHV-list) lists the tame concealed algebras \cite{Bo5,HaVo}.
\section{Staircase algebras}\label{SectStaircase} Let $n\in\mathbf{N}$ and let $\lambda\vdash n$ be an increasing partition of $n$. Let $Y(\lambda)$ be the Young diagram of $\lambda$, that is, the box-diagram of which the $i$-th row contains $\lambda_i$ boxes. We denote by $\lambda^{T}$ the transposed increasing partition given by the columns of the Young diagram (from right to left). Starting with $(1,1)$ in the bottom left corner, we number the boxes of $\lambda$ by $(i,j)$ increasing $i$ from bottom to top and $j$ from left to right.\\[1ex] Corresponding to $Y(\lambda)$, let us define the quiver $\mathcal{Q}(\lambda)$: its vertices are given by the tuples $(i,j)$ appearing in $Y(\lambda)$; the arrows are given by all horizontal arrows $\alpha_{i,j}: (i,j) \rightarrow (i-1,j)$ and all vertical arrows $\beta_{i,j}: (i,j) \rightarrow (i,j-1)$, whenever all these vertices are defined. The quiver $\mathcal{Q}(\lambda)$ can be easily visualized by turning the Young diagram $90^{o}$ anti-clockwise and drawing arrows from left to right and from top to bottom.\\[1ex] Let us define the path algebra $\mathcal{A}(\lambda)=K\mathcal{Q}(\lambda)/I$ with relations given by $I:=I(\lambda)$, which is the $2$-sided admissible ideal generated by all commutativity relations in the appearing squares in $\mathcal{Q}(\lambda)$ which are, if defined, of the form $\beta_{i,j}\alpha_{i,j+1} - \alpha_{i-1,j+1}\beta_{i,j+1}$. Since $I$ is admissible and $\mathcal{Q}(\lambda)$ is connected, $\mathcal{A}(\lambda)$ is a basic, connected, finite-dimensional $K$-algebra.\\[1ex] We call $n$ the size of $\mathcal{A}(\lambda)$ and $l:=l(\lambda)$ the length of $\mathcal{A}(\lambda)$, if $\lambda=(\lambda_1,...,\lambda_l)$. For easier notation, we merge similar entries of $\lambda$ by potencies, for example $(1,1,2,2,2,6,8,8)=: (1^2,2^3,6,8^2)$. Then the length is given by the number of entries in original notation; the number of entries in potency-notation is called the number of steps $s:=s(\lambda)$ of $\mathcal{A}(\lambda)$. Since the quivers $\mathcal{Q}(\lambda)$ look like staircases, the following definition is reasonable. \begin{definition}\label{defstaircase} Let $\mathcal{A}$ be a finite-dimensional $K$-algebra. It is called \begin{itemize} \item $n$-staircase algebra, if there is an increasing partition $\lambda\vdash n$, such that $\mathcal{A}\cong \mathcal{A}(\lambda)$. \item staircase algebra, if there exists a natural number $n$, such that $\mathcal{A}$ is an $n$-staircase algebra. \item $m$-step, if the number of steps equals $m$. \end{itemize} \end{definition} We denote the Tits quadratic form by $q_{\lambda}:=q_{\mathcal{A}(\lambda)}$ and the Auslander-Reiten quiver by $\Gamma(\lambda):=\Gamma_{\mathcal{A}(\lambda)}$.\\[1ex] Let us consider an example to illustrate these definitions before discussing staircase algebras and their properties in detail. \begin{example}\label{exStaircase}
Let $n=7$ and $\lambda=(1,1,2,3)$ and its Young diagram drawn in Example \ref{gradedExample}. Its quiver $\mathcal{Q}(\lambda)$ is given by \begin{center}\small\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=1.81em, column sep=1.8em, text height=0.9ex, text depth=0.1ex] { & & & (1,3) & & & & & \bullet \\
& & (2,2) & (1,2)&\hat{=} & & & \bullet & \bullet \\ (4,1) & (3,1) & (2,1) & (1,1) & &\bullet &\bullet & \bullet & \bullet\\
}; \path[->]
(m-2-3) edge node[above]{$\alpha_{2,2}$} (m-2-4) (m-2-3) edge node[right]{$\beta_{2,2}$} (m-3-3) (m-1-4) edge node[right]{$\beta_{1,3}$} (m-2-4) (m-2-4) edge node[right]{$\beta_{1,2}$} (m-3-4) (m-3-1) edge node[above]{$\alpha_{4,1}$} (m-3-2) (m-3-2) edge node[above]{$\alpha_{3,1}$} (m-3-3) (m-3-3) edge node[above]{$\alpha_{2,1}$} (m-3-4) (m-2-8) edge (m-2-9) (m-2-8) edge (m-3-8) (m-1-9) edge (m-2-9) (m-2-9) edge (m-3-9) (m-3-6) edge (m-3-7) (m-3-7) edge (m-3-8) (m-3-8) edge (m-3-9); \path[-] (m-2-8) edge[dotted] (m-3-9) ;\end{tikzpicture}\end{center} The $7$-staircase algebra $\mathcal{A}(\lambda)$ of $3$ steps and of length $4$ is defined by \[\mathcal{A}(\lambda)= K\mathcal{Q}(\lambda)/(\beta_{1,2}\alpha_{2,2} - \alpha_{2,1}\beta_{2,2}) .\] \end{example}
\subsection{Link to graded nilpotent pairs}\label{link} Each graded nilpotent pair $(\varphi, \psi)$ together with a bigraded vector space $V$ as in \ref{gradNilpTheory} can be considered as a representation $M:=M(\varphi, \psi,V)$ of $\mathcal{A}(\lambda(V))$ in a natural way. We denote $\underline{\dim}V:= \underline{\dim}_{\mathcal{A}(\lambda(V))}M$ which depends on the bigrading on $V$, but not on $(\varphi, \psi)$.
\begin{example} Consider the setup of Example \ref{gradedExample}. Then the $\mathcal{A}((1^2,2,4))$-representation $M(\varphi, \psi, V)$ is given by \begin{center}\small\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=1.81em, column sep=1.8em, text height=0.9ex, text depth=0.1ex] { & & & K \\
& & K^3 & K^2 \\
K & K & K^2 & 0 \\
}; \path[->] (m-1-4) edge node[right]{$\psi_{1,3}$} (m-2-4) (m-2-3) edge node[above]{$\varphi_{2,2}$} (m-2-4) (m-2-3) edge node[left]{$\psi_{2,2}$} (m-3-3) (m-2-4) edge (m-3-4) (m-3-1) edge node[below]{$\varphi_{4,1}$} (m-3-2) (m-3-2) edge node[below]{$\varphi_{3,1}$} (m-3-3) (m-3-3) edge (m-3-4) (m-2-3) edge[-,dotted] (m-3-4) ;\end{tikzpicture}\end{center} \end{example}
Representation-theoretically speaking, the graded nilpotent pairs of $V$ are encoded in the representation variety $R_{\underline{\dim}V}\mathcal{A}(\lambda(V))$. Therefore, certain criteria can be translated from the representation theory of finite-dimensional algebras straight away. \begin{theorem}\label{gradedCriterionLambda} Let $\lambda$ be a partition. Modulo Levi-base change, there are only finitely many $\lambda$-graded nilpotent pairs if and only if $\mathcal{A}(\lambda)$ is of finite representation type. Otherwise, there are at most one-parameter families of non-decomposable graded nilpotent pairs if and only if $\mathcal{A}(\lambda)$ is tame. \end{theorem} \begin{lemma}\label{gradedFiniteDim} The number of graded nilpotent pairs (up to Levi-base change) of the fixed bigraded vector space $V$ is finite if and only if $R_{\underline{\dim}V}\mathcal{A}(\lambda(V))$ admits only finitely many $\GL_{\underline{\dim}V}$-orbits. \end{lemma} \begin{corollary} The equivalent conditions of Lemma \ref{gradedFiniteDim} hold true if and only if the number of isomorphism classes of representations in $\rep_K\mathcal{A}(\lambda(V))(\underline{\dim} V)$ is finite. \end{corollary} These criteria motivate the classification of representation types of staircase algebras in Section \ref{SectRepTypes} and certain further considerations in Section \ref{SectCasestudy}. To start, we study general properties of staircase algebras. \subsection{General properties}\label{properties} For $(i,j)\in \mathcal{Q}(\lambda)_0$, let $S(i,j)$ be the standard simple representation at the vertex $(i,j)$ of $\mathcal{A}(\lambda)$. The (isomorphism classes of the) projective indecomposables of $\mathcal{A}(\lambda)$ are parametrized by $P(i,j)$, $(i,j)\in \mathcal{Q}(\lambda)_0$, which are given by
\[P(i,j)_{k,l}=\left\lbrace \begin{array}{ll} K & {\rm if}~ k\leq i ~{\rm and}~ l\leq j, \\ 0 & {\rm otherwise}.
\end{array}
\right. \] together with identity and zero maps, accordingly. \begin{proposition}\label{basicproperties} The algebra $\mathcal{A}(\lambda)$ is triangular, fulfills the separation condition and is strongly simply connected. \end{proposition} \begin{proof} Since $\mathcal{Q}(\lambda)$ does not contain oriented cycles, $\mathcal{A}(\lambda)$ is triangular.\\ The radical of every projective indecomposable is indecomposable, such that each projective indecomposable has separated radical and $\mathcal{A}(\lambda)$ fulfills the separation condition.\\ Let $\lambda$ be an increasing partition and consider a convex subcategory $C$ of $A(\lambda)$. We aim to show that $C$ fulfills the separation condition.
The radical of every projective $P$ of the vertex $(i,j)$ is either indecomposable or decomposes into exactly two indecomposables. The latter is the case if and only if $(i-1,j-1)$ is not a vertex of the quiver $\mathcal{Q}$ of $C$. Let $\mathcal{Q}'$ be the subquiver of $\mathcal{Q}$ which is obtained by deleting all start vertices of all paths ending in $(i,j)$. It is clear that $\mathcal{Q}'$ decomposes into two disjoint quivers, these correspond to the supports of the two indecomposable direct summands of $\rad P$. Thus, $C$ fulfills the separation condition. The algebra $\mathcal{A}(\lambda)$ is, thus, strongly simply connected. \end{proof} We approach the module categories by general techniques. Since Proposition \ref{basicproperties} states that $\mathcal{A}(\lambda)$ fulfills the separation condition, the following lemma follows from Section \ref{reptheory}). \begin{lemma}\label{existencepreproj} The algebra $\mathcal{A}(\lambda)$ admits a preprojective component $\Pi(\lambda):= \Pi_{\mathcal{A}(\lambda)}$. \end{lemma} The knowledge of the orbit type, that is, of the type of the orbit quiver $\Upsilon(\lambda):=\Upsilon_{\mathcal{A}(\lambda)}$ gives first clues about the corresponding representation types, which we classify in Theorem \ref{reptype}. \begin{lemma}\label{orbittype} The orbit type of $\mathcal{A}(\lambda)$ is \begin{enumerate} \item $A_n$, if $\lambda=(1^k,n-k)$ for some $k$, \item $D_n$, if $\lambda\in\{(1^{n-4},2^2), (2,n-2)\}$, \item $E_6$, if $\lambda\in\{(1,2,3), (2^3), (3^2)\}$, \item $E_7$, if $\lambda\in\{(1^2,2,3), (1,2,4), (1,2^3), (3,4), (1,3^2), (2^2,3)\}$, \item $E_8$, if $\lambda\in\{(1^3,2,3), (1,2,5), (1^2,2^3), (3,5), (1^2,3^2), (2^2,4), (2^4), (4^2)\}$, \item $\widetilde{E_7}$, if $\lambda\in\{(1,3,4), (1,2^2,3), (2,3^2), (1^2,2,4)\}$; and if $\lambda\in\{(3^3), (1,4^2), (2^3,3)\}$. \item $\widetilde{E_8}$, if $\lambda\in\{(3,6), (4,5), (1,2,6), (2^2,5), (1^3,3^2), (1,2^4), (1^4,2,3), (1^3,2^3)\}$ ; and if $\lambda=(5^2)$. \end{enumerate} In every other case, the orbit type is wild. Note that for every listed case except for $\lambda\in\{(3^3), (1,4^2), (2^3,3), (5^2)\}$, every indecomposable projective indecomposable comes up in the preprojective component. In the Euclidean cases, no injective indecomposable is contained in the preprojective component. \end{lemma} \begin{proof} The Auslander-Reiten quivers of every staircase algebra appearing in 1. to 5., that is, of every staircase algebra of Dynkin orbit type, are depicted in the Appendix \ref{appendix}. The orbit type can be read off directly.\\[1ex] Let $n\in\{8,9\}$, then the orbit quivers listed in 6. and 7. can easily be calculated by knitting the beginning of the preprojective components, since every projective indecomposable comes up. Every remaining case is seen to be of of wild orbit type by the same method. There are two exceptions of this procedure:\\
In case $\lambda=(3^3)$ the claim follows from knitting - but one has to realize that the projective indecomposable $P(3,3)$ is injective as well and does not appear in the preprojective component, since this case is representation-infinite, see Lemma \ref{reptypel3}.\\
In case $\lambda = (1,4^2) = (2^3,3)^T$ knitting yields the claim, but it must be proved that $P(2,4)$ is not preprojective. One method to show this fact is to knit the Auslander-Reiten quiver $\Gamma_{3,1}$ restricted to the vertex $(3,1)$; it is cyclic and there are indecomposables $U_1,..,U_8$, such that every $\tau^{-}$-translation of these will be non-zero-dimensional at the vertex $(3,1)$. Thus, it suffices to calculate the Auslander-Reiten quiver until $U_1,..,U_8$ appear and realizes that the radical of $P(2,4)$ does not appear.\\[1ex] For $n\geq 10$, every case except $(5^2)$ is of wild orbit type. The case $(5^2)$ is of type $\widetilde{E_8}$, which can be seen by knitting its preprojective component. Again, one of the projective indecomposables is injective and does not appear in the preprojective component - since this case is representation-infinite by Lemma \ref{reptypel3}. \end{proof}
\section{Representation types}\label{SectRepTypes} \subsection{Reductions}\label{reductions} In order to classify the representation types of all staircase algebra, it is of great value to have certain reduction statements available. \begin{lemma}\label{reductioncomp}
Let $\mathcal{A}$ be a convex subcategory of $\mathcal{A}'$. Then
\begin{enumerate} \item ... if $\mathcal{A}$ is tame, then $\mathcal{A}'$ is tame or wild.
\item ... if $\mathcal{A}$ is wild, then $\mathcal{A}'$ is wild. \item ... if $\mathcal{A}'$ has finite representation type, $\mathcal{A}$ has finite representation type. \item ... if $\mathcal{A}'$ is tame, then $\mathcal{A}$ is tame or of finite representation type. \end{enumerate}
In particular, if $\lambda \leq \lambda'$ is a subpartition, then these facts hold true for $\mathcal{A}=\mathcal{A}(\lambda)$ and $\mathcal{A}'=\mathcal{A}(\lambda')$.
\end{lemma} \begin{proof}
The claim follows from general representation theory of quivers with relations \cite{ASS} by restricting in a natural way or expanding with zero. \end{proof} \begin{lemma}\label{reductionsymm} For each partition $\lambda$, the representation type of $\mathcal{A}(\lambda)$ and $\mathcal{A}(\lambda^{T})$ is the same. \end{lemma}
\subsection{Classification} We provide a complete classification of the representation type of a staircase algebra. In order to assure a nice structure of the presentation, we begin by classifying the representation types of staircase algebras of length $2$ in Lemma \ref{reptypel2} and of length $3$ in Lemma \ref{reptypel3}, since most of the staircase algebras of finite and tame representation type come up in these contexts. We then generalize the results to arbitrary staircase algebras in Theorem \ref{reptype} which completes the classification. \begin{lemma}\label{reptypel2}
A staircase algebra $\mathcal{A}(\lambda)$ of length $2$, that is, $\lambda=(\lambda_1,\lambda_2)\vdash n$, is
\begin{itemize}
\item of finite representation type if and only if $n\leq 8$ or $\lambda_1\in\{1,2\}$.
\item tame concealed if and only if $\lambda=(3,6)$.
\item tame, but not tame concealed, if and only if $\lambda\in\{(4,5), (5^2)\}$. \end{itemize} Otherwise, $\mathcal{A}(\lambda)$ is of wild representation type. \end{lemma}
\begin{proof} The following table shows the structure of our proof; the marked ones show which cases need to be proved in order to classify every remaining non-marked case due to reductions via Lemma \ref{reductioncomp} and Lemma \ref{reductionsymm} (W=Wild, T=Tame, TC= Tame concealed, F=Finite):
\begin{center}
\begin{tabular}{l||l|l|l|l|l|l|l|l|l|l|} \hline \vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{lightviolet}\vdots &\cellcolor{lightviolet}\vdots&\cellcolor{lightviolet}\vdots&\cellcolor{lightviolet}\vdots\\ \hline 6 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{lightviolet}W &\cellcolor{lightviolet}W &\cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 5 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{dlightblue}\textbf{T} & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 4 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{dlightgreen}\textbf{F} & \cellcolor{dlightblue}\textbf{T} & \cellcolor{dlightviolet}\textbf{W} & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 3 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{dlightgreen}\textbf{F} & \cellcolor{dlightblue} \textbf{TC} & \cellcolor{dlightviolet}\textbf{W} & \cellcolor{lightviolet}W &\cellcolor{lightviolet} \cellcolor{lightviolet}$\cdots$\\ \hline 2 & \cellcolor{grey}x & \cellcolor{lightgreen}F &\cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F &\cellcolor{dlightgreen}\textbf{F} & \cellcolor{dlightgreen}\textbf{F}& \cellcolor{dlightgreen}\textbf{F}& \cellcolor{dlightgreen}$\cdots$\\ \hline 1 & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F &\cellcolor{lightgreen}F & \cellcolor{lightgreen}F &\cellcolor{lightgreen}F& \cellcolor{lightgreen}F& \cellcolor{dlightgreen}$\cdots$\\ \hline\hline $\lambda_1/\lambda_2$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & $\cdots$ \end{tabular} \end{center} Representation-finite cases: Let $\lambda_1=1$, then $\mathcal{Q}(\lambda)$ is of type $A_n$ which is representation-finite.
For the remaining maximal finite cases, the Auslander-Reiten quivers (or their symmetric versions), are attached in the Appendix \ref{appendix}. In particular, if $\lambda_1=2$, then the Auslander-Reiten quiver $\Gamma(\lambda)$ for $\lambda=(2,6)$ is attached and it is easy to see that the quiver stays finite, if $\lambda_2$ increases.\\[1ex]
Representation-infinite cases: The case $(3,6)$ is tame concealed by \cite{HaVo}. \\[1ex] The cases $\lambda\in\{(5^2),(4,5)\}$ are tame, since the algebras $\mathcal{A}(\lambda)$ do not contain a hypercritical convex subcategory (Unger's list \cite{Un}) by Theorem \ref{criterionTame} (see Lemma \ref{MaxTameStraightForw3} for a straight-away proof of their tameness). They are not minimal tame and thus not tame concealed \cite{Ri}, since they contain a tame concealed subquiver of the BHV-list \cite{BGRS}: \begin{center}\small\begin{tikzpicture}[descr/.style={fill=white}] \matrix (m) [matrix of math nodes, row sep=0.8em, column sep=1.0em, text height=0.8ex, text depth=0.1ex] { &\bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \\}; \path[-] (m-1-2) edge (m-1-3) (m-1-3) edge (m-1-4) (m-1-4) edge (m-1-5) (m-2-1) edge (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge (m-2-4) (m-1-2) edge (m-2-2) (m-1-3) edge (m-2-3) (m-1-4) edge (m-2-4) (m-1-2) edge[-,dotted] (m-2-3) (m-1-3) edge[-,dotted] (m-2-4) ;\end{tikzpicture}\end{center} The cases $\lambda=(4,6)$ and $\lambda'=(3,7)$ are wild by Corollary \ref{criterionWild}, since \[q_{\lambda}\left(\begin{array}{cccccc} &&2& 4 &4 &2\\
1& 2 &4 &4& 2& 0 \end{array}\right) = q_{\lambda'}\left(\begin{array}{cccccccc} &&&&4& 6 &4 \\ 1& 2 &4 &6& 8& 6 & 2 \end{array}\right) = -1 \] and explicit $2$-parameter families are presented in Remark \ref{explicit2Param}. \end{proof}
\begin{lemma}\label{reptypel3}
A staircase algebra $\mathcal{A}(\lambda)$ of length $3$, that is, $\lambda=(\lambda_1,\lambda_2,\lambda_3)\vdash n$, is \begin{itemize} \item of finite representation type if and only if $n\leq 7$ or $\lambda\in\{(1,1,\lambda_3), (1,2,5), (2^2,4)\}$. \item tame concealed if and only if $\lambda\in\{(1,3,4), (1,2,6), (2^2,5), (1^3,3^2)\}$. \item tame, but not tame concealed, if and only if $\lambda\in\{(1,4^2), (2,3^2), (2^3,3), (3^3)\}$. \end{itemize} Otherwise, $\mathcal{A}(\lambda)$ is of wild representation type.
\end{lemma}
\begin{proof} Whenever $\lambda_1\geq 3$, either $\lambda=(3^3)$, or $\mathcal{A}(\lambda)$ is of wild representation type: the case $\lambda=(2,3,4)$ is wild by Corollary \ref{criterionWild} \[q_{\lambda}\left(\begin{array}{cccc} & &2 &2\\ & 2 &4 &2\\ 1 &3& 3& 1 \end{array}\right) = -1. \]
If $\lambda=(3^3)$, then $\mathcal{A}(\lambda)$ is not wild, since it does not contain a hypercritical convex subcategory (Unger's list \cite{Un}). See Lemma \ref{MaxTameStraightForw3} for a straight forward proof of its tameness. It is not tame concealed, since it contains the tame concealed algebra \begin{center}\small\begin{tikzpicture}[descr/.style={fill=white}] \matrix (m) [matrix of math nodes, row sep=0.8em, column sep=1.0em, text height=0.8ex, text depth=0.1ex] { & \bullet & \bullet\\
\bullet & \bullet & \bullet\\ \bullet & \bullet & \\}; \path[->] (m-1-2) edge[=] (m-1-3) (m-2-1) edge (m-2-2) (m-2-1) edge (m-3-1) (m-2-2) edge (m-2-3) (m-2-2) edge (m-3-2) (m-3-1) edge (m-3-2) (m-1-2) edge (m-2-2) (m-1-3) edge (m-2-3) (m-1-2) edge[-,dotted] (m-2-3) (m-2-1) edge[-,dotted] (m-3-2) ;\end{tikzpicture}\end{center} By reduction via Lemma \ref{reductioncomp} we only consider $\lambda_1\in\{1,2\}$. The following tables visualize all cases and it suffices to prove the marked ones by Lemma \ref{reductioncomp}. \begin{center}
\begin{tabular}{|l||l|l|l|l|l|l|l|l|l|} \hline $\lambda_1=1$ & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{lightviolet}\vdots &\cellcolor{lightviolet}\vdots &\cellcolor{lightviolet}\vdots &\cellcolor{lightviolet}\vdots\\ \hline 6 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{lightviolet}W &\cellcolor{lightviolet}W &\cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 5 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 4 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{dlightblue}\textbf{T} & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 3 & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{dlightgreen}\textbf{F} & \cellcolor{dlightblue}\textbf{TC} & \cellcolor{dlightviolet}\textbf{W} & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 2 & \cellcolor{grey}x & \cellcolor{lightgreen}F &\cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{dlightgreen}\textbf{F} &\cellcolor{dlightblue}\textbf{TC} & \cellcolor{dlightviolet}\textbf{W}& \cellcolor{lightviolet}W& \cellcolor{lightviolet}$\cdots$\\ \hline 1 & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F & \cellcolor{lightgreen}F &\cellcolor{lightgreen}F & \cellcolor{lightgreen}F &\cellcolor{lightgreen}F& \cellcolor{dlightgreen}\textbf{F}& \cellcolor{dlightgreen}$\cdots$\\ \hline \hline $\lambda_1=2$ &\cellcolor{grey}\vdots& \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{grey}\vdots & \cellcolor{lightviolet}\vdots &\cellcolor{lightviolet}\vdots&\cellcolor{lightviolet}\vdots&\cellcolor{lightviolet}\vdots\\ \hline 4 &\cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{grey}x & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 3 &\cellcolor{grey}x &\cellcolor{grey}x & \cellcolor{dlightblue}\textbf{T} & \cellcolor{dlightviolet}\textbf{W} & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}W & \cellcolor{lightviolet}$\cdots$\\ \hline 2 &\cellcolor{grey}x &\cellcolor{lightgreen}F &\cellcolor{lightgreen}F & \cellcolor{dlightgreen}\textbf{F} & \cellcolor{dlightblue}\textbf{TC} &\cellcolor{dlightviolet}\textbf{W} & \cellcolor{lightviolet}W& \cellcolor{lightviolet}W& \cellcolor{lightviolet}$\cdots$\\ \hline $\lambda_2/\lambda_3$ & 1& 2 & 3 & 4 & 5 & 6 & 7 & 8 & $\cdots$\\ \hline \end{tabular} \end{center}
Representation-finite cases: Either $\lambda_2=1$, then we arrive at an $A_n$-classification. For the remaining finite cases, the Auslander-Reiten quivers or their symmetric versions (by Lemma \ref{reductionsymm}) are depicted in the Appendix \ref{appendix}.\\[1ex] Representation-infinite cases: \\[1ex] Every marked tame concealed case is tame concealed by the BHV-list \cite{HaVo}. \\[1ex] The case $\lambda=(1,4^2)$ is not wild by Theorem \ref{criterionTame}, since the algebra $\mathcal{A}(\lambda)$ does not contain any hypercritical convex subcategory (Unger's list \cite{Un}). It is representation-infinite and not tame concealed, that is, minimal tame, since it contains the case $(1,3,4)$ which is tame.\\[1ex] The case $\lambda=(2,3^2)$ is tame, but not tame concealed since $\mathcal{A}(3^3)$ is tame and since it contains the above depicted convex subcategory.\\[1ex] For every minimal wild case, by Corollary \ref{criterionWild} we provide dimension vectors $v$ which fulfill $q_{\lambda}(v)\leq -1$: \begin{center}
\begin{tabular}{|c|c|c|} \hline $\small\begin{array}{cccc}
& &1\\ & &2\\
& 2& 4\\
& 4& 4\\ 2 & 4& 2\\ \end{array}$& $\small\begin{array}{cccc} & &1\\ & &2\\ & &4\\ & &6\\
&&8\\
& 6&10\\ 4 & 8& 6\\ \end{array}$& $\small\begin{array}{ccc} & &1\\ & &2\\ & &4\\
& & 6\\ 2 & 6& 8\\ 4 & 6& 4\\ \end{array}$\\\hline $(1,3,5)$& $(1,2,7)$& $(2^2,6)$\\ \hline
\end{tabular}
\end{center} The case $\lambda=(2,3,4)$ has been shown to be wild above. \end{proof}
We are able to deduce all remaining cases now which leads to a complete classification of the representation types of all staircase algebras. \begin{theorem}\label{reptype} A staircase algebra $\mathcal{A}(\lambda)$ is \begin{itemize} \item representation-finite if and only if one of the following holds true: \begin{enumerate}
\item\label{1stcase} $\lambda\in\{(n),~ (1^k,n-k),~ (2,n-2),~ (1^{n-4},2^2)\}$ for some $k\leq n$, \item\label{2ndcase} $n\leq 8$ and $\lambda\notin\{(1,3,4),~ (2,3^2),~ (1,2^2,3),~ (1^2,2,4)\}$. \end{enumerate} \item tame concealed if and only if $\lambda$ comes up in the following list:\\ $(3,6)$, $(1,2,6)$, $(1,3,4)$, $(2^2,5)$, $(1^2,2,4)$, $(1,2^2,3)$, $(1^3,3^2)$, $(1^3,2^3)$, $(1^4,2,3)$. \item tame, but not tame concealed if and only if $\lambda$ comes up in the following list:\\ $(4,5)$, $(5^2)$, $(1,4^2)$, $(2,3^2)$, $(3^3)$, $(2^3,3)$, $(1,2^4)$, $(2^5)$. \end{itemize}
Otherwise, $\mathcal{A}(\lambda)$ is of wild representation type. \end{theorem} \begin{proof}
All listed representation-finite cases are in fact of finite representation type: Let $\lambda=(1^k,n-k)$ for some $k$, then we obtain an $A_n$-classification problem, thus, representation-finiteness \cite{Ga1}. Given the two symmetric cases $\lambda=(2,n-2)=(1^{n-4},2^2)^T$, finiteness follows from Lemma \ref{reptypel2}. For every remaining finite case, the Auslander-Reiten quivers can be found in the Appendix \ref{appendix}.\\[1ex]
Assume, $l(\lambda)\leq 3$, then the claimed classification follows from Lemma \ref{reptypel2} and Lemma \ref{reptypel3}. Thus, let without loss of generality $\lambda=(\lambda_1,..., \lambda_k)$, such that $k\geq 4$; and $\lambda_{k-1}>1$ (otherwise we arrive at a finite case).\\[1ex] If $\lambda_{k-1}\geq 4$, then $\mathcal{A}(\lambda)$ is of wild representation type by Lemma \ref{reductioncomp}: The case $\mathcal{A}(1,1,3,4)$ is wild by Corollary \ref{criterionWild}, since
\[q_{\lambda}\left(\begin{array}{cccc}
& & &2\\ & &2 &4\\ & &4 &4\\ 1 &2& 4& 2 \end{array}\right) = -1. \] If $\lambda_{k-1}=2$, then $\lambda$ is of the form $(1,...,1,2,...,2,x)$ for some $x$. \begin{itemize} \item If $x\in\{2,3\}$, then the transpose of every case has been considered in Lemma \ref{reptypel2} or Lemma \ref{reptypel3}. \item If $x\geq 4$, then \begin{itemize} \item for $\lambda_{k-2}=2$ the algebra is wild due to reduction by Lemma \ref{reductioncomp} since $\mathcal{A}((1,1,3,4)^T)=\mathcal{A}(1,2,2,4)$ is wild via symmetry of Lemma \ref{reductionsymm}. \item if $\lambda_{k-2}=1$ and $x=4$, the algebra $\mathcal{A}(\lambda)$ is tame concealed \cite{HaVo}. \item for $\lambda_{k-2}=1$ and $x\geq 5$ the algebra is wild by Corollary \ref{criterionWild}, since
\[q_{\lambda}\left(\begin{array}{cccc}
& & &1\\
& & &2\\ & & &4\\ & &4 &6\\ 2 &4& 6& 4 \end{array}\right) = -1. \] \end{itemize} \end{itemize} If $\lambda_{k-1}=3$, then (since $\mathcal{A}(1,1,3,4)$ is wild) $\mathcal{A}(\lambda)$ is wild whenever $\lambda_k\geq 4$. If $\lambda_k=3$, the transpose of $\lambda$ is of length $3$ and has, thus, been considered in Lemma \ref{reptypel3}. \end{proof}
\begin{corollary}\label{blockstalg} The tensor algebra $KA_m\otimes_K KA_l$ is of finite representation type if and only if $(m,l)\in\{(1,l), (m,1), (2,2), (2,3), (3,2), (4,2), (2,4)\}$. It is tame exactly if $(m,l)\in\{(2,5), (5,2), (3,3)\}$ and wild, otherwise. \end{corollary} \begin{proof} The tensor algebra $KA_m\otimes_K KA_l$ equals the $1$-step staircase algebra $\mathcal{A}(k^l)$, thus, the proof follows from Theorem \ref{reptype}. \end{proof} The hierarchy of cases is depicted in the Appendix \ref{hierarchy}. Note that the representation type of the staircase algebra $\mathcal{A}((2^k))$ has been examined - it is already known by Asashiba; and by Escolar and Hiraoka \cite{EH}.\\[1ex]
By Lemma \ref{orbittype}, Theorem \ref{reptype} yields the following classification via orbit types. \begin{lemma}\label{classificationType} $\mathcal{A}(\lambda)$ is of finite / tame / wild representation type if and only if its orbit quiver is Dynkin / Euclidean / wild. \end{lemma} These results were obtained in the $\GL_n$-setup. We think that it should be possible to generalize them to arbitrary classical Lie types by methods similar to \cite{HiRoe}. \section{Properties depending on representation types}\label{SectCasestudy} We have a closer look at the representation theory of staircase algebras now and divide our considerations by representation types, which are known from Theorem \ref{reptype}. \subsection{Finite cases}\label{finiteCases} All finite cases are listed in the table, together with their orbit quiver $\Upsilon(\lambda)$ (see Lemma \ref{orbittype}) and the link to their Auslander-Reiten quiver $\Gamma(\lambda)$ in the Appendix \ref{appendix}). In the latter, the corresponding maximal dimension vectors are marked. Since $\Gamma(\lambda)$ always contains a sincere representation, each such algebra $\mathcal{A}(\lambda)$ is tilted Dynkin and the orbit quiver indicates the frame. By Theorem \ref{criterionFinite} up to isomorphism, every indecomposable appears in $\Gamma(\lambda)$; such that they can be constructed directly or by means of methods from Auslander-Reiten Theory \cite{ASS}. This way, a complete representative system of indecomposable modules is obtained for each finite case. Clearly, all orbits of a fixed dimension vector can be deduced from these by Krull-Remak-Schmidt. By results of Bongartz \cite{Bo2}, the explicit calculation of orbit closures can nicely be done. \begin{center}
\begin{tabular}[h]{|c|c|c|c||c|c|c|c|} \hline $n$ & $\lambda$ & $\Upsilon(\lambda)$ & $\Gamma(\lambda)$ & $n$ & $\lambda$ & $\Upsilon(\lambda)$ & $\Gamma(\lambda)$\\ \hline
$n$& $\begin{array}{c} (1^{n-k},k) \end{array}$
& $A_n$ & -
\\ \hline
$n$& $\begin{array}{c} (1^{n-4},2^2) \\ (2,n-2)\end{array}$
& $D_n$ & \ref{ark26}&$7$&$\begin{array}{c} (1,2,4) \\ (1^2,2,3)\end{array}$&
$E_7$ & \ref{ark124}\\ \hline
$8$& $\begin{array}{c} (1,2,5) \\ (1^3,2,3)\end{array}$ & $E_8$ & \ref{ark125}&$7$ &$\begin{array}{c} (3,4) \\ (1,2^3)\end{array}$&
$E_7$ & \ref{ark34}
\\ \hline
$8$ &$\begin{array}{c}
(3,5)\\ (1^2,2^3)\end{array}$ & $E_8$ & \ref{ark35}&$7$& $\begin{array}{c} (1,3^2) \\ (2^2,3)\end{array}$ & $E_7$ & \ref{ark133}
\\ \hline
$8$&$\begin{array}{c} (1^2,3^2) \\ (2^2,4)\end{array}$ & $E_8$ & \ref{ark224}&$6$& $\begin{array}{c}
(1,2,3) \end{array}$
& $E_6$ & \ref{ark123}
\\ \hline
$8$
&$\begin{array}{c}
(2^4), (4^2)\end{array}$
& $E_8$ & \ref{ark2222}& $6$ &$\begin{array}{c} (2^3), (3^2)\end{array}$
& $E_6$ & \ref{ark222}
\\ \hline \end{tabular} \end{center} \subsection{Tame concealed algebras}\label{TC} Given a tame concealed algebra $\mathcal{A}(\lambda)$, we know that there is an algebra $\mathcal{B}$ and a preprojective $\mathcal{B}$-tilting module $T$, such that $\End_{\mathcal{B}}(T) =\mathcal{A}(\lambda)$. \begin{center} \includegraphics[trim=130 485 185 125,clip,width=290pt]{TC_small.pdf} \end{center} The table shows every such algebra together with some information. Any tame concealed algebra has a unique one-parameter family of indecomposable modules $X$ with $\End(X) = K$, $\Ext^1(X,X) =K$, and the dimension vector of these modules is the minimal positive nullroot (see \cite{HaVo}). It generates the radical of $q_{\lambda}$. The orbit quiver $\Upsilon(\lambda)$ equals the frame of the tilted algebra, that is, the Gabriel quiver of $\mathcal{B}$ and the tubular type is the same as the tubular type of $\mathcal{B}$ \cite{Ri}. Each tilting module is given by a direct sum of preprojective indecomposables and there is a stable separating tubular $\mathbf{P}_1K$-family of the tubular type of the algebra which separates the preprojective and preinjective component. In these components, all indecomposables come up (up to isomorphism). The regular component equals the $\Hom_{\mathcal{B}}(T,\_)$-translation of the regular component of the Euclidean quiver \cite{Ri}. The simple regular modules are those at the mouths of the tubes; and they have explicitly been calculated for $\widetilde{E_7}$ and $\widetilde{E_8}$ in \cite{DlR2}. Thus, all simple regular modules and all tubes are obtained. Note that by results of Bongartz \cite{Bo6}, orbit closures can be calculated.
\subsection{Non-concealed tame algebras}\label{TameMax} The non-maximal non-concealed staircase algebras are corresponding to the partitions $\lambda=(2, 3^2)$, $\lambda'=(4,5)$ and $\lambda'^T=(1,2^4)$. Both are Euclidean tilted by Theorem \ref{criterionTameTilted} and Corollary \ref{criterionTiltedSincere} and their orbit type ($\widetilde{E_7}$ for $\lambda$, $\widetilde{E_8}$ for $\lambda'$) equals their frame.
The tilting module is given by a direct sum of preprojective indecomposables plus at least one non-preprojective module, since the algebras are not tame concealed. Each contains a tame concealed subcategory and the radical of the quadratic form is always generated by the induced minimal nullroot, since these algebras are Euclidean tilted and, thus, their quadratic form is isometric to a quadratic form of a quiver. By Lemma \ref{criterionTameTilted} the quadratic form is non-negative and there is no quiver of non-negative quadratic form with $2$-dimensional radical.\\[1ex]
Concerning the maximal tame staircase algebras, we prove tameness without using the list of hypercritical algebras \cite{Un}, now. We know from Theorem \ref{criterionTame} that $\mathcal{A}(\lambda)$ is tame if and only if the quadratic form $q_{\lambda}$ is weakly non-negative. \begin{lemma}\label{MaxTameStraightForw3} Let $\lambda=(3^3)$ and $\lambda'\in\{(5^2), (2^5)\}$, then $q_{\lambda}$ and $q_{\lambda'}$ are non-negative and \[\rad q_{\lambda} =\left\langle u:=\left(\begin{array}{ccc} 1&1&0\\ 1&0&-1\\ 0&-1&-1 \end{array} \right),~v:=\left(\begin{array}{ccc} 0&1&1\\ 1&2&1\\ 1&1&0 \end{array} \right)\right\rangle.\] \[\rad q_{\lambda'} =\left\langle \left(\begin{array}{ccccc} 2&3&2&0 &-1\\ 1&0&-2&-3&-2 \end{array} \right),~\left(\begin{array}{ccccc} 0&1&2 &2&1\\ 1&2&2&1&0 \end{array} \right)\right\rangle.\] In particular, $\mathcal{A}(\lambda)$ and $\mathcal{A}(\lambda')$ are tame and non-tilted. \end{lemma} \begin{proof}
Let $\lambda'':=(2,3^2)$, then $\mathcal{A}(\lambda'')$ is tilted by Lemma \ref{criterionTiltedSincere}. It is Euclidean tilted, since its preprojective component is of type $\widetilde{E_7}$ ("contains a $\widetilde{E_7}$-slice") and contains all projective indecomposables. By Lemma \ref{criterionTameTilted}, $q_{\lambda''}$ is non-negative.\\[1ex] Note that $u$ is orthogonal to every other vector concerning the symmetric form $q_{\lambda}(\_,\_)$; in particular, $q_{\lambda}(u)=0$.\\[1ex] Let $x\in \mathbf{R}^9= \mathbf{R}^{\mathcal{Q}(\lambda)_0}$, then $\tilde{x}:=x-x_1u \in\mathbf{R}^8\cong \mathbf{R}^{\mathcal{Q}(\lambda'')_0}$ and $q_{\lambda}(x)=q_{\lambda''}(\tilde{x})\geq 0$. If $x\in \rad q_{\lambda}$, then $\widetilde{x}\in\rad q_{\lambda''}$, and, thus, $\widetilde{x}=\xi\cdot v$ for some $\xi$ as has been shown before.\\
Since $q_{\lambda}$ is non-negative, $\mathcal{A}(\lambda)$ is tame by Lemma \ref{criterionTame}. If $\mathcal{A}(\lambda)$ was tilted, its quadratic form would be isometric to a quadratic form of a quiver. But there is no such non-negative form with $2$-dimensional radical.\\
The proof for $\lambda'$ can be done analogously. \end{proof} Tameness of $(1,4^2)=(2^3,3)^T$ follows from tameness of $(5,5)$ by Lemma \ref{reductioncomp}, since $q_{(1,4^2)} = q_{(4,5)}$. Note that the radical of the quadratic form $q_{\lambda}$ for $\lambda=(1,4,4)$ is $1$-dimensional, since the algebra $\mathcal{A}(4,5)$ is tilted. The orbit type of the algebras $\mathcal{A}(1,4^2)$, $\mathcal{A}(2^3,3)$ and $\mathcal{A}(3^3)$ is $\widetilde{E_7}$ and the orbit type of $\mathcal{A}(5^2)$ and $\mathcal{A}(2^5)$ is $\widetilde{E_8}$. \\[1ex]
By Lemma \ref{criterionTameTilted} and Lemma \ref{MaxTameStraightForw3}, we have proved that the Tits form of every tame algebra $\mathcal{A}(\lambda)$ is non-negative.
\begin{lemma}\label{finiteGrowth} Let $\mathcal{A}(\lambda)$ be tame, then it has finite growth. \end{lemma} \begin{proof} Let $\mathcal{A}(\lambda)$ be a tame staircase algebras for $\lambda\notin\{(3^3),(2^5),(5^2)\}$. Then $\rad q_{\lambda}$ is $1$-dimensional whence Proposition \ref{criterionFinGrowth} yields the claim. Let $\lambda\in\{(3^3),(2^5),(5^2)\}$, then we know explicit generators of $\rad q_{\lambda}$ from Lemma \ref{MaxTameStraightForw3}. Clearly, $\corank^0 q_{\lambda}\leq 1$, such that the claim follows from Proposition \ref{criterionFinGrowth}, as well. \end{proof} \subsection{Minimal wild cases}\label{minWildCases} The minimal wild cases can be found in the graphic in Appendix \ref{hierarchy}. Some of them are hypercritical (see \cite{Un}), namely $(1,3,5)$, $(1^2,2,5)$, $(3,7)$, $(2^2,6)$, $(1,2,7)$ and their transposes. Furthermore, all orbit quivers of the minimal wild cases are of extended Euclidean type $\widetilde{\widetilde{E_7}}$ or $\widetilde{\widetilde{E_8}}$.\\[1ex]
Let $n\geq 2$, then concrete $n$-parameter families can be produced nicely. Let $\mathcal{A}(\lambda)$ be a minimal wild staircase algebra and let $\mathcal{A}(\lambda')$ be a convex subcategory which is tame concealed and is obtained from $\mathcal{A}(\lambda)$ by cancelling a source vertex $x$ in which one arrow $\alpha: x\rightarrow y$ starts. \\[1ex] Let $M'$ be a preprojective indecomposable $\mathcal{A}(\lambda')$-representation, such that $\dim_K M'_y=n+1$. Then the $\mathcal{A}(\lambda)$-representations defined by $M_i =M'_i$ if $i\neq x$ and $M_x=K$ together with the induced maps of $M'$ and the embedding \[M_{\alpha}: K\xrightarrow{\left(\begin{array}{c} a_1\\ \vdots\\ a_{n+1}\\ \end{array}\right)} M_x\] gives a $\mathbf{P}^{n}$-family of pairwise non-isomorphic indecomposables. \begin{remark}\label{explicit2Param} Explicit $2$-parameter families are induced by these dimension vectors:
\begin{tabular}{|c|c|c|c|c|} \hline
\begin{footnotesize}\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=0.2em, column sep=0.1em, text height=0.79ex, text depth=0.1ex] { & 2& \textbf{3}\\
2 & 5& 7\\
\textbf{3}& 7& \\
}; \end{tikzpicture} \end{footnotesize}&
\begin{footnotesize}\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=0.2em, column sep=0.1em, text height=0.79ex, text depth=0.1ex] {&&& \textbf{3}\\
&2& 5& 6 \\ \textbf{3}& 6 & 7& 4\\
}; \end{tikzpicture} \end{footnotesize}
& \begin{footnotesize}\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=0.2em, column sep=0.1em, text height=0.79ex, text depth=0.1ex] { & && 5& 8& 6& \\ \textbf{3}& 6& 9& 12& 10& 4\\}; \end{tikzpicture} \end{footnotesize} & \begin{footnotesize}\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=0.2em, column sep=0.1em, text height=0.79ex, text depth=0.1ex] {&&& \textbf{3}\\
&&& 6\\
&& 5& 9\\ \textbf{3} & 6 & 9& 7\\
}; \end{tikzpicture} \end{footnotesize}&
\begin{footnotesize}\begin{tikzpicture} \matrix (m) [matrix of math nodes, row sep=0.2em, column sep=0.1em, text height=0.79ex, text depth=0.1ex] { & && & & 6& \\ & && & 8& 12& \\ \textbf{3}& 6& 9& 12& 15& 10\\}; \end{tikzpicture} \end{footnotesize} \\\hline \end{tabular} (the marked entries show, where an embedding of $K$ leads to a $2$-parameter family). This way, for each wild staircase algebra, a $2$-parameter family can be constructed. \end{remark} \section{Results for graded nilpotent pairs}\label{SectGradedNilp} We obtain certain results for graded nilpotent pairs, which immediately follow from our examinations of staircase algebras. Let $\lambda$ be a partition and let $V$ be a bigraded finite-dimensional vector space of shape $\lambda$.
\begin{theorem}\label{gradedLambda} The number of $\lambda$-graded nilpotent pairs (up to Levi-base change) is finite if and only if \begin{enumerate}
\item $\lambda\in\{(n),~ (1^k,n-k),~ (2,n-2),~ (1^{n-4},2^2)\}$ for some $k\leq n$, \item $n\leq 8$ and $\lambda\notin\{(1,3,4),~ (2,3^2),~ (1,2^2,3),~ (1^2,2,4)\}$. \end{enumerate} \end{theorem} \begin{proof} Follows from Theorem \ref{gradedCriterionLambda} and Theorem \ref{reptype}: The staircase algebra $\mathcal{A}(\lambda)$ has finite representation type if and only if the number of isomorphism classes of representations is finite. The latter correspond to $\lambda$-graded nilpotent pairs (up to Levi-base change) bijectively. \end{proof} If $\mathcal{A}(\lambda)$ is tame, then there are only one-parameter families of $\lambda$-graded nilpotent pairs (up to Levi-base change). \begin{lemma}\label{gradedTame} Assume that $\lambda\in\{(1,3,4),~ (1,2^2,3),~ (1^2,2,4),~(3,6),~(1^3,2^3),~(1,2,6),$\\ $(1^4,2,3),~(2^2,5),~(1^3,3^2)\}$. Then there are only finitely many graded nilpotent pairs on $V$ (modulo base change in the homogeneous components) if and only if $\underline{\dim}V$ does not contain a minimal nullroot as in \ref{TC}. \end{lemma}
\begin{proof} If $\mathcal{A}(\lambda)$ is tame concealed, then the number of isomorphism classes of a fixed dimension vector $\underline{d}$ is infinite if and only if $\underline{d}$ does not contain a minimal nullroot. Thus, the claim follows from Theorem \ref{reptype}. \end{proof}
\begin{lemma}\label{gradedInfinite} If $\underline{\dim}V$ contains a minimal nullroot (see Section \ref{SectCasestudy} for the explicit list), then the number of graded nilpotent pairs on $V$ is (up to isomorphism) infinite. \end{lemma}
\appendix
\section{Auslander-Reiten quivers}\label{appendix}
\subsection[The case (3,4)]{The case $\lambda=(3,4)$}\label{ark34}
\begin{center} \includegraphics[trim=122 520 155 130, clip,width=320pt]{ark34.pdf} \end{center}
\subsection[The case (2,6)]{The case $\lambda=(2,6)$}\label{ark26} Generalizes easily to $\lambda=(1,...,1,2,2)$ and $\lambda=(2,n-2)$.
\begin{center} \includegraphics[trim=112 440 135 130, clip,width=310pt, height=140pt]{ark26.pdf} \end{center}
\subsection[The case (1,2,3)]{The case $\lambda=(1,2,3)$}\label{ark123}
\begin{center} \includegraphics[trim=118 555 235 125, clip,width=250pt,height=90pt]{ark123.pdf} \end{center}
\subsection[The case (2,2,2)]{The case $\lambda=(2,2,2)$}\label{ark222}
\begin{center} \includegraphics[trim=138 585 234 102, clip,width=200pt, height=90pt]{ark222.pdf} \end{center}
\subsection[The case (1,3,3)]{The case $\lambda=(1,3,3)$}\label{ark133}
\begin{center} \includegraphics[trim=118 520 100 130, clip,width=320pt, height=110pt]{ark133.pdf} \end{center}
\subsection[The case (3,5)]{The case $\lambda=(3,5)$}\label{ark35}
\begin{center} \includegraphics[trim=8 480 0 135, clip,width=380pt, height=100pt]{ark35.pdf} \end{center}
\subsection[The case (1,2,4)]{The case $\lambda=(1,2,4)$}\label{ark124}
\begin{center} \includegraphics[trim=118 520 135 130, clip,width=320pt, height=89pt]{ark124.pdf} \end{center}
\subsection[The case (1,2,5)]{The case $\lambda=(1,2,5)$}\label{ark125}
\begin{center} \includegraphics[trim=8 480 0 135,width=380pt, height=98pt]{ark125.pdf} \end{center}
\subsection[The case (2,2,4)]{The case $\lambda=(2^2,4)$}\label{ark224}
\begin{center} \includegraphics[trim=8 480 0 139,clip,width=380pt, height=98pt]{ark224.pdf} \end{center}
\subsection[The case (2,2,2,2)]{$\lambda = (2^4)$ or $\lambda = (4^2)$}\label{ark2222}
\begin{center} \includegraphics[trim=58 480 0 147, clip,width=380pt, height=98pt]{ark44.pdf} \end{center}
\section{Hierarchy of representation types}\label{hierarchy} \begin{center} \includegraphics[trim=133 155 97 125, clip,width=380pt]{visual.pdf} \end{center}
\end{document} | arXiv |
\begin{document}
\title[FRACTIONAL INTEGRAL INEQUALITIES FOR CONVEX AND CONCAVE FUNCTIONS]{ FRACTIONAL INTEGRAL INEQUALITIES VIA ATANGANA-BALEANU OPERATORS FOR CONVEX AND CONCAVE FUNCTIONS} \author[A.O. Akdemir]{Ahmet Ocak Akdemir$^1$} \address{$^1 $DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ARTS, A\v{G }RI UNIVERSITY, A\v{G}RI, TURKEY} \email{[email protected]} \author[A. Karao\v{g}lan]{Ali Karao\v{g}lan$^2$} \address{$^2$DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ARTS, ORDU UNIVERSITY, ORDU, TURKEY} \email{[email protected]} \author[M.A. Ragusa]{Maria Alessandra Ragusa$^3$} \address{$^3$DIPARTIMENTO DI MATEMATICA E INFORMATICA, UNIVERSIT\'A DI CATANIA, CATANIA, ITALY - RUDN UNIVERSITY, MOSCOW, RUSSIA} \email{[email protected]} \author[E. Set]{Erhan Set$^{4}$ } \address{$^1$DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ARTS, ORDU UNIVERSITY, ORDU, TURKEY} \email{[email protected]} \subjclass[2010]{26A33, 26A51, 26D10. } \keywords{Convex functions, H\"older inequality, Young inequality, power mean inequality, Atangana-Baleanu fractional derivative, Atangana-Baleanu fractinal derivative} \dedicatory{} \maketitle
\begin{abstract} Recently, many fractional integral operators were introduced by different mathematicians. One of these fractional operators, Atangana-Baleanu fractional integral operator, was defined by Atangana and Baleanu in \cite{ABC}. In this study, firstly, a new identity by using Atangana-Baleanu fractional integral operators are proved. Then, new fractional integral inequalities have been obtained for convex and concave functions with the help of this identity and some certain integral inequalities. \end{abstract}
\section{Introduction}
Mathematics is a tool that serves pure and applied sciences with its deep-rooted history as old as human history, and sheds light on how to express and then solve problems. Mathematics uses various concepts and their relations with each other while performing this task. By defining spaces and algebraic structures built on spaces, mathematics creates structures that contribute to human life and nature. The concept of function is one of the basic structures of mathematics, and many researchers have focused on new function classes and made efforts to classify the space of functions. One of the types of functions defined as a product of this intense effort is the convex function, which has applications in statistics, inequality theory, convex programming and numerical analysis. This interesting class of functions is defined as follows.\newline
\begin{defn} The mapping $f:[\theta _{1},\theta _{2}]\subseteq \mathbb{R}\rightarrow \mathbb{R}$, is said to be convex if \begin{equation} f(\lambda x+(1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y) \label{1} \end{equation} is valid for all $x,y\in \lbrack \theta _{1},\theta _{2}]$ and $\lambda \in \lbrack 0,1]$. \end{defn}
Many inequalities have been obtained by using this unique function type and varieties in inequality theory, which is one of the most used areas of convex functions. We will continue by introducing the Hermite-Hadamard inequality that generate limits on the mean value of a convex function and the famous Bullen inequality as follows. \newline Assume that $f:I\subseteq
\mathbb{R}
\rightarrow
\mathbb{R}
$ is a convex mapping defined on the interval $I$ of $\mathbb{R}$ where $ \theta _{1}<b.$ The following statement; \begin{equation} f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \leq \frac{1}{\kappa _{2}-\theta _{1}}\int\limits_{\theta _{1}}^{\theta _{2}}f(x)dx\leq \frac{ f(\theta _{1})+f(\theta _{2})}{2} \label{HH} \end{equation} holds and known as Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if $f$ is concave. \newline The Bullen's integral inequality can be presented as \begin{equation*} \frac{1}{\theta _{2}-\theta _{1}}\int_{\theta _{1}}^{\theta _{2}}f\left( x\right) dx\leq \frac{1}{2}\left[ f\left( \frac{\theta _{1}+\theta _{2}}{2} \right) +\frac{f\left( \theta _{1}\right) +f\left( \theta _{2}\right) }{2} \right] \end{equation*} where $f:I\subset
\mathbb{R}
\rightarrow
\mathbb{R}
$ is a convex mapping on the interval $I$ of $\mathbb{R}$ where $\kappa _{1},\theta _{2}\in I$ with $\theta _{1}<\theta _{2}$.\newline To provide detail information on convexity, let us consider some earlier studies that have been performed by many researchers. In \cite{c6}, Jensen introduced the concept of convex function to the literature for the first time and drew attention to the fact that it seems to be the basis of the concept of incremental function. In \cite{c8}, Beckenbach has mentioned about the concept of convexity and emphasized several features of this useful function class. In \cite{c7}, the authors have focused the relations between convexity and Hermite-Hadamard's inequality. This study has led many researchers to the link between convexity and integral inequalities, which has guided studies in this field. Based on these studies, many papers have been produced for different kinds of convex functions. In \cite{c2}, Akdemir et al. have proved several new integral inequalities for geometric-arithmetic convex functions via a new integral identity. Several new Hadamard's type integral inequalities have been established with applications to special means by Kavurmaci et al. in \cite{c3}. Therefore, a similar argument has been carried out by Zhang et al. but now for $s-$ geometrically convex functions in \cite{c5}. On all of these, Xi et al. have extended the challenge to $m-$ and $(\alpha ,m)-$convex functions by providing Hadamard type inequalities in \cite{c4}.\newline Although fractional analysis has been known since ancient times, it has recently become a more popular subject in mathematical analysis and applied mathematics. The adventure that started with the question of whether the solution will exist if the order is fractional in a differential equation, has developed with many derivative and integral operators. By defining the derivative and integral operators in fractional order, the researchers who aimed to propose more effective solutions to the solution of physical phenomena have turned to new operators with general and strong kernels over time. This orientation has provided mathematics and applied sciences several operators with kernel structures that differ in terms of locality and singularity, as well as generalized operators with memory effect properties. The struggle that started with the question of how the order in the differential equation being a fraction would have consequences, has now evolved into the problem of how to explain physical phenomena and find the most effective fractional operators that will provide effective solutions to real world problems. Let us introduce some fractional derivative and integral operators that have broken ground in fractional analysis and have proven their effectiveness in different fields by using by many researchers. \newline We will remember the Caputo-Fabrizio derivative operators. Also, we would like to note that the functions belong to Hilbert spaces denoted by $H^{1}(0,\theta _{2})$
\begin{defn} \cite{CF} Let $f\in H^{1}(0,\theta _{2})$, $\theta _{2}>\theta _{1}$, $ \alpha \in \lbrack 0,1]$ then, the definition of the new Caputo fractional derivative is: \begin{equation} ^{CF}D^{\alpha }f(\tau_{1} )=\frac{M(\alpha )}{1-\alpha }\int_{\kappa _{1}}^{\tau_{1} }f^{\prime }(s)exp\left[ -\frac{\alpha }{(1-\alpha )} (\tau_{1} -s)\right] ds \label{tymfg89} \end{equation} where $M(\alpha )$ is normalization function. \end{defn}
Depends on this interesting fractional derivative operator, the authors have defined the Caputo-Fabrizio fractional integral operator as follows.
\begin{defn} \cite{AD} Let $f\in H^{1}(0,\theta _{2})$, $\theta _{2}>\theta _{1}$, $ \alpha \in \lbrack 0,1]$ then, the definition of the left and right side of Caputo-Fabrizio fractional integral is: \begin{equation*} \left( _{:::\theta _{1}}^{CF}I^{\alpha }\right) (\tau_{1} )=\frac{1-\alpha }{ B(\alpha )}f(\tau_{1} )+\frac{\alpha }{B(\alpha )}\int_{\theta _{1}}^{\tau_{1} }f(y)dy, \end{equation*} and \begin{equation*} \left( ^{CF}I_{\theta _{2}}^{\alpha }\right) (\tau_{1} )=\frac{1-\alpha }{ B(\alpha )}f(\tau_{1} )+\frac{\alpha }{B(\alpha )}\int_{\tau_{1} }^{\kappa _{2}}f(y)dy \end{equation*} where $B(\alpha )$ is normalization function. \end{defn}
The Caputo-Fabrizio fractional derivative, which is used in dynamical systems, physical phenomena, disease models and many other fields, is a highly functional operator by definition, but has a deficiency in terms of not meeting the initial conditions in the special case $\alpha=1$. The improvement to eliminate this deficiency has been provided by the new derivative operator developed by Atangana-Baleanu, which has versions in the sense of Caputo and Riemann. In the sequel of this paper, we will denote the normalization function with $B(\alpha)$ that the same properties with the $ M(\alpha)$ which defined in Caputo-Fabrizio definition.
\begin{defn} \cite{ABC} Let $f\in H^{1}(\theta _{1},\theta _{2})$, $\theta _{2}>\kappa _{1}$, $\alpha \in \lbrack 0,1]$ then, the definition of the new fractional derivative is given: \begin{equation} _{::::::\theta _{1}}^{ABC}D_{\tau_{1} }^{\alpha }\left[ f(\tau_{1} )\right] = \frac{B(\alpha )}{1-\alpha }\int_{a}^{\tau_{1} }f^{\prime }(x)E_{\alpha } \left[ -\alpha \frac{(\tau_{1} -x)^{\alpha }}{(1-\alpha )}\right] dx. \label{abc} \end{equation} \end{defn}
\begin{defn} \cite{ABC} Let $f\in H^{1}(\theta _{1},\theta _{2})$, $\theta _{2}>\kappa _{1}$, $\alpha \in \lbrack 0,1]$ then, the definition of the new fractional derivative is given: \begin{equation} _{:::::::\theta _{1}}^{ABR}D_{\tau_{1} }^{\alpha }\left[ f(\tau_{1} )\right] =\frac{B(\alpha )}{1-\alpha }\frac{d}{d\tau_{1} }\int_{\theta _{1}}^{\tau_{1} }f(x)E_{\alpha }\left[ -\alpha \frac{(\tau_{1} -x)^{\alpha } }{(1-\alpha )}\right] dx. \label{abr} \end{equation} \qquad \end{defn}
Equations (\ref{abc}) and (\ref{abr}) have a non-local kernel. Also in equation (\ref{abr}) when the function is constant we get zero.
The associated fractional integral operator has been defined by Atangana-Baleanu as follows.
\begin{defn} \cite{ABC} The fractional integral associate to the new fractional derivative with non-local kernel of a function $f\in H^{1}(\kappa _{1},\theta _{2})$ as defined: \begin{equation*} _{::::\theta _{1}}^{AB}I_{{}}^{\alpha }\left\{ f(\tau_{1} )\right\} =\frac{ 1-\alpha }{B(\alpha )}f(\tau_{1} )+\frac{\alpha }{B(\alpha )\Gamma (\alpha )} \int_{\theta _{1}}^{\tau_{1} }f(y)(\tau_{1} -y)^{\alpha -1}dy \end{equation*} where $\theta _{2}>\theta _{1},\alpha \in \lbrack 0,1].$ \end{defn}
In \cite{TD}, Abdeljawad and Baleanu introduced right hand side of integral operator as following; the right fractional new integral with ML kernel of order $\alpha \in \lbrack 0,1]$ is defined by \begin{equation*} \left( ^{AB}I_{\theta _{2}}^{\alpha }\right) \left\{ f(\tau_{1} )\right\} = \frac{1-\alpha }{B(\alpha )}f(\tau_{1} )+\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\tau_{1} }^{\theta _{2}}f(y)(y-\tau_{1} )^{\alpha -1}dy. \end{equation*} In \cite{AD}, Abdeljawad and Baleanu has presented some new results based on fractional order derivatives and their discrete versions. Conformable integral operators have been defined by Abdeljawad in \cite{13}. This useful operator has been used to prove some new integral inequalities in \cite{15}. Another important fractional operator -Riemann-Liouville fractional integral operators- have been used to provide some new Simpson type integral inequalities in \cite{11}. Ekinci and \"{O}zdemir have proved several generalizations by using Riemann-Liouville fractional integral operators in \cite{a8} and the authors have established some similar results with this operator in \cite{e1}. In \cite{b11}, Akdemir et al. have presented some new variants of celebrated Chebyshev inequality via generalized fractional integral operators. The argument has been proceed with the study of Rashid et al. (see \cite{a1}) that involves new investigations related to generalized $k-$fractional integral operators. In \cite{a2}, Rashid et al. have presented some motivated findings that extend the argument to the Hilbert spaces. For more information related to different kinds of fractional operators, we recommend to consider \cite{8}. The applications of fractional operators have been demonsrated by several researchers, we suggest to see the papers \cite{e2}-\cite{e4}.\newline The main motivation of this paper is to prove an integral identity that includes the Atangana-Baleanu integral operator and to provide some new Bullen type integral inequalities for differentiable convex and concave functions with the help of this integral identity. Some special cases are also considered.
\section{Main Results}
We will start with a new integral identity that will be used the proofs of our main findings:
\begin{lem} \label{ghd65} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$. Then we have the following identity for Atangana-Baleanu fractional integral operators \begin{eqnarray*} &&\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2} \right) \right] \\ &=&\int_{0}^{1}\left( (1-\tau _{1})^{\alpha }-\tau _{1}^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2} \theta _{2}\right) d\tau _{1} \\ &&+\int_{0}^{1}\left( \tau _{1}^{\alpha }-(1-\tau _{1})^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2} \kappa _{1}\right) d\tau _{1} \end{eqnarray*} where $\alpha ,\tau _{1}\in \lbrack 0,1]$, $\Gamma (.)$ is Gamma function and $B(\alpha )$ is normalization function.
\begin{proof} By adding $I_{1}$ and $I_{2}$, we have \begin{eqnarray*} I_{1}+I_{2} &=&\int_{0}^{1}\left( (1-\tau _{1})^{\alpha }-\tau _{1}^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1} }{2}\kappa _{2}\right) d\tau _{1} \\ &&+\int_{0}^{1}\left( \tau _{1}^{\alpha }-(1-\tau _{1})^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2} \theta _{1}\right) d\tau _{1}. \end{eqnarray*} By using integration, we have \begin{eqnarray} I_{1} &=&\int_{0}^{1}\left( (1-\tau _{1})^{\alpha }-\tau _{1}^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1} }{2}\kappa _{2}\right) d\tau _{1} \notag \label{gh56+} \\ &=&\frac{\left( (1-\tau _{1})^{\alpha }-\tau _{1}^{\alpha }\right) f\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right)
d\tau _{1}}{\frac{\theta _{1}-\theta _{2}}{2}}\bigg|_{1}^{0} \notag \\ &&-\frac{2\alpha }{\kappa _{2}-\theta _{1}}\int_{0}^{1}\left( (1-\tau _{1})^{\alpha -1}+\tau _{1}^{\alpha -1}\right) f\left( \frac{1+\tau _{1}}{2} \theta _{1}+\frac{1-\tau _{1}}{2}\kappa _{2}\right) d\tau _{1} \notag \\ &=&-\frac{2}{\theta _{1}-\theta _{2}}f(\theta _{1})-\frac{2}{\kappa _{1}-\theta _{2}}f(\frac{\theta _{1}+\theta _{2}}{2}) \notag \\ &&-\frac{2\alpha }{\kappa _{2}-\theta _{1}}\int_{0}^{1}(1-\tau _{1})^{\alpha -1}f\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) d\tau _{1} \notag \\ &&-\frac{2\alpha }{\theta _{2}-\theta _{1}}\int_{0}^{1}\tau _{1}^{\alpha -1}f\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) d\tau _{1} \notag \\ &=&\frac{2}{\theta _{2}-\theta _{1}}\left( f(\theta _{1})+f(\frac{\kappa _{1}+\theta _{2}}{2})\right) -\frac{2^{\alpha +1}\alpha }{(\kappa _{2}-\theta _{1})^{\alpha +1}}\int_{\theta _{1}}^{\frac{\theta _{1}+\kappa _{2}}{2}}\left( x-\theta _{1}\right) ^{\alpha -1}f(x)dx \notag \\ &&-\frac{2^{\alpha +1}\alpha }{(\theta _{2}-\theta _{1})^{\alpha +1}} \int_{\theta _{1}}^{\frac{\theta _{1}+\theta _{2}}{2}}\left( \frac{\kappa _{1}+\theta _{2}}{2}-x\right) ^{\alpha -1}f(x)dx. \notag \\ && \end{eqnarray} Multiplying both side of \eqref{gh56+} identity by $\frac{(\kappa _{2}-\theta _{1})^{\alpha +1}}{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}$, we have \begin{eqnarray} &&\frac{(\theta _{2}-\theta _{1})^{\alpha +1}}{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}I_{1} \notag \label{lk98n} \\ &=&\frac{(\theta _{2}-\theta _{1})^{\alpha }}{2^{\alpha }B(\alpha )\Gamma (\alpha )}\left( f(\theta _{1})+f(\frac{\theta _{1}+\kappa _{2}}{2})\right) - \frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\kappa _{1}}^{\frac{\theta _{1}+\theta _{2}}{2}}\left( x-\theta _{1}\right) ^{\alpha -1}f(x)dx \notag \\ &&-\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\theta _{1}}^{\frac{ \theta _{1}+\theta _{2}}{2}}\left( \frac{\theta _{1}+\theta _{2}}{2} -x\right) ^{\alpha -1}f(x)dx. \notag \\ && \end{eqnarray} Similarly, by using integration, we get \begin{eqnarray} I_{2} &=&\int_{0}^{1}\left( \tau _{1}^{\alpha }-(1-\tau _{1})^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1} }{2}\kappa _{1}\right) d\tau _{1} \notag \label{349f} \\ &=&\frac{\left( \tau _{1}^{\alpha }-(1-\tau _{1})^{\alpha }\right) f\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right)
d\tau _{1}}{\frac{\theta _{2}-\theta _{1}}{2}}\bigg|_{1}^{0} \notag \\ &&-\frac{2\alpha }{\kappa _{2}-\theta _{1}}\int_{0}^{1}\left( \tau _{1}^{\alpha -1}+(1-\tau _{1})^{\alpha -1}\right) f\left( \frac{1+\tau _{1}}{ 2}\theta _{2}+\frac{1-\tau _{1}}{2}\kappa _{1}\right) d\tau _{1} \notag \\ &=&\frac{2}{\theta _{2}-\theta _{1}}\left( f(\theta _{2})+f(\frac{\kappa _{1}+\theta _{2}}{2})\right) -\frac{2^{\alpha +1}\alpha }{(\kappa _{2}-\theta _{1})^{\alpha +1}}\int_{\frac{\theta _{1}+\theta _{2}}{2} }^{\theta _{2}}\left( x-\frac{\theta _{1}+\theta _{2}}{2}\right) ^{\alpha -1}f(x)dx \notag \\ &&-\frac{2^{\alpha +1}\alpha }{(\theta _{2}-\theta _{1})^{\alpha +1}}\int_{ \frac{\theta _{1}+\theta _{2}}{2}}^{\theta _{2}}\left( \theta _{2}-x\right) ^{\alpha -1}f(x)dx. \notag \\ && \end{eqnarray} Multiplying both side of \eqref{349f} identity by $\frac{(\theta _{2}-\kappa _{1})^{\alpha +1}}{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}$, we get \begin{eqnarray} &&\frac{(\theta _{2}-\theta _{1})^{\alpha +1}}{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}I_{2} \notag \label{67dd0} \\ &=&\frac{(\theta _{2}-\theta _{1})^{\alpha }}{2^{\alpha }B(\alpha )\Gamma (\alpha )}\left( f(\theta _{2})+f(\frac{\theta _{1}+\kappa _{2}}{2})\right) - \frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\frac{\theta _{1}+\theta _{2} }{2}}^{\theta _{2}}\left( x-\frac{\theta _{1}+\kappa _{2}}{2}\right) ^{\alpha -1}f(x)dx \notag \\ &&-\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\frac{\theta _{1}+\kappa _{2}}{2}}^{\theta _{2}}\left( \theta _{2}-x\right) ^{\alpha -1}f(x)dx. \notag \\ && \end{eqnarray} By adding identity \eqref{lk98n} and \eqref{67dd0}, we obtain \begin{eqnarray*} &&\frac{(\theta _{2}-\theta _{1})^{\alpha +1}}{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}[I_{1}+I_{2}] \\ &=&\frac{(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha }\Gamma (\alpha )}{2^{\alpha }B(\alpha )\Gamma (\alpha )}\left[ f(\theta _{1})+f( \frac{\theta _{1}+\theta _{2}}{2})\right] \\ &&-\frac{1-\alpha }{B(\alpha )}f(\theta _{1})-\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\theta _{1}}^{\frac{\theta _{1}+\theta _{2}}{2} }\left( x-\theta _{1}\right) ^{\alpha -1}f(x)dx \\ &&-\frac{1-\alpha }{B(\alpha )}f\left( \frac{\theta _{1}+\theta _{2}}{2} \right) -\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\theta _{1}}^{\frac{ \theta _{1}+\theta _{2}}{2}}\left( \frac{\theta _{1}+\theta _{2}}{2} -x\right) ^{\alpha -1}f(x)dx \\ &+&\frac{(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha }\Gamma (\alpha )}{2^{\alpha }B(\alpha )\Gamma (\alpha )}\left[ f(\theta _{2})+f( \frac{\theta _{1}+\theta _{2}}{2})\right] \\ &&-\frac{1-\alpha }{B(\alpha )}f(\theta _{2})-\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\frac{\kappa _{1}+\theta _{2}}{2}}^{\theta _{2}}\left( \theta _{2}-x\right) ^{\alpha -1}f(x)dx \\ &&-\frac{1-\alpha }{B(\alpha )}f\left( \frac{\theta _{1}+\theta _{2}}{2} \right) -\frac{\alpha }{B(\alpha )\Gamma (\alpha )}\int_{\frac{\kappa _{1}+\theta _{2}}{2}}^{\theta _{2}}\left( x-\frac{\theta _{1}+\theta _{2}}{2} \right) ^{\alpha -1}f(x)dx. \end{eqnarray*} Using the definition of Atangana-Baleanu fractional integral operators, we get \begin{eqnarray*} &&\frac{(\theta _{2}-\theta _{1})^{\alpha +1}}{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}\Bigg[\int_{0}^{1}\left( (1-\tau _{1})^{\alpha }-\tau _{1}^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1} }{2}\theta _{2}\right) d\tau _{1} \\ &&+\int_{0}^{1}\left( \tau _{1}^{\alpha }-(1-\tau _{1})^{\alpha }\right) f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2} \theta _{1}\right) d\tau _{1}\Bigg] \\ &=&\frac{(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha }\Gamma (\alpha )}{2^{\alpha }B(\alpha )\Gamma (\alpha )}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\kappa _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\theta _{1}+\theta _{2} }{2}\right) +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2} \right) \right] . \end{eqnarray*} \end{proof} \end{lem}
\begin{thm} \label{ttr4h} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$ and $f^{\prime }\in L_{1}[\theta _{1},\theta _{2}]$. If $
|f^{\prime }|$ is a convex function, we have the following inequality for Atangana-Baleanu fractional integral operators \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{2\left[ \left\vert f^{\prime }(\theta _{1})\right\vert +\left\vert f^{\prime }(\theta _{2})\right\vert \right] }{\alpha +1} \end{eqnarray*} where $\alpha \in \lbrack 0,1]$, $B(\alpha )$ is normalization function.
\begin{proof} By using Lemma \ref{ghd65}, we can write \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1} \\ &&+\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}. \end{eqnarray*}
By using convexity of $|f^{\prime }|$, we get \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\int_{0}^{1}(1-\tau _{1})^{\alpha }\left[ \frac{1+\tau _{1}}{2} \left\vert f^{\prime }(\theta _{1})\right\vert +\frac{1-\tau _{1}}{2} \left\vert f^{\prime }(\theta _{2})\right\vert \right] d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\theta _{1})\right\vert +\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\theta _{2})\right\vert \right] d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\theta _{2})\right\vert +\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{1})\right\vert \right] d\tau _{1} \\ &&+\int_{0}^{1}(1-\tau _{1})^{\alpha }\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\theta _{2})\right\vert +\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\theta _{1})\right\vert \right] d\tau _{1}. \end{eqnarray*} By computing the above integral, we obtain \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{2\left[ \left\vert f^{\prime }(\theta _{1})\right\vert +\left\vert f^{\prime }(\theta _{2})\right\vert \right] }{\alpha +1} \end{eqnarray*} and the proof is completed. \end{proof} \end{thm}
\begin{crly} In Theorem \ref{ttr4h}, if we choose $\alpha =1$ we obtain \begin{eqnarray*}
&&\bigg|\frac{f(\theta _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\kappa _{2}}{2}\right) }{\theta _{2}-\theta _{1}}-\frac{4}{(\theta _{2}-\kappa _{1})^{2}}\int_{\theta _{1}}^{\theta _{2}}f(x)dx\bigg| \\ &\leq &\frac{\left\vert f^{\prime }(\theta _{1})\right\vert +\left\vert f^{\prime }(\kappa _{2})\right\vert }{2}. \end{eqnarray*} \end{crly}
\begin{thm} \label{dfgre89} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$ and $f^{\prime }\in L_{1}[\theta _{1},\theta _{2}]$. If $
|f^{\prime }|^{q}$ is a convex function, then we have the following inequality for Atangana-Baleanu fractional integral operators: \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{2}{(\alpha p+1)^{\frac{1}{p}}}\left[ \left( \frac{3\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}}{4}\right) ^{\frac{1}{q}}+\left( \frac{3\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}}{4}\right) ^{\frac{1}{q}}\right] \end{eqnarray*} where $p^{-1}+q^{-1}=1$, $\alpha \in \lbrack 0,1]$, $q>1$, $B(\alpha )$ is normalization function.
\begin{proof} By using the identity that is given in Lemma \ref{ghd65}, we have \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1}+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}+\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}. \end{eqnarray*} By applying H\"{o}lder inequality, we have \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}\right) ^{\frac{1 }{p}}\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2} \kappa _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}}. \end{eqnarray*}
By using convexity of $|f^{\prime }|^{q}$, we obtain \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}\right) ^{\frac{1 }{p}}\left( \int_{0}^{1}\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}+\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}\right] dv\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}+\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}+\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left[ \frac{1+\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}+\frac{1-\tau _{1}}{2}\left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}}. \end{eqnarray*} By calculating the integrals that is in the above inequalities, we get desired result. \end{proof} \end{thm}
\begin{crly} In Theorem \ref{dfgre89}, if we choose $\alpha =1$ we obtain \begin{eqnarray*}
&&\bigg|\frac{f(\theta _{1})+f(\theta _{2})+2f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) }{\theta _{2}-\theta _{1}}-\frac{4}{(\kappa _{2}-\theta _{1})^{2}}\int_{\theta _{1}}^{\theta _{2}}f(x)dx\bigg| \\ &\leq &\frac{1}{(p+1)^{\frac{1}{p}}}\bigg[\left( \frac{3\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}}{4}\right) ^{\frac{1}{q}} \\ &&+\left( \frac{3\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}}{4}\right) ^{\frac{1 }{q}}\bigg]. \end{eqnarray*} \end{crly}
\begin{thm} \label{908tty6} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$ and $f^{\prime }\in L_{1}[\theta _{1},\theta _{2}]$. If $
|f^{\prime }|^{q}$ is a convex function, then we have the following inequality for Atangana-Baleanu fractional integral operators \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \frac{1}{\alpha +1}\right) ^{1-\frac{1}{q}}\Bigg[\left( \frac{ \alpha +3}{2(\alpha +1)(\alpha +2)}\left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}+\frac{1}{2(\alpha +2)}\left\vert f^{\prime }(\kappa _{2})\right\vert ^{q}\right) ^{\frac{1}{q}} \\ &&+\left( \frac{2\alpha +3}{2(\alpha +1)(\alpha +2)}\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\frac{1}{2(\alpha +1)(\alpha +2)}\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}\right) ^{\frac{1}{q}} \\ &&+\left( \frac{2\alpha +3}{2(\alpha +1)(\alpha +2)}\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\frac{1}{2(\alpha +1)(\alpha +2)}\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}\right) ^{\frac{1}{q}} \\ &&+\left( \frac{\alpha +3}{2(\alpha +1)(\alpha +2)}\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\frac{1}{2(\alpha +2)}\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}\right) ^{\frac{1}{q}}\Bigg] \end{eqnarray*} where $\alpha \in \lbrack 0,1]$, $q\geq 1$, $B(\alpha )$ is normalization function.
\begin{proof} By Lemma \ref{ghd65}, we get \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1}+\int_{0}^{1}t^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\kappa _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}+\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}. \end{eqnarray*} By applying power mean inequality, we get \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}\right) ^{1-\frac{ 1}{q}}\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}\right) ^{1-\frac{1}{q} }\left( \int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}\right) ^{1-\frac{1}{q} }\left( \int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}\right) ^{1-\frac{1}{q }}\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}}. \end{eqnarray*}
By using convexity of $|f^{\prime }|^{q}$, we obtain \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}\right) ^{1-\frac{ 1}{q}}\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }\left[ \frac{1+\tau _{1}}{2} \left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\frac{1-\tau _{1}}{2} \left\vert f^{\prime }(\theta _{2})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}\right) ^{1-\frac{1}{q} }\left( \int_{0}^{1}\tau _{1}^{\alpha }\left[ \frac{1+\tau _{1}}{2} \left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\frac{1-\tau _{1}}{2} \left\vert f^{\prime }(\theta _{2})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}\right) ^{1-\frac{1}{q} }\left( \int_{0}^{1}\tau _{1}^{\alpha }\left[ \frac{1+\tau _{1}}{2} \left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\frac{1-\tau _{1}}{2} \left\vert f^{\prime }(\theta _{1})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}\right) ^{1-\frac{1}{q }}\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }\left[ \frac{1+\tau _{1}}{2} \left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\frac{1-\tau _{1}}{2} \left\vert f^{\prime }(\theta _{1})\right\vert ^{q}\right] d\tau _{1}\right) ^{\frac{1}{q}} \end{eqnarray*} By computing the above integrals, the proof is completed. \end{proof} \end{thm}
\begin{crly} In Theorem \ref{908tty6}, if we choose $\alpha =1$ we obtain \begin{eqnarray*}
&&\bigg|\frac{2\left[ f(\theta _{1})+f(\theta _{2})+2f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right] }{\theta _{2}-\theta _{1}}-\frac{8}{
(\theta _{2}-\theta _{1})^{2}}\int_{\theta _{1}}^{\theta _{2}}f(x)dx\bigg| \\ &\leq &\left( \frac{1}{2}\right) ^{1-\frac{1}{q}}\Bigg[\left( \frac{ 2\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}}{6}\right) ^{\frac{1}{q}}+\left( \frac{ 5\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}}{12}\right) ^{\frac{1}{q}} \\ &&+\left( \frac{5\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}}{12}\right) ^{\frac{ 1}{q}}+\left( \frac{2\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}}{6}\right) ^{\frac{1 }{q}}\Bigg]. \end{eqnarray*} \end{crly}
\begin{thm} \label{9888um} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$ and $f^{\prime }\in L_{1}[\theta _{1},\theta _{2}].$ If $
|f^{\prime }|^{q}$ is a convex function, then we have the following inequality for Atangana-Baleanu fractional integral operators \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{4}{p(\alpha p+1)}+\frac{2\left[ \left\vert f^{\prime }(\kappa _{1})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{2})\right\vert ^{q} \right] }{q} \end{eqnarray*} where $p^{-1}+q^{-1}=1$, $\alpha \in \lbrack 0,1]$, $q>1$, $B(\alpha )$ is normalization function.
\begin{proof} By using identity that is given in Lemma \ref{ghd65}, we get \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{ 1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1}+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert d\tau _{1} \\ &&+\int_{0}^{1}\tau _{1}^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}+\int_{0}^{1}(1-\tau _{1})^{\alpha }\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert d\tau _{1}. \end{eqnarray*} By using the Young inequality as $xy\leq \frac{1}{p}x^{p}+\frac{1}{q}y^{q}$ \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{1}{p}\int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}+\frac{1}{q} \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+ \frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1} \\ &&+\frac{1}{p}\int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}+\frac{1}{q} \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+ \frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1} \\ &&+\frac{1}{p}\int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}+\frac{1}{q} \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+ \frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1} \\ &&+\frac{1}{p}\int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}+\frac{1}{q} \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+ \frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}. \end{eqnarray*}
By using convexity of $|f^{\prime }|^{q}$ and by a simple computation, we have the desired result. \end{proof} \end{thm}
\begin{crly} In Theorem \ref{9888um}, if we choose $\alpha =1$ we obtain \begin{eqnarray*}
&&\bigg|\frac{f(\theta _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\kappa _{2}}{2}\right) }{\theta _{2}-\theta _{1}}-\frac{4}{(\theta _{2}-\kappa _{1})^{2}}\int_{\theta _{1}}^{\theta _{2}}f(x)dx\bigg| \\ &\leq &\frac{2}{p^{2}+p}+\frac{\left\vert f^{\prime }(\theta _{1})\right\vert ^{q}+\left\vert f^{\prime }(\theta _{2})\right\vert ^{q}}{q} . \end{eqnarray*} \end{crly}
\begin{thm} \label{k?y67f4} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$ and $f^{\prime }\in L_{1}[\theta _{1},\theta _{2}]$. If $
|f^{\prime }|$ is a concave for $q>1$, then we have \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \frac{1}{\alpha +1}\right) \Bigg[\left\vert f^{\prime }\left( \frac{\theta _{1}(\alpha +3)+\theta _{2}(\alpha +1)}{2(\alpha +2)}\right) \right\vert +\left\vert f^{\prime }\left( \frac{\theta _{1}(2\alpha +3)+\theta _{2}}{2(\alpha +2)}\right) \right\vert \\ &&+\left\vert f^{\prime }\left( \frac{\theta _{2}(2\alpha +3)+\theta _{1}}{ 2(\alpha +2)}\right) \right\vert +\left\vert f^{\prime }\left( \frac{\kappa _{2}(\alpha +3)+\theta _{1}(\alpha +1)}{2(\alpha +2)}\right) \right\vert \Bigg] \end{eqnarray*} where $\alpha \in \lbrack 0,1]$ and $B(\alpha )$ is normalization function.
\begin{proof} From Lemma \ref{ghd65} and the Jensen integral inequality, we have \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}\right) \left\vert f^{\prime }\left( \frac{\int_{0}^{1}(1-\tau _{1})^{\alpha }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) d\tau _{1}}{\int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}}\right) \right\vert \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}\right) \left\vert f^{\prime }\left( \frac{\int_{0}^{1}\tau _{1}^{\alpha }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) d\tau _{1}}{ \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}}\right) \right\vert \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}\right) \left\vert f^{\prime }\left( \frac{\int_{0}^{1}\tau _{1}^{\alpha }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) d\tau _{1}}{ \int_{0}^{1}\tau _{1}^{\alpha }d\tau _{1}}\right) \right\vert \\ &&+\left( \int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}\right) \left\vert f^{\prime }\left( \frac{\int_{0}^{1}(1-\tau _{1})^{\alpha }\left( \frac{ 1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) d\tau _{1} }{\int_{0}^{1}(1-\tau _{1})^{\alpha }d\tau _{1}}\right) \right\vert . \end{eqnarray*} By computing the above integrals we have \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \frac{1}{\alpha +1}\right) \Bigg[\left\vert f^{\prime }\left( \frac{\theta _{1}(\alpha +3)+\theta _{2}(\alpha +1)}{2(\alpha +2)}\right) \right\vert +\left\vert f^{\prime }\left( \frac{\theta _{1}(2\alpha +3)+\theta _{2}}{2(\alpha +2)}\right) \right\vert \\ &&+\left\vert f^{\prime }\left( \frac{\theta _{2}(2\alpha +3)+\theta _{1}}{ 2(\alpha +2)}\right) \right\vert +\left\vert f^{\prime }\left( \frac{\kappa _{2}(\alpha +3)+\theta _{1}(\alpha +1)}{2(\alpha +2)}\right) \right\vert \Bigg]. \end{eqnarray*} So, the proof is completed. \end{proof} \end{thm}
\begin{crly} In Theorem \ref{k?y67f4}, if we choose $\alpha =1$ we obtain \begin{eqnarray*}
&&\bigg|\frac{f(\theta _{1})+f(\theta _{2})+2f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) }{\theta _{2}-\theta _{1}}-\frac{4}{(\kappa _{2}-\theta _{1})^{2}}\int_{\theta _{1}}^{\theta _{2}}f(x)dx\bigg| \\ &\leq &\left( \frac{1}{4}\right) \Bigg[\left\vert f^{\prime }\left( \frac{ 2\theta _{1}+\theta _{2}}{3}\right) \right\vert +\left\vert f^{\prime }\left( \frac{5\theta _{1}+\theta _{2}}{6}\right) \right\vert \\ &&+\left\vert f^{\prime }\left( \frac{5\theta _{2}+\theta _{1}}{6}\right) \right\vert +\left\vert f^{\prime }\left( \frac{2\theta _{2}+\theta _{1}}{3} \right) \right\vert \Bigg]. \end{eqnarray*} \end{crly}
\begin{thm} \label{ght56d} Let $f:[\theta _{1},\theta _{2}]\rightarrow \mathbb{R}$ be differentiable function on $(\theta _{1},\theta _{2})$ with $\kappa _{1}<\theta _{2}$ and $f^{\prime }\in L_{1}[\theta _{1},\theta _{2}]$. If $
|f^{\prime }|^{q}$ is a concave function, we have \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{2}{(\alpha p+1)^{\frac{1}{p}}}\left[ \left\vert f^{\prime }\left( \frac{3\theta _{1}+\theta _{2}}{4}\right) \right\vert +\left\vert f^{\prime }\left( \frac{3\theta _{2}+\theta _{1}}{4}\right) \right\vert \right] \end{eqnarray*} where $p^{-1}+q^{-1}=1$, $\alpha \in \lbrack 0,1]$, $q>1$.
\begin{proof} By using the Lemma \ref{ghd65} and H\"{o}lder integral inequality, we can write \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\left( \int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}\right) ^{\frac{1 }{p}}\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2} \kappa _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}\tau _{1}^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}} \\ &&+\left( \int_{0}^{1}(1-\tau _{1})^{\alpha p}d\tau _{1}\right) ^{\frac{1}{p} }\left( \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+\frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\right) ^{\frac{1}{q}}. \end{eqnarray*}
By using concavity of $|f^{\prime }|^{q}$ and Jensen integral inequality, we get \begin{eqnarray*} &&\int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{1}+ \frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1} \\ &=&\int_{0}^{1}\tau _{1}^{0}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2 }\theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) \right\vert ^{q}d\tau _{1} \\ &\leq &\left( \int_{0}^{1}\tau _{1}^{0}d\tau _{1}\right) \left\vert f^{\prime }\left( \frac{\int_{0}^{1}\tau _{1}^{0}\left( \frac{1+\tau _{1}}{2} \theta _{1}+\frac{1-\tau _{1}}{2}\theta _{2}\right) d\tau _{1}}{ \int_{0}^{1}\tau _{1}^{0}d\tau _{1}}\right) \right\vert ^{q} \\ &=&\left\vert f^{\prime }\left( \frac{3\theta _{1}+\theta _{2}}{4}\right) \right\vert ^{q}. \end{eqnarray*} Similarly \begin{equation*} \int_{0}^{1}\left\vert f^{\prime }\left( \frac{1+\tau _{1}}{2}\theta _{2}+ \frac{1-\tau _{1}}{2}\theta _{1}\right) \right\vert ^{q}d\tau _{1}\leq \left\vert f^{\prime }\left( \frac{3\theta _{2}+\theta _{1}}{4}\right) \right\vert ^{q} \end{equation*} so, we obtain \begin{eqnarray*}
&&\bigg|\frac{2(\theta _{2}-\theta _{1})^{\alpha }+(1-\alpha )2^{\alpha +1}\Gamma (\alpha )}{(\theta _{2}-\theta _{1})^{\alpha +1}}\left[ f(\kappa _{1})+f(\theta _{2})+2f\left( \frac{\theta _{1}+\theta _{2}}{2}\right) \right] \\ &&-\frac{2^{\alpha +1}B(\alpha )\Gamma (\alpha )}{(\theta _{2}-\kappa _{1})^{\alpha +1}}\left[ ^{AB}I_{\frac{\theta _{1}+\theta _{2}}{2}}^{\alpha }f(\theta _{1})+_{::::\theta _{1}}^{AB}I^{\alpha }f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) \right. \\ &&\left. +_{::\frac{\theta _{1}+\theta _{2}}{2}}^{AB}I^{\alpha }f(\theta _{2})+^{AB}I_{\theta _{2}}^{\alpha }f\left( \frac{\theta _{1}+\theta _{2}}{2}
\right) \right] \bigg| \\ &\leq &\frac{2}{(\alpha p+1)^{\frac{1}{p}}}\left[ \left\vert f^{\prime }\left( \frac{3\theta _{1}+\theta _{2}}{4}\right) \right\vert +\left\vert f^{\prime }\left( \frac{3\theta _{2}+\theta _{1}}{4}\right) \right\vert \right] . \end{eqnarray*} \end{proof} \end{thm}
\begin{crly} In Theorem \ref{ght56d}, if we choose $\alpha =1$ we obtain \begin{eqnarray*}
&&\bigg|\frac{f(\theta _{1})+f(\theta _{2})+2f\left( \frac{\kappa _{1}+\theta _{2}}{2}\right) }{\theta _{2}-\theta _{1}}-\frac{4}{(\kappa _{2}-\theta _{1})^{2}}\int_{\theta _{1}}^{\theta _{2}}f(x)dx\bigg| \\ &\leq &\frac{1}{(p+1)^{\frac{1}{p}}}\Bigg[\left\vert f^{\prime }\left( \frac{ 3\theta _{1}+\theta _{2}}{4}\right) \right\vert +\left\vert f^{\prime }\left( \frac{3\theta _{2}+\theta _{1}}{4}\right) \right\vert \Bigg]. \end{eqnarray*} \end{crly}
\section{Conclusion}
In this study, an integral identity including Atangana-Baleanu integral operators has been proved. Some integral inequalities are established by using H\"older inequality, Power-mean inequality, Young inequality and convex functions with the help of Lemma 2.1 which has the potential to produce Bullen type inequalities. Some special cases of the results in this general form have been pointed out. Researchers can establish new equations such as the integral identity in the study and reach similar inequalities of these equality-based inequalities.
\section{Acknowledgments}
The publication has been prepared with the support of GNAMPA 2019 and the RUDN University Strategic Academic Leadership Program.
\section*{Funding}
GNAMPA 2019 and the RUDN University Strategic Academic Leadership Program.
\section*{Availability of data and materials}
Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current study.
\section*{Competing interests}
The authors declares that there is no conflict of interests regarding the publication of this paper.
\section*{Author's contributions}
All authors jointly worked on the results and they read and approved the final manuscript.
\end{document} | arXiv |
\begin{document}
\title{Single-point position and transition defects in continuous time quantum walks} \author{Zhi-Jian Li$^{1,\ast}$ \& J. B. Wang$^{2,\dag}$} \affiliation{{$^{1}$Institute of Theoretical Physics, Shanxi University, Taiyuan, 030006, China}\\ {$^{2}$School of Physics, The University of Western Australia, WA 6009, Australia}\\ {$^\ast$ Email: [email protected] and $^\dag$Email: [email protected]}} \begin{abstract} We present a detailed analysis of continuous time quantum walks (CTQW) with both position and transition defects defined at a single point in the line. Analytical solutions of both traveling waves and bound states are obtained, which provide valuable insight into the dynamics of CTQW. The number of bound states is found to be critically dependent on the defect parameters, and the localized probability peaks can be readily obtained by projecting the state vector of CTQW on to these bound states. The interference between two bound states are also observed in the case of a transition defect. The spreading of CTQW probability over the line can be finely tuned by varying the position and transition defect parameters, offering the possibility of precision quantum control of the system. \end{abstract} \maketitle \section{Introduction} Compared to the classical random walk, which is a memoryless Markov process, a quantum walk is unitary and time-reversible~\cite{discrete,continuous}. It exhibits markedly different behavior due to superposition, interference, and quantum correlations. For instance, a quantum walk can propagate quadratically faster than its classical counterpart and result in a probability distribution vastly different from the classically expected Gaussian distribution~\cite{kempe}. Quantum walks have become useful tools for modeling and analyzing the behavior of quantum systems, for simulating biological processes such as energy transfer in photosynthesis~\cite {photosynthesis}, for studying quantum phenomena such as perfect state transfer~\cite{statetransfer}, Anderson localization~\cite{localization} and topological phases~\cite{topology}, as well as for developing novel quantum algorithms in quantum information processing~\cite{algorithms,childs}. Experimentally, quantum walks have been implemented in a variety of systems, such as nuclear magnetic resonance~\cite{nmr}, trapped ions and trapped cold neutral atoms ~\cite{ion,atom}, single photons in bulk ~\cite{photon}, fiber optics~\cite{fiber}, and coupled waveguide arrays~\cite{waveguide}.
With the physical implementation of quantum walks comes the issue of disorder and decoherence. The effects of decoherence and disorder on the quantum walks have been extensively studied, for example, their transition to classical random walks under the influence of decoherence~\cite {dec1,dec2,dec3}. Static and dynamic disorder also alters quantum walks from ballistic spread to localization through a disruption of the interference pattern~\cite{dis1,dis2,dis3,dis4,dis5,dis6}. Recently, W\'{o}jcik et al.~ \cite{wojcik}, Li et al.~\cite{we1} and Zhang et al.~\cite{xue} investigated the localization property of one-dimensional discrete time quantum walks (DTQW) with a single-point phase defect. Motes et al.~\cite{motes} use a bit-flip coin at a boundary to introduce the position defects and find the walker escapes dramatically faster through the boundary. For continuous time quantum walk with defects, although a precursor work by Koster and Slater~ \cite{koster} has explored quantitatively the limiting case of a single diagonal defect in a one-dimensional molecular crystal using a nearest-neighbor tight-binding model, an analytic derivation is absent to provide insight for the prevalence results relying on numerical methods. Li et al. \cite{we1} and Izaac et al.\cite{we2} have compared similar behaviors between CTQWs and DTQWs with single- and double-point defects. In this paper, we extend these works to include not only position defects but also transition defects in continuous time quantum walks, presenting analytical solutions of both traveling waves or bound states of CTQWs in position space. Here, the bound state means that the quantum walk is localized in one region of the position space with zero probability in the limit of asymptotic infinity. We use its analytical expression to discuss the associated eigenstate localization.
\section{Results}
\subsection{The single-point defect model of CTQW}
The continuous time quantum walk was first posited by Farhi and Gutmann~\cite {continuous}, as a quantization of the corresponding classical continuous time random walk. In CTQWs, classical probabilities are replaced by quantum probability amplitudes, with the system evolving as per the Schr\"{o}dinger equation in discrete space, rather than the Markovian master equation~\cite {wang}. To illustrate, we consider a classical continuous time random walk on the discrete graph $G(V$,$E)$ described by two sets $V$ and $E$. The set $ V$ is composed of the unordered nodes $j$ and the set $E$ includes the edges $e_{jk}=(j,k)$ connecting the node $j$ to the node $k$. The transition rate matrix $H$ is defined as \begin{equation} \begin{array}{c} H_{jk}=\left\{ \begin{array}{ll} \gamma _{jk} & \text{for $j\neq k$ and }e_{jk}\in E \\ 0 & \text{for $j\neq k$ and }e_{jk}\notin E \\ -\varepsilon _{j} & \text{for $j=k$} \end{array} \right. \end{array} \label{eq:h0} \end{equation} where $\gamma _{jk}$ is the probability per unit time for making a transition from node $k$ to node $j$. For the probability to be conservative, the constraint \begin{equation} \varepsilon _{j}=\sum_{k,k\neq j}^{N}\gamma _{jk}, \label{eqn.trans-mat-conservative} \end{equation} is required, where $N$ is the total number of nodes in the graph. If the transition rates between any two connected nodes are the same, i.e. $\gamma _{jk}=\gamma $, the diagonal element $\varepsilon _{j}=d_{j}\gamma $ with $ d_{j}$ denoting the degree of the node $j$ or the number of sites connected to node $j$. The state of the random walker is fully described by the probability distribution vector $\mathbf{P}(t)$, with its time evolution governed by the master equation \begin{equation} \frac{d\mathbf{P}(t)}{dt}=H\mathbf{P}(t), \end{equation} which has the formal solution $\mathbf{P}(t)=e^{Ht}\mathbf{P}(0)$.
Extending the above description to the quantum realm involves replacing the real valued probability distribution vector $\mathbf{P}(t)$ with a complex valued wave function $|\psi (t)\rangle$ and adding the complex notation $-i $ to the evolution exponent, namely \begin{equation}
|\psi (t)\rangle =e^{-iHt}|\psi (0)\rangle . \label{eq:se} \end{equation}
The quantum transition matrix $H$, often referred to as the system Hamiltonian, is required to be Hermitian instead of being constrained by Eq.~(\ref{eqn.trans-mat-conservative}). Consequently, the above time evolution is unitary, guaranteeing that the norm of $|\psi (t)\rangle $ is conserved under a CTQW. Let $\mathbf{j}$ be the position operator with eigenvector $|j\rangle$. The system state vector can be expanded in the position Hilbert space with basis $\left\{ |j\rangle \right\}$, $|\psi
(t)\rangle =\sum_{j}a_{j}(t)|j\rangle$ where $a_{j}(t)=\langle j|\psi (t)\rangle $ represents the probability amplitude of the walker being found at node $j$ at time $t$. The resulting probability distribution is given by $
P_{j}=|a_{j}(t)|^{2}=|\langle j|\psi (t)\rangle |^{2}$.
For a CTQW on a uniform infinite line, its Hamiltonian can be expressed as \begin{equation} H_{0}=\varepsilon \sum\limits_{j}\left\vert j\right\rangle \left\langle j\right\vert -\gamma \sum\limits_{j}\left( \left\vert j+1\right\rangle \left\langle j\right\vert +\left\vert j-1\right\rangle \left\langle j\right\vert \right) . \end{equation} Here, each node is connected to its neighboring nodes by a constant transition rate $\gamma $, and each node has a constant potential energy $ \varepsilon $. Now we introduce two types of single-point defects in this model, one being a position defect that has a different potential energy $ \alpha $ at node $j_{d}$ and the other as a transition defect, where a distinctive transition rate $\beta $ is assigned. Without loss of generality, we assume that the parameters $\varepsilon$, $\gamma$ , $\alpha$ and $\beta$ are reals. To account for these defects, the system Hamiltonian is modified as \begin{equation} H=H_{0}+H_{1}+H_{2}, \end{equation} with \begin{eqnarray}
H_{1} &=&\alpha |j_{d}\rangle \langle j_{d}|, \\
H_{2} &=&-\beta \left( |j_{d}\rangle \langle j_{d}+1|+|j_{d}+1\rangle
\langle j_{d}|+|j_{d}\rangle \langle |j_{d}-1|+|j_{d}-1\rangle \langle j_{d}|\right) . \end{eqnarray} The position energy at the defect node $j_{d}$ is $\varepsilon +\alpha $ and the transition rate between it and its neighboring nodes is $\gamma +\beta $.
\subsection{Eigen problem of the model Hamiltonian}
The Hamiltonian of the CTQW on an uniform infinite line is invariant under spatial translation. Consider the discrete translational operator $\mathbf{T}
_{n}$, which acts on the node states such that $\mathbf{T}_{n}|j\rangle
=|j+n\rangle $. This operator is unitary, and as such can be written in the form $\mathbf{T}_{n}=e^{i\mathbf{k}n} $, where $\mathbf{k}$ is an Hermitian operator and the generator of the translation. In the case where the Hamiltonian is invariant under spatial translation, the Hermiticity of $
\mathbf{k}$ indicates that its eigenstates $|k\rangle
=\sum_{j}e^{ikj}|j\rangle $ form a complete orthonormal basis, satisfying the eigenvalue equation $H_{0}|k\rangle =\left( \varepsilon -2\gamma \cos {k}
\right) |k\rangle $, where $0\leq k\leq \pi $. The addition of a defect breaks the translational symmetry of the system, which results in an emergence of localized eigenstates of the corresponding quantum walk. The eigenstates of CTQW on a infinite line with a single-point defect can be obtained by solving a set of recurrence equations as the following.
Expanding the eigenstate $\left\vert \psi \right\rangle $ of $H$ in the position space as $\left\vert \psi \right\rangle =\sum\limits_{j}C_{j}\left\vert j\right\rangle$ and substituting it into the eigen equation $\left\langle j\right\vert H\left\vert \psi \right\rangle
=\lambda \left\langle j|\psi \right\rangle$ with eigenvalue $\lambda $, we get a set of recurrence equations about $C_{j}$ \begin{eqnarray} &&\gamma\text{ }C_{j+1}-\left( \varepsilon -\lambda \right) C_{j}+\gamma C_{j-1} =0\text{ \ \ \ \ \ \ \ for }j\neq j_{d},\text{ }j_{d}\pm 1~, \label{Cj} \\ &&\left( \gamma +\beta \right) C_{j_{d}+1}+\left( \gamma +\beta \right) C_{j_{d}-1}-\left( \varepsilon +\alpha -\lambda \right) C_{j_{d}} =0~, \text{ \ \ \ \ \ \ } \label{cz1} \\ &&\left( \gamma +\beta \right) C_{j_{d}}+\gamma C_{j_{d}+2}-\left( \varepsilon -\lambda \right) C_{j_{d}+1} =0~,\text{ } \label{c0} \\ &&\left( \gamma +\beta \right) C_{j_{d}}+\gamma C_{j_{d}-2}-\left( \varepsilon -\lambda \right) C_{j_{d}-1} =0~. \label{cf1} \end{eqnarray} The general solution of Eq.~(\ref{Cj}) is \begin{equation} C_{j}=Ay^{(j-j_{d})}+By^{-(j-j_{d})}\text{ \ \ \ \ \ for }j\neq j_{d},\text{ }j_{d}\pm 1~, \label{gs} \end{equation} where $A$ and $B$ are arbitrary constants, and $y$ satisfies the following equation \begin{equation} \left( y-\frac{\varepsilon -\lambda }{\gamma }-\frac{1}{y}\right) =0 \label{yeq} \end{equation} Solving the above equation yields $y_{\pm }=\frac{\left( \varepsilon -\lambda \right) \pm \sqrt{-4\gamma ^{2}+\left( \varepsilon -\lambda \right) ^{2}}}{2\gamma }~.$ It can be easily shown that $y_{+}=y_{-}^{-1}$, and therefore we only need to substitute $y=y_{+}$ into Eq.~(\ref{gs}) as our general solution.
Due to the reflection symmetry of the underlying potential with defects at a single node $j=j_{d}$, the system eigenvectors in position space must possess either an odd or even parity at the defect node. In the case of odd parity, i.e. $ C_{j}=-C_{-j+2j_{d}}$, we let $C_{j}=sign(j-j_{d})\left( Ay^{\left\vert j-j_{d}\right\vert }+By^{-\left\vert j-j_{d}\right\vert }\right) $, Substituting this into Eqs.(\ref{Cj}-\ref{cf1}) and using Eq.(\ref{yeq}), we obtain the coefficients as \begin{equation} B=-A,\text{ \ }C_{j_{d}}=0,\text{ \ and ~}C_{j_{d}+1}=-C_{j_{d}-1}=A\frac{ \sqrt{-4\gamma ^{2}+(\varepsilon -\lambda )^{2}}}{\gamma }~. \label{codd} \end{equation} In the case of even parity, i.e. $C_{j}=C_{-j+2j_{d}}$, we let $ C_{j}=Ay^{\left\vert j-j_{d}\right\vert }+By^{-\left\vert j-j_{d}\right\vert }$ and the coefficients are \begin{equation} B=f(\lambda )A,\text{ \ }C_{j_{d}}=\frac{\gamma }{\gamma +\beta } (1+f(\lambda ))A,\text{ \ }C_{j_{d}+1}=C_{j_{d}-1}=\frac{\gamma (\alpha +\varepsilon -\lambda )}{2(\gamma +\beta )^{2}}(1+f(\lambda ))A~, \label{ceven} \end{equation} where \begin{equation} f(\lambda )=\frac{-\left( \alpha +\varepsilon -\lambda \right) \gamma ^{2}+\left( \gamma +\beta \right) ^{2}\left( \varepsilon -\lambda +\sqrt{ -4\gamma ^{2}+\left( \varepsilon -\lambda \right) ^{2}}\right) }{\left( \alpha +\varepsilon -\lambda \right) \gamma ^{2}-\left( \gamma +\beta \right) ^{2}\left( \varepsilon -\lambda -\sqrt{-4\gamma ^{2}+\left( \varepsilon -\lambda \right) ^{2}}\right) }~. \label{fffff} \end{equation} The arbitrary constant A in Eqs. (\ref{codd}) and (\ref{ceven}) will be determined by the normalized condition of the state vector.
The eigen vectors are traveling waves or bound states in position space are modulated by the module value of y, which depends on the eigenvalues of the system. When $\lambda \in \left[ \varepsilon -2\left\vert \gamma \right\vert ,\varepsilon +2\left\vert \gamma \right\vert \right]$, $\left\vert y\right\vert =1$ and we can set $y=e^{ik}.$ The solution given by Eq.~(\ref {gs}) is thus a traveling wave, and the corresponding eigenvalue $\lambda $ can be obtained from Eq.~(\ref{yeq}) \begin{equation} \lambda =\lambda _{k}=\varepsilon -2\gamma \cos(k)~, \label{tev} \end{equation} where $k$ is analogous to the wave number of free particle in period lattice. Substituting Eq.~(\ref{tev}) into Eq.~(\ref{codd}) and Eq.(\ref {ceven}) , we get$\ $the normalized odd-parity traveling eigenvector \begin{equation} \left\vert \psi _{k}^{o}\right\rangle =\frac{i}{\sqrt{\pi }} \sum\limits_{j}\sin \left[ k\left( j-j_{d}\right) \right] \left\vert j\right\rangle ~. \label{evodd} \end{equation} and even-parity traveling eigenvector \begin{eqnarray} \left\vert \psi _{k}^{e}\right\rangle &=&\frac{1+f(\lambda _{k})}{\sqrt{4\pi -\left[ 2+f(\lambda _{k})+f^{\ast }(\lambda _{k})\right] \left[ 1-\left( \frac{\gamma }{\gamma +\beta }\right) ^{2}\right] }} \notag \\ &&\left[ \frac{-\beta }{\gamma +\beta }\left\vert j_{d}\right\rangle +\sum\limits_{j}\left( \cos (k\left\vert j-j_{d}\right\vert )+i\frac{ 1-f(\lambda _{k})}{1+f(\lambda _{k})}\sin (k\left\vert j-j_{d}\right\vert )\right) \left\vert j\right\rangle \right] ~, \label{eveven} \end{eqnarray} respectively, where \begin{equation} f(\lambda _{k})=\frac{2i\left( \gamma +\beta \right) ^{2}\sin (k)-\left[ \gamma \alpha -2\beta (2\gamma +\beta )\cos (k)\right] }{2i\left( \gamma +\beta \right) ^{2}\sin (k)+\left[ \gamma \alpha -2\beta (2\gamma +\beta )\cos (k)\right] }~. \label{fff} \end{equation} We note that odd-parity traveling eigenvector is independent on the defect parameters $\alpha$ and $\beta$, just like on the uniform infinite lattice line traveling with constant amplitude. It is very different for the even-parity traveling eigenvector, in which the wave traveling towards right and the wave traveling towards left have different amplitudes and they are inversion symmetry about the defect position. The amplitudes are adjusted not only by the defect parameters but also by the wave number $k$. If only $ \beta =0$, the the even-parity traveling eigenvector reduces to $\left\vert \psi _{k}^{e}\right\rangle =\frac{1}{2\sqrt{\pi }}\frac{4\gamma \sin (k)}{ -i\alpha +2\gamma \sin (k)}\sum_{j}\left[ \cos (k\left\vert j-j_{d}\right\vert )+\frac{\alpha \csc (k)}{2\gamma }\sin (k\left\vert j-j_{d}\right\vert )\right] \left\vert j\right\rangle $ as given by Izaac et al.~\cite{we2}. If both $\beta =0$ and $\alpha =0$, it comes back to the free case $\left\vert \psi _{k}^{e}\right\rangle =\frac{1}{\sqrt{\pi }} \sum_{j}\cos (k\left\vert j-j_{d}\right\vert )\left\vert j\right\rangle .$
When $\lambda <\varepsilon -2\left\vert \gamma \right\vert,$ we have $ \left\vert y\right\vert >1$, and when $\lambda >\varepsilon +2\left\vert \gamma \right\vert ,$ $\left\vert y\right\vert <1$. For Eq.(\ref{gs}) being convergent at the infinity, either $A$ or $B$ must be zero. In the case of odd-parity, there is no physical solution for $C_{j}$ due to the requirement $B=-A$. However, for the case of even parity, if $f^{sign(1-\left\vert y\right\vert )}(\lambda )=0$, Eq.~(\ref{gs}) can be reduced to $ C_{j}=Ay^{sign(1-\left\vert y\right\vert )\left\vert j-j_{d}\right\vert }$ \ $(j\neq j_{d}$, $j_{d}\pm 1)$, the bound eigenvector exists, and the corresponding bound eigenvalues $\lambda _{b}$ can be obtained from solving the equation $f^{sign(1-\left\vert y\right\vert )}(\lambda )=0$ as \begin{equation} \lambda _{b}=\lambda _{\pm }=\varepsilon +\frac{\beta (2\gamma +\beta )\alpha }{(\gamma +2\beta )^{2}-2\beta ^{2}}\pm \frac{(\gamma +\beta )^{2}}{ (\gamma +2\beta )^{2}-2\beta ^{2}}\sqrt{4(\gamma +2\beta )^{2}-8\beta ^{2}+\alpha ^{2}}~. \label{eqbound} \end{equation} In this case the system has zero, one, or two bound eigenstates, dependent on the value range of the parameters $\varepsilon $, $\gamma $, $\alpha $ and $\beta $ to satisfy with $\left\vert y\right\vert >1$ or $\left\vert y\right\vert <1$. Other coefficients in Eq.~(\ref{ceven}) are found to be $ C_{j_{d}}=\frac{\gamma }{\gamma +\beta }A$ and $C_{j_{d}+1}=C_{j_{d}-1}= \frac{\gamma (\alpha +\varepsilon -\lambda _{b})}{2(\gamma +\beta )^{2}}A.$ Finally, the normalized bound eigenvector with even parity can be written as \begin{equation} \left\vert \psi ^{b}\right\rangle =A_{b}\left[ \sum\limits_{j\neq j_{d}, \text{ }j_{d}\pm 1}y^{sign(1-\left\vert y\right\vert )\left\vert j-j_{d}\right\vert }\left\vert j\right\rangle +\frac{\gamma }{\gamma +\beta } \left\vert j_{d}\right\rangle +\frac{\gamma (\alpha +\varepsilon -\lambda _{b})}{2(\gamma +\beta )^{2}}\left( \left\vert j_{d}+1\right\rangle +\left\vert j_{d}-1\right\rangle \right) \right] \label{evbound} \end{equation} with \begin{equation*} A_{b}=\left[{2\frac{y^{sign(1-\left\vert y\right\vert )4}}{ 1-y^{sign(1-\left\vert y\right\vert )2}}+\left( \frac{\gamma }{\gamma +\beta }\right) ^{2}+2\left( \frac{\gamma (\alpha +\varepsilon -\lambda _{b})}{ 2(\gamma +\beta )^{2}}\right) ^{2}}\right]^{-1/2}. \end{equation*} Its distribution on the position space is centered at the defect node, and exponentially decays with increasing of the distance from defect node. The height of the center peak and the decaying rate are determined by the strength of the defect.
Using the orthogonality relations of the sine and cosine functions, it can be easily shown that
$\left\langle \psi _{k}^{o}|\psi _{k}^{e}\right\rangle =0,\left\langle \psi _{k}^{o}|\psi ^{b}\right\rangle =0,\left\langle \psi _{k}^{e}|\psi ^{b}\right\rangle =0$ for all values of $0\leq k\leq \pi ,$ and $I=\sum_{b}\left\vert \psi ^{b}\right\rangle \left\langle \psi ^{b}\right\vert+\int_{0}^{\pi }dk\left( \left\vert \psi _{k}^{o}\right\rangle \left\langle \psi _{k}^{o}\right\vert +\left\vert \psi _{k}^{e}\right\rangle \left\langle \psi _{k}^{e}\right\vert \right).$
That is to say, the eigenvectors obtained above remain orthonormal with respect to each other and they form a complete set of basis. Consequently, the time-evolution of an arbitrary initial state $\left\vert \psi (0)\right\rangle $ can be constructed in an integral form as \begin{equation} \left\vert \psi (t)\right\rangle =e^{-iHt}\left\vert \psi (0)\right\rangle = \left[ \int_{0}^{\pi }dke^{-i\lambda _{k}t}\left( \left\vert \psi _{k}^{o}\right\rangle \left\langle \psi _{k}^{o}\right\vert +\left\vert \psi _{k}^{e}\right\rangle \left\langle \psi _{k}^{e}\right\vert \right) +\sum_{b}e^{-i\lambda _{b}t}\left\vert \psi ^{b}\right\rangle \left\langle \psi ^{b}\right\vert \right] \left\vert \psi (0)\right\rangle . \label{timeevolution} \end{equation} We have verified numerically in the following calculation that the integral result given by the above equation is completely consistent with that obtained by taking the matrix exponential of the Hamiltonian directly from Eq.~(\ref{eq:se}).
\subsection{The effect of a position defect}
Choosing the parameter values $\varepsilon =2, \gamma =1$ and $\beta =0$, we firstly examine the effects of a position defect on the quantum walk. In this case, there is always one bound state as long as $\alpha\neq 0$. The bound eigen energy $\lambda _{b}$ as a function of $\alpha$ is shown in Fig.1, in which $\lambda _{b}=\lambda _{+}>\varepsilon +2\gamma$ if $\alpha >0$ or $\lambda _{b}=\lambda _{-}<\varepsilon -2\gamma$ if $\alpha <0$. \begin{figure}
\caption{The variation of bound energy with the strength of position defect.}
\end{figure}
The left panel of Fig.2 shows the CTQW probability distribution at $t=30$, given that the quantum walk initially starts at the origin $j_{0}=0$, the strength of defect $\alpha =3$, and the defect position $j_{d}=0,1,2,5$, respectively. \begin{figure}
\caption{ Left panel: the probability distribution of CTQW with a single-point position defect when $t=30$, $\alpha=3$, $j_{0}=0$ and $ j_{d}=0,1,2,5$; Right panel: the probability at the defect position as a function of position defect strength.}
\end{figure} If a defect is located at the initial position $j_{d}=j_{0}$, a large sharp peak appears at this position (see Fig.2(a)) and its height remains largely unchanged with time. For comparison, the dashed line depicts the probability distribution of the free quantum walk without the defect. When the defect position is the nearest to the initial position of CTQW, i.e. $\left\vert j_{d}-j_{0}\right\vert =1$, the probability distribution also has a small peak localized at the defect position (see Fig.2(b)). However, when the defect position deviates away the initial position more a little, i.e. $\left\vert j_{d}-j_{0}\right\vert >1$, the CTQW probability at the defect position decrease rapidly to a minimum (see Fig.2(c)(d)). This phenomenon is related to the bound state induced by the presence of a single defect. It can be readily illustrated through decomposed form of the CTQW probability at the defect position \begin{eqnarray}
P_{j_{d}}=\left\vert \int_{0}^{\pi }dke^{-i\lambda _{k}t}\left\langle j_{d}|\psi _{k}^{o}\right\rangle \left\langle \psi _{k}^{o}|j_{0}\right\rangle
+\int_{0}^{\pi }dke^{-i\lambda _{k}t}\left\langle j_{d}|\psi _{k}^{e}\right\rangle \left\langle \psi _{k}^{e}|j_{0}\right\rangle
+e^{-i\lambda _{b}t}\left\langle j_{d}|\psi ^{b}\right\rangle
\left\langle \psi ^{b}|j_{0}\right\rangle \right\vert ^{2}. \label{allprob_jd} \end{eqnarray}
The first term in the sign of absolute value is zero forever due to $\left\langle j_{d}| \psi_{k}^{o}\right\rangle =0 $ in Eq.(\ref{evodd}). With the changes of $j_{d}-j_{0}$, the probability deriving from the second term has larger amplitudes at the tails of its distribution, just similar to the probability distribution of the free quantum walk induced by the interference of traveling waves. Unlike that, the probability deriving from the third term is mainly localized around $j_{d}-j_{0}=0$. Compared with the third term, the second term can be neglected when the distance between the initial position and the defect position is not too large. So Eq.(\ref{allprob_jd}) can be approximated as \begin{eqnarray}
P_{j_{d}}\approx\left\vert \left\langle \psi ^{b}|j_{d}\right\rangle \right\vert
^{2}\left\vert \left\langle \psi ^{b}|j_{0}\right\rangle \right\vert ^{2}, \label{prob_jd} \end{eqnarray} which is the combined projections of the initial position state $\left\vert j_{0}\right\rangle $ and defect position state $\left\vert j_{d}\right\rangle $ onto the bound eigenstate $\left\vert \psi ^{b}\right\rangle $. This approximation may be weakly depend on the defect parameter values and evolution time, but under our choosing parameter values they are at least different from two orders of magnitude. The height of the large sharp peak in Fig.2(a), calculating from Eq.(\ref{allprob_jd}), is 0.692427, and the height of the smaller peak in Fig.2(b) is 0.0637546, which almost agree with the approximate results from Eq.(\ref{prob_jd}) $P_{j_{d}}=\left( A_{b}\frac{\gamma }{\gamma +\beta }\right) ^{4}=0.692308$ and $P_{j_{d}}=\left( A_{b}\frac{ \gamma }{\gamma +\beta }\right) ^{2}\left( A_{b}\frac{\gamma (\alpha +\varepsilon -\lambda _{b})}{2(\gamma +\beta )^{2}}\right) ^{2}=0.063466$, respectively. Therefore, the spike in the probability distribution at the defect position can be regarded as a fingerprint of this bound state, which can be termed as eigen-localization. When $\left\vert j_{d}-j_{0}\right\vert >1$, $P_{j_{d}}=\left( A_{b}\frac{\gamma }{\gamma +\beta }\right) ^{2}\left( A_{b}y^{sign(1-\left\vert y\right\vert )\left\vert j_{0}-j_{d}\right\vert }\right) ^{2}$ in Eq.(\ref{prob_jd}) decrease exponentially with the increase of distance $\left\vert j_{d}-j_{0}\right\vert $ and the approximation becomes invalid. From Fig.2(b)-(d), it is also observed that the CTQW is largely reflected by the defect with a small probability of transmission. Prior to encountering the defect, the CTQW is free and evolves symmetrically in both the left and right direction. Once the part moving in the right direction meets the defect, it will be largely reflected and move towards the left. As a result, two envelopes appear on the left side of the defect position and they overlap each other resulting in a complex interference pattern, as shown in Fig.2(c) and (d).
The right panel of Fig.2 shows the CTQW probability distribution at the defect position $j_{d}=0, 1, 2, 5$, respectively, as a function of the defect strength $\alpha$ at $t=30$. It is shown that, although the bound energy is less than the traveling-wave energy when $\alpha <0$ and greater when $\alpha >0$, the probability at the defect position is symmetric about $ \alpha =0$. That is to say, CTQW treats the single-point position defect exactly the same regardless of it being a potential barrier or a potential well. When the CTQW starts from the defect position, the probability amplitude at the defect position increases monotonically with the strength of the defect potential (see Fig.2(e)). The stronger the defect potential, the larger the probability amplitude, with the CTQW largely localized at the defect position. When the CTQW does not start from the defect position, the probability at the defect position is not monotonic but rather increases firstly and then decreases with increasing defect strength $\alpha $. It tends to zero for the stronger defect strength.
In addition, Fig.2(a) shows that, besides a large peak at the origin, two smaller peaks are also observed at the tails of probability distribution, as the same locations as the ballistic peaks of the free quantum walk. Even when the CTQW starts from the left of the defect and it is largely reflected, as shown in Fig.2(b)-(d), the probability distribution still has a smaller peak on the right tail. For illustrating how a single-point position defect affect the CTQW spreads on the line, we plot the variation of CTQW's standard deviation $\sigma =\sqrt{ \left\langle \psi (t)\right\vert \mathbf{j}^{2}\left\vert \psi (t)\right\rangle -\left\langle \psi (t)\right\vert \mathbf{j}\left\vert \psi (t)\right\rangle^{2}}$ with time $t$ in Fig.3, which demonstrates predominantly a linear relationship regardless of being localized or reflected by the defect. \begin{figure}
\caption{The standard deviation of CTQW with a single-point position defect as a function of time.}
\end{figure} However, the spreading speed (given by the slope of standard deviation with time) is dependent on the position of the defect. The appearance of defect makes the standard deviation less than that of a defect free CTQW. As expected, for the case $j_{d}=j_{0}$ the spreading speed is the least due to strong localization. The pink dash-dot-dot line of $\left\vert j_{d}-j_{0}\right\vert =5$ clearly shows that the CTQW spreads like a free QW at the beginning, but when it encounters the defect the spreading speed starts to decrease. In general, the larger the distance $\left\vert j_{d}-j_{0}\right\vert $, the greater the spreading speed. As an exception, we observe a much higher spreading speed for the case $\left\vert j_{d}-j_{0}\right\vert =1$ (the red dotted line in Fig.3) due to the large reflected peak at the far left end, indicating strong interference and resonance for this special case.
\subsection{The effect of transition defect}
In this section, we focus on the effect of a single-point transition defect on the spreading properties of CTQW. We choose the parameters $\varepsilon =2, \gamma =1$ and $\alpha=0$, the bound energy as a function of transition defect strength is shown in Fig.4. When $\left\vert \gamma +\beta \right\vert \leq 1$ (i.e. $-2 \leq \beta \leq 0) $, no bound eigenstate exists, or else there are two bound states. \begin{figure}
\caption{The variation of bound energy with the strength of the transition defect.}
\end{figure} When the defect is located at the initial position ($j_{d}=j_{0}=0$), the resulting probability distribution over the discrete position space at time $ t=30$ is shown in Fig.5. \begin{figure}
\caption{The probability distribution of CTQW with a single-point transition defect when $t=30$, $j_{d}=j_{0}=0$, and $\beta=-0.9,-0.5,0.5,2$.}
\end{figure} Some important features to note: (1) if $(\gamma +\beta )=0$, the initial position is disconnected from its neighbors and consequently the CTQW stays at the initial position; (2) as $\left\vert \gamma +\beta \right\vert $ deviates slightly from zero, the residual effect of the disconnection still shows and the probability distribution has a peak at the initial position (see Fig.5(a)); this peak decreases with time, which distinguishes it from the localized peak induced by eigen bound state; (3) as $\left\vert \gamma +\beta \right\vert $ increases until it approaches 1, the CTQW spreads in a similar way as a free QW since there is no bound state yet (see Fig.5(b)); and (4) when $\left\vert \gamma +\beta \right\vert >1$ (e.g. $ \beta $ = 0.5 and 2, as shown in Fig.5 (c) and (d) respectively), the transition defect induces two bound states surrounding the defect, resulting in a large probability in the vicinity of the defect position due to eigen-localization.
Unlike the position defect induced localization where the maximum of probability is always at the defect position, the maximum probability induced by a transition defect may also be at the defect neighbors (see the insert in Fig.5(c)(d)), which is resulted by interference between the two bound states. Neglecting the contribution from traveling eigen state, the localization probability around defect position $\left( j_{d}=j_{0}\right) $ can be approximately expressed by \begin{eqnarray} P_{j_{d}} &=&\left( \frac{\gamma }{\gamma +\beta }\right) ^{4}\left[ A_{+}^{4}+A_{-}^{4}+2\cos [(\lambda _{+}-\lambda _{-})t]\left( A_{+}A_{-}\right) ^{2}\right],\label{tpd1} \\ P_{j_{d}\pm 1} &=&\left( \frac{\gamma }{\gamma +\beta }\right) ^{4}\left[ A_{+}^{4}\left( \frac{\alpha +\varepsilon -\lambda _{+}}{2(\gamma +\beta )} \right) ^{2}+A_{-}^{4}\left( \frac{\alpha +\varepsilon -\lambda _{-}}{ 2(\gamma +\beta )}\right) ^{2}\right. \notag \\ &&\left. +2\cos [(\lambda _{+}-\lambda _{-})t]A_{+}^{2}A_{-}^{2}\frac{ (\alpha +\varepsilon -\lambda _{+})}{2(\gamma +\beta )}\frac{(\alpha +\varepsilon -\lambda _{-})}{2(\gamma +\beta )}\right]\label{tpd2}. \end{eqnarray} The last terms in the square brackets of the above equations represent the interference between the two bound states. The values of Eqs.(\ref{tpd1}) and (\ref{tpd2}) are approximately equal to the peak values in Fig.5(c) and (d), fully indicating that these peaks are the eigen localization. In Fig.6, we plot the localized probability at defect position as a function of the transition defect strength $\beta $ when $j_{d}=j_{0}$. \begin{figure}
\caption{The probability at defect position as a function of the transition defect strength when $t=30$ and $j_{d}=j_{0}$.}
\end{figure} The oscillatory behavior in the range of $\left\vert \gamma +\beta \right\vert >1$ displays clearly the coherent effect between the two bound states. Similar oscillation also occurs for the probabilities at the neighbors of the defect position. When $\beta =-\gamma =-1$, complete disconnection between the initial position and its neighbors, we have $ P_{j_{d}}=1$. Smooth variation of $P_{j_{d}}$ with the small deviation from $ \beta =-1$ indicates the disconnection effect remains.
The influence of a transition defect on the spread speed of CTQW is shown in Fig.7 through the variation of its standard deviation with time. \begin{figure}
\caption{The standard deviation of CTQW with a single-point transition defect as a function of time.}
\end{figure} One particular interesting case is $\beta =-0.5$, where the spreading speed is significantly larger than that of a defect free CTQW, due to constructive interference caused by the defect. In general, however, the transition defect reduces the spreading speed due to eigen-localization and transition defect trapping. Also, when $\left\vert \gamma +\beta \right\vert $ deviates slightly from zero (e.g. $\beta =-0.9$), the variation of standard deviation is clearly non-linear. This is because the residual disconnection effect decreases with time, as the probability remaining at the initial position decreases, and correspondingly the spreading speed increases.
When the CTQW does not start from the defect position, i.e., $j_{d}\neq j_{0}=0$, Fig.8 presents the probability distribution at time $t=30$. \begin{figure}
\caption{The probability distribution of CTQW with a single-point transition defect when $t=30$, $j_{0}=0$, $j_{d}=1,2,5$ and $\beta=-0.5,0.5$.}
\end{figure} The left panel, with $\beta=-0.5$ and thus no bound state existing, shows that the CTQW wave-packet is largely reflected with a smaller transmission peak observed at the same locations as the ballistic peaks of the free quantum walk. The right panel is the situation for $\beta =0.5$, where two bound state exist. If the defect position is the nearest to the initial position of the CTQW, $j_{d}=j_{0}+1$, the eigen localization induced by two bound states accumulates the probability in the vicinity of the defect position and displays strong eigen-localisation (see Fig.8(b)). Only considering the projections of the bound eigenvectors, we have $P_{j_{d}=1}=0.003$ and $P_{j_{d}+1=2}=P_{j_{d}-1=0}=0.209$, which is nearly equal to the coordinate values in Fig.8(b). If the defect position goes away from the initial position, $\left\vert j_{d}-j_{0}\right\vert >1$, the factor $y^{sign(1-\left\vert y\right\vert )\left\vert j-j_{d}\right\vert}y^{sign(1-\left\vert y\right\vert
)\left\vert j_{d}-j_{0}\right\vert}$ in combinated projection $\left\langle j|\psi ^{b}\right\rangle
\left\langle \psi ^{b}|j_{0}\right\rangle $ makes the eigen-localization probability decay exponentially with increasing distance $\left\vert j_{d}-j_{0}\right\vert$.
\section{Discussion}
We have introduced a new form of defects in continuous time quantum walks, namely a single-point transition defect. A complete set of analytical eigenvectors in position space for CTQW on the line with a single -point position defect and a single-point transition defect is obtained. While the system containing only a single-point position defect has one bound state, the system possessing a single-point transition defect has zero, one, or two bound states dependent on the transition defect parameters. With these bound eigenstate solutions we are able to understand the detailed dynamical properties of CTQW, including transmission, reflection and localization. We found that the induced localization at the defect position is determined by the combined projections of the initial position state $\left\vert j_{0}\right\rangle $ and defect position state $\left\vert j_{d}\right\rangle $ onto the bound eigenstate $\left\vert \psi ^{b}\right\rangle $. Also, the coherent effect between two bound eigenstates can be identified through the oscillating eigen localization for the case of single-point transition defect. We present a particularly interesting case where, due to constructive interference caused by the defect, the spreading speed is significantly larger than that of a defect free CTQW. This study provides another way of controlling the scattering properties of quantum walks by introducing transition defects besides the previously studied position defects.
This kind of eigenstate localization is different from the Anderson localization of CTQWs. The Hamiltonian in the Anderson model are randomly chosen， whereas the Hamiltonian under our consideration is deterministic. The propagation behavior for a system which exhibits Anderson localization is that for any initial state and an arbitrary number of time steps, and the probability to find the particle at a position is upper bounded by an almost exponentially decaying function in the distance from its initial position. The eigenstate localization for our model considered depend strongly on the initial state of the quantum walker, more precisely on the distance between the defect position and the initial position. In fact, there are initial states such that the propagation behavior is ballistic in the sense that the variance of the particle's position distribution grows quadratically with time. The single-point defects in our model, as a local modification, can be regarded as a perturbation of a translationally invariant Hamiltonian and such perturbations generically generate bound eigenvectors. The peak in the probability distribution, occurring around the defect, can be understood as eigen-localization, which should be also allowed for high dimensions.
\end{document} | arXiv |
A generalization of Posner's theorem on generalized derivations in rings
N. U. Rehman 1
E. K. Sögütcü 2
H. M. Alnoghashi 1
1 Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India.
2 Department of Mathematics, Sivas Cumhuriyet University, Faculty of Science, Sivas, Turkey.
10.30504/jims.2022.335190.1059
In this paper, we generalize the Posner's theorem on generalized derivations in rings as follows: Let $\mathscr{A}$ be an arbitrary ring, $\mathscr{I}$ a non-zero ideal, $\mathscr{T}$ is a prime ideal of $\mathscr{A}$ such that $\mathscr{T}\subset \mathscr{I},$ and $\psi $ be a non-zero generalized derivation associated with a non-zero derivation $\rho $ of $\mathscr{A}.$ If one of the following conditions is satisfied: (i) $[\psi (x),x]\in \mathscr{T},$ (ii) $[[\psi (x),x],y]\in \mathscr{T},$ (iii) $\overline{[\psi (x),x]}\in \mathscr{Z}(\mathscr{A}/\mathscr{T})$ and (iv) $\overline{[[\psi (x),x],y]}\in \mathscr{Z}(\mathscr{A}/\mathscr{T})$ $\forall $ $x,y\in \mathscr{I},$ then $\rho (\mathscr{A})\subseteq \mathscr{T}$ or $\mathscr{A}/mathscr{T}$ is commutative. At the example, it is given that the hypothesis of the theorems are necessary.
Prime ideal
commutativity
generalized derivations
16-XX Associative rings and algebras
[1] F. A. Almahdi, A. Mamouni and M. Tamekkante, A generalization of Posner's theorem on derivations in rings, Indian J. Pure Appl. Math. 51 (2020), no. 1, 187–194.
[2] R. Awtar, Lie structure in prime rings with derivations, Publ. Math. Debrecen 31 (1984), no. 3-4, 209–215.
[3] N. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955) 19–22.
[4] M. Hongan, N. Rehman and H. Alnoghashi, Differential identities on ideals in prime rings, Afr. Mat. 33 (2022), no.3, 11 pages.
[5] P. H. Lee and T. K. Lee, Lie ideals of prime rings with derivations, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 1, 75–80.
[6] A. Mamouni, L. Oukhtite and M. Zerra, On derivations involving prime ideals and commutativity in rings, São Paulo J. Math. Sci. 14 (2020), no. 2, 675–688.
[7] H. E. Mir, A. Mamouni and L. Oukhtite, Commutativity with algebraic identities involving prime ideals, Commun. Korean Math. Soc. 35 (2020), no. 3, 723–731.
[8] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093–1100.
[9] N. Rehman and H. Alnoghashi, Commutativity of prime rings with generalized derivations and anti-automorphisms, Georgian Math. J. 29 (2022), no. 4, 583–594.
[10] N. Rehman and H. Alnoghashi, T -commuting generalized derivations on ideals and semi-prime ideal-II, Mat. Stud. 57 (2022), no. 1, 98–110.
[11] N. Rehman, H. Alnoghashi, and M. Hongan, A note on generalized derivations on prime ideals, J. Algebra Relat. Topics 10 (2022), no. 1, 159–169.
[12] N. Rehman, H. Alnoghashi and A. Boua, Identities in a prime ideal of a ring involving generalized derivations, Kyungpook Math. J. 61 (2021), no. 4, 727–735.
[13] N. Rehman, M. Hongan and H. Alnoghashi, On generalized derivations involving prime ideals, Rend. Circ. Mat. Palermo (2) 71 (2022), no. 2, 601–609.
[14] N. Rehman and H. Alnoghashi, T -Commuting generalized derivations on ideals and semi-prime ideal, Bol. Soc. Parana. Mat. accepted, 14 Pages.
This issue is in progress but all papers are fully citable
Pages 1-9
Rehman, N. U., Sögütcü, E. K., & Alnoghashi, H. M. (2022). A generalization of Posner's theorem on generalized derivations in rings. Journal of the Iranian Mathematical Society, 3(1), 1-9. doi: 10.30504/jims.2022.335190.1059
N. U. Rehman; E. K. Sögütcü; H. M. Alnoghashi. "A generalization of Posner's theorem on generalized derivations in rings". Journal of the Iranian Mathematical Society, 3, 1, 2022, 1-9. doi: 10.30504/jims.2022.335190.1059
Rehman, N. U., Sögütcü, E. K., Alnoghashi, H. M. (2022). 'A generalization of Posner's theorem on generalized derivations in rings', Journal of the Iranian Mathematical Society, 3(1), pp. 1-9. doi: 10.30504/jims.2022.335190.1059
Rehman, N. U., Sögütcü, E. K., Alnoghashi, H. M. A generalization of Posner's theorem on generalized derivations in rings. Journal of the Iranian Mathematical Society, 2022; 3(1): 1-9. doi: 10.30504/jims.2022.335190.1059
zbMATH 2023-01-16
MathScinet 2022-12-31
Scopus 2022-12-12 | CommonCrawl |
Example transpose 3x3 matrix of a
Symmetric Matrix & Skew Symmetric Matrix Definition
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European Journal of Sustainable Development Research
Issue 1 - In Progress
2020 - Volume 4 Issue 1
Fossil Fuel Substitution with Renewables for Electricity Generation – Effects on Sustainability Goals
Efstathios E. Michaelides 1 *
1 Dept. of Engineering, TCU, Fort Worth, TX, 76132, USA
Full Text (PDF)
European Journal of Sustainable Development Research, 2020 - Volume 4 Issue 1, Article No: em0111
https://doi.org/10.29333/ejosdr/6344
Published Online: 07 Dec 2019
In-text citation: (Michaelides, 2020)
Reference: Michaelides, E. E. (2020). Fossil Fuel Substitution with Renewables for Electricity Generation – Effects on Sustainability Goals. European Journal of Sustainable Development Research, 4(1), em0111. https://doi.org/10.29333/ejosdr/6344
In-text citation: (1), (2), (3), etc.
Reference: Michaelides EE. Fossil Fuel Substitution with Renewables for Electricity Generation – Effects on Sustainability Goals. EUR J SUSTAIN DEV RES. 2020;4(1):em0111. https://doi.org/10.29333/ejosdr/6344
AMA 10th edition
Reference: Michaelides EE. Fossil Fuel Substitution with Renewables for Electricity Generation – Effects on Sustainability Goals. EUR J SUSTAIN DEV RES. 2020;4(1), em0111. https://doi.org/10.29333/ejosdr/6344
Reference: Michaelides, Efstathios E.. "Fossil Fuel Substitution with Renewables for Electricity Generation – Effects on Sustainability Goals". European Journal of Sustainable Development Research 2020 4 no. 1 (2020): em0111. https://doi.org/10.29333/ejosdr/6344
Reference: Michaelides, Efstathios E. "Fossil Fuel Substitution with Renewables for Electricity Generation – Effects on Sustainability Goals". European Journal of Sustainable Development Research, vol. 4, no. 1, 2020, em0111. https://doi.org/10.29333/ejosdr/6344
The substitution of fossil fuel power plants with renewable units will lead to a profound reduction of CO2 emissions and will assist in evading the Global Climate Change. Using demand data from the electricity grid of Texas, this paper develops a scenario for the substitution of coal, at first, and of all fossil fuel power plants, secondly, in the entire State of Texas. Meeting the electricity demand of the grid with renewables, makes it necessary to develop significant energy storage capacity in addition to the renewable – wind and solar – installations. Because of the lower pant capacity factors of wind and solar units, the calculations show that significantly higher renewable power capacity must be built than the current capacity of the fossil fuel units to be substituted. Also, that a substantial fraction of the generated energy is lost in the storage-recovery processes. All these factors are expected to increase the price of electricity paid by the consumers. Ahis will have an impact on the less affluent segments of the population and – if ignored by national energy policies – will affect the goal to reduce inequality within and among countries.
sustainable development solar energy wind energy fossil substitution electricty prices
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As the global society becomes more industrialized and urbanized, the demand for electric power grows substantially faster than other types of energy. In the period 1978 to 2017 the annual growth of electricity demand was 4.9%, while that the Total Primary Energy Supply (TPES) growth was 1.9%. In the second decade of the 21st century the global electricity demand doubles every 14.5 years and all indications are that it will continue increasing with a similar rate until 2050 (International Energy Agency, 2018; US-Energy Information Administration, 2016; Michaelides, 2018). A significant contribution to the strong growth of the global demand for electric power and the global trends of power demand is a consequence of the increasing use of air-conditioning (a/c) systems during the summer season (Worland, 2018). With more than 81% of the global population residing in the geographical zone between the two 45o latitudes (north and south) where a/c is desirable or essential in the summer, the economic development of several nations in this geographic zone has also ushered the accelerated use of a/c and the significant electricity consumption during the summer months.
Approximately 39% of the global electric energy production is generated from coal and another 23% from natural gas, with the contribution of coal continuously increasing since the beginning of the 21st century (International Energy Agency, 2018). The combustion of the two fossil fuels – and especially the combustion of coal – emits large quantities of CO2 in the atmosphere. CO2 is the most abundant Greenhouse Gas (GHG), and the principal contributor to the average global temperature rise and the Global Climate Change (GCC). Coal power plants produce approximately 1.15 tons of CO2 per MWh and natural gas turbines produce (on the average) approximately 0.75 tons of CO2 per MWh (Michaelides, 2018). The global electric power generation industry contributed approximately 40% of the 38 billion tons of CO2 that was produced in 2016 with all the emissions being point-source emissions (International Energy Agency, 2018; US-Energy Information Administration, 2016). It becomes apparent that the most effective way to reduce the global GHG emissions and elude the GCC is the substitution of fossil fuel power plants – starting with the more offensive coal power plants – with renewable energy sources, primarily with wind and solar energy, which are available in all the regions of the globe. Such initiatives to reduce the CO2 emissions by curtailing the use of fossil fuels for the generation of electricity have been adopted in several regions of the globe, and especially within the European Union.
Increasing the contribution of nuclear energy – a largely non-CO2 emitting energy source – in the mix of electric power generation is another option. However, because of problems associated with the disposal and long-term custody of nuclear wastes, there are very few initiatives around the world to "go nuclear." Hydroelectric power and geothermal energy are clean and effective solutions, but not available in all the regions of the globe. In addition, most of the prime hydroelectric and geothermal resources have been already utilized (Michaelides, 2016, 2018) for the generation of electricity in the OECD and most developing countries. This leaves solar and wind power – two abundant but diffuse energy sources – as the principal alternatives for the substitution of fossil fuel power plants.
The main principle of sustainable development is the ability of a society to ensure that it meets the needs for the present, without compromising the ability of future generations to meet their own needs (Brundtland, 1987). This general principle, which was coined by the Brundtland Commission, has generated sets of guidelines for nations to achieve higher standards of living (development), while taking under consideration the interrelationships between people, resources, and the natural environment. Guidelines for sustainable development recognize that economic development is what humans do as they attempt to improve their conditions within their habitat; and that the environmental, social, and economic factors, which affect all nations and human societies, are highly interconnected and inseparable.
The economic development of nations has always been accompanied by the significant increase of energy consumption and, especially, of electricity. Table 1 shows the increase of the GDP of the entire world as well as of five nations in the period 2001 to 2016, adjusted for purchasing power parity (PPP); the corresponding TPES growth; and the electric energy production growth in the same time period (International Energy Agency, 2003, 2018).
Table 1. GDP (adjusted for PPP), TPES and electricity consumption of the world and in five nations, in the years 2001 to 2016. Data from International Energy Agency (2018) and International Energy Agency (2003)
2001 GDP, $US billion
Annual Growth, %
2001 TPES, Quads
2016, TPES, Quads
2001, Electricity, TWh,
2016, Electrcity, TWh,
Two significant trends are apparent in the data of Table 1:
The economic development of nations (GDP growth) is accompanied by significant growth in TPES consumption and electric energy generation.
The rate of electric energy consumption growth significantly surpasses that of TPES. This trend extends to most nations during the period of the first two decades of the 21st century.
It must be noted that one of the unintended consequences of higher electricity generation in the period 2001-2016 is the increased combustion of coal, which was the primary contributing factor to the global CO2 emissions growth – from 25,406 billion tons of CO2 in 2001 to 38,100 billion tons of CO2 in 2018 (Boden et al., 2012; Michaelides, 2018).
Electricity is transmitted from the generating power plants to the consumers via a vast network of transformers and cables that are parts of the regional electricity grids. These grids do not store electric energy. At any instant, the consumers' demand for electric power is matched by the electricity supply from the generating stations. Of the latter, the very large base-load units primarily use Rankine cycles with nuclear or coal as their fuels; intermediate- and peak-load power plants essentially operate with Brayton and Diesel cycles (or with combined cycles) and use gaseous hydrocarbons or liquid petroleum products. These power plants are available to produce power at will and their operation closely follows the regional power demand.1
The vast majority of power plants operating with Rankine cycles are base-load units that cannot be switched on and off frequently. Power generation in such units may be slightly reduced – to approximately 80% of their rated capacity by steam throttling. Frequent, large scale power fluctuations and stoppages are avoided in these power plants because they damage the machinery. Nuclear and coal-fired steam power plants are base-load units that are placed in the grid for almost continuous operation. The diurnal fluctuations of electric power demand are primarily met with gas turbines and, wherever available, with hydroelectric units (El-Wakil, 1984; Michaelides, 2018).
This paper aims at communicating some of the important consequences of fossil fuel power plant substitution with wind and solar units. The paper offers insights on the demand-supply mismatch; the need for energy storage; the evolution of the so called "duck demand curve;" the effects of adding nuclear capacity to balance the required energy storage and plant capacity; and the effect of fossil fuel substitution on the price of electricity and four sustainability goals. While particular reference to the ERCOT electricity grid in Texas is made (for which a great deal of data are available and which is typical of a grid in a region with high a/c utilization), the conclusions of the paper are general and may be extended to most electric power grids where substitution with renewable electricity sources is anticipated.
Demand and Supply Mismatch
Solar and wind are abundant and environmentally friendly forms of energy, but they are very diffuse and not always available when electric power is demanded. Solar is periodically variable and wind power is almost intermittent. As a consequence, the substitution of electricity generated by fossil fuels with the renewable wind and solar energy sources encounters important limitations that are chiefly characterized by supply and demand mismatch.
The first limitation is inadequate supply: During the early evening hours of a hot summer day, when there is widespread use of the a/c systems in most equatorial and temperate regions of the globe, there is only very low solar irradiance and the feeble wind currents are not sufficient to satisfy the (still high) electricity demand. The second limitation is related to oversupply: It is axiomatic that photovoltaic (PV) and thermal solar installations only produce electric power during the daylight hours. When the region of an electricity grid produces a high fraction of its total annual electric energy by solar installations, there is high solar electricity generation during the morning hours. The data on the demand side show that, during the early daylight hours (from the early morning until noon), the demand for electricity is lower because the ambient temperature is not high enough for the a/c units to operate continuously. The high power production from the solar systems in combination with the moderate electricity demand by the households, causes a supply-demand mismatch, which is manifested in the so called U-shaped demand curve, or "duck curve" (Freeman et al., 2016; Weber, 2016). The mismatch becomes significant and affects the entire electricity grid, when the penetration of PV-generated energy in the electricity market exceeds approximately 30% (Leonard et al., 2020). The non-renewable demand reduction levels are significant to negatively affect the operation of the base-load units that utilize Rankine cycles and are designed to continuously produce electric power.
Figure 1a. Power demand and supply in the North Texas region when 20% and 30% of the annual electric energy is supplied by the wind
Figure 1 shows this demand-supply mismatch when a relatively large fraction of the annual electric energy is supplied by wind and solar power in ERCOT2 the independent electricity grid that supplies most of Texas (ERCOT, 2018). Figure 1a shows the power demand in this grid on May 4 2018 – a windy day – as well as the electric power production if sufficient wind capacity were installed in the region to supply 20% and 30% of the annually consumed electric energy in the region (in addition to the wind turbines that currently satisfy part of the demand in the ERCOT region). The power output of the four nuclear reactors in the State (4,985 MW) was added in the renewable power supply, because these units must operate continuously. It is observed that, at the 30% wind energy supply, the power produced by the wind turbines would be higher than the demand from midnight until 9:00 am. Figure 1b also depicts the power demand in the ERCOT region on April 16 2018 – a clear day with a great deal of sunshine– as well as the electric power production if sufficient PV panels were installed in the region to supply 20% and 30% of the annually consumed electric energy in the region. As with Figure 1a, the supply side includes the output of the four nuclear base-load units (4,985 MW). It is observed again that, when the solar irradiance generates 30% of the annual energy demand, then the generated power by the PV cells would be significantly higher than the demand between the hours 9:00 am and 5:00 pm. Supply-demand mismatch, such as the one shown in the lower part of the Figure generates the (now infamous) duck curve (Freeman et al., 2016; Michaelides, 2019a; Weber, 2016). The large-scale construction of Zero-Energy Buildings (ZEBs) – buildings that generate annually as much electric energy as they consume using PV cells and rely on the electricity grid for all shortfalls in the nighttime – has the same effect as the one depicted in Figure 1b. Grid-Independent Buildings (GIBs) avoid this problem by storing the surplus energy to be used when solar irradiance is not available (Leonard & Michaelides, 2018).
Figure 1b. Power demand and supply in the North Texas region when 20% and 30% of the annual electric energy is supplied by the sun
When the electric power demand is higher than the supply, the deficit is currently generated by intermediate- and peak-power units. However, when supply exceeds demand, the excess electric power in the grid must be either dissipated (and wasted) or stored to be used during another time period (El-Wakil, 1984; Leonard et al., 2020; Weber, 2016). In order to avoid the supply-demand mismatch the authorities in charge of the electricity grid may pursue a combination of the following:
Offer to the consumers incentives to control and adjust the electric power demand – a very difficult task to accomplish in a market-oriented economy.
Reduce the number of base-load units (primarily coal and nuclear) or completely eliminate them in favor of units that may be switched on and off rapidly. This entails significant investment in peak power units. In addition, the peak-power units work on natural gas or diesel and produce CO2.
Install utility-level storage capacity. This may also include chilled water storage that supplies the a/c demand of buildings.
All three options necessitate significant investment in equipment by the electricity production and distribution corporations, a cost that will eventually be passed to the consumers and will increase the cost of the electric power.
Energy and Sustainability Goals
Within the broad guidelines of sustainable development of the 1987 report by the U.N. World Commission on Environment and Development (Brundtland ,1987), the U.N. General Assembly approved in 2015 the Sustainable Development Goals (SDGs), a set of seventeen statements/goals that may be used for the drafting of development plans and policies in the near and intermediate future (Rosen, 2019). While all the goals are interconnected and one may see the relevance of energy availability in all of the goals, two of them are directly connected and affect the global electric power generation industry:
Ensure access to affordable, reliable, sustainable, and modern energy for all.
Take urgent action to combat climate change and its impacts.
Since energy availability and consumption are determinants of the economic development of states and regions, two additional goals are indirectly related to the affordable energy supply:
End poverty in all its forms everywhere.
Reduce inequality within and among countries.
Because climate change is inextricably connected to the production of CO2 in combustion processes, it is rational to conclude that the first two SDGs endorse the production of energy from non-fossil fuel resources, primarily from renewable sources. This will include the substitution of coal (firstly) and natural gas (secondly) in electricity generating power plants, which are point sources of CO2 emissions and currently account for approximately 40% of the global CO2 emissions.
The substitution of fossil fuel power plants with renewable sources does not happen on a one-to-one basis and depends on the type of the renewable energy source. Hydroelectric energy is the best alternative for this substitution, because hydrostatic energy is essentially stored in the water reservoirs to be used – almost at will – when the demand arises. Countries such as Norway and Canada, with a great deal of untapped hydroelectric potential, are in a very good position to substitute fossil fuels with renewable hydropower for the generation of electricity. However, wind is an intermittent energy source and the sun is a periodically variable source. Wind turbines are at standstill on calm days and solar cells do not produce any power in the nighttime. For the reliable supply of electric power, energy must be stored to satisfy the demand on calm evenings and nights, when the generation of electric power by wind and solar is almost zero.
The storage and subsequent re-generation of electricity entails thermodynamic irreversibilities. Significantly more energy must be generated by wind and solar energy to be stored and satisfy the demand on windless nights. The energy storage systems are built with additional capital expenses that, eventually, will be reflected in the cost of electricity. One of the aims of this paper is the calculation of the additional energy production and the needed energy storage requirements for the substitution of the coal and natural gas power plants within the ERCOT region. As most of the other regions in the OECD countries, the ERCOT region does not possess significant additional hydroelectric resources – beyond the currently utilized 186 MW capacity that generates 0.2% of the electricity for the grid – and must rely on solar and wind power for the substitution of the fossil fuel units.
Power Generation and Demand in the ERCOT system
Figure 2 depicts the electric power demand in ERCOT during three typical workdays of the year 2018: January 17, March 21 and July 11. The effect of the a/c demand in the summer is evident in this Figure where the daily peak power increases from 41,000 MW during the winter day and 38,000 MW during the day in the spring to more than 66,000 MW in the summer. The small difference between the winter and spring days is primarily due to the auxiliary equipment (fans and blowers) that operate with the natural gas furnaces, which provide heating to most of the buildings in the area. A salient feature of Figure 2 is that the maximum power demand in the summer occurs late in the afternoon, when the sun is setting and solar irradiance is far from its maximum. The Figure also shows that the variability of the electric power is significantly higher in the summer than in the spring and the winter: the ratio of the daily maximum to the daily minimum power demand is 1.27 for the day in January; 1.34 for the day in March and 1.77 for the day in July.
Figure 2. ERCOT electricity demand on three typical workdays of the year
The grid is supplied by a large number of electric power generating units of all types including: nuclear, coal, natural gas, biomass, wind, solar PV, and hydroelectric. Figure 3 shows the primary energy sources that generated the annual electric energy in the ERCOT interconnect during the period 2006-2018. It is observed that natural gas and coal supplied most of the electric energy for the grid during this period. In the last year, 2018, the contribution of these two sources was approximately 69.3% of the total generated electricity, with coal supplying 24.5% and natural gas 44.8% (ERCOT, 2018). The "other" energy sources include PV solar, which currently supplies less than 0.6% of the total annual energy. It is apparent that, at present, solar energy does not significantly contribute to the production of electricity in the grid. On the contrary, wind power produced approximately 18.6% of the total electric energy in 2018; it has surpassed the generation from nuclear energy and it is projected to surpass the coal-generated electricity by 2024. The maximum electric power demand during 2018 in the grid was 73,308 MW (on July 19) and the minimum demand 27,139 MW (on April 23) (ERCOT, 2018). It is estimated (Michaelides, 2019a) that the total CO2 emissions in 2018 from all the coal power plants was approximately 107 million tons; and from the gas units and additional 125 million tons.
Figure 3. The energy sources mix that supplied electric energy in ERCOT in the period 2006-2018
It must be noted that the total installed capacity for wind generation in ERCOT at the end of 2018 was 22,607 MW, while that of nuclear units was 4,985 MW. The latter generated power almost continuously and accounted for 41.125 TWh with aggregate capacity factor 94.5%, while the wind generators that rely on the intermittent wind power generated 69.796 TWh with aggregate capacity factor 35.2% (ERCOT, 2018).
Substitution of Fossil Fuels with Renewables
CO2 emissions by coal power plants amount to approximately 1.15 kg per kWh of generated electric energy, the highest emissions of all fossil fuel power plants. The corresponding emissions for natural gas units are approximately 0.75 kg of CO2 per kWh generated. Both numbers depend on the composition of the coal or natural gas as well as the thermal efficiency of the unit (Michaelides, 2018). Given the contributions of CO2 to GCC, and because coal units emit the highest amount of CO2 per kWh, it is sensible for electricity grids to first substitute the coal units in favor of renewables and other forms of primary energy (e.g. nuclear) that do not contribute to the GCC. Following the retirement of the coal power plants it would be advisable to retire the units that burn other fossil fuels – the natural gas units and the diesel-engine units.
The data of Figures 1 and 2 shows that this substitution is not simply achieved by constructing more solar and wind farms. When the contribution of wind and solar energy exceeds approximately 25% of the annual energy production, there are several time periods during a year, when the power produced by the renewable sources exceeds the demand. In such cases, a number of the other power plants must be shut down or the excess electric power must be dissipated and wasted. This limitation becomes particularly acute during windy weather when the wind power generation is at its peak – often during the early morning hours when the electricity demand is close to its daily minimum level. During 2016 the ERCOT grid recorded 74 hours when the spot prices of wind-generated electricity were negative because of excess wind power supply.
Shutting down base-load steam units that operate on the Rankine cycle for a few hours during a day is unrealistic, because the units need several hours to come back online. This becomes an insurmountable problem for the nuclear power plants in the USA, which utilize boron shimming for the control of nuclear reactivity and thermal power. The larger coal-fired units also require 4-6 hours to start from cold, but their output may be reduced by about 20% via steam throttling. Such time lags are high enough to preclude the Ranking cycle units from closely following the diurnal electric power demand. For this reason, the base-load power plants operate continuously, at fairly constant power levels. A glance at the data of electric power demand proves that, if the numbers of wind and solar energy units increase to produce an additional 25% of the total annual energy in the ERCOT region, the entire electric power generation system must be modified to maintain its flexibility and reliability.
A solution to this problem is to replace the inflexible steam power plants (coal and nuclear) with gas turbines and diesel engines that are flexible enough to follow the diurnal variability of the electric power demand and the production of the renewable energy units. However, this option has significant disadvantages:
It excludes generation from the nuclear units that currently produce very cheap electricity without any CO2 emissions and do not contribute to the GCC problem. This option also eliminates the construction of new zero-CO2 emitting nuclear power plants.
The natural gas and diesel power plants are fossil fuel units and emit CO2.
A second option is to develop storage systems throughout the region: the systems store the excess energy generated by the renewable sources and use this energy at a later time. The option may be extended from the coal units' substitution to the other fossil fuel power generation units in order to achieve totally carbon-free electric power generation. This option has the additional beneficial effect of eliminating the inefficient peak-power units – older gas turbines with very low efficiency and capacity factors –that are only used for short, high-demand time periods.
The following energy storage systems are practicable for the large-scale energy storage, which is necessary for the ERCOT grid:
Pumped Hydroelectric Systems (PHSs) (Michaelides, 2018; Weber, 2016).
Hydrogen supplied to fuel cells, which are Direct Energy Conversion devices and are not subjected to the Carnot limitations (Bockris, 2002; Michaelides, 2012; Weber, 2016).
Chemical batteries (Batteries, 2017; Michaelides, 2018; Weber, 2016).
PHS storage is not a feasible option in the ERCOT region, because the entire region is relatively flat and does not have the river valleys and high-altitude water reservoirs that are necessary for PHSs. Chemical storage in batteries and hydrogen is feasible and the storage systems may be distributed throughout the region, with small districts and groups of buildings developing shared capacity.
It is noted that the specific energies of lithium batteries, lead batteries, and hydrogen are approximately 0.4 kWh/kg, 0.03 kWh/kg and 31.2 kWh/kg (Batteries, 2017; Michaelides, 2018). Battery storage systems are by far more massive than hydrogen systems. For short term storage (e.g. the diurnal cycle of 24 hours) solid-state batteries have higher round trip efficiencies than the hydrogen-fuel-cell combination storage. However, all batteries suffer from internal power drift (they get discharged) in the long term and this makes them unsuitable for seasonal (e.g. from spring when winds are high, to summer) energy storage. This leaves hydrogen as the most suitable energy storage medium in the region.
It must be noted that the technology for hydrogen utilization as an energy carrier is sufficiently advanced for the hydrogen storage systems to be applied to larger regions: The electrolysis processes for the production of hydrogen is simple, its technology is advanced, and the method has been used for more than two centuries. Hydrogen fuel cell technology – albeit not as advanced as that of electrolysis – has also been used for a long time and is currently used in urban transportation. In the first two decades of the 21st century, several commercial automotive corporations manufacture hydrogen fuel cell cars and buses that utilize hydrogen as a clean fuel in storage tanks at pressures up to 700 bar (Michaelides, 2018).
The System and Governing Equations
Figure 4 is the schematic diagram of the main components of a statewide system that would generate, store, and distribute renewable electric power in ERCOT. A large number of PV systems (solar farms) and wind turbines throughout the State ystemsle energy generation system generate electric power from solar irradiance and the wind. When there is sufficient demand to absorb the renewable energy, the generated power is directly fed to the grid. When the demand is low, part of the energy is supplied to the grid and the rest is diverted to water electrolysis systems that generate hydrogen gas, which is locally stored in hydrogen tanks under pressure. During periods of high power demand, the chemical energy of hydrogen is converted to electricity in fuel cells and is fed back to the grid via dc-to-ac power inverters.
Figure 4. Schematic diagram of the renewable energy generation system
For the calculation of the parameters that would enable generation from renewables and storage in the ERCOT region, an hour-by-hour simulation was performed using the 2018 electric power demand data. Since wind and solar are the main renewable energy sources in the region, the total renewable power generation during the hour of the year, i, is the sum of the power generated by the wind and the solar installations:
\[E_{\text{Pi}} = E_{\text{WPi}} + E_{\text{SPi}}\] (1)
At any hour, i, the power to be stored or retrieved from storage is equal to the difference between generation and demand:
\[\delta E_{\text{Si}} = E_{\text{Pi}} - E_{\text{Di}}\] (2)
Because of thermodynamic irreversibilities in the energy conversion processes, part of the electric energy produced and diverted to storage is lost and is not converted to the chemical energy of hydrogen. On the other end of the storage process, part of the chemical energy of hydrogen is lost during its conversion to electricity. These energy losses, which are significant in all energy storage systems, are taken into account by the efficiencies of the electrolytic process, ηel, and of the fuel cells, ηfc. Accordingly, the stored energy (energy storage level) in the (i+1)st hour becomes:
\[\begin{matrix} E_{Si + 1} = E_{\text{Si}} + \left( \delta E_{\text{Si}} \right)\eta_{\text{el}}\text{ if }E_{\text{Pi}} \geq E_{\text{Di}} \\ E_{Si + i} = E_{\text{Si}} - \left( \delta E_{\text{Si}} \right)/\eta_{\text{fc}}\text{ if }E_{\text{Pi}} < E_{\text{Di}} \\ \end{matrix}\] (3)
where ESi is the energy storage level at the previous hour, i. The values of the efficiencies used in this study are: \(\eta_{\text{el}}\) = 78% (Mazloomi et al., 2012), and \(\eta_{\text{fc}}\) = 75% (US-DOE Hydrogen Fuel Cell factsheet, 2006). Small amounts of energy are also dissipated in the other equipment of Figure 4: the efficiencies of the Maximum Power Point Trackers (MPPTs) are higher than 95% (Haeberlin et al. 2006, 2017); modern dc to ac inverters also exhibit efficiencies that are higher than 95%. For the calculations these efficiencies are lumped with the efficiencies of electrolysis and of the fuel cells.
The entire generation-storage system must be robust and flexible enough to satisfy the diurnal, seasonal, and annual power demand fluctuations, which become higher during the summer months, when the a/c demand is high in the entire region served by ERCOT. For the reliability of the entire grid system, when all fossil fuel power plants are substituted by renewables, it was stipulated that the energy storage tanks must always have enough capacity to supply the entire grid for a minimum of ten days. If there is a system failure, malfunction or adverse weather conditions that temporarily reduce the amount of renewable energy produced, then the grid operators will have ample time to purchase hydrogen, or divert energy from a different geographical area to ensure that the electricity supply in the grid will continue uninterrupted. As a result of this constraint the stored energy in the entire system is significantly higher than zero during all hours of the year.
Storage and Additional Energy Needs
Calculations were first performed for the substitution of the coal power plants in the ERCOT grid system, which produce significantly more CO2 (and other pollutants) per kWh generated than the gas turbines, and are widely considered to be the first power units to be substituted. Simulations were performed for every hour of the year 2018, using the power demand data of ERCOT (Ercot, n.d.). In 2018 the several coal power plants within the area supplied by the grid produced 93.249 TWh of electricity (24.8% of the total generation). The calculations included several combinations of wind and solar generation and the results are depicted in Figure 5. The ordinate of the figure is the percentage of wind (100% minus the ordinate value is the percentage of solar) power used for the substitution of the coal power plants. The two abscissas of the figure show: on the right the storage capacity, in m3, that is necessary for the substitution at the given power mix of wind and solar; and on the left the electric energy, in MWh, that must be produced by the PV installations and the wind installations, as well as the energy lost during the storage and re-generation processes. It is observed that minimum storage capacity is required (and also minimum energy is lost in the storage-regeneration processes) when the additional renewable energy is generated in approximately equal parts from solar and from wind. The flat energy loss curve denotes that minimum energy losses also occur when the mix of the additional wind and solar energy are approximately equal. Figure 5 also shows that, if all the additional energy is generated by the wind – by continuing the current trends of adding wind farms throughout Texas, – then 254.4 million MWh must be produced by the new wind farms (of which 89.7 million MWh would be lost in the storage-regeneration processes) and a storage capacity of 10.3 million m3 must be built.
Figure 5. Energy produced by wind and solar and storage capacity for the substitution of the coal power plants
The second set of calculations was performed for the substitution of all the fossil fuel powered units in ERCOT, coal as well as natural gas power plants. The coal power plants produced approximately 24.8% of the electricity demand in 2018 and natural gas turbines produced an additional 44.5%. Figure 6 depicts the storage requirements for this substitution; the additional energy that must be generated from wind and solar; and the annual energy lost in the storage-regeneration system. It is observed that, for the entire substitution of the fossil fuel units, the minimum storage capacity (29.8 million m3) occurs when only 20% of the additional renewable energy is generated from wind and the remaining 80% from solar. This happens because energy generated from solar installations is better correlated with the diurnal power demand, and especially with the power peaks on summer days that are due to a/c usage. The minimum energy lost (31.5 million MWh) corresponds to approximately 75% wind power and 25% solar power mix for the additional energy that needs to be produced annually. With this mix of renewable energy generation, the required storage capacity is 41.7 million m3.
Table 2 gives the additional power of PV and wind turbines that (together with the storage system) would generate the additional energy to substitute the coal power plants and all the fossil fuel plants. In the latter case the minima of lost energy and storage capacity do not coincide, as it is evident in Figure 6. The data for the 100% substitution by wind are offered because, at present, the generation of power by wind turbines is by far less expensive than generation by PV cells, even when the governmental subsidies are taken into account. As a consequence, there is a great deal of investment and strong growth in the wind power installed capacity.
Table 2. Additional renewable power needs for the substitution of fossil-fueled power plants
PV Power, MW
Wind Power, MW
Storage capacity, m3
Substitution of coal units with minimum storage
Substitution of coal units with 100% wind power
Substitution of all fossil fuel units with minimum storage
Substitution of all fossil fuel units at minimum energy lost
Substitution of all fossil fuel units with 100% wind power
Figure 6. Energy produced by wind and solar and storage capacity for the substitution of all fossil fuel power plants
It must be noted that, because the two renewable energy sources do not produce power continuously and because of the energy lost in the storage-regeneration process, significantly more solar and wind capacity must be installed than the substituted fossil fuel units. For example, the 80,700 MW combined installed capacity in the first line of Table 2 would substitute approximately 14,000 MW of coal-fired units. Because of the low energy generation capacity factors (approximately 36% for wind and 22% for solar) the substitution of fossil fueled units with renewable units does not happen on one-to-one basis of rated capacity. The additional installed capacity together with the installation costs of the storage systems would add to the price of electricity and will, eventually, be passed to the consumers.
Addition of Nuclear Generating Capacity
A glance at Table 2 proves that the required storage capacity and the combined wind and solar power installations are very high and this would require substantial capital investment. A viable method to significantly decrease the installed renewable power and the storage capacity is to increase the energy generation contribution of nuclear power within the ERCOT grid by constructing new nuclear power plants, which do not emit CO2. Since the nuclear power plants are base-load units, effectively the construction of more nuclear reactors would balance the decommissioning of the coal power plants – also base-load units. Table 3 shows some of the results for the substitution of all fossil fuel power plants with wind and solar renewable sources when the installed nuclear capacity is 4,985 MW (the current capacity); 9,970 MW (twice the current capacity); and 19,940 MW (four times the current capacity).
Table 3. Additional renewable power needs and storage capacity when the nuclear capacity is doubled and quadrupled
Minimum energy lost at 4,985 MW nuclear capacity
With 100% wind power and 4,985 MW nuclear capacity
Minimum energy lost at 19,940 MW nuclear capacity
With 100% wind power and 19,940 MW nuclear capacity
It is observed in Table 3 that both the storage requirements and the renewables installed capacity decrease significantly when the nuclear units are increased. Because the nuclear power plants are base-load plants, the additional nuclear capacity is significantly lower than the capacity of the renewable sources they would substitute. It may be seen in Table 3 that the addition of 4,985 nuclear capacity alleviates the addition of 16,500 MW of wind and solar combined installed capacity at the minimum energy lost option; and 17,100 MW of wind capacity at the 100% wind option.
Effects on Prices, Incomes and Sustainability Goals
Prices and costs are estimated using the theories of economics, which is a social science based on empirical principles and not natural laws. Unlike the principles of thermodynamics that are natural laws, the principles of economics emanate from correlations of empirical data; they are not permanent; and they heavily depend on the time of observations and the type of economy they were made – e.g. the energy price mechanisms are different in free-market economies, regulated economies, and centrally-planned economies. All energy prices (including the cost of power installations) exhibit high variability over long time periods, and are laden with significant uncertainty (Michaelides, 2017, 2018). For this reason, future prices and costs should be viewed as estimates with significant variability, rather than accurate calculations. A second source of variability of the renewable energy units is the regional variation of solar irradiance and wind velocity: because of the lower irradiance in the northern regions, the same PV cells would produce only 54% of the annual energy in Duluth Minnesota, than in Dallas, Texas. The lesser energy production from the same (and equally priced) PV systems would be reflected in higher prices of electricity, expressed as $/kWh.
A complication that often enters price calculations is that, at present, renewable energy is subsidized by the governments. However, when renewables produce a high fraction of the electricity in a region the subsidies expire. As a result, the current price of electricity production from renewables may not reflect the price after all the subsidies expire. Wind power is currently subsidized in Texas by $23/MWh for the first ten years of power production by the wind farms. New wind farm subsidies in ERCOT will not be available in 2020, because the technology is considered mature to survive on its own. Both solar and wind power production tax credits and subsidies will be phased out by 2025 (Rhodes, 2019) unless extended by a new law. Similarly, the renewable energy subsidies will be phased out in other countries when solar and wind power generate a sizable fraction of the electricity. On the opposite end of power generation, several studies that calculate the cost of electricity from coal and natural gas include a cost of carbon abatement or the cost for carbon capture and sequestration, environmental costs for the cleanup of CO2, and other pollution products (Lazard, 2016). Since, at least at the present time, these costs are not born by the electricity generation corporations in any form, they will not be considered in this study.
The average retail price of electric energy paid by the residential sector in the ERCOT region in 2019 is $0.108/kWh including all applicable taxes (tcap, 2018). Of this, approximately $0.05/kWh accounts for the energy transmission cost, which is born by the consumers. The prices for the commercial and industrial sectors are significantly lower, with the average of all sectors being $0.086/kWh (tcap, 2018). Because the ERCOT region is "deregulated," several competing corporations offer their own plans to residential customers. The retail prices in these plans depend on the specific area, where electricity is consumed: in the Houston area the retail residential prices are in the range $0.089-$0.150, while in the Dallas-Fort Worth Metroplex the retail residential prices are in the $0.079-$0.140 range (tcap, 2018). These residential prices are lower than the average USA electricity prices and significantly lower than the retail residential prices in most of the industrialized countries in the world (International Energy Agency, 2019). The low electricity prices have helped residents afford a/c systems that are now necessary in all households and businesses.
A glance at the previous sections proves that the substitution of fossil fuel power plants with renewables will entail the following expenses, which in a market economy would be passed to the consumers:
Cost of construction of the renewable installations (e.g. wind and solar farms). As it becomes apparent from Tables 2 and 3, the installed capacity of renewables is 3-5 times higher than the capacity of the units they substitute.
Cost of the energy storage systems and of the energy lost in the storage-regeneration process. Several sources of the life-cycle costs of energy storage systems were assessed and compared in Zakeri and Syri (2015). It becomes apparent in this and similar studies that storage costs will add significantly to the price for electricity paid by the consumers, even if the energy storage is accomplished in electric vehicle batteries (Michaelides, 2019b).
Now, let us assume that the substitution of the fossil fueled power plants becomes a reality. This will trigger the following effects:
The CO2 emissions in the electricity generation sector of Texas (and also of the entire USA) will plummet, a desirable effect.
The amount of the generated electric energy will increase because of the irreversibilities in the storage-regeneration process. This effect is not desirable, but it can be accommodated by the society at large.
The price of electricity in the region will increase, very likely significantly because of items A and B, above. With the nuclear capacity unchanged; with all fossil fuel generating plants decommissioned; and with the estimated costs for solar and wind power generation without subsidies (Ferreira et al., 2018; Hoffmann, 2006; Zakeri & Syri, 2015) and including storage and transmission, the estimates of the future electricity price for the consumers are in the range $0.30/kWh-$0.45/kWh, a significant increase over the current electricity prices.
The sharp increase of electricity prices – by a factor between 2.8 and 4.2 – will have a significant effect on the population that would have to pay more for their electricity consumption, in particular during the hot summer months. Residents in the low income brackets may not be able to afford complete a/c services for their households and this may become a health hazard, especially for the older and the infirm; residents in the intermediate income brackets will spend a significantly higher fraction of their income on electricity and will have less disposable income for other necessities or pleasure activities; high-income residents will be able to afford the electricity price increases albeit at a reduction of their disposable income. The net result of these expected effects is an increase in the disposable income inequality of citizens in the region and this is incongruent with the goal of reducing inequality within and among the nations – the 4th goal in the section on "Energy and Sustainability Goals." A projection of the expected increase of electricity prices to other nations, also leads to the conclusion that, with a fully renewable energy economy, it will become more expensive for citizens of less affluent nations to enjoy the fruits of a/c at home and higher mechanization at work. As a consequence, the goal of reducing poverty in all its forms may suffer. An additional effect on the economy of nations is that, as a result of the reduction of the disposable income of a large fraction of the population, other economic sectors will suffer and national economies as a whole will shrink (Samuelson & Nordhaus, 2009).
The higher prices of electricity may lead to a reduction of its demand, which would partly alleviate increased prices. If prices increase significantly, the demand by the consumers may decrease. However, one must take into account that, for most citizens, electricity is a necessity and its price elasticity is very low. Most energy-saving projects – e.g. increasing the buildings' insulation and installation of more efficient a/c systems – are capital intensive and are not readily affordable by citizens in the lower income brackets. As societies move toward a "greener" mix for the production of electricity, they must ensure that low-interest loans and other subsidies become available to low-income homeowners for energy conservation measures.
The correlation of higher electricity prices and income inequality is similar to the effect of a carbon tax on the incomes of the citizens. Two recent studies (Goulder et al., 2019; Williams III et al., 2015) determined the effects of a carbon tax on the incomes of USA citizens in five income categories when all the tax is returned (recycled) to the citizenry: as lump-sum payments to the lower incomes; as capital tax relief; and as labor tax relief. Table 4 shows these effects for the five income quintiles (Williams et al., 2015) with the 1st quintile being at the lowest income.
Table 4. Effect of a carbon tax with three kinds of tax recycling in the USA ( Williams III et al., 2015)
Lump-Sum Rebate
Capital Tax Rebate
Labor Tax Rebate
It is observed in Table 4 that if the carbon tax is not recycled back to the citizenry with direct payments to the lowest incomes, the increase of the energy prices will have a significant and negative effect on the lower income citizens. The same conclusion was reached in Goulder et al. (2019), whose authors determined that anything other than lump-sum rebates to the lower income citizens would constitute a regressive taxation. Clearly, the substitution of fossil fuels with renewables or a carbon avoidance tax, which is designed to promote the substitution of fossil fuels with renewables, must be accompanied by public policy measures that would protect the incomes and the lifestyles of the poorer citizens in a society.
The substitution of coal and natural gas electric power plants with renewable energy is desirable for the avoidance of GCC. However, and because solar energy is periodically variable and wind power is intermittent, this substitution will not be realized without the development of infrastructure for the storage of energy. The substitution of, firstly coal and, secondly, natural gas from the electricity production mix in the ERCOT electricity grid is feasible as an engineering project, but entails the installation of significantly higher wind and PV capacity; high energy storage capacity; and significant energy losses in the storage-recovery process. Given the terrain of the region, hydrogen storage is the most suitable energy storage method, with hydrogen produced by electrolysis, stored locally in tanks and converted to electricity with fuel cells. The addition of more nuclear capacity is beneficial because it reduces significantly the requirements for storage and the additional installed capacity.
The installation of significantly higher generating capacity, the development of energy storage systems, and the thermodynamic irreversibilities associated with the energy storage and recovery processes, are very likely to increase significantly the price of electricity for the consumers. This is an undesirable outcome that affects disproportionately the less affluent members of the society and is incongruent with the goals to reduce poverty and reduce inequalities. It would be advisable that attention be given to this undesirable outcome by governments, which may adopt public policies, price structures, and subsidies that that would make electricity affordable for all.
This research was partly supported by the Tex Moncrief Chair of Engineering at TCU.
Hydroelectric power plants may be used either as base-load or intermediate-load units.↩
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\begin{abstract} Two meromorphic functions $ f $ and $ g $ are said to share a value $ s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $ g(z)-s $ have the same set of zeros counting multiplicities (ignoring multiplicities). We say that a meromorphic function $ f $ share $ s\in\mathbb{C} $ partially $ CM $ with a meromorphic function $ g $ if $ E(s,f)\subseteq E(s,g)$. It is easy to see that the condition ``partially shared values $ CM $" is more general than the condition ``shared value $ CM $". With the idea of partially shared values, in this paper, we prove some uniqueness results between non-constant meromorphic functions and their shifts or generalized differences. We exhibit some examples to show that the result of {Charak \emph{et al.}} \cite{Cha & Kor & Kum-2016} is not true for $k=2 $ or $ k=3 $. We find some gaps in proof of the result of {Lin} \emph{et al.} \cite{Lin & Lin & Wu}, and we not only correct them but also generalize their result in a more convenient way. A number of examples have been exhibited to validate certain claim of the main results of this paper and also to show that some of the conditions are sharp. In the end, we have posed some open questions for further investigation of the main result of the paper. \end{abstract}
\maketitle
\section{Introduction}
We assume that the reader is familiar with the elementary Nevanlinna theory, for detailed information, we refer the reader \cite{Goldberg,Hay-1964,Laine-1993}. Meromorphic functions considered in this paper are always non-constant, unless otherwise specified.
For such a function $f$ and $a\in\mathbb{\overline C}=:\mathbb{C}\cup\{\infty\}$, each $z$ with $f(z)=a$ will be called $a$-point of $f$. We will use here some standard definitions and basic notations from this theory. In particular by $N(r,a;f)$ ($\overline N(r,a;f)$), we denote the counting function (reduced counting function) of $a$-points of meromorphic functions $f$, $T(r,f)$ is the Nevanlinna characteristic function of $f$ and $S(r,f)$ is used to denote each functions which is of smaller order than $T(r,f)$ when $r\rightarrow \infty$. We also denote $ \mathbb{C^{*}} $ by $\mathbb{C^{*}}:=\mathbb{C}\smallsetminus\{0\}$.
\par For a meromorphic function $ f $, the order $ \rho(f) $
and the hyper order $ \rho_2(f) $ of $ f $ are defined respectively by
\begin{eqnarray*}
\rho(f)=\limsup_{r\rightarrow\infty}\frac{\log^{+}T(r,f)}{\log
r}\;\;\mbox{and}\;\;\rho_2(f)=\limsup_{r\rightarrow\infty}\frac{\log^{+}\log^{+}T(r,f)}{\log
r}. \end{eqnarray*}
\par For $ a\in\mathbb{C}\cup\{\infty\} $, we also define \begin{eqnarray*}
\Theta(a;f)=1-\limsup_{r\rightarrow +\infty}\frac{\overline
N\left(r,{1}/{(f-a)}\right)}{T(r,f)}.\end{eqnarray*}
\par We denote $ \mathcal{S}(f) $ as the family of all meromorphic functions $ s $ for which $ T(r,s)=o(T(r,f)) $, where $ r\rightarrow\infty $ outside of a possible exceptional set of finite logarithmic measure. Moreover, we also include all constant functions in $ \mathcal{S}(f) $, and let $\hat{\mathcal{S}}(f)=\mathcal{S}(f)\cup\{\infty\} $. For $
s\in\hat{\mathcal{S}}(f) $, we say that two meromorphic functions $ f $ and $ g$ share $ s $ $ CM $ when $ f(z)-s $ and $ g(z)-s $ have the same zeros with the same multiplicities. If multiplicities are not taking into account, then we say that $ f $ and $ g $ share $ s $ $ IM $.
\par In addition, we denote $ \overline E(s,f) $ by the set of zero of $ f-s $, where a zero is counted only once in the set, and by the set $\overline E_{k)}(s,f) $, we understand a set of zeros of $ f-s $ with multiplicity $ p \leq k $, where a zero with
multiplicity $ p $ is counted only once in the set. Similarly, we denote the reduced counting function corresponding to $ \overline E_{k)}(s,f) $ as $
\overline N_{k)}\left(r,1/(f-s)\right) $.
\par In the uniqueness theory of meromorphic functions, the the famous classical results are the five-point, resp. four-point, uniqueness theorems due to Nevanlinna \cite{Nevanlinna-1929}. The five-point theorems states that if two meromorphic functions $f$, $g$
share five distinct values in the extended complex plane $IM$, then $f\equiv g$. The beauty of this
result lies in the fact that there is no counterpart of this result in case of real valued functions. On the other hand, four-point theorem states that if two meromorphic functions $f,\;g$ share four
distinct values in the extended complex plane $CM$, then
$f\equiv T\circ g$, where $T$ is a M$\ddot{o}$bius transformation.
\par Clearly, these results initiated the study of uniqueness of two meromorphic functions $f$ and $g$. The study of such uniqueness theory becomes more interesting if the function $g$ has some expressions in terms of $f$.
\par Next we explain the following definition which will be required in the sequel.
\begin{defi}
Let $f$ and $g$ be two meromorphic functions such that $f$ and $g$ share the
value $a$ with weight $k$ where $a\in\mathbb{C}\cup\{\infty\}$. We denote by
$\overline N_E^{(k+1}\left(r,1/(f-a)\right)$ the counting function
of those $a$-points of $f$ and $g$ where $p=q\geq k+1$, each point in this
counting function counted only once.
\end{defi}\par
In what follows, let $c$ be a non-zero constant. For a meromorphic
function $f$, let us denote its shift $I_{c}f$ and difference operators
$\Delta_{c}f$, respectively, by $I_{c}f(z)=f(z+c)$ and
$\Delta_{c}f(z)=(I_{c}-1)f(z)=f(z+c)-f(z).$
\par Recently an increasing amount of
interests have been found among the researchers to find results which are the difference analogue of Nevanlinna theory. For finite ordered meromorphic functions, {Halburd and Korhonen} \cite{Hal & Kor-JMMA-2006}, and {Chiang and Feng} \cite{Chi & Fen-2008} developed independently parallel difference version of the famous Nevanlinna theory. As applications of this theory, we refer the reader to see the articles in case of set sharing problems (see, for example \cite{Ahamed-SUBB-2019,Aha-TJA-2021,Ban-Aha-Filomat-2019,Ban-Aha-MS-2020,Che-Che-BMMSS-2012,Zha-JMMA-2010}), finding solutions to the Fermat-type difference equations (see e.g. \cite{Aha-JCMA-2021,Liu-AM-2012,Cao-MJM-2018}), Nevanlinna theory of the Askey–Wilson divided difference operators (see e.g. \cite{Chiang-Feng-AM-2010}), meromorphic solutions to the difference equations of Malmquist type (see e.g. \cite{Lu-BAMS-2016}) and references therein.
\par Regarding periodicity of meromorphic functions, {Heittokangas} et. al. \cite{Hei & Kor & Lai & Rie-CVTA-2001,Hei & Kor &
Lai & Rie-JMMA-2009} have considered the problem of value sharing for shifts of meromorphic functions and obtained the following result.
\begin{theoA}\cite{Hei & Kor & Lai & Rie-CVTA-2001}
Let $ f $ be a meromorphic function of finite order, and let $
c\in\mathbb{C^{*}} $. If $ f(z) $ and $ f(z+c) $ share three distinct periodic
functions $ s_1, s_2, s_3\in\hat{\mathcal{S}}(f) $ with period $ c $ $ CM $,
then $ f(z)\equiv f(z+c) $ for all $ z\in\mathbb{C} $.
\end{theoA}
\par In $ 2009 $, Heittokangas \emph{et al.} \cite{Hei & Kor & Lai & Rie-JMMA-2009} improved {Theorem A} by replacing ``sharing three small functions $ CM $" by ``$ 2\; CM + 1\; IM $" and obtained the following result.
\begin{theoB}\cite{Hei & Kor & Lai & Rie-JMMA-2009}
Let $ f $ be a meromorphic function of finite order, and let $
c\in\mathbb{C^{*}} $. Let $ s_1, s_2, s_3\in\hat{\mathcal{S}}(f) $ be three
distict periodic function with period $ c $. If $ f(z) $ and $ f(z+c) $ share $
s_1, s_2\in\hat{\mathcal{S}}(f) $ $ CM $ and $ s_3 $ $ IM $, then $ f(z)\equiv
f(z+c) $ for all $ z\in\mathbb{C} $.
\end{theoB}\par
In $ 2014 $, {Halburd} \emph{et al.} \cite{Hal & Kor & Toh-TAMS-2014} extended some results in this direction to meromorphic functions $ f $ whose hyper-order $ \rho_2(f)$ less than one. One may get much more information from \cite{Aha-JCMA-2021,Aha-TJA-2021,Che & Lin-2016,Hei & Kor & Lai & Rie-CVTA-2001,Hei & Kor & Lai &
Rie-JMMA-2009,Liu-JMMA-2009,Liu & Yan-AM-2009} and the references therein, about the relationship between a
meromorphic function $ f(z) $ and it shift $ f(z+c) $.
\par In $ 2016 $, {Li and Yi} \cite{Li & Yi-BKMS-2016} obtained a uniqueness result of meromorphic functions $ f $ sharing four values with their shifts $ f(z+c) $.
\begin{theoC}\cite{Li & Yi-BKMS-2016}
Let $ f $ be a non-constant meromorphic function of hyper-order $ \rho_2(f)<1 $
and $ c\in\mathbb{C^{*}} $. Suppose that $ f $ and $ f(z+c) $ share $ 0$, $1$,
$\eta $ $ IM $, and share $ \infty $ $ CM $, where $ \eta $ is a finite value
such that $ \eta\neq 0, 1 $. Then $ f(z)\equiv f(z+c) $ for all $ z\in\mathbb{C}
$.
\end{theoC} We now recall here the definition of partially shared values by two meromorphic functions $ f $ and $ g $. \begin{defi}\cite{Chen-CMFT-2018}
Let $ f $ and $ G $ be non-constant meromorphic functions and $ s\in\mathbb{C}\cup\{\infty\} $. Denote the set of all zeros of $ f-s $ by $ E(s,f) $, where a zero of multiplicity $ m $ is counted $ m $ times. If $ E(s,f)\subset E(s,g) $, then we say that $ f $ and $ g $ partially share the value $ s $ $ CM $. Note that $ E(s,f)=E(s,g) $ is equivalent to $ f $ and $ g $ share the value $ s $ $ CM $. Therefore, it is easy to see that the condition ``partially shared values $ CM $" is more general than the condition ``shared value $ CM $". \end{defi} \par In addition, let $ \overline E(s,f) $ denote the set of zeros of $ f-s $, where a zero is counted only once in the set, and $ \overline E_{k)}(s,f) $ denote the set of zeros of $ f-s $ with multiplicity $ l\leq k $, where a zero with multiplicity $ l $ is counted only once in the set. The reduced counting function corresponding to to $ \overline E_{k)}(s,f) $ are denoted by $ \overline N_{k)}(r,1/(f-s)) $.
\par Charak \emph{et al.} \cite{Cha & Kor & Kum-2016} gave the following definition of partial sharing.
\begin{defi}\cite{Cha & Kor & Kum-2016}
We say that a meromorphic function $ f $ share $ s\in\hat{\mathcal{S}} $
partially with a meromorphic function $ g $ if $ \overline E(s,f)\subseteq \overline E(s,g)
$, where $ \overline E(s,f) $ is the set of zeros of $ f(z)-s(z) $, where each zero is counted only once.
\end{defi}
\par Let $ f $ and $ g $ be two non-constant meromorphic functions and $
s(z)\in\hat{\mathcal{S}}(f)\cap\hat{\mathcal{S}}(g) $. We denote by $ \overline
N_0(r,s;f,g ) $ the counting function of common solutions of $ f(z)-s(z)=0 $ and
$ g(z)-s(z)=0 $, each counted only once. Put \begin{eqnarray*} \overline N_{12}(r,s;f,g)=\overline
N\left(r,\frac{1}{f-s}\right)+\overline N\left(r,\frac{1}{g-s}\right)- 2\overline
N_{0}(r,s;f,g). \end{eqnarray*}
It is easy to see that $ \overline N_{12}(r,s;f,g) $ denoted the counting
function of distinct solutions of the simultaneous equations $ f(z)-s(z)=0 $ and $ g(z)-s(z)=0 $.
\par In $ 2016 $, \textit{Charak} \emph{et al.} \cite{Cha & Kor & Kum-2016} introduced the above notion of partial sharing of values and applying this notion of sharing, they have obtained the following interesting result.
\begin{theoD}\cite{Cha & Kor & Kum-2016}
Let $ f $ be a non-constant meromorphic function of hyper order $ \rho_2(f)<1
$, and $ c\in\mathbb{C^{*}} $. Let $ s_1, s_2, s_3, s_4\in\hat{\mathcal{S}}(f) $
be four distinct periodic functions with period $ c $. If $ \delta(s,f)>0 $ for
some $ s\in\hat{\mathcal{S}}(f) $ and \begin{eqnarray*} \overline E(s_{j},f)\subseteq\overline E(s_{j},
f(z+c)), \;\;\; \text{j=1, 2, 3, 4,} \end{eqnarray*} then $ f(z)=f(z+c) $ for all $
z\in\mathbb{C} $.
\end{theoD}
\par In $ 2018 $, \textit{Lin} \emph{et al.} \cite{Lin & Lin & Wu} investigated further on the result of \textit{Charak} \emph{et al.} \cite{Cha & Kor &
Kum-2016} replacing the condition ``partially shared value $ \overline
E(s,f)\subseteq\overline E(s,f(z+c)) $" by the condition ``truncated partially shared
value $ \overline E_{k)}(s,f)\subseteq\overline E_{k)}(s,f(z+c)) $", $ k $ is a positive integer. By the following example, \textit{Lin} et. al. \cite{Lin & Lin & Wu} have shown that the result of \emph{Charak} et. al. \cite{Cha & Kor & Kum-2016} is not be true for $ k=1 $ if truncated partially shared values is considered.
\begin{exm}\cite{Lin & Lin & Wu}
Let $ f(z)={2e^z}/{(e^{2z}+1)} $ and $ c=\pi i $, $ s_1=1 $, $
s_2=-1 $, $ s_3=0 $, $ s_4=\infty $ and $ k=1 $. It is easy to see that $ f(z+\pi i)=-{2e^z}/{(e^{2z}+1)}$ and $ f(z) $ satisfies all the other conditions of {Theorem D}, but $
f(z)\not\equiv f(z+c) $.
\end{exm}\par
However, after a careful investigation, we find that {Theorem D} is not valid in fact for each positive integer $k $ although $ f(z) $ and $ f(z+c) $ share value $ s\in\{s_1, s_2, s_3, s_4\} $ $ CM $. We give here only two examples for $ k=2 $ and $ k=3 $.
\begin{exm}
Let $ f(z)={\left(ae^z(e^{2z}+3)\right)}/{\left(3e^{2z}+1\right)} $, $ c=\pi i $ and $
s_1=a $, $ s_2=-a $, where $ a\in\mathbb{C^{*}} $, $ s_3=0 $, $ s_4=\infty $ and
$ k_1=2=k_2 $. It i easy to see that $ f(z+\pi
i)=-{\left(ae^z(e^{2z}+3)\right)}/{\left(3e^{2z}+1\right)} $ and $ f(z) $ satisfies all
the conditions of {Theorem D}, but $ f(z)\not\equiv f(z+c) $.
\end{exm}
\begin{exm}
Let $ f(z)={\left(4ae^z(e^{2z}+1)\right)}/{\left(e^{4z}+6e^{2z}+1\right)} $ and $ c=\pi i
$, $ s_1=a $, $ s_2=-a $, where $ a\in\mathbb{C^{*}} $, $ s_3=0 $, $ s_4=\infty
$ and $ k_1=3=k_2 $. Then clearly $ f(z+\pi
i)=-{\left(4ae^z(e^{2z}+1)\right)}/{\left(e^{4z}+6e^{2z}+1\right)} $ and $ f(z) $
satisfies all the conditions of {Theorem D}, but $ f(z)\not\equiv f(z+c) $.
\end{exm}
\par In $ 2018 $, \textit{Lin} \emph{et al.} \cite{Lin & Lin & Wu} established the following result considering partially sharing values.
\begin{theoE}\cite{Lin & Lin & Wu}
Let $ f $ be a non-constant meromorphic function of hyper-order $ \rho_2(f)<1 $
and $ c\in\mathbb{C^{*}} $. Let $ k_1, k_2 $ be two positive integers, and let $
s_1, s_2\in\mathcal{S}(f)\cup\{0\} $, and $ s_3, s_4\in\hat{\mathcal{S}}(f) $ be
four distinct periodic functions with period $ c $ such that $ f $ and $ f(z+c)
$ share $ s_3, s_4 $ $ CM $ and \begin{eqnarray*} \overline E_{k_{j})}(s_{j},f)\subseteq \overline
E_{k_{j})}(s_{j},f(z+c)),\;\; j=1, 2. \end{eqnarray*} If $
\Theta(0,f)+\Theta(\infty;f)>{2}/{(k+1)} $, where $ k=\min\{k_1,
k_2\} $, then $ f(z)\equiv f(z+c) $ for all $ z\in\mathbb{C} $.
\end{theoE} \par
\par As a consequence of {Theorem E}, \textit{Lin} \emph{et al.} \cite{Lin & Lin & Wu} obtained the following result.
\begin{theoF} \cite{Lin & Lin & Wu}
Let $ f $ be a non-constant meromorphic function of hyper order $ \rho_2(f)<1
$, $ \Theta(\infty,f)=1 $ and $ c\in\mathbb{C^{*}} $. Let $ s_1, s_2,
s_3\in\mathcal{S}(f) $ be three distinct periodic functions with period $ c $
such that $ f(z) $ and $ f(z+c) $ share $ s_3 $ $ CM $ and \begin{eqnarray*} \overline
E_{k)}(s_j,f)\subseteq \overline E_{k)}(s_j,f(z+c)),\;\; j=1, 2. \end{eqnarray*} If $ k\geq 2 $,
then $ f(z)\equiv f(z+c) $ for all $ z\in\mathbb{C} $.
\end{theoF}
\textit{Lin} \emph{et al.} \cite{Lin & Lin & Wu} have showed that number ``$ k= 2 $" is sharp for the function $ f(z)=\sin z $ and $ c=\pi $. It is easy to see that $ f(z+c) $ and $ f(z) $ share the value $ 0 $ $ CM $ and $ \overline E_{1)}(1,f(z))= \overline E_{1)}(1,f(z+c))=\phi $ and $ \overline E_{1)}(-1,f(z))= \overline E_{1)}(-1,f(z+c))=\phi $ but $ f(z+c)\not\equiv f(z) $. Since Theorem F is true for $ k\geq 2 $, hence \textit{Lin} \emph{et al.} \cite{Lin & Lin & Wu} investigated further to explore the situation when $ k=1 $ and obtained the result.
\begin{theoG} \cite{Lin & Lin & Wu}
Let $ f $ be a non-constant meromorphic function of hyper order $ \rho_2(f)<1
$, $ \Theta(\infty,f)=1 $ and $ c\in\mathbb{C^{*}} $. Let $ s_1, s_2,
s_3\in\mathcal{S}(f) $ be three distinct periodic functions with period $ c $
such that $ f(z) $ and $ f(z+c) $ share $ s_3 $ $ CM $ and \begin{eqnarray*} \overline
E_{1)}(s_j,f)\subseteq \overline E_{1)}(s_j,f(z+c)),\;\; j=1, 2. \end{eqnarray*} Then $
f(z)\equiv f(z+c) $ or $ f(z)\equiv - f(z+c) $ for all $ z\in\mathbb{C} $.
Moreover, the later occurs only if $ s_1+s_2=2s_3 .$
\end{theoG} \begin{rem} We find in the proof of \cite[Theorem 1.6]{Lin & Lin & Wu}, \textit{Lin} \emph{et al.} made a mistake. In {Theorem 1.6}, they have obtained $ f(z+c)\equiv -f(z) $ as one of the conclusion under the condition $s_1+s_2=2s_3 $, where correct one it will be $ f(z+c)\equiv -f(z)+2s_3 $. One can easily understand it from the following explanation. In \cite[Proof of Theorem 1.6, page - 476]{Lin & Lin & Wu} the authors have obtained $ \alpha=-1 $, where $ \alpha $, the way they have defined, finally will be numerically equal with ${\left(f(z+c)-s_3\right)}/{\left(f(z)-s_3\right)}=\alpha $, when $ s_1+s_2=2s_3 $. Hence after combining, it is easy to see that ${\left(f(z+c)-s_3\right)}{\left(f(z)-s_3\right)}=-1 $ and this implies that $ f(z+c)\equiv -f(z)+2s_3.$
\end{rem} \par In this paper, taking care of these points. our aim is to extend the above results with certain suitable setting. Henceforth, for a meromorphic function $ f $ and $ c\in\mathbb{C^{*}} $, we recall here (see \cite{Aha & RM & 2019}) $ \mathcal{L}_c(f): =c_1f(z+c)+c_0f(z) $, where $ c_1 (\neq 0), c_0\in\mathbb{C} $. Clearly, $\mathcal{L}_c(f) $ is a generalization of shift $ f(z+c) $ as well as the difference operator $ \Delta_{c}f $.
\par To give a correct version of the result of Lin \emph{et al.} with a general setting, we are mainly interested to find the affirmative answers of the following questions.
\begin{ques}
Is it possible to extend $ f(z+c) $ upto $ \mathcal{L}_c(f) $, in all the above mentioned results?
\end{ques}
\begin{ques}
Can we obtained a similar result of Theorem E, replacing the condition $
\Theta(0;f)+\Theta(\infty;f)>{2}/{(k+1)} $, where $ k=\min\{k_1,
k_2\} $ by a more general one?
\end{ques} If the answers of the above question are found to be affirmative, then it is natural to raise the following questions.
\begin{ques}
Is the new general condition, so far obtained, sharp?
\end{ques}
\begin{ques}
Can we find the class of all the meromorphic function which satisfies the difference equation $ \mathcal{L}_c(f)\equiv f $?
\end{ques} Answering the above questions is the main objective of this paper. We organize the paper as follows: In section 2, we state the main results of this paper and exhibit several examples pertinent with the different issues regarding the main results. In section 3, key lemmas are stated and some of them are proved. Section 4 is devoted specially to prove the main results of this paper. In section 5, some questions have raised for further investigations on the main results of this paper. \section{Main Results}
We prove the following result generalizing that of \textit{Lin} \emph{et al.} \cite{Lin & Lin & Wu}.
\begin{theo}\label{th2.1}
Let $ f $ be a non-constant meromorphic function of hyper order $ \rho_2(f)<1 $ and $ c, c_1\in\mathbb{C^{*}} $. Let $ k_1 $, $ k_2 $ be two positive integers, and $ s_1 $, $ s_2\in\mathcal{S}\smallsetminus\{0\} $, $ s_3 $, $
s_4\in\hat{\mathcal{S}}(f) $ be four distinct periodic functions with period $ c $ such that $ f $ and $ \mathcal{L}_c(f) $ share $ s_3, $ $ s_4 $ $ CM $ and
\begin{eqnarray*} \overline E_{k_{j})}(s_j,f)\subseteq \overline E_{k_{j})}(s_j,\mathcal{L}_c(f)),\;\;
j=1, 2. \end{eqnarray*} If \begin{eqnarray*}
\Theta(0;f)+\Theta(\infty;f)>\frac{1}{k_1+1}+\frac{1}{k_2+1}, \end{eqnarray*} then $\mathcal{L}_c(f)\equiv f $. Furthermore, $ f $ assumes the following form \begin{eqnarray*}
f(z)=\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z) ,\end{eqnarray*} where $ g(z) $ is a meromorphic function such that $ g(z+c)=g(z) $, for all $ z\in\mathbb{C} $.
\end{theo}
\begin{rem}
The following examples show that the condition\begin{eqnarray*}
\Theta(0;f)+\Theta(\infty;f)>\frac{1}{k_1+1}+\frac{1}{k_2+1}\end{eqnarray*} in
{Theorem \ref{th2.1}} is sharp. \end{rem}
\begin{exm}
Let $ f(z)={\left(ae^z(e^{2z}+3)\right)}/{\left(3e^{2z}+1\right)} $, $ c=\pi i $ and $
s_1=a $, $ s_2=-a $, where $ a\in\mathbb{C^{*}} $, $ s_3=0 $, $ s_4=\infty $ and
$ k_1=2=k_2 $. It is easy to see that $ \mathcal{L}_{\pi
i}(f)=-{\left(ae^z(e^{2z}+3)\right)}/{\left(3e^{2z}+1\right)} $, where $ c_1=c_0+1 $, $ c_0, c_1\in\mathbb{C^{*}} $, and $ f(z) $ satisfies all the conditions of {Theorem \ref{th2.1}} and \begin{eqnarray*}
\Theta(0;f)+\Theta(\infty;f)=\frac{2}{3}=\frac{1}{k_1+1}+\frac{1}{k_2+1},\end{eqnarray*}
\par where $ \Theta(0,f)={1}/{3}=\Theta(\infty,f) $, but $
\mathcal{L}_{\pi i}(f)\not\equiv f $.
\end{exm}
\begin{exm}
Let $ f(z)={\left(4ae^z(e^{2z}+1)\right)}/{\left(e^{4z}+6e^{2z}+1\right)} $, $ c=\pi i
$, $ s_1=a $, $ s_2=-a $, where $ a\in\mathbb{C^{*}} $, $ s_3=0 $, $ s_4=\infty
$ and $ k_1=3=k_2 $. Then clearly $ \mathcal{L}_{\pi
i}(f)=-{\left(4ae^z(e^{2z}+1)\right)}/{\left(e^{4z}+6e^{2z}+1\right)} $, where $
c_1=c_0+1 $, $ c_0, c_1\in\mathbb{C^{*}} $, and $ f(z) $ satisfies all the
conditions of \emph{Theorem \ref{th2.1}} and \begin{eqnarray*}
\Theta(0;f)+\Theta(\infty;f)=\frac{1}{2}=\frac{1}{k_1+1}+\frac{1}{k_2+1},\end{eqnarray*}
\par where $ \Theta(0,f)={1}/{2}$, $\Theta(\infty,f)=0 $ but we see that $ \mathcal{L}_{\pi i}(f)\not\equiv f $.
\end{exm}
\par As the consequences of {Theorem \ref{th2.1}}, we prove the following result.
\begin{theo}\label{th2.2}
Let $ f $ be a non-constant meromorphic function of hyper-order $ \rho_2(f)<1
$, $ \Theta(\infty,f)=1 $ and $ c, c_1\in\mathbb{C^{*}} $. Let $ s_1, s_2,
s_3\in\hat{\mathcal{S}}(f) $ be three distinct periodic functions with period $
c $ such that $ f $ and $ \mathcal{L}_c(f) $ share $ s_3 $ $ CM $ and \begin{eqnarray*} \overline
E_{k_{j})}(s_j,f)\subseteq \overline E_{k_{j})}(s_j,\mathcal{L}_c(f)),\;\; j=1, 2.
\end{eqnarray*} If $ k_1, k_2\geq 2 $, then $ \mathcal{L}_c(f)\equiv f $. Furthermore, $ f $
assumes the following form \begin{eqnarray*}
f(z)=\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z) ,\end{eqnarray*} where
$ g(z) $ is a meromorphic function such that $ g(z+c)=g(z) $, for all $
z\in\mathbb{C} $.
\end{theo} \par The following example shows that, the number $ k_1=2=k_2 $ is sharp in {Theorem \ref{th2.2}}.
\begin{exm}\label{ex2.1}
We consider $ f(z)=a\cos z $, where $ a\in\mathbb{C^{*}} $, $s_1=a, s_2=-a$ and
$ s_3=0 $. We choose $ \mathcal{L}_{\pi}(f)=c_1f(z+\pi)+c_0f(z) $, where $
c_1,\; c_0\in\mathbb{C^{*}} $ with $ c_1=c_0+1 $. Clearly $ f $ and $
\mathcal{L}_{\pi}(f) $ share $ s_3 $ $ CM $, $ \Theta(\infty,f)=1 $, $ \overline
E_{1)}(a,f)=\phi=\overline E_{1)}(a,\mathcal{L}_{\pi}(f)) $ and $ \overline
E_{1)}(-a,f)=\phi=\overline E_{1)}(-a,\mathcal{L}_{\pi}(f)) $, but $ f(z)\not\equiv
\mathcal{L}_{\pi}(f).$
\end{exm}
\par Naturally, we are interested to find what happens, when $ k_1=1=k_2 $, and hence we obtain the following result.
\begin{theo}\label{th2.3}
Let $ f $ be a non-constant meromorphic function of hyper-order $ \rho_2(f)<1
$ with $ \Theta(\infty,f)=1 $ and $ c, c_1\in\mathbb{C^{*}} $. Let $ s_1, s_2,
s_3\in\hat{\mathcal{S}}(f) $ be three distinct periodic functions with period $
c $ such that $ f $ and $ \mathcal{L}_c(f) $ share $ s_3 $ $ CM $ and \begin{eqnarray*} \overline
E_{1)}(s_j,f)\subseteq \overline E_{1)}(s_j,\mathcal{L}_c(f)),\;\; j=1, 2. \end{eqnarray*} Then
$ \mathcal{L}_c(f)\equiv f $ or $ \mathcal{L}_c(f)\equiv -f+2s_3 $.Furthermore,
\begin{enumerate}
\item[(i)] If $ \mathcal{L}_c(f)\equiv f $, then
\begin{eqnarray*} f(z)=\left(\frac{
1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z).\end{eqnarray*}
\item[(ii)] If $ \mathcal{L}_c(f)\equiv -f+2s_3 $, then \begin{eqnarray*}
f(z)=\left(\frac{-1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z)+2s_3,\;\;\text{for
all}\; z\in\mathbb{C},\end{eqnarray*}
\end{enumerate}
where $ g(z) $ is a meromorphic function such that $ g(z+c)=g(z) $ Moreover, $
\mathcal{L}_c(f)\equiv -f+2s_3 $ occurs only if $ s_1+s_2=2s_3 $.
\end{theo}
\begin{rem}
We see that {Theorems \ref{th2.1}, \ref{th2.2}}\; and {\ref{th2.3}}\;
directly improved, respectively, {Theorems E, F} and {G}.
\end{rem}
\begin{rem}
We see from {Example \ref{ex2.1}}\; that, in {Theorem
\ref{th2.3}},\; the possibility $ \mathcal{L}_c(f)\equiv -f+2s_3 $ could be occurred.
\end{rem}\par The following example shows that the restrictions on the growth of
$ f $ in our above results are necessary and sharp.
\begin{exm}
Let $ f(z)=e^{g(z)} $, where $ g(z) $ is an entire function with $ \rho(g)=1 $, and hence for $ c_1={1}/{2}=c_0 $, it is easy to see that $
\mathcal{L}_{\pi}(z)={\left(e^{2g(z)}+1\right)}/{2e^{g(z)}} $. We
choose $ s_1=1,\; s_2=-1 $ and $ s_3=\infty $. Clearly $ \Theta(\infty,f)=1 $, $
\rho_2(f)=1 $, $ f $ and $ \mathcal{L}_{\pi}(f) $ share $ s_3 $ $ CM $ and $ \overline
E_{1)}(1,f)\subseteq\overline E_{1)}(1,\mathcal{L}_{\pi}(z)) $ and $ \overline
E_{1)}(-1,f)\subseteq\overline E_{1)}(-1,\mathcal{L}_{\pi}(z)) $ but we see that
neither $ \mathcal{L}_{\pi}(z)\not\equiv f $ nor $
\mathcal{L}_{\pi}(z)\not\equiv -f+2s_3 $. Also the function has not the specific
form.
\end{exm} \par
\begin{rem}
The next example shows that the condition $ \Theta(\infty,f)=1 $ in {Theorem \ref{th2.3}} can not be omitted.
\end{rem}
\begin{exm}
Let $ f(z)=1/ \cos z $, $c_1=1$, $ c_0=0 $, $ s_1=1 $, $ s_2=-1 $ and $ s_3=0
$. Clearly $ \Theta(\infty,f)=0 $, $ f $ and $ \mathcal{L}_{3\pi/2}(f) $ share $
s_3 $ $ CM $, $ \overline E_{1)}(1,f)\subseteq\overline E_{1)}(1,\mathcal{L}_{3\pi/2}(z)) $
and $ \overline E_{1)}(-1,f)\subseteq\overline E_{1)}(-1,\mathcal{L}_{3\pi/2}(z)) $.
However, one may observe that neither $ \mathcal{L}_{3\pi/2}(z)\not\equiv f $ nor $
\mathcal{L}_{3\pi/2}(z)\not\equiv -f+2s_3 $. Also the function has not the specific
form.
\end{exm}
\section{Key lemmas}
In this section, we present some necessary lemmas which will play key role to prove the main results. For a non-zero complex number $ c $ and for integers $ n\geq 1 $, we define the higher order difference operators $ \Delta_c^nf:=\Delta_c^{n-1}(\Delta_c f) $.
\begin{lem}\cite{Yan-FE-1980}\label{lem3.1}
Let $ c\in\mathbb{C} $, $ n\in\mathbb{N} $, let $ f $ be a meromorphic function
of finite order. Then any small periodic function $ a\equiv
a(z)\in\mathcal{S}(f) $ \begin{eqnarray*}
m\left(r,\frac{\Delta_c^nf}{f(z)-a(z)}\right)=S(r,f), \end{eqnarray*} where the exponential set associated with $ S(r,f) $ is of at most finite logarithmic
measure.
\end{lem}
\begin{lem}\cite{Moh-FFA-1971,Val-BSM-59}\label{lem3.2}
If $ \mathcal{R}(f) $ is rational in $ f $ and has small meromorphic
coefficients, then \begin{eqnarray*} T(r,\mathcal{R}(f))=\deg_f(\mathcal{R})T(r,f)+S(r,f).
\end{eqnarray*}
\end{lem}
\begin{lem}\cite{Yan & Yi-2003}\label{lem3.3}
Suppose that $ h $ is a non-constant entire function such that $ f(z)=e^{h(z)}
$, then $ \rho_2(f)=\rho(h) $.
\end{lem}
\par In \cite{Chi & Fen-2008,Hal & Kor-JMMA-2006}, the first difference analogue
of the lemma on the logarithmic derivative was proved and for the hyper-order $
\rho_2(f)<1 $, the following is the extension, see \cite{Hal & Kor &
Toh-TAMS-2014}.
\begin{lem}\cite{Hal & Kor & Toh-TAMS-2014}\label{lem3.4}
Let $ f $ be a non-constant finite order meromorphic function and $
c\in\mathbb{C} $. If $ c $ is of finite order, then \begin{eqnarray*}
m\left(r,\frac{f(z+c)}{f(z)}\right)=O\left(\frac{\log r}{r} T(r,f)\right) \end{eqnarray*}
for all $ r $ outside of a set $ E $ with zero logarithmic density. If the hyper
order $ \rho_2(f)<1 $, then for each $ \epsilon>0 $, we have \begin{eqnarray*}
m\left(r,\frac{f(z+c)}{f(z)}\right)=0\left(\frac{T(r,f)}{r^{1-\rho_2-\epsilon}}\right)
\end{eqnarray*} for all $ r $ outside of a set of finite logarithmic measure.
\end{lem}
\begin{lem}\cite{Yamanoi-AM-2004}\label{lem3.5}
Let $ f $ be a non-constant meromorphic function, $ s_j\in\hat{\mathcal{S}}(f)
$, $ j=1, 2, ..., q,\;$ $ (q\geq 3) $. Then for any positive real number $
\epsilon $, we have \begin{eqnarray*} (q-2-\epsilon)T(r,f)\leq\sum_{j=1}^{q}\overline
N\left(r,\frac{1}{f-s_j}\right),\; r\not\in E, \end{eqnarray*} where $ E\subset [0,\infty)
$ and satisfies $\displaystyle \int_{E}d\log \log r<\infty $.
\end{lem} We now prove the following lemma, a similar proof of this lemma can also be found in \cite{Aha & RM & 2019}.
\begin{lem}\label{lem3.6}
Let $ f $ be a non-constant meromorphic function such that \begin{eqnarray*} \overline
E(s_j,f)\subseteq \overline E(s_j,c_1f(z+c)+c_0f(z)),\;\; j=1, 2,\end{eqnarray*} where $ s_1,
s_2\in\mathcal{S}(f) $, $ c,\; c_0,\; c_1(\neq 0)\in\mathbb{C^{*}} $, then $ f $
is not a rational.
\end{lem}
\begin{proof} We wish to prove this lemma by the method of contradiction. Let $ f $ be a rational function. Then $ f(z)={P(z)}/{Q(z)} $ where $ P $ and $ Q $ are two polynomials relatively prime to each other and $ P(z)Q(z)\not\equiv 0 $. Hence \begin{eqnarray}\label{e1.1} E(0,P)\cap E(0,Q)=\phi \end{eqnarray} It is easy to see that\begin{eqnarray*}
c_1f(z+c)+c_0f(z)&=&c_1\frac{P(z+c)}{Q(z+c)}+c_0\frac{P(z)}{Q(z)}\\&=&\frac{c_1P(z+c)Q(z)+c_0P(z)Q(z+c)}{Q(z+c)Q(z)}\\&=&\frac{P_1(z)}{Q_{1}(z)},\text{(say)} \end{eqnarray*} where $ P_1 $ and $ Q_1 $ are two relatively prime polynomials and $ P_1(z)Q_1(z)\not\equiv 0 $.
\par Since $ \overline
E(s_1,f)\subseteq\overline E(s_1,c_1f(z+c)+c_0f(z)) $ and $ f $ is a rational function, there must exists a polynomial $ h(z) $ such that \begin{eqnarray*}
c_1f(z+c)+c_0f(z)-s_1=(f-s_1)h(z), \end{eqnarray*} which can be re-written as \begin{eqnarray}\label{e2.1}
\frac{c_1P(z+c)Q(z)+c_0P(z)Q(z+c)}{Q(z+c)Q(z)}-s_1\equiv
\left(\frac{P(z)}{Q(z)}-s_1\right)h(z).\end{eqnarray}
We now discuss the following cases:\\
\noindent{\bf{Case 1.}} Let $ P(z) $ is non-constant.\par Then by the
\textit{Fundamental Theorem of Algebra}, there exists $ z_0\in\mathbb{C} $ such
that $ P(z_0)=0 $. Then it follows from (\ref{e2.1}) that \begin{eqnarray}\label{e2.3}
c_1\frac{P(z_0+c)}{Q(z_0+c)}\equiv(1-h(z_0))s_1^{0}, \end{eqnarray} where $
s_1^{0}=s_1(z_0) $.\par
\noindent{\bf{Subcase 1.1.}} Let $ z_0\in\mathbb{C} $ be such that $ s_1(z_0)=0
$.\par Then from (\ref{e2.3}), it is easy to see that $ P(z_0+c)=0 $. Then we can deduce
from (\ref{e1.1}) that $ P(z_0+mc)=0 $ for all positive integer $ m $. However,
this is impossible, and hence we conclude that the polynomial $ P(z) $ is a
non-zero constant.\par
\noindent{\bf{Subcase 1.2.}} Let $ z_0\in\mathbb{C} $ be such that $
s_1(z_0)\neq 0 $. \par Then from (\ref{e2.3}), we obtain that \begin{eqnarray*}
P(z_0+c)\equiv\frac{s_1^{0}}{c_1}(1-h(z_0))Q(z_0+c).\end{eqnarray*}\par Next proceeding
exactly same way as done in above, we obtain \begin{eqnarray}\label{e2.4}
P(z_0+mc)\equiv\frac{s_1^{0}}{c_1}(1-h(z_0))Q(z_0+mc).\end{eqnarray}\par In view of (\ref{e2.3})
and (\ref{e2.4}), a simple computation shows that \begin{eqnarray*}
\frac{P(z_0+c)}{Q(z_0+c)}=\frac{P(z_0+mc)}{Q(z_0+mc)}\;\; \text{for all
positive integers $ m $,} \end{eqnarray*} which contradicts the fact that $ E(0,P)\cap
E(0,Q)=\phi $.\par
Therefore, we see that $ f(z) $ takes the form $
f(z)={\eta}/{Q(z)} $, where $ P(z)=\eta=\text{constant}\; (\neq
0).$\par
\noindent{\bf{Case 2.}} Let $ Q(z) $ be non-zero constant.\par Now
\begin{eqnarray}\label{e2.5} c_1f(z+c)+c_0f(z)&=&\frac{c_1\eta\;Q(z)+c_0\eta\;
Q(z+c)}{Q(z+c)Q(z)}.\end{eqnarray}\par Since $ E(s_2,f)=E(s_2,c_1f(z+c)+c_0f(z)) $ then
there exists a polynomial $ h_1(z) $ such that $
c_1f(z+c)+c_0f(z)-s_2=(f-s_2)h_1(z),$ which can be written as \begin{eqnarray}\label{e2.6} c_1\;Q(z)+c_0
Q(z+c)\equiv \frac{\eta-s_2Q(z)}{d}h_1(z)Q(z+c).\end{eqnarray}\par Since $ Q(z) $ and hence $ Q(z+c) $ be non-constant polynomials, hence by the \textit{Fundamental Theorem of Algebra}, there exist $ z_0 $ and $ z_1 $ such that $ Q(z_0)=0=Q(z_1+c) $.
\par \noindent{\bf{Subcase 2.1.}} When $ Q(z_0)=0 $, then from (\ref{e2.6}), we
see that $ h_1(z_0)=-{c_0}/{\eta} $, which is absurd.\par
\par \noindent{\bf{Subcase 2.2.}} When $ Q(z_1+c)=0 $, then from (\ref{e2.6}),
we get $ Q(z_1)=0 $, which is not possible.\par
Thus we conclude that $ Q(z) $ is a non-zero constant, say $ \eta_2 $. Thus we have
$ f(z)={\eta}/{\eta_2} $, a constant, which is a contradiction. This completes the proof.
\end{proof}
\begin{lem}\cite{Hal & Kor & Toh-TAMS-2014}\label{lem3.7}
Let $ T: [0,+\infty]\rightarrow [0,+\infty] $ be a non-decreasing continuous
function, and let $ s\in (0,+\infty) $. If the hyper-order of $ T $ is strictly
less than one, i.e., \begin{eqnarray*} \limsup_{r\rightarrow +\infty}\frac{\log ^{+}\log
^{+} T(r)}{\log r}=\rho_2 <1,\end{eqnarray*} then \begin{eqnarray*}
T(r+s)=T(r)+o\left(\frac{T(r)}{r^{1-\rho_2-\epsilon}}\right),\end{eqnarray*} where $
\epsilon>0 $ and $ r\rightarrow\infty $, outside of a set of finite logarithmic
measure.
\end{lem}
\section{Proofs of the main results}
In this section, we give the proofs of our main results.
\begin{proof}[Proof of Theorem \ref{th2.1}]
First of all we suppose that $ s_j\in\mathbb{C} $, $ j=1, 2, 3, 4 $. By the
assumption of the theorem, $ f(z) $ and $ \mathcal{L}_c(f)=c_1f(z+c)+c_0f(z) $
share $ s_3 $, $ s_4 $ $ CM $, hence we must have \begin{eqnarray}\label{e3.1}
\frac{\left(f-s_3\right)\left(\mathcal{L}_c(f)-s_4\right)}{\left(f-s_4\right)\left(\mathcal{L}_c(f)-s_3\right)}=e^{h(z)},
\end{eqnarray} where $ h(z) $ is an entire function with $ \rho(h)<1 $ by Lemma \ref{lem3.3}. In view of Lemma \ref{lem3.4}, we obtain \begin{eqnarray*}
T\left(r,e^h\right)=S(r,f). \end{eqnarray*}\par Therefore with the help of Lemma
\ref{lem3.2}, we obtained \begin{eqnarray*} T(r,\mathcal{L}_c(f))=T(r,f)+S(r,f). \end{eqnarray*}
\par Next we suppose that $ z_0\in\overline E_{k)}(s_1,f)\cup\overline E_{k)}(s_2,f) $. Then from (\ref{e3.1}), one may easily deduce that $ e^{h(z_0)}=1 $. For the sake of
convenience, we set $ \gamma :=e^{h(z)} $ and \begin{eqnarray*} S(r)
:=S(r,\mathcal{L}(f))=S(r,f).\end{eqnarray*} \par We now split the
problem into two cases.\par
\noindent{\bf Case 1.} Let $ e^{h(z)}\neq 1 $.\par A simple computation shows that
that \begin{eqnarray}\label{e3.2} \overline N_{k_1)} \left(r,\frac{1}{f-s_1}\right)&\leq&
N\left(r,\frac{1}{\gamma -1}\right)\leq T(r,\gamma)+O(1)\leq S(r) \end{eqnarray} and
\begin{eqnarray}\label{e3.3} \overline N_{k_2)} \left(r,\frac{1}{f-s_2}\right)&\leq&
N\left(r,\frac{1}{\gamma -1}\right)\leq T(r,\gamma)+O(1)\leq S(r). \end{eqnarray} \par
Without loss of generality, we may assume that $ s_3 $, $ s_4
\in\mathcal{S}(f)\smallsetminus\{0\}$. By Lemma \ref{lem3.5}, for \begin{eqnarray*}
\epsilon\in\left(0,\frac{1}{3}\left(\Theta(0;f)+\Theta(\infty;f)\right)-\frac{1}{k_1+1}-\frac{1}{k_2+1}\right),
\end{eqnarray*} we obtain \begin{eqnarray}\label{e3.4} (4-\epsilon) T(r,f)\leq \overline N(r,f)+\overline
N\left(r,\frac{1}{f}\right)+\sum_{j=1}^{4}\overline
N\left(r,\frac{1}{f-s_j}\right)+S(r,f). \end{eqnarray}\par With the help of (\ref{e3.2})
and (\ref{e3.3}), it follows from (\ref{e3.4}) that \begin{eqnarray*} (2-\epsilon) T(r,f)\leq
\overline N(r,f)+\overline N\left(r,\frac{1}{f}\right)+\sum_{j=1}^{2}\overline
N_{(k_j+1}\left(r,\frac{1}{f-s_j}\right)+S(r,f) \end{eqnarray*} which gives \begin{eqnarray*}
\Theta(0;f)+\Theta(\infty;f)\leq\frac{1}{k_1+1}+\frac{1}{k_2+1} \end{eqnarray*} which
contradicts \begin{eqnarray*} \Theta(0;f)+\Theta(\infty;f)>\frac{1}{k_1+1}+\frac{1}{k_2+1}.
\end{eqnarray*}
\noindent{\bf Case 2.} Therefore, we have $ e^{h(z)}\equiv 1 $ and hence \begin{eqnarray*}
\frac{(f-s_3)(\mathcal{L}_c(f)-s_4)}{(f-s_4)(\mathcal{L}_c(f)-s_3)}=1, \end{eqnarray*}
on simplification, we obtain $ \mathcal{L}_c(f)\equiv f(z) $, for all $
z\in\mathbb{C} $.\par We are now to find the class of all the
meromorphic functions satisfying the difference equation $ \mathcal{L}_c(f)\equiv f $. By assumption of the result, and using {Lemma \ref{lem3.6}}, it is easy to see that $ f $
is not a rational function. Therefore $ f(z) $ must be a transcendental meromorphic function. \par We also see that $ f(z) $ and $ f(z+c) $ are related by \begin{eqnarray}\label{e3.5}
f(z+c)=\left(\frac{1-c_0}{c_1}\right)f(z). \end{eqnarray} \par Let $ f_1(z) $ and $ f_2(z)
$ be two solutions of (\ref{e3.5}) (see \cite{Aha & RM & 2019} for more details). Then it is easy to see that\begin{eqnarray}\label{e3.6}
f_1(z+c)=\left(\frac{1-c_0}{c_1}\right)f_1(z)\\\label{e3.7}
f_2(z+c)=\left(\frac{1-c_0}{c_1}\right)f_2(z). \end{eqnarray} \par We set $
h(z):=f_1(z)/f_2(z) $. Then in view of (\ref{e3.6}) and (\ref{e3.7}), we
obtain \begin{eqnarray*}
h(z+c)=\frac{f_1(z+c)}{f_2(z+c)}=\displaystyle\frac{\displaystyle\frac{1-c_0}{c_1}f_1(z)}{\displaystyle\frac{1-c_0}{c_1}f_2(z)}=\frac{f_1(z)}{f_2(z)}=h(z),\end{eqnarray*}
for all $ z\in\mathbb{C} $. Therefore, it is easy to verify that $$f_2(z)=\displaystyle\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g_2(z),$$ where $ g_2(z) $ is a meromorphic function with $ g_2(z+c)=g_2(z) $, is a solution of (\ref{e3.5}). Hence, it is also easy to verify that $ f_1(z)=f_2(z)h(z) $, a solution of (\ref{e3.5}). Thus the linear combination \begin{eqnarray*} a_1f_1(z)+a_2f_2(z)&=&
\displaystyle\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}\left(a_1h(z)+a_2\right)g_2(z)\\&=&\displaystyle\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}\sigma
(z),\end{eqnarray*} where $ \sigma(z)=\left(a_1h(z)+a_2\right)g_2(z) $ is such that $
\sigma(z+c)=\sigma(z) $, for all $ z\in\mathbb{C} $, is the general solution of
(\ref{e3.5}). Hence, the precise form of $ f(z) $ is \begin{eqnarray*}
f(z)=\displaystyle\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z),
\end{eqnarray*} where $ g(z) $ is a meromorphic function with $ g(z+c)=g(z) $, for all $ z\in\mathbb{C} $.
\par This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th2.3}]
Let us suppose that $ g(z) $ is the canonical product of the poles of $ f $. Then by Lemma \ref{lem3.4}, we obtained \begin{eqnarray}\label{e4.8}
m\left(r,\frac{g(z+c)}{g(z)}\right)=S(r,f)=S(r,f). \end{eqnarray}\par Since $ \Theta(\infty;f)=1 $, hence it is easy to see that \begin{eqnarray*} \limsup_{r\rightarrow +\infty}\frac{\overline
N(r,f)}{T(r,f)}=0. \end{eqnarray*}\par Therefore it follows from (\ref{e4.8}) that\begin{eqnarray}\label{e4.9} T\left(r,\frac{g(z+c)}{g(z)}\right)=S(r,f). \end{eqnarray}\par Since $ f
$ and $ \mathcal{L}_c(f) $ share $ s_3 $ $ CM $, by
Lemma \ref{lem3.3}, we obtain
\begin{eqnarray}\label{e4.10}
\frac{\mathcal{L}_c(f)-s_3}{f-s_3}=e^{\mathcal{H}(z)}\frac{g(z)}{g(z+c)}, \end{eqnarray}
where $ \mathcal{H}(z) $ is an entire function, with $ \rho(\mathcal{H})<1 $. By Lemma \ref{lem3.4}, we also obtain
\begin{eqnarray}\label{e4.11} T\left(r,e^{\mathcal{H}(z)}\frac{g(z)}{g(z+c)}\right)=S(r,f).
\end{eqnarray}\par Therefore, by Lemma \ref{lem3.2} and (\ref{e4.11}), a simple computation shows that $
T(r,\mathcal{L}_c(f))=T(r,f)+S(r,f). $ For the sake convenience, we set \begin{eqnarray*}
\beta:=e^{\mathcal{H}(z)}\frac{g(z)}{g(z+c)}\;\;\text{and}\;\;
S(r):=S(r,\mathcal{L}_c(f))=S(r,f). \end{eqnarray*}\par If $ \mathcal{L}_c(f)\not\equiv
f(z) $. i.e., if $ \beta\neq 1 $, then with the help of (\ref{e4.10}) and from the assumption, we obtain \begin{eqnarray}\label{e4.12} \overline
N_{1)}\left(r,\frac{1}{f-s_1}\right)&\leq& N\left(r,\frac{1}{\beta
-1}\right)\leq T(r,\beta)+O(1)= S(r). \end{eqnarray} and \begin{eqnarray}\label{e4.13} \overline
N_{1)}\left(r,\frac{1}{f-s_2}\right)&\leq& N\left(r,\frac{1}{\beta
-1}\right)\leq T(r,\beta)+O(1)= S(r). \end{eqnarray} By Lemma \ref{lem3.7}, and using
(\ref{e4.12}) and (\ref{e4.13}), we easily obtain \begin{eqnarray}\label{e4.14} \overline
N_{1)}\left(r,\frac{1}{\mathcal{L}_c(f)-s_1}\right)\leq \overline
N_{1)}\left(r,\frac{1}{f-s_1}\right)+S(r)=S(r). \end{eqnarray} and \begin{eqnarray}\label{e4.15} \overline
N_{1)}\left(r,\frac{1}{\mathcal{L}_c(f)-s_2}\right)\leq \overline
N_{1)}\left(r,\frac{1}{f-s_1}\right)+S(r)=S(r). \end{eqnarray}\par On the other hand, it follows from
(\ref{e4.10}) that \begin{eqnarray}\label{e4.16} \mathcal{L}_c(f)-s_1&=&(s_3-s_1)+\beta\;
(f-s_3)\\&=&\beta\;\left(f-\frac{s_1+(\beta -1)s_3}{\beta}\right)\nonumber. \end{eqnarray}
\par Similarly, we obtain \begin{eqnarray}\label{e4.17}
\mathcal{L}_c(f)-s_2=\beta\;\left(f-\frac{s_2+(\beta
-1)s_3}{\beta}\right).\end{eqnarray}\par It is easy to see that \begin{eqnarray}\label{e4.18}
N\left(r,\frac{1}{\mathcal{L}_c(f)-s_1}\right)=N\left(r,\frac{1}{f-\frac{s_1+(\beta
-1)s_3}{\beta}}\right)+S(r). \end{eqnarray} and \begin{eqnarray}\label{e4.19}
N\left(r,\frac{1}{\mathcal{L}_c(f)-s_2}\right)=N\left(r,\frac{1}{f-\frac{s_2+(\beta
-1)s_3}{\beta}}\right)+S(r). \end{eqnarray}\par Now our aim is to deal with the following three cases.\vphantom{1mm}\\
\noindent{\bfseries{Case 1.}} Suppose that $ \left({\left((\beta-1)s_3+s_1\right)}/{\beta}\right)\neq s_2 $.\par Since $ \left({\left((\beta
-1)s_3+s_1\right)}/{\beta}\right)\neq s_1 $ and $ \Theta(\infty;f)=1 $, hence by Lemma
\ref{lem3.5} for $ \epsilon\in \left(0,{1}/{2}\right) $, it
follows from (\ref{e4.10}), (\ref{e4.12}), (\ref{e4.13}), (\ref{e4.14}) and
(\ref{e4.18}) that \begin{eqnarray} && (2-\epsilon) T(r,f)\\&\leq& \overline N(r,f)+\overline
N\left(r,\frac{1}{f}\right)+\overline N\left(r,\frac{1}{f-s_2}\right)+\overline
N\left(r,\frac{1}{f-\frac{(\beta -1)s_3+s_1}{\beta}}\right)\nonumber\\&\leq& \overline
N_{(2}\left(r,\frac{1}{f-s_1}\right)+\overline
N_{(2}\left(r,\frac{1}{f-s_2}\right)+\overline
N_{(2}\left(r,\frac{1}{\mathcal{L}_c(f)-s_1}\right)\nonumber\\&\leq&\frac{1}{2}
T(r,f)+\frac{1}{2} T(r,f)+\frac{1}{2} T(r,f)+S(r)\nonumber\\&=&\frac{3}{2}
T(r,f)+S(r,f)\nonumber, \end{eqnarray} which is a contradiction.
\\
\noindent{\bfseries{Case 2.}} Suppose that $ \left({\left((\beta
-1)s_3+s_2\right)}/{\beta}\right)\neq s_1 $.\par Since $ \left({\left((\beta
-1)s_3+s_2\right)}/{\beta}\right)\neq s_2 $ and $ \Theta(\infty;f)=1 $, hence proceeding
exactly same way as done {Case 1}, we arrive at a contradiction.\par Therefore,
we must have $ \mathcal{L}_c(f)\equiv f $, and hence following the proof of {Theorem \ref{th2.1}}, we obtain the precise form of the function.
\\ \noindent{\bfseries{Case 3.}} Suppose that \begin{eqnarray*} \frac{(\beta
-1)s_3+s_2}{\beta}=s_1 \end{eqnarray*} and \begin{eqnarray*} \frac{(\beta -1)s_3+s_1}{\beta}=s_2
.\end{eqnarray*}\par An elementary calculation shows that $ \beta =-1 $, so that $ 2s_3=s_1+s_2 $. Thus from (\ref{e4.10}), we have $ \mathcal{L}_c(f)\equiv -f(z)+2s_3 $ and by the same argument used in the previous cases, it is not hard to show that $ f(z) $ will take the form \begin{eqnarray*}
f(z)=\left(\frac{-1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z)+2s_3,\;\;\text{for all}\; z\in\mathbb{C}, \end{eqnarray*} where $ g(z) $ is a meromorphic function with period $ c $. This completes the proof.
\end{proof}
\section{Concluding remarks and open question}
\par Let us suppose that $ \mathcal{L}_c(f)\equiv f $, where $ f $ is a non-constant meromorphic functions. Since $ f $ can not be rational function (see \cite{Aha & RM & 2019} for detailed information), hence $ f $ must be transcendental and hence $ f(z) $ takes the precise form \begin{eqnarray*}
f(z)=\left(\frac{1-c_0}{c_1}\right)^{\displaystyle\frac{z}{c}}g(z), \end{eqnarray*} where $ g(z) $ is a meromorphic periodic function $ c $. We can write $ f(z)=\alpha^{\frac{z}{c}}g(z), $ where $ \alpha $ is a root of the equation $ c_1z+c_0=1 $.
\par For further generalization, we define $
\mathcal{L}_c^n(f):=c_nf(z+nc)+\cdots+c_1f(z+c)+c_0f(z) $ (see \cite{Ban & Aha & JCMA-2020} for details), where $ c_n(\neq 0), c_1, c_0\in\mathbb{C} $. For particular values of the constants $ c_j=(-1)^{n-j}\binom nj $ for $ j=0, 1, \ldots, n $, it is easy to see that $\mathcal{L}_c^n(f)=\Delta_{c}^n(f). $ One can check that $ f(z)=2^{^{\frac{z}{c}}}g(z) $, where $ g $ is a
meromorphic function of period $ c $, solves the difference equation $ \Delta_{c}^n(f)\equiv f $. We are mainly interested to find the precise form of the function $ f $ when it solves the difference equation $ \mathcal{L}_c^n(f)\equiv f $. However, we conjecture the following.\\
\noindent{\bf Conjecture:} Let $ f $ be a meromorphic function such that $ \mathcal{L}_c^n(f)\equiv f $,
then $ f $ assumes the form $$ f(z)=\alpha_n^{{z}/{c}}g_n(z)+\cdots+\alpha_1^{{z}/{c}}g_1(z), $$
where $ g_j $ $ (j=1, 2, \ldots, n) $ are meromorphic functions of period $ c $, and $ \alpha_j $ $ (j=1, 2, \ldots, n) $ are roots of the equation $ c_nz^n+\cdots+c_1z+c_0=1 $.
\par Based on the above discussions, we now pose the following question for future investigations on the main results of the paper. \begin{ques} Keeping all other conditions intact, for a meromorphic function $ f $, is it possible to get a corresponding result of {Theorems \ref{th2.1}, \ref{th2.2}} and {\ref{th2.3}}\; for $ \mathcal{L}_c^n(f) $? \end{ques}
\end{document} | arXiv |
Donaldson–Thomas theory
In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998). Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.
String theory
Fundamental objects
• String
• Cosmic string
• Brane
• D-brane
Perturbative theory
• Bosonic
• Superstring (Type I, Type II, Heterotic)
Non-perturbative results
• S-duality
• T-duality
• U-duality
• M-theory
• F-theory
• AdS/CFT correspondence
Phenomenology
• Phenomenology
• Cosmology
• Landscape
Mathematics
• Geometric Langlands correspondence
• Mirror symmetry
• Monstrous moonshine
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Related concepts
• Theory of everything
• Conformal field theory
• Quantum gravity
• Supersymmetry
• Supergravity
• Twistor string theory
• N = 4 supersymmetric Yang–Mills theory
• Kaluza–Klein theory
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Theorists
• Aganagić
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• Yau
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• Zaslow
• Zumino
• Zwiebach
• History
• Glossary
Donaldson–Thomas theory is physically motivated by certain BPS states that occur in string and gauge theory[1]pg 5. This is due to the fact the invariants depend on a stability condition on the derived category $D^{b}({\mathcal {M}})$ of the moduli spaces being studied. Essentially, these stability conditions correspond to points in the Kahler moduli space of a Calabi-Yau manifold, as considered in mirror symmetry, and the resulting subcategory ${\mathcal {P}}\subset D^{b}({\mathcal {M}})$ is the category of BPS states for the corresponding SCFT.
Definition and examples
The basic idea of Gromov–Witten invariants is to probe the geometry of a space by studying pseudoholomorphic maps from Riemann surfaces to a smooth target. The moduli stack of all such maps admits a virtual fundamental class, and intersection theory on this stack yields numerical invariants that can often contain enumerative information. In similar spirit, the approach of Donaldson–Thomas theory is to study curves in an algebraic three-fold by their equations. More accurately, by studying ideal sheaves on a space. This moduli space also admits a virtual fundamental class and yields certain numerical invariants that are enumerative.
Whereas in Gromov–Witten theory, maps are allowed to be multiple covers and collapsed components of the domain curve, Donaldson–Thomas theory allows for nilpotent information contained in the sheaves, however, these are integer valued invariants. There are deep conjectures due to Davesh Maulik, Andrei Okounkov, Nikita Nekrasov and Rahul Pandharipande, proved in increasing generality, that Gromov–Witten and Donaldson–Thomas theories of algebraic three-folds are actually equivalent.[2] More concretely, their generating functions are equal after an appropriate change of variables. For Calabi–Yau threefolds, the Donaldson–Thomas invariants can be formulated as weighted Euler characteristic on the moduli space. There have also been recent connections between these invariants, the motivic Hall algebra, and the ring of functions on the quantum torus.
• The moduli space of lines on the quintic threefold is a discrete set of 2875 points. The virtual number of points is the actual number of points, and hence the Donaldson–Thomas invariant of this moduli space is the integer 2875.
• Similarly, the Donaldson–Thomas invariant of the moduli space of conics on the quintic is 609250.
Definition
For a Calabi-Yau threefold $Y$[3][4] and a fixed cohomology class $\alpha \in H^{\text{even}}(Y,\mathbb {Q} )$ there is an associated moduli stack ${\mathcal {M}}(Y,\alpha )$ of coherent sheaves with Chern character $c({\mathcal {E}})=\alpha $. In general, this is a non-separated Artin stack of infinite type which is difficult to define numerical invariants upon it. Instead, there are open substacks ${\mathcal {M}}^{\sigma }(Y,\alpha )$ parametrizing such coherent sheaves ${\mathcal {E}}$ which have a stability condition $\sigma $ imposed upon them, i.e. $\sigma $-stable sheaves. These moduli stacks have much nicer properties, such as being separated of finite type. The only technical difficulty is they can have bad singularities due to the existence of obstructions of deformations of a fixed sheaf. In particular
${\begin{aligned}T_{[{\mathcal {E}}]}{\mathcal {M}}^{\sigma }(Y,\alpha )&\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}})\\{\text{Ob}}_{[{\mathcal {E}}]}({\mathcal {M}}^{\sigma }(Y,\alpha ))&\cong {\text{Ext}}^{2}({\mathcal {E}},{\mathcal {E}})\end{aligned}}$
Now because $Y$ is Calabi-Yau, Serre duality implies
${\text{Ext}}^{2}({\mathcal {E}},{\mathcal {E}})\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}}\otimes \omega _{Y})^{\vee }\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}})^{\vee }$
which gives a perfect obstruction theory of dimension 0. In particular, this implies the associated virtual fundamental class
$[{\mathcal {M}}^{\sigma }(Y,\alpha )]^{vir}\in H_{0}({\mathcal {M}}^{\sigma }(Y,\alpha ),\mathbb {Z} )$
is in homological degree $0$. We can then define the DT invariant as
$\int _{[{\mathcal {M}}^{\sigma }(Y,\alpha )]^{vir}}1$
which depends upon the stability condition $\sigma $ and the cohomology class $\alpha $. It was proved by Thomas that for a smooth family $Y_{t}$ the invariant defined above does not change. At the outset researchers chose the Gieseker stability condition, but other DT-invariants in recent years have been studied based on other stability conditions, leading to wall-crossing formulas.[5]
Facts
• The Donaldson–Thomas invariant of the moduli space M is equal to the weighted Euler characteristic of M. The weight function associates to every point in M an analogue of the Milnor number of a hyperplane singularity.
Generalizations
• Instead of moduli spaces of sheaves, one considers moduli spaces of derived category objects. That gives the Pandharipande–Thomas invariants that count stable pairs of a Calabi–Yau 3-fold.
• Instead of integer valued invariants, one considers motivic invariants.
See also
• Enumerative geometry
• Gromov–Witten invariant
• Hilbert scheme
• Quantum cohomology
References
1. Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories". arXiv:math/0212237.
2. Maulik, D.; Nekrasov, N.; Okounkov, A.; Pandharipande, R. (2006). "Gromov–Witten theory and Donaldson–Thomas theory, I". Compositio Mathematica. 142 (5): 1263–1285. arXiv:math/0312059. doi:10.1112/S0010437X06002302. S2CID 5760317.
3. Szendroi, Balazs (2016-04-27). "Cohomological Donaldson-Thomas theory". arXiv:1503.07349 [math.AG].
4. Thomas, R. P. (2001-06-11). "A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations". arXiv:math/9806111. {{cite journal}}: Cite journal requires |journal= (help)
5. Kontsevich, Maxim; Soibelman, Yan (2008-11-16). "Stability structures, motivic Donaldson-Thomas invariants and cluster transformations". arXiv:0811.2435. {{cite journal}}: Cite journal requires |journal= (help)
• Donaldson, Simon K.; Thomas, Richard P. (1998), "Gauge theory in higher dimensions", in Huggett, S. A.; Mason, L. J.; Tod, K. P.; Tsou, S. T.; Woodhouse, N. M. J. (eds.), The geometric universe (Oxford, 1996), Oxford University Press, pp. 31–47, ISBN 978-0-19-850059-9, MR 1634503
• Kontsevich, Maxim (2007), Donaldson–Thomas invariants (PDF), Mathematische Arbeitstagung, Bonn{{citation}}: CS1 maint: location missing publisher (link)
| Wikipedia |
Calculus of Variations and Partial Differential Equations
June 2017 , 56:74 | Cite as
Multiple positive solutions of the stationary Keller–Segel system
Denis Bonheure
Jean-Baptiste Casteras
Benedetta Noris
First Online: 28 April 2017
We consider the stationary Keller–Segel equation
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta v+v=\lambda e^v, \quad v>0 \quad &{} \text {in }\Omega ,\\ \partial _\nu v=0 &{}\text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a ball. In the regime \(\lambda \rightarrow 0\), we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given \(n\in \mathbb {N}_0\), we build a solution having multiple layers at \(r_1,\ldots ,r_n\) by which we mean that the solutions concentrate on the spheres of radii \(r_i\) as \(\lambda \rightarrow 0\) (for all \(i=1,\ldots ,n\)). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of \(\Omega \) as \(\lambda \rightarrow 0\). Instead they satisfy an optimal partition problem in the limit.
Mathematics Subject Classification
35J25 35B05 35B09 35B25 35B32 35B40
Communicated by A. Malchiodi.
D. Bonheure & J. B. Casteras are supported by INRIA – Team MEPHYSTO, MIS F.4508.14 (FNRS), PDR T.1110.14F (FNRS); J. B. Casteras is supported by the Belgian Fonds de la Recherche Scientifique – FNRS; D. Bonheure & B. Noris are partially supported by the Project ERC Advanced Grant 2013 No. 339958: "Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT" and by ARC AUWB-2012-12/17-ULB1-IAPAS.
Appendix 1: A Green's function in dimension 2
In this appendix we prove the analogous of [3, Proposition 2.1] in the 2-dimensional case (the case \(N=2\) is not treated in [3]).
Lemma 6.1
There exist two positive, linearly independent solutions \(\zeta \in C^2 ((0,1])\) and \(\xi \in C^2 ([0,1])\) of the equation
$$\begin{aligned} -u^{\prime \prime }-\dfrac{1}{r}u^\prime +u=0\ in\ (0,1), \end{aligned}$$
$$\begin{aligned} \xi ^\prime (0)=\zeta ^\prime (1)=0,\quad r(\xi ^\prime (r) \zeta (r) - \xi (r) \zeta ^\prime (r))=1, \ \forall r\in (0,1]. \end{aligned}$$
Moreover, \(\xi \) is bounded and increasing in [0, 1], \(\zeta \) is decreasing in (0, 1] and
$$\begin{aligned} \xi (0)=1,\quad \lim _{r\rightarrow 0^+}\dfrac{\zeta (r)}{-\ln r}=1,\quad \lim _{r\rightarrow 0^+ }(-r\zeta ^\prime (r))=1. \end{aligned}$$
As a consequence, the Green function defined in (1.11) (for \(N=2\)) can be written as follows
$$\begin{aligned} G(r,s)=\left\{ \begin{array}{ll} s^{N-1}\xi (r)\zeta (s)\quad \text {for }r\le s \\ s^{N-1}\xi (s)\zeta (r)\quad \text {for }r> s. \end{array}\right. \end{aligned}$$
Let \(\xi (r):= I_0 (r)\) be the modified Bessel function of the first kind (see (10.25) of [28]). It is well-known that \(\xi \) is positive, bounded, increasing, and that \(\xi ^\prime (0)=0,\ \xi (0)=1\). Let \(s=\ln r\), \(s\in (-\infty ,0]\) and \(\varphi (s)= \xi (e^s)\). In this new variable, we have that
$$\begin{aligned} {\left\{ \begin{array}{ll} \varphi ^{\prime \prime }+\varphi e^{2s}=0 \quad \text { in } (-\infty ,0)\\ \displaystyle \lim _{s\rightarrow -\infty } \varphi (s)=1, \quad \displaystyle \lim _{s\rightarrow -\infty } \varphi ^\prime (s)=0,\\ \varphi (0)=\xi (1)>0, \quad \varphi ^\prime (0)=\xi ^\prime (1)>0. \end{array}\right. } \end{aligned}$$
We set
$$\begin{aligned} \psi (s) := \varphi (s) \left\{ \dfrac{1}{\varphi (0) \varphi ^\prime (0)}+ \int _s^0 \dfrac{1}{\varphi ^2 (t)}dt \right\} . \end{aligned}$$
By direct calculations one can check that \(\psi ^\prime (0)=0\),
$$\begin{aligned} -\psi ^{\prime \prime }+e^{2s}\psi =0, \text { and } \varphi ^\prime (s) \psi (s) -\psi ^\prime (s) \varphi (s) =1, \end{aligned}$$
for every \(s\in (-\infty ,0)\). We also have \(\psi ^\prime (s)= - \int _s^0 \psi (t) e^{2t} dt <0\). Moreover, the relation \(\displaystyle \lim _{s\rightarrow -\infty } \varphi (s)=1\) implies
$$\begin{aligned} \lim _{s\rightarrow -\infty } \psi ^\prime (s)=-1, \end{aligned}$$
and using L'hospital rule,
$$\begin{aligned} \lim _{s\rightarrow -\infty } \dfrac{\psi (s)}{s}=\lim _{s\rightarrow -\infty }\psi ^\prime (s)=-1. \end{aligned}$$
Thus letting \(\zeta (r):= \psi (\ln r)\), we have all the claimed properties.
Finally, in order to prove (6.2), we have to show that, for all \(\varphi \in C^\infty ([0,1])\),
$$\begin{aligned} \int _0^1 \left( \frac{\partial G}{\partial r}(r,s) \varphi '(r)+ G(r,s)\varphi (r) \right) r^{N-1} \,dr =s^{N-1} \varphi (s). \end{aligned}$$
By the defintion G in (6.2), the left hand side of (6.3) rewrites as
$$\begin{aligned} \int _0^s \left( \xi '(r)\varphi '(r) + \xi (r) \varphi (r) \right) \zeta (s) s^{N-1} r^{N-1} \,dr + \int _s^1 \left( \zeta '(r)\varphi '(r) + \zeta (r) \varphi (r) \right) \xi (s) s^{N-1} r^{N-1} \,dr. \end{aligned}$$
We integrate by parts and we use the equation satisfied by \(\xi \) and \(\zeta \) and the respective boundary conditions, to obtain
$$\begin{aligned}&\int _0^s s^{N-1} \zeta (s) \varphi (r) \left[ -\left( r^{N-1}\xi '(r)\right) '+\xi (r)r^{N-1} \right] \,dr + s^{2N-2}\xi '(s)\zeta (s)\varphi (s) \\&\quad \quad +\int _s^1 s^{N-1} \xi (s) \varphi (r) \left[ -\left( r^{N-1}\zeta '(r)\right) '+\zeta (r)r^{N-1} \right] \,dr - s^{2N-2}\xi (s)\zeta '(s)\varphi (s) \\&\quad =s^{2N-2}\xi '(s)\zeta (s)\varphi (s) - s^{2N-2}\xi (s)\zeta '(s)\varphi (s). \end{aligned}$$
Then using (6.1) we obtain \(s^{N-1} \varphi (s)\), so that (6.3) is proved. \(\square \)
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1.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium
Bonheure, D., Casteras, JB. & Noris, B. Calc. Var. (2017) 56: 74. https://doi.org/10.1007/s00526-017-1163-3
Accepted 30 March 2017
First Online 28 April 2017 | CommonCrawl |
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April 2017, 37(4): 2103-2113. doi: 10.3934/dcds.2017090
On the local C1, α solution of ideal magneto-hydrodynamical equations
Shu-Guang Shao 1,2, , Shu Wang 1, , Wen-Qing Xu 1, and Yu-Li Ge 2,,
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China
* Corresponding author: Yu-Li Ge, Email:[email protected]
Received June 2016 Revised November 2016 Published December 2016
This paper is devoted to the study of the two-dimensional andthree-dimensional ideal incompressible magneto-hydrodynamic (MHD)equations in which the Faraday law is inviscid. We consider thelocal existence and uniqueness of classical solutions for the MHDsystem in Hölder space when the general initial data belongs to$C^{1,α}(\mathbb{R}^n)$ for $n=2$ and $n=3$.
Keywords: Ideal MHD equations, local C1, α solution, Hölder space.
Mathematics Subject Classification: Primary:35Q35, 35Q60;Secondary:35Q30.
Citation: Shu-Guang Shao, Shu Wang, Wen-Qing Xu, Yu-Li Ge. On the local C1, α solution of ideal magneto-hydrodynamical equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090
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Shu-Guang Shao Shu Wang Wen-Qing Xu Yu-Li Ge
Article outline | CommonCrawl |
Singular double-phase systems with variable growth for the Baouendi-Grushin operator
Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data
Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues
Yuncherl Choi 1, , Taeyoung Ha 2, , Jongmin Han 3,4,, , Sewoong Kim 5, and Doo Seok Lee 6,
Ingenium College of Liberal Arts, Kwangwoon University, Seoul 01891, Korea
Division of Medical Mathematics, National Institute for Mathematical Sciences, Daejeon 34047, Korea
Department of Mathematics, Kyung Hee University, Seoul 02447, Korea
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Korea
Samsung Fire & Marine Insurance, Seoul 04523, Korea
Department of Undergraduate Studies, Daegu Gyeongbuk Institute of Science and Technology, Daegu 42988, Korea
* Corresponding author: Jongmin Han
Received September 2020 Published September 2021 Early access February 2021
Figure(8) / Table(4)
In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers $ {\lambda}_0 $ and $ {\lambda}_1 $ for the control parameter $ {\lambda} $ in the equation. Motivated by [9], we assume that $ {\lambda}_0< {\lambda}_1 $ and the linearized operator at the trivial solution has multiple critical eigenvalues $ \beta_N^+ $ and $ \beta_{N+1}^+ $. Then, we show that as $ {\lambda} $ passes through $ {\lambda}_0 $, the trivial solution bifurcates to an $ S^1 $-attractor $ {\mathcal A}_N $. We verify that $ {\mathcal A}_N $ consists of eight steady state solutions and orbits connecting them. We compute the leading coefficients of each steady state solution via the center manifold analysis. We also give numerical results to explain the main theorem.
Keywords: Brusselator model, dynamic phase transition, attractor bifurcation, center manifold function.
Mathematics Subject Classification: Primary: 35B32, 35B41; Secondary: 35K40.
Citation: Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035
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Figure 1. Examples of Structure of $ {\mathcal A}_N $ in Table 1, 2 and 3
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Figure 2. Examples of Structure of $ {\mathcal A}_N $ in Table 4
Figure 3. Case (ⅰ) of (4.3) and $ N = 4 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^+(x) $ and (b) $ v_h(x,t) \to v_1^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $
Figure 4. Case (ⅱ) of (4.3) and $ N = 4 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_2^+(x) $ and (b) $ v_h(x,t) \to v_2^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $
Figure 5. Case (ⅲ) of (4.3) and $ N = 4 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^+(x) $ and (b) $ v_h(x,t) \to v_1^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $
Figure 6. Case (ⅰ) of (4.3) and $ N = 8 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^-(x) $ and (b) $ v_h(x,t) \to v_1^-(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $
Figure 7. Case (ⅱ) of (4.3) and $ N = 8 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^+(x) $ and (b) $ v_h(x,t) \to v_2^+(x) $. With $ w (x,0) = w_2(x) $, (c) $ u_h(x,t) \to u_1^-(x) $ and (d) $ v_h(x,t) \to v_1^-(x) $
Figure 8. Case (ⅲ) of (4.3) and $ N = 8 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_2^+(x) $ and (b) $ v_h(x,t) \to v_2^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^-(x) $ and (d) $ v_h(x,t) \to v_1^-(x) $
Table 1. Stability for $ k = 2 $
subcases $ w_1^+ $ $ w_1^- $ $ w_2^+ $ $ w_2^- $
(ⅰ-1) stable saddle $ \times $ $ \times $
(ⅰ-2) saddle stable $ \times $ $ \times $
(ⅰ-3) $ \times $ $ \times $ stable saddle
(ⅰ-4) $ \times $ $ \times $ saddle stable
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subcases $ w_1^\pm $ $ w_2^\pm $ $ w_3^\pm $ $ w_4^\pm $
(ⅱ-1) stable saddle $ \times $ $ \times $
(ⅱ-2) saddle stable $ \times $ $ \times $
(ⅱ-3) $ \times $ $ \times $ stable saddle
(ⅱ-4) $ \times $ $ \times $ saddle stable
subcases $ w_1^+ $ $ w_1^- $ $ w_2^+ $ $ w_2^- $ $ w_3^\pm $ $ w_4^\pm $
(ⅲ-1) stable saddle $ \times $ $ \times $ saddle stable
(ⅲ-2) saddle stable $ \times $ $ \times $ stable saddle
(ⅲ-3) $ \times $ $ \times $ stable saddle saddle stable
(ⅲ-4) $ \times $ $ \times $ saddle stable stable saddle
(ⅳ-1) stable stable saddle saddle
(ⅳ-2) saddle saddle stable stable
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\begin{document}
\title{Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation}
\titlerunning{Advective Allen-Cahn equation}
\author{ Murat Uzunca\inst{1} \and B\"{u}lent Karas\"{o}zen\inst{2} \and Ay\c{s}e Sar{\i}ayd{\i}n Filibelio\u{g}lu \inst{1}}
\authorrunning{Uzunca et al.}
\institute{ Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey, {\tt [email protected]}, {\tt [email protected]} \and Department of Mathematics \& Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey, {\tt [email protected]} }
\maketitle
\begin{abstract}
We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with non-divergence-free velocity fields. Numerical simulations for convection dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension. \end{abstract}
\section{Introduction}
Interfacial dynamics has great importance in modeling of multi phase flow in material sciences, and binary fluids flow movement. We consider the Allen-Cahn equation with advection as a model of diffuse interface for two phase flows \cite{liu12} \begin{equation} \label{advac}
\frac{\partial u}{\partial t} = \mathcal{L} u - \frac{1}{\epsilon}f(u)\quad \hbox{in } \Omega \times (0,T] \end{equation} under homogeneous Neumann boundary conditions, where $\mathcal{L}$ denotes the linear operator related to the diffusion and advection parts of the system, i.e. $\mathcal{L}u=\epsilon\Delta u - \nabla\cdot (\mathbf{V}u)$. The term $f(u)=F'(u)=2u(1 - u)(1 -2u)$ characterizes the cubic bistable nonlinearity with the double--well potential $F(u)$ of the two phases, and $\epsilon$ describes the surface tension. We consider a prescribed fixed velocity field $\mathbf{V}=(V_1, V_2)^T$. In coupled incompressible fluid mechanics and diffusive interface models, the velocity field satisfies the Navier--Stokes equations \cite{liu12}, and therefore is divergence free, i.e. $\nabla \cdot \mathbf{V} = 0$. We consider in this work the non-divergence-free velocity fields which are either expanding ($\nabla \cdot \mathbf{V}> 0$) or sheering ($\nabla \cdot \mathbf{V} < 0$) as in \cite{liu12}.
The advective Allen-Cahn equation \eqref{advac} describes the diffuse interface dynamics associated with surface energies, and has two different time scales; the small surface tension, and the convection time scale. Both time scales cause computational stiffness \cite{liu12}. The dynamics of surface tension in two-phase fluids are studied numerically by the level--set algorithm method and the diffuse interface method \cite{liu12}.
In this work we apply the adaptive method of time layers (AMOT) \cite{deuflhard12}, or adaptive Rothe method, where the advective Allen-Cahn equation \eqref{advac} is discretized first in time then in space, in contrast to the usual method of lines approach. Hereby spatial discretization is considered as a perturbation of the time integration. AMOT was applied to linear and nonlinear partial differential equations using linearly implicit time integrators in several papers \cite{bornemann90,deuflhard12,frank11,lang93,lang01,lang01a}. We have chosen the linearly three stage Rosenbrock (ROS3P) method \cite{lang01} as the time integrator. ROS3P solver is third order in time, L-stable and can efficiently deal with the large stiff systems arising from the discretization of \eqref{advac}. It does not show any order reduction in time in contrast to other Rosenbrock methods of order higher than two \cite{lang01}. Unlike the fully implicit schemes, it requires only the solution of three linear systems per time step with the same coefficient matrix. In non-stationary models, the potential internal/boundary layers moves as the time progresses. The time step-sizes have to be adapted properly to resolve these layers accurately. The simple embedded a posteriori error estimator as the difference of second and third order ROS3P solvers allow the construction of an efficient adaptive time integrator. To resolve the sharp layers and oscillations in advection-dominated regimes, we apply symmetric interior penalty Galerkin (SIPG) method \cite{arnold01,riviere08}, as a stable space discretization in the family of discontinuous Galerkin (dG) methods. Further, we apply the adaptive SIPG method in space with the residual-based a posteriori error estimator \cite{schotzau09,uzunca14adg} to handle unphysical oscillations. The spatial mesh is refined or coarsened locally to obtain an accurate approximation with less degree of freedoms (DoFs) and less computational time. We show in numerical experiments that the proposed time-space algorithm AMOT is capable of damping the oscillations which may vary as the time progresses.
The paper is organized as follows. In Section~\ref{dg} we give the fully discrete formulation of the advective AC model \eqref{advac}. The time-space adaptive algorithm is described in Section~\ref{adap}. In Section~\ref{numeric}, results of numerical experiments for convection dominated expanding and sheering flows are presented.
\section{Time-Space Discretization} \label{dg}
In this section we apply the method of time layers to discretize the model \eqref{advac} in time. The resulting sequence of elliptic problems are discretized by the SIPG method at each time step. We consider the partition of time interval $[0,T]$, as $I_k=(t^{k-1},t^k]$ with time step-sizes $\tau_k=t^{k}-t^{k-1}$, $k=1,2, \ldots, J$. The approximate solution at the time $t=t^k$ is denoted by $u^k\approx u(t^k)$. We apply the 3-stage Rosenbrock solver ROS3P \cite{lang01} with an embedded error estimator in time: \begin{equation}\label{ros_system} \left(\frac{1}{\gamma\tau_k}- J^{k-1}\right)K_i = \mathcal{L}\left( z^i\right) - f\left(z^i\right) + \sum_{j=1}^{i-1} \frac{c_{ij}}{\tau_k}K_j, \quad j=1,2,3, \end{equation} where $z^i=u^{k-1} +\sum_{j=1}^{i-1} a_{ij}K_j$ and $J^{k-1}:=J(u^{k-1})$ is the Jacobian $J(u)=\partial_u({\mathcal L}u-f(u))$ at $u^{k-1}$. The second order solution $\hat{u}^k$ and the third order solution $u^k$ are given by \begin{equation}\label{sol} \begin{aligned} \hat{u}^k &= u^{k-1} + \hat{m}_1K_1+\hat{m}_2K_2+\hat{m}_3K_3 ,\\ u^k &= u^{k-1} + m_1K_1+m_2K_2+m_3K_3 \end{aligned} \end{equation} with the same stage vectors $K_i$. For the derivation of ROS3P solver and for parameter values, we refer to \cite{lang01}. The difference of the solutions $u^k$ and $\hat{u}^k$ is used as an error indicator in the time-adaptivity. Due to the linearly implicit nature of the Rosenbrock methods, the stage vectors $K_i$ in \eqref{ros_system} are solved using linear systems with the same coefficient matrix, which increases the computational efficiency in time integration of nonlinear PDEs \cite{deuflhard12,lang01a}.
The semi-discrete systems \eqref{ros_system} are discretized in space by the SIPG method with upwinding for the convective term \cite{ayuso09}. On the time interval $I_n=(t^{k-1},t^k]$, we consider a family ${\mathcal{T}}_h^k$ of shape regular elements (triangles) $E\in{\mathcal{T}}_h^k$, and we denote the initial one by ${\mathcal{T}}_h^0$. Here, the mesh ${\mathcal{T}}_h^k$ is obtained by local refinement/coarsening of the mesh ${\mathcal{T}}_h^{k-1}$ from the previous time step. Then, with the dG finite element space $V_h^k:=V_h(\mathcal{T}_h^k)$, on an individual time step $I_n=(t^{k-1},t^k]$, in time ROS3P and in space SIPG discretized fully discrete scheme for the model \eqref{advac} reads as: for all $v_h^k\in V_h^k$, find $u_h^k$ (or $\hat{u}_h^k$) in \eqref{sol} with $K_i\in V_h^k$, $i=1,2,3$, satisfying \begin{equation}\label{fully} \left( \left(\frac{1}{\gamma\tau_k}- J_h^{k-1}\right)K_i,v_h^k\right) = -a_h(z_h^i,v_h^k) - b_{h}(z_h^i, v_h^k) + \left( \sum_{j=1}^{i-1} \frac{c_{ij}}{\tau_k}K_j,v_h^k \right), \end{equation} where $z_h^i=u_h^{k-1} +\sum_{j=1}^{i-1} a_{ij}K_j$, $J_h^{k-1}=J(u_h^{k-1})$ and $(\cdot , \cdot)$ stands for the discrete inner product $(\cdot , \cdot)_{L^2(\mathcal{T}_h^k)}$. The forms $a_h(u_h^k,v_h^k)$ and $b_h(u_h^k,v_h^k)$ are the bilinear and linear forms given by \begin{align*} a_{h}(u_h^k, v_h^k)=& \sum \limits_{E \in {\mathcal{T}}_h^k} \int_{E} \epsilon \nabla u_h^k\cdot\nabla v_h^k dx + \sum \limits_{E \in {\mathcal{T}}_h^k} \int_{E} (\mathbf{V}\cdot \nabla u_h^k + (\nabla\cdot\mathbf{V}) u_h^k) v_h^k dx \\ &+ \sum \limits_{E \in {\mathcal{T}}_h^k}\int_{\partial E^-\setminus\Gamma_h^- } \mathbf{V}\cdot \mathbf{n}_E ((u_{h}^{out})^k-(u_{h}^{in})^k) v_h^k ds \\ &- \sum \limits_{E \in {\mathcal{T}}_h^k} \int_{\partial E^-\cap \Gamma_h^{-}} \mathbf{V}\cdot \mathbf{n}_E (u_{h}^{in})^k v_h^k ds + \sum \limits_{ e \in \Gamma_h^k}\frac{\sigma \epsilon}{h_{e}} \int_{e} [u_h^k]\cdot[v_h^k] ds \\ &- \sum \limits_{ e \in \Gamma_h^k} \int_{e} ( \{\epsilon \nabla v_h^k \}\cdot[u_h^k] + \{\epsilon \nabla u_h^k \}\cdot [v_h^k] )ds,\\ b_{h}(u_h^k, v_h^k) =& \sum \limits_{E \in {\mathcal{T}}_h^k} \int_{K} \frac{1}{\epsilon}f(u_h^k) v_h^k dx, \end{align*} where $u_{h}^{out}$ and $u_{h}^{in}$ denote the traces on an edge from outside and inside of an element $E$, respectively, $h_e$ is the length of an edge $e$, $\Gamma_h^k$ is the set of interior edges, $\partial E^-$ and $\Gamma_h^-$ are the sets of inflow boundary edges of an element $E\in{\mathcal{T}}_h^k$ and on the boundary $\partial\Omega$, respectively. The parameter $\sigma$ is called the penalty parameter to penalize the jumps in dG schemes, and $[\cdot]$ and $\{\cdot\}$ stand as the jump and average operators, respectively \cite{riviere08}.
\section{Adaptive Method of Time Layers (AMOT)} \label{adap}
The goal of AMOT is on each time step $I_k=(t^{k-1},t^k]$ adjusting the time step-size and the spatial mesh adaptively. To do this, AMOT aims to bound the total error $\|u(t^k)-\hat{u}_h^k\|$ by suitable temporal and spatial estimators, where $u(t^k)$ is the true solution of the continuous model \eqref{advac} and $\hat{u}_h^k$ is the $2^{nd}$ order (in time) discrete solution of the fully discrete system \eqref{fully} on ${\mathcal{T}}_h^{k-1}$, at the time $t=t^k$. In order to define the temporal and spatial estimators separately, we replace the true solution $u(t^k)$ by its best available approximation $\overline{u_h^{k,+}}$ which is the $3^{rd}$ order (in time) discrete solution of the fully discrete system \eqref{fully} on an auxiliary very fine mesh $\overline{{\mathcal{T}}_h^k}\supset{\mathcal{T}}_h^{k-1}$, and we add and subtract the term $u_h^k$ which is the $3^{rd}$ order (in time) discrete solution of the fully discrete system \eqref{fully} on ${\mathcal{T}}_h^{k-1}$ at the time $t=t^k$. Then, similar to \cite[Sec. 9.2]{deuflhard12}, we get \begin{equation}\label{error} \begin{aligned}
\|u(t^k)-\hat{u}_h^k\|_{L^2({\mathcal{T}}_{h}^{k-1})} & \approx \|\overline{u_h^{k,+}}-\hat{u}_h^k\|_{L^2({\mathcal{T}}_{h}^{k-1})} = \|\overline{u_h^{k,+}}-u_h^k+u_h^k-\hat{u}_h^k\|_{L^2({\mathcal{T}}_{h}^{k-1})} \\
& \leq \underbrace{\|\overline{u_h^{k,+}}-u_h^k\|_{L^2({\mathcal{T}}_{h}^{k-1})}}_{:=\varepsilon_S} + \underbrace{\|u_h^k-\hat{u}_h^k\|_{L^2({\mathcal{T}}_{h}^{k-1})}}_{:=\varepsilon_T}\\ & \leq TOL_S + TOL_T \leq TOL \end{aligned} \end{equation} for a user prescribed tolerance $TOL$, and further, we set $TOL_T=\alpha TOL$ and $TOL_S=(1-\alpha) TOL$ for user defined $0<\alpha <1$. In \eqref{error}, the term $\varepsilon_T$ controls the temporal adjustment, while the term $\varepsilon_S$ controls the acceptance of spatial mesh. Note that the temporal error estimator $\varepsilon_T$ is nothing but the difference of the order 2 and order 3 (in time) solutions of the fully discrete system \eqref{fully} on ${\mathcal{T}}_h^{k-1}$ at the time $t=t^k$. As a result, on each time step $I_k$, AMOT starts on the spatial mesh ${\mathcal{T}}_h^{k-1}$ with the adjustment of the time step-size $\tau_k$ according to the acceptance relation \cite{deuflhard12} \begin{equation}\label{tau} \tau^* = \sqrt[3]{\frac{\rho TOL_T}{\varepsilon_T}}\tau_k \end{equation} with a safety factor $\rho\approx 0.9$, and the computed time step-size $\tau^*$ is accepted if $\varepsilon_T\leq TOL_T$.
\begin{algorithm} \caption{AMOT Algorithm on a single time step $I_k=(t^{k-1},t^k]$} \textbf{Input:} $u_h^{k-1}$, $\tau^*$, ${\mathcal{T}}_h^{k-1}$, $TOL_S$, $TOL_T$ \\ \textbf{Output:} $u_h^{k,+}$, $\tau_{k}$, $\tau^*$, ${\mathcal{T}}_h^k$ \begin{algorithmic} \STATE {\bf do}\\
\qquad $\tau_k=\tau^*$\\
\qquad compute $u_h^k$ and $\hat{u}_h^k$ on ${\mathcal{T}}_h^{k-1}$\\
\qquad {\bf if} $\varepsilon_T>TOL_T$\\
\qquad \qquad compute new step-size $\tau^*$ according to \eqref{tau}\\
\qquad {\bf end if}\\
\qquad compute error indicator $\eta$ and construct the auxiliary fine mesh $\overline{{\mathcal{T}}_h^k}$\\
\qquad compute the best available approximation $\overline{u_h^{k,+}}$ on $\overline{{\mathcal{T}}_h^k}$\\
\qquad {\bf if} $\varepsilon_S>TOL_S$\\
\qquad \qquad refine elements $E\in {\mathcal{T}}_h^{k-1}$ with $(\varepsilon_S)_E>0.005\times TOL_S$\\
\qquad \qquad coarsen elements $E\in {\mathcal{T}}_h^{k-1}$ with $(\varepsilon_S)_E<10^{-13}$\\
\qquad \qquad construct the new spatial mesh ${\mathcal{T}}_h^k$\\
\qquad {\bf end if} \\
\qquad compute $u_h^{k,+}$ on ${\mathcal{T}}_h^k$ \STATE {\bf until} $\varepsilon_T\leq TOL_T$ and $\varepsilon_S\leq TOL_S$ \end{algorithmic} \label{algorithm} \end{algorithm}
After time step-size adjustment, AMOT continues with the refinement and coarsening of the spatial mesh ${\mathcal{T}}_h^{k-1}$ to obtain the new spatial mesh ${\mathcal{T}}_h^k$ according to the spatial estimator $\varepsilon_S=\sum_{E} (\varepsilon_S)_E$ in \eqref{error}, where the local elements $E\in{\mathcal{T}}_h^{k-1}$ are refined for large $(\varepsilon_S)_E$ and the ones are coarsened for small $(\varepsilon_S)_E$. To match the elements $E\in{\mathcal{T}}_h^{k-1}$ to be refined, we check the condition $(\varepsilon_S)_E>0.005\times TOL_S$, while we check the condition $(\varepsilon_S)_E<10^{-13}$ to match the elements $E\in{\mathcal{T}}_h^{k-1}$ to be coarsened. Yet, to compute the spatial estimator $\varepsilon_S$, we need the best available approximation $\overline{u_h^{k,+}}$ which is the solution of the discrete system \eqref{fully} on a very fine auxiliary mesh $\overline{{\mathcal{T}}_h^k}\supset{\mathcal{T}}_h^{k-1}$. To construct the auxiliary fine mesh $\overline{{\mathcal{T}}_h^k}$, one needs a local error indicator to match the elements $E\in{\mathcal{T}}_h^{k-1}$ to be refined. We use residual-based error indicator \cite{schotzau09} \begin{equation}\label{ind} \eta =\left( \sum \limits_{E \in {\mathcal{T}}_h^{k-1}}\eta_E^2\right)^{1/2} \; , \quad \eta_E^2= \eta_{E_R}^2 + \eta_{E_0}^2 + \eta_{E_{\partial}}^2, \end{equation} where $\eta_{E_R}$ denote the cell residuals \begin{equation*}
\eta_{E_R}^2 = \lambda_E^2\left\| \frac{u_h^k-u_h^{k-1}}{\tau_k} - \epsilon\Delta u_h^k + \nabla\cdot (\mathbf{V}u_h^k) + \frac{1}{\epsilon}f(u_h^k)\right\|_{L^2(E)}^2, \end{equation*} for a weight function $\lambda_E$, while $\eta_{E_0}$ and $\eta_{E_{\partial}}$ stand for the edge residuals coming from the jump of the numerical solution on the interior and Neumann boundary edges, respectively, see \cite{schotzau09,uzunca14adg} for details. Using the local error indicators $\eta_E$ in \eqref{ind}, we construct the auxiliary fine mesh $\overline{{\mathcal{T}}_h^k}$ by refining the elements $E\in M_E\subset{\mathcal{T}}_h^{k-1}$, where the set $M_E$ is determined by the bulk criterion $$ \sum \limits_{E \in M_E}\eta_E^2 \geq \theta\sum \limits_{E \in {\mathcal{T}}_h^{k-1}}\eta_E^2. $$ for a user prescribed $0<\theta <1$, where larger $\theta$ leads to more elements to be refined. In our simulations we take $\theta =0.9$ since we need a very fine auxiliary mesh. The AMOT procedure, see Algorithm~\ref{algorithm}, continues until the temporal and spatial acceptance conditions $\varepsilon_S\leq TOL_S$ and $\varepsilon_T\leq TOL_T$ are satisfied.
\section{Numerical Experiments} \label{numeric}
In this section, we demonstrate the accuracy and efficiency of the proposed AMOT for expanding and sheering flow examples. In all examples, we set the tolerance $TOL=0.001$, the parameter $\alpha =0.5$ and the diffusion coefficient $\epsilon =0.01$. The spatial domain is taken as $\Omega =[-1,1]^2$ and the time interval is $[0,06]$. For the SIPG discretization we use piecewise discontinuous linear polynomials. Numerical solutions on uniform meshes in space are computed with the constant time step $\tau =0.001$ and using a $64\times 64$ uniform spatial mesh with DoFs $24576$.
\subsection{Sheering Flow} \label{ex2}
We consider \eqref{advac} with the sheering velocity field $\mathbf{V}=(0,-100 x)$, and with the initial condition as $1$ on $[-0.1,0.1]^2$ otherwise $0$ \cite{liu12}. In Fig.~\ref{plot1}, left, the unphysical oscillations of the solution on uniform mesh can be clearly seen. The oscillations are damped out by the AMOT algorithm, in Fig.~\ref{plot1}, middle, and adaptive mesh is concentrated in the region where the sharp layers occur.
\begin{figure}
\caption{Sheering Flow: Solution profiles at final time obtained by uniform (left) and adaptive (middle) schemes, and adaptive mesh at final time (right)}
\label{plot1}
\end{figure}
The refinement and coarsening of AMOT algorithm works well as shown in Fig.~\ref{tdt1}, right. The mesh becomes finer at the very beginning and then, gets coarser around $t=0.02$ as the size of the interior layer becomes smaller due to the sheering and the time step-size increases monotonically, Fig.~\ref{tdt1}, left.
\begin{figure}
\caption{Sheering Flow: Evolution of time step size (left) and DoFs (right) }
\label{tdt1}
\end{figure}
\subsection{Expanding Flow } \label{ex1}
As the second example, we consider the expanding velocity field $\mathbf{V}=(10 x,10 y)$. The initial condition is taken as $1$ in the square $[-0.3,0.3]^2$ and $0$ otherwise \cite{liu12}. The unphysical oscillations are damped again in Fig.~\ref{plot2}, middle. The mesh refined slightly and time step-size increases at the beginning, and then refinement and coarsening proceed simultaneously, Fig.~\ref{tdt2}. Time step-size slightly decreases after $t=0.02$ following refinement/coarsening.
\begin{figure}
\caption{Expanding Flow: Solution profiles at final time obtained by uniform (left) and adaptive (middle) schemes, and adaptive mesh at final time ( right)}
\label{plot2}
\end{figure}
\begin{figure}
\caption{Expanding flow: Evolution of time step sizes (left) and DoFs (right) }
\label{tdt2}
\end{figure}
\section*{Acknowledgments} This work has been partially supported METU Research Fund Project BAP-07-05-2013-004 and by Scientific Human Resources Development Program (\"OYP) of the Turkish Higher Education Council (Y\"OK).
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi
\end{document} | arXiv |
Ferdinand Karl Schweikart
Ferdinand Karl Schweikart (1780–1857) was a German jurist and amateur mathematician who developed an astral geometry before the discovery of non-Euclidean geometry.
Ferdinand Karl Schweikart
Born(1780-02-28)28 February 1780
Erbach im Odenwald, Hesse, now Germany
Died17 August 1857(1857-08-17) (aged 77)
Königsberg, Prussia, now Russia
Alma materUniversity of Marburg
Scientific career
FieldsMathematics, Jurisprudence
InstitutionsUniversity of Marburg
University of Königsberg
Life and work
Schweikart, son of an attorney in Hesse, was educated in the school of his town. He went to the high school in Hanau and Waldeck before entering in 1796 to study law in the university of Marburg, where he attended lectures of the mathematics professor J.K.F. Hauff.[1] He was awarded a doctorate in law at the university of Jena in 1798.
After practicing as a lawyer for a few years in Erbach, he was, from 1803 to 1807, instructor of the youngest prince of Hohenlohe-Ingelfingen.[2] From 1809, he was university professor of jurisprudence successively at the universities of Giessen (1809-1812), Kharkiv (1812-1816), Marburg (1816-1821) and Königsberg (1821 afterwards).[3]
But Schweikart is best remembered for his works on mathematics: in 1807 he published Die Theorie der Parallellinien, nebst dem Vorschlage ihrer Verbannung aus der Geometrie (The theory of parallel lines, along with the suggestions of their banishment from geometry).[4] Then, in 1818 he wrote to Gauss, through his student Christian Ludwig Gerling, about a new geometry, called by him as astral geometry, where the sum of the angles of a triangle was less than 180º (as in hyperbolic geometry).[5] He influenced the work of his nephew, the mathematician Franz Taurinus.
References
1. Halsted 1896, p. 105.
2. Winter 1891, p. 358.
3. Meyer, 1909. Meyers Großes Konversations-Lexikon.
4. Bardi 2009, p. 127.
5. Gray 2006, p. 66.
Bibliography
• Bardi, Jason Socrates (2009). The fifth postulate: how unraveling a two-thousand-year-old mystery unraveled the universe. Wiley. ISBN 978-0-470-14909-6.
• Gray, Jeremy (2006). "Gauss and Non-Euclidean Geometry". In András Prékopa; Emil Molnár (eds.). Non-Euclidean Geometries. Mathematics and its Applications. Vol. 581. Springer. pp. 61–80. doi:10.1007/0-387-29555-0_2. ISBN 978-0-387-29554-1. S2CID 55674427.
• Halsted, George Bruce (1896). "Subconscious Pangeometry". The Monist. 7 (1): 100–106. doi:10.5840/monist1896713. ISSN 0026-9662.
• Winter, Georg (1891). "Schweikart, Ferdinand Karl". Allgemeine Deutsche Biographie (in German). Historischen Kommission bei der Bayerischen Akademie der Wissenschaften. p. 358.
External links
• "Schweikart". Meyers Großes Konversations-Lexikon. 1909. Retrieved October 12, 2018.
Authority control
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• ISNI
• VIAF
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• Germany
• Netherlands
People
• Deutsche Biographie
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| Wikipedia |
Pharmacokinetic model of unfractionated heparin during and after cardiopulmonary bypass in cardiac surgery
Zaishen Jia1,
Ganzhong Tian2,
Yupeng Ren2,
Zhiquan Sun1,
Wei Lu2 &
Xiaotong Hou1
Journal of Translational Medicine volume 13, Article number: 45 (2015) Cite this article
Unfractionated heparin (UFH) is widely used as a reversible anti-coagulant in cardiopulmonary bypass (CPB). However, the pharmacokinetic characteristics of UFH in CPB surgeries remain unknown because of the lack of means to directly determine plasma UFH concentrations. The aim of this study was to establish a pharmacokinetic model to predict plasma UFH concentrations at the end of CPB for optimal neutralization with protamine sulfate.
Forty-one patients undergoing CPB during cardiac surgery were enrolled in this observational clinical study of UFH pharmacokinetics. Patients received intravenous injections of UFH, and plasma anti-FIIa activity was measured with commercial anti-FIIa assay kits. A population pharmacokinetic model was established by using nonlinear mixed-effects modeling (NONMEM) software and validated by visual predictive check and Bootstrap analyses. Estimated parameters in the final model were used to simulate additional protamine administration after cardiac surgery in order to eliminate heparin rebound. Plans for postoperative protamine intravenous injections and infusions were quantitatively compared and evaluated during the simulation.
A two-compartment pharmacokinetic model with first-order elimination provided the best fit. Subsequent simulation of postoperative protamine administration suggested that a lower-dose protamine infusion over 24 h may provide better elimination and prevent heparin rebound than bolus injection and other infusion regimens that have higher infusion rates and shorter duration.
A two-compartment model accurately reflects the pharmacokinetics of UFH in Chinese patients during CPB and can be used to explain postoperative heparin rebound after protamine neutralization. Simulations suggest a 24-h protamine infusion is more effective for heparin rebound prevention than a 6-h protamine infusion.
Unfractionated heparin (UFH) is an anionic mixture of highly sulfated linear glucosamine-glycans with varying molecular weights (3–30 kD) [1,2]. The anti-coagulation effect of heparin is dependent upon binding with the serine protease inhibitor anti-thrombin III (ATIII) [3]. Binding with ATIII increases the inhibitory activity of ATIII against both thrombin (FIIa) and factor Xa (FXa) and other serine proteases in the coagulation cascade by over 1000-fold [2,4]. Human FIIa is much more sensitive than FXa to inhibition mediated by the heparin-ATIII complex, so the anti-FIIa assay may have a smaller lower limit of quantitation (LLOQ) [5]. We measured anti-FIIa activity as an index to quantify plasma UFH levels in humans [6].
The UFH half-life is 1–2 h in human plasma, depending on dose [2], as higher doses produce a prolonged half-life due to the mechanism of plasma clearance, which involves rapid distribution via UFH binding to plasma proteins and receptors on endothelial cells and macrophages, followed by slower elimination through the kidneys [5]. Thus, UFH pharmacokinetics may include a "peripheral process" by which the UFH molecule is converted from the free to the bound state. Moreover, plasma UFH concentrations may exhibit larger inter-individual variability than other anti-coagulation drugs.
UFH is widely utilized in cardiac surgery to achieve adequate anti-coagulation and to restore normal hemostasis during and after CPB. UFH anti-coagulation can be reversed through formation of a stable complex with the highly cationic protamine sulfate, although its use as a 'neutralizer' of UFH carries some risk. Rapid administration of protamine sulfate can cause life-threatening hemodynamic disturbances such as systemic arterial hypotension and pulmonary hypertension, along with histamine release and hypoxemia, especially at the end of CPB when the myocardium is recovering from ischemic insult [2,7,8]. Complete reversal of UFH anti-coagulation is typically achieved with an excessive dose of protamine sulfate, although this has been associated with increased bleeding and inhibition of platelet glycoprotein Ib-von Willebrand factor, increased expression of P-selectin, blockade of calcium-release channels, and negative inotropic effects [2,9]. Accurate protamine dosing requires an understanding of real plasma UFH levels. However, the pharmacokinetic profile of UFH is unknown for Asian patients undergoing CPB.
Since 1962, Hyun and other researchers have used the phrase "heparin rebound" to describe the reappearance of UHF in circulating blood even after a dose of excess protamine sulfate [10,11]. Teoh et al. postulated that heparin rebound could be due to a portion of UFH administered during surgery that remains protein-bound and does not form a complex with protamine sulfate; these complexes dissociate slowly to produce anticoagulant effects [12]. No quantitative studies of this phenomenon have been published.
The objective of this study was to quantitate the pharmacokinetic characteristics of UFH in Chinese patients undergoing CPB, characterize the correlation between heparin rebound and UFH pharmacokinetics, and establish a pharmacokinetic model. The model can be used to simulate and optimize protamine sulfate dosing in order to minimize the side effects of protamine sulfate and restore normal hemostasis with minimal post-operative bleeding and blood transfusion.
After subjects with dysfunctions of the kidney, liver, or blood coagulation were excluded, 41 study patients underwent CPB during cardiac surgery (Table 1). The study protocol was approved by the Ethics Committee of Capital Medical University (Beijing, China). All subjects provided written informed consent prior to enrollment.
Table 1 Baseline characteristics of patients
Subjects had received an initial intra-venous injection of UFH (Changzhou Qianhong Biopharma, Changzhou, China) at 375 IU/kg (3 mg/kg; 1 mg = 125 IU) before CPB. Ten minutes after the initial dose, the first blood sample was collected from the jugular vein. The CPB pipelines were primed with 1500 mL of balanced solution and a second UFH dose (1 mg/kg). The CPB flow was maintained at 2.2–2.4 L/min/m2 with gravity siphon venous drainage. The temperature of the CPB system drifted to 32°C. The targeted mean perfusion pressure was 50–70 mmHg. Myocardial preservation was achieved with blood cardioplegic solution; ultrafiltration was avoided throughout the operation. A series of irregular UFH intravenous injections were administered, depending on each patient's blood coagulation activity during CPB.
Before UFH neutralization with protamine sulfate, sampling was performed at 30-min intervals. All samples were collected in 3-mL vacuum tubes buffered with sodium citrate and stored immediately at 4°C. Stored blood samples were centrifuged (2000 × g, 15 min) within 24 h of collection to remove platelets and hemocytes. Platelet-poor plasma (PPP) was then collected and stored at −80°C. The actual UFH dosing time and dosages of were recorded carefully.
After UFH neutralization, the sampling schedule was 2, 4, 8, 12, and 24 h. Actual sampling times deviated slightly from the schedule, so only the actual sampling times were documented.
Sample dilution and determination of anti-FIIa activity
The actual anti-FIIa activity in plasma during CPB is much larger than the measurable range (0.0–0.6 IU/mL for the anti-FIIa assay). Thus, samples collected before protamine neutralization were diluted with Normal Pooled Platelet-poor plasma (NPPPP) at 1:29. NPPPP was derived from PPP collected from 20 randomly selected healthy adults. Twenty parts of plasma were centrifuged separately (2000 x g, 15 min, at room temperature) to remove platelets, then were mixed into a single-part NPPPP.
Anti-FIIa activities in plasma were determined as reported by Falkon et al. [13,14]. All assays were performed on an ACL-TOP automated coagulation assay platform (Instrumentation Laboratory, Orangeburg, NY). The Heparin Chromogenic Activity Kit 820 (American Diagnostica, Stamford, CT) was used for the anti-FIIa assay.
Modeling strategy
NONMEM (version VII, level 2.0; Icon Development Solutions, Hanover, MD) was used to establish the pharmacokinetic model, with gFORTRAN (version 4.0) as the FORTRAN compiler and platform R (version 2.15.0) as the statistical and plotting software. All modeling and simulation procedures were performed on an operative platform for NONMEM known as "Perl speaks NONMEM" (PsN; version 3.4.0). First-order conditional estimation (FOCE) was the chosen algorithm. Results with p < 0.01 were considered significant.
Our data involved a series of post-neutralization points indicating heparin rebound, which cannot be explained by a mono-compartment model. Thus, the base model that provided the best data fit was a two-compartment model with first-order elimination and multiple irregular intravenous administration of UFH (Figure 1).
Two-compartment model with intravenous injection and first-order elimination. X UFH-C and X UFH-P represent the amount of UFH in central and peripheral compartments respectively; V UFH-C and V UFH-P are the apparent volumes of distribution for the central and peripheral compartments respectively.
The base model was a typical two-compartment model with multiple dosing during surgery. However, due to injection of excessive protamine sulfate, the amount of UFH in the central compartment was supposed to be instantly cleared. Thus, X UFH-C was set to 0 at that time, whereas the amount of UFH in the peripheral compartment remained unaffected. Distribution to the central compartment followed the inverse pattern, as the amount of UFH in the central compartment increased from 0. These assumptions were used to describe UFH neutralization with excess protamine sulfate and heparin rebound after neutralization.
In Figure 1, X UFH-C and X UFH-P are represent the amounts of UFH in central and peripheral compartments, whereas V UFH-C and V UFH-P and are the apparent volumes of distribution for the central and peripheral compartments, respectively; K 10 is the first-order elimination rate constant; K 12 and K 21 and are the first-order rate constants of the transportations of UFH from the central to the peripheral compartment and from the peripheral to the central compartment, respectively.
The original differential equations corresponding to the two-compartment model with single intravenous administration and first-order elimination are given in Eq. 1 and Eq. 2.
$$ \frac{d{X}_{UFH-C}}{dt}={K}_{21}{X}_{UFH-P}-{K}_{12}{X}_{UFH-C}-{K}_{10}{X}_{UFH-C} $$
$$ \frac{d{X}_{UFH-P}}{dt}=-{K}_{21}{X}_{UFH-P}+{K}_{12}{X}_{UFH-C} $$
The initial condition: t = 0, X UFH − C = D inj , X UFH − P = 0, where t is time, and D inj is the injected dose of UFH. To be more intuitive, we excluded the algebraic steps, and the Laplace-transformed equations of integral form are given by Eq. 3, 4, 5, 6.
$$ \alpha =\frac{\left(\left({K}_{10}+{K}_{12}+{K}_{21}\right)+\sqrt{{\left({K}_{10}+{K}_{12}+{K}_{21}\right)}^2-4{K}_{10}{K}_{21}}\right)}{2} $$
$$ \beta =\frac{\left(\left({K}_{10}+{K}_{12}+{K}_{21}\right)-\sqrt{{\left({K}_{10}+{K}_{12}+{K}_{21}\right)}^2-4{K}_{10}{K}_{21}}\right)}{2} $$
$$ {X}_{UFH-C}={f}_1(t)=\frac{\left(\alpha -{K}_{21}\right){D}_{inj}{e}^{-\alpha t}+\left({K}_{21}-\beta \right){D}_{inj}{e}^{-\beta t}}{\alpha -\beta } $$
$$ {X}_{UFH-P}={f}_2(t)=\frac{-{K}_{12}{D}_{inj}{e}^{-\alpha t}+{K}_{12}{D}_{inj}{e}^{-\beta t}}{\alpha -\beta } $$
Thus, we have the following equation representing the time-sequence function of the plasma UFH level, shown in Eq. 7:
$$ {C}_{UFH-C}={f}_3(t)=\frac{\left(\alpha -{K}_{21}\right){D}_{inj}{e}^{-\alpha t}+\left({K}_{21}-\beta \right){D}_{inj}{e}^{-\beta t}}{\left(\alpha -\beta \right)\left(V/F\right)} $$
Irregular intra-venous injection of UFH could be administered when there was a risk of blood coagulation during CPB. Thus, the equations representing the model of a single intravenous injection of UFH (Eq. 5, 6, 7) were updated for multiple irregular intravenous injections of UFH, which is given by Eq. 8–10 a according to the "superposition principle" of linear systems.
$$ {X}_{UFH-C}={g}_1(t)={\displaystyle \sum_{i=1}^n}\frac{\left(\alpha -{K}_{21}\right){D}_i{e}^{-\alpha \left(t-{\tau}_i\right)}+\left({K}_{21}-\beta \right){D}_i{e}^{-\beta \left(t-{\tau}_i\right)}}{\alpha -\beta } $$
$$ {C}_{UFH-C}={g}_3(t)={\displaystyle \sum_{i=1}^n}\frac{\left(\alpha -{K}_{21}\right){D}_i{e}^{-\alpha \left(t-{\tau}_i\right)}+\left({K}_{21}-\beta \right){D}_i{e}^{-\beta \left(t-{\tau}_i\right)}}{\left(\alpha -\beta \right)\left(V/F\right)} $$
Where τ i is the time of the i th administration of UFH; D i is the dose of the i th administration of UFH; and the maximum number of UFH doses is counted as n.
The amount of UFH in the central compartment must clear instantly when protamine sulfate is administered for neutralization. To describe such a situation after neutralization or the post-CPB pharmacokinetics of UFH mathematically, the initial conditions of the differential equations (Eq. 1–2) mentioned above were reset to t = T neu , X UFH − C = 0, X UFH − P = g 2(T neu ), where T neu represents the time of neutralization using protamine sulfate, and the function g 2 represents Eq. 9. Then Eq. 5–7, representing the amounts of UFH in the central and peripheral compartments as well as the plasma UFH level, were updated under new initial conditions, resulting in Eq. 11, 12, 13:
$$ {X}_{UFH-C}={h}_1(t)=\frac{g_2\left({T}_{neu}\right)\left(\alpha -{K}_{21}\right)\left(\beta -{K}_{21}\right)\left[{e}^{\left(\beta -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}-{e}^{\left(\alpha -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}\right]}{K_{12}\left(\alpha -\beta \right)} $$
$$ {X}_{UFH-P}={h}_2(t)=\frac{g_2\left({T}_{neu}\right)\left[\left(\alpha -{K}_{21}\right){e}^{\left(\alpha -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}-\left(\beta -{K}_{21}\right){e}^{\left(\beta -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}\right]}{\alpha -\beta } $$
$$ {C}_{UFH-C}={h}_3(t)=\frac{g_2\left({T}_{neu}\right)\left(\alpha -{K}_{21}\right)\left(\beta -{K}_{21}\right)\left[{e}^{\left(\beta -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}-{e}^{\left(\alpha -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}\right]}{K_{12}\left(\alpha -\beta \right)\left(V/F\right)} $$
Thus, the overall time sequence function of plasma UFH during and after CPB should combine Eq. 10 and Eq. 13, as follows: and together, as described below:
$$ {\mathrm{C}}_t=\left\{\begin{array}{c}\hfill {g}_3(t)={\displaystyle \sum_{i=1}^n}\frac{\left(\alpha -{K}_{21}\right){D}_i{e}^{-\alpha \left(t-{\tau}_i\right)}+\left({K}_{21}-\beta \right){D}_i{e}^{-\beta \left(t-{\tau}_i\right)}}{\left(\alpha -\beta \right)\left(V/F\right)} when\ 0\le t<{T}_{neu}\hfill \\ {}\hfill {h}_3(t)=\frac{g_2\left({T}_{neu}\right)\left(\alpha -{K}_{21}\right)\left(\beta -{K}_{21}\right)\left[{e}^{\left(\beta -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}-{e}^{\left(\alpha -{K}_{12}-{K}_{21}-{K}_{10}\right)\left(t-{T}_{neu}\right)}\right]}{K_{12}\left(\alpha -\beta \right)\left(V/F\right)}\ when t\ge {T}_{neu}\hfill \end{array}\right. $$
As shown in Eq. 14, the pharmacokinetics of UFH comprised two mathematically different stages. To have a simple iteration step and a short computation time and, most importantly, to set the amount of UFH in the central compartment as 0 at the time of neutralization, the Laplace-transformed equation (Eq. 14) along with the $PRED block in NONMEM was used in the computation process.
During computation, h 1–3(t) and g 1–3(t) were calculated simultaneously, and their real-time values documented automatically by NONMEM. At the time of neutralization, the value of X UFH − C was instantly set to 0, whereas the value of X UFH − P at that moment was documented automatically as the new initial condition of the inverse distribution of UFH from the peripheral compartment towards the central compartment.
Population pharmacokinetic modeling involves random effects and fixed effects [15]. In this study, the random effects were interindividual and residual. The inter-individual random effects were analyzed using an exponential model (Eq. 15), whereas the residual random effects were evaluated using a hybrid model (Eq. 16):
$$ {P}_i={P}_{pop}\cdotp {e}^{\eta_i} $$
Where P i is the pharmacokinetic parameter of the i th individual; P pop is the typical population parameter; η i is the inter-individual variability of the i th individual, following a normal distribution of N(0, ω 2)
$$ \left\{\begin{array}{c}\hfill {C}_{pred}=\frac{X_{UFH-C}}{V_{UFH-C}}\hfill \\ {}\hfill {C}_{obs}={C}_{pred}\cdotp \cdotp \left(1+{\varepsilon}_1\right)+{\varepsilon}_2\hfill \end{array}\right. $$
In Eq. 16, C obs is the observed plasma concentration, Cpred is the predicted plasma concentration, and ε 1 and ε 2 define the proportional error and additional error, respectively, following a normal distribution of N(0, σ 1 2) and N(0, σ 2 2).
Continuous fixed effects (age, body weight) were analyzed in the pharmacokinetic model in a linear manner:
$$ {P}_i={P}_{pop}\cdotp \cdotp \left(1\pm {\theta}_{COV}\cdotp \cdotp \left(\overline{COV}-CO{V}_i\right)\right)\cdotp \cdotp {e}^{\eta_i} $$
Where P i and P pop are the individual and population pharmacokinetic parameters, θ COV is the influence coefficient of the given fixed effect, and \( \overline{COV} \) and \( \underset{\bar{\mkern6mu}}{CO{V}_i} \) are the mean and individual values of the fixed effect.
Discontinuous fixed effects (e.g. sex) were analyzed in the pharmacokinetic model in a conditional manner:
$$ {P}_i={P}_{pop}\cdotp {e}^{\eta_i}\cdotp {\theta}^{GNDR} $$
Where GNDR is the value of represents sex (0 for male, 1 for female). θ is the influence coefficient of sex. θ GNDR equals 1 for males and θ when it is for females.
Covariate analysis was performed by a stepwise regression known as "forward inclusion" and "backward elimination". In forward modeling, all covariates were added to the base model, one after the other. Then, all covariates with a decrease in the objective function value (OFV) over 6.63 (x 2 distribution with 1df for p < 0.01) were listed in descending order according to the decrease in OFV. All remaining covariates were again added to the base model in order. If the OFV reduction was over 6.63 (p < 0.01), the covariate was retained. Otherwise, it was ruled out until no further reduction of the OFV occurred (full model).
In backward modeling, all covariates in the full model were removed one at a time. Only covariates with sufficient contributions to the prediction of the pharmacokinetic model were retained based on an increase in the OFV greater than 10.83 (p < 0.001). Otherwise, the covariate was ruled out until no further increase of the OFV occurred (final model).
The basic goodness-of-fit plots, including population predicted concentration (PRED) vs. observed concentration (OBS) plot, individual predicted concentration (IPRED) vs. OBS plot, conditional weighted residuals (CWRES) vs. PRED plot, and CWRES vs. time plot, were used to evaluate the final model.
The bootstrap method was used to evaluate accuracy and stability. The original dataset was re-sampled randomly 1000 times, producing 1000 new datasets. The final model was recalculated for the new datasets, and the median value and 90% confidence intervals of the recalculated model parameters compared to the final model.
The visual predictive check (VPC) method was used to evaluate the accuracy and predictive ability of the final model. The final model with the original dataset was simulated 1000 times with different random seeds, and the 5%, 50%, and 95% fractiles along with the 90% confidence interval of 1000-fold simulation compared to the observed data.
We also simulated the administration of excess protamine via bolus injection or infusions to determine which provides minimal heparin rebound after CPB. Simulations were performed with NONMEM and the results compared to identify the optimal treatment for heparin rebound.
Thirty-two patients completed the clinical study. Total administered UFH dose was about 33000 IU/surgery. The change in anti-FIIa during CPB and 24 h after the end of CPB is shown in Figure 2.
Anti-F IIa activity vs . Time. A, Anti-FIIa activity vs. time for blood samples obtained during CPB. B, anti-FIIa activity vs. time for blood samples obtained no less than24 h after the end of CPB (n = 32). In both plots, the time at the start of CPB was set to 0.
The model was parameterized in terms of volume of distribution and clearance rather than rate constants. The inter-individual random effect was evaluated with an exponential error model and the intra-individual random effect was evaluated with a hybrid model, involving additive and proportional errors. The fixed effects of covariates were tested (age, body weight, sex) with forward modeling and backward elimination methods. None of the tested covariates significantly decreased the objective function, and thus did not improve the fit. The goodness-of-fit plots of the final model are shown in Figure 3; estimates of the pharmacokinetic parameters of the final model along with the results of Bootstrap analyses are listed in Table 2.
Basic goodness-of-fit plots of the final population pharmacokinetic model. A: observed vs. population predicted concentrations. B: observed vs. individual predicted concentrations. C: conditional weighted residuals vs. population predicted concentrations. D: conditional weighted residuals vs. time after first dose. The red line represents the linear fit by the ordinary least square (OLS) method.
Table 2 Estimates of the parameters of the final model and results of Bootstrap analyses
The final model was validated with Bootstrap and VPC. Bootstrap analyses showed a success rate of 57.3% (573 out of 1000 were successful in covariance steps, whereas 982 out of 1000 were successful in minimization). Parameter distribution in Bootstrap analyses is summarized in Figure 4 and the VPC result shown in Figure 5. Simulated plasma Anti-FIIa activities of a hypothetical individual whose pharmacokinetic parameters were identical to the parameter estimations in our final model are shown in Figure 6.
Distributions of the OFV and key parameters in bootstrap analyses. A: OFV; B: CL/F; C: VUFH-C in Bootstrap analyses; D: Q/F; E: VUFH-P.
VPC plots of the final model. A: before protamine neutralization; B: post-CPB VPC. Blue-colored areas are the 90% CI and pink-colored areas are the 50% CI for simulated concentrations. The red line represents the median of the observed concentration; the dashed lines represent the upper and lower 90% CI of observed concentrations.
Simulated plots of infusion times and a protamine infusion rate of 25 mg/h. A: 6-h infusion; B: 8-h infusion; C: 12-h infusion; D: 16-h infusion; E: 24-h infusion. Blue lines are simulated concentrations of UFH without follow-up protamine infusions or doses. Red lines denote simulated concentrations of UFH over different protamine infusion times.
The results suggested that UFH (375 IU/kg) administered during CPB follows a two-compartment distribution and first-order elimination curve with an approximate initial half-life of 90 min. Median plateau anti-FIIa activity during CPB was 2–19 IU/mL (Figure 5), which was within the therapeutic range of UFH during CPB [16]. Thus, UFH levels in CPB may exhibit very large interindividual differences. The median anti-FIIa level at the end of CPB was 4.8 IU/mL and was neutralized with a mean 2.04 h after the start of CPB. A heparin-rebound peak of 0.04 IU/mL was attained 8 h after the end of CPB and was maintained above 0.02 IU/mL 24 h after neutralization, evidence of heparin rebound.
Based on patient plasma UFH levels, a population pharmacokinetic model was established using NONMEM. According to the goodness-of-fit plots in Figure 3, CWRES seems to be distributed randomly between −4 to +4 during and after CPB, and the population predictions vs. observations were distributed along the y = x line, suggesting that our final model was not biased and was consistent.
Continuous infusion of protamine (25 mg/h) after CPB can reduce the severity of post-operative heparin rebound [4]. Thus, for a better illustration of our final model and to test its clinical utility, we simulated the plasma Anti-FIIa activities of a hypothetical individual whose pharmacokinetic parameters were identical to the parameter estimations in our final model. That is, under hypothetical situations, continuous protamine infusions at 25 mg/h lasting 6–24 h were administered after CPB. The result of this simulation was similar to prior reports (Figure 6) [4]. A comparison of plots A and E suggest a 6-h infusion would decrease the plasma heparin levels caused by heparin rebound. A 24-h infusion would provide even better control.
Most studies on UFH pharmacokinetics have been based on measuring anti-coagulation activity as represented by the activated partial thromboplastin time or activated coagulation time in plasma or whole blood. Such methods are unstable, have poor reproducibility across assays and, most importantly, can be used only to monitor plasma UFH levels during or after CPB [17-20]. In our study, patient plasma levels of UFH were monitored during and after CPB using an established anti-FIIa assay [21,22]. Heparin rebound could be caused by the inverse distribution of UFH from the peripheral compartment to the central compartment. Therefore, additional administration of protamine after CPB could reduce the intensity and duration of post-CPB heparin rebound.
To our knowledge, this is the first report of a UFH pharmacokinetic model for CPB surgery; the model can predict UFH concentrations at the end of CPB to guide optimal neutralization with protamine sulfate. Study limitations include the fact that only anti-FIIa activity was monitored and UFH metabolites were not studied, which means the roles of LMWH in heparin rebound and during CPB remain unknown. In addition, the study patients were given several concomitant drugs that may also have affected coagulation.
A two-compartment model demonstrates the precise pharmacokinetics of UFH in Chinese patients during CPB and explains the postoperative heparin rebound after protamine neutralization. A 24-h protamine infusion is more effective than the 6-h infusion method for reducing plasma heparin levels caused by heparin rebound.
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This study was supported by grants from the Research Fund of Capital Medical Development and the National Natural Science Foundation of China (No. 81270327, 81470528).
Department of Extracorporeal Circulation, Center for Cardiac Intensive Care, Beijing Anzhen Hospital, Capital Medical University, Beijing Institute of Heart Lung and Blood Vessel Disease, No. 2 Anzhen Road, Chaoyang District, Beijing, 100029, China
Zaishen Jia
, Zhiquan Sun
& Xiaotong Hou
Department of Pharmaceutics, School of Pharmaceutical Science, Peking University Health Science Centre, No.38 Xueyuan Road, Haidian District, Beijing, 100191, China
Ganzhong Tian
, Yupeng Ren
& Wei Lu
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Correspondence to Xiaotong Hou.
The authors declare that they have no competing of interests.
ZJ: experimental design; quality control; sample collection; and theoretical guidance as a clinical physician. GT: experimental design; collection and determination of samples; data reduction; and computational modeling. YR: computational modeling. ZS: quality control. WL: experimental design computational modeling. XH: experimental design; quality control and theoretical guidance. All authors read and approved the final manuscript.
Authors' information
Declaration of co-authorship: Zaishen Jia and Ganzhong Tian contributed equally to this work. ZaishenJia: experimental design, quality control, sample collection, and theoretical guidance as a clinical physician. Ganzhong Tian: experimental design, collection and determination of samples, data reduction, and computational modeling.
Zaishen Jia and Ganzhong Tian contributed equally to this work.
Jia, Z., Tian, G., Ren, Y. et al. Pharmacokinetic model of unfractionated heparin during and after cardiopulmonary bypass in cardiac surgery. J Transl Med 13, 45 (2015) doi:10.1186/s12967-015-0404-5
Cardiopulmonary bypass
Pharmacokinetic model
Unfractionated heparin | CommonCrawl |
\begin{document}
\begin{abstract} In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(\pi x_k)$ for $N \to \infty$, where the numbers $\omega=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $\omega$, thereby improving earlier results obtained by Hlawka in 1969. Furthermore, we consider the special cases when the points $\omega$ are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues. \end{abstract}
\date{} \maketitle
\section{Introduction and statement of the results} \label{sect_1}
Let $f$ be a function $f:[0,1] \mapsto \mathbb{R}_{0}^{+}$ and $(x_{k})_{k \geq 1}$ be a sequence of numbers in the unit interval. Much work was done on analyzing so-called {\em Weyl sums} of the form $S_{N} := \sum^{N}_{k=1} f(x_{k})$, and on the convergence behavior of $\frac{1}{N} S_{N}$ to $\int^{1}_{0} f(x) \rd x$. See for example \cite{b,c,a,d}. It is the aim of this paper to propagate the analysis of corresponding ``{\em Weyl products}'' $$ P_{N} := \prod^{N}_{k=1} f(x_{k}), $$ in particular with respect to their asymptotic behavior for $N \rightarrow \infty$.
Note that, formally, studying products $P_{N}$ in fact is just a special case of studying $S_{N}$, since $$ \log P_{N} = \sum^{N}_{k=1} \log f(x_k), $$ unless $f(x)= 0$ for some $x \in [0,1]$. Thus we will concentrate on functions $f$ for which $f(0) =0$ (and possibly also $f(1) = 0)$.\\
Assuming an even distribution of the sequence $(x_{k})_{k \geq 1}$, one expects $\frac{1}{N} \sum^{N}_{k=1} \log f(x_k)$ to tend to the integral $\int^{1}_{0} \log f(x) \rd x$ if this exists. That means, very roughly, that we expect $$ \prod^{N}_{k=1} f(x_{k}) \approx \left({\rm e}^{\int^{1}_{0} \log f(x) \rd x}\right)^{N}, $$ which we can rewrite as $$ \prod^{N}_{k=1} S_{f} \, f(x_k) \approx 1, \qquad \text{where} \quad S_{f} := {\rm e}^{- \int^{1}_{0} \log f(x) \rd x}. $$ Hence it makes sense to study the asymptotic behavior of the normalized product $$ \prod^{N}_{k=1} S_{f} f(x_k) \ ~\mbox{ rather than } \ ~\prod^{N}_{k=1} f(x_k). $$ A special example of such products played an important role in \cite{A-H-L} in the context of pseudorandomness properties of the Thue--Morse sequence, where \emph{lacunary} trigonometric products of the form $$ \prod^{N}_{k=1} 2 \sin(\pi 2^{k} \alpha) $$ for $\alpha \in \mathbb{R}$ were analyzed. It was shown there that for almost all $\alpha$ and all $\varepsilon >0$ we have \begin{equation} \label{tm1}
\prod^{N}_{k=1} |2 \sin( \pi 2^{k} \alpha)| \leq \exp\left((\pi + \varepsilon ) \sqrt{N \log \log N}\right) \end{equation} for all sufficiently large $N$ and \begin{equation} \label{tm2}
\prod^{N}_{k=1} |2 \sin( \pi 2^{k} \alpha) | \geq \exp \left((\pi-\varepsilon)\sqrt{N \log \log N}\right) \end{equation} for infinitely many $N$.\\
In the present paper we restrict ourselves to $f(x) = \sin(\pi x)$ and we will extend the analysis of such products to other types of sequences $(x_k)_{k\geq1}$. In particular we will consider two well-known types of uniformly distributed sequences, namely the van der Corput sequence $(x_k)_{k \geq 1}$ and the Kronecker sequence $(\{k \alpha\})_{k \geq 1}$ with irrational $\alpha \in [0,1]$. Furthermore, we will determine the typical behavior of $$ \prod^{N}_{k=1} 2 \sin(\pi x_{k}), $$ that is, the almost sure order of this product for ``random'' sequences $\left(x_{k}\right)_{k \geq 1}$ in a suitable probabilistic model.\\
Such sine-products and estimates for such products play an important role in many different fields of mathematics. We just mention a few of them: interpolation theory (see \cite{l,m}), partition theory (see \cite{sud,wright}), Pad\'e approximation (see \cite{lub staff}), KAM theory and $q$-series (see \cite{celso,hidet,knill,knillfolk,kuznet}), analytic continuation of Dirichlet series (see \cite{Knill+Les,wagner}), and many more.\\
All our results use methods from uniform distribution theory and discrepancy theory, so we will introduce some of the basic notions from these subjects. Let $x_1, \dots, x_N$ be numbers in $[0,1]$. Their \emph{star-discrepancy} is defined as $$
D_{N}^{*}=D_N^*(x_{1}, \ldots, x_{N}) = \sup_{a\in [0,1]} \left|\frac{A_{N}(a)}{N} -a \right|, $$ where $A_{N} (a) := \# \left\{1 \leq n \leq N \ : \ x_n \in [0,a)\right\}$. An infinite sequence $(x_k)_{k \geq 1}$ in $[0,1]$ is called \emph{uniformly distributed modulo one} (u.d. mod 1) if for all $a \in [0,1]$ we have $$ \lim_{N \to \infty} \frac{A_N(a)}{N} = a, $$ or, equivalently, $$ \lim_{N \to \infty} D_N^* = 0. $$ For more basic information on uniform distribution theory and discrepancy, we refer to \cite{drmotatichy,e}.\\
Now we come to our new results. First we will give general estimates for products $\prod^{N}_{k=1} 2 \sin(\pi x_{k})$ in terms of the star-discrepancy $D_{N}^{*}$ of $(x_k)_{1 \leq k \leq N}$. A similar result in a weaker form was obtained by Hlawka \cite{l} (see also \cite{m}).
\begin{theorem} \label{th_kh} Let $\left(x_{k}\right)_{k \geq 1}$ be a sequence of real numbers from $[0,1]$ which is u.d. mod 1. Then for all sufficiently large $N$ we have \begin{equation} \label{equ_a} \prod^{N}_{k=1} 2 \sin (\pi x_{k}) \leq \left(\frac{N}{\Delta_{N}}\right)^{2 \Delta_{N}}, \end{equation} where $\Delta_{N} := ND_{N}^{*}$. \end{theorem}
Concerning the quality of Theorem~\ref{th_kh}, consider the case when $(x_{k})_{k \geq 1}$ is a low-discrepancy sequence such as the van der Corput sequence (which is treated in Theorem~\ref{th_a} below). Then $\Delta_{N} = \mathcal{O}\left(\log N\right)$, and Theorem~\ref{th_kh} gives \begin{equation} \label{equ_b} \prod^{N}_{k=1} 2 \sin (\pi x_{k}) \leq N^{\gamma \log N} \end{equation} for some $\gamma \in \mathbb{R}^+$ and all sufficiently large $N$. Stronger asymptotic bounds are provided by Theorem~\ref{th_a} below; thus, Theorem \ref{th_kh} does not provide a sharp upper bound in this case.\\
As another example, let $x_k=k/(N+1)$ for $k=1,2,\ldots,N$. This point set has star-discrepancy $D_N^*=1/(N+1)$, and hence the general estimate \eqref{equ_a} gives \begin{equation} \label{lhp} \prod^{N}_{k=1} 2 \sin\left(\pi \frac{k}{N+1}\right) \leq (N+1)^2. \end{equation} On the other hand, the product on the left-hand side of \eqref{lhp} is well known to be exactly $N+1$ (see also Lemma~\ref{lem_c_fritz} below). Thus, the general estimate from Theorem \ref{th_kh} has an additional factor $N$ in comparison with the correct order in this case, which is quite close to optimality.\\
As already mentioned above, Hlawka~\cite{l,m} studied similar questions in connection with interpolation of analytic functions on the complex unit disc. There he considered products of the form $$ \omega_{N}(z)= \prod^{N}_{k=1}(z-\xi_{k})^{2}, $$ where $\xi_{k}$ are points on the unit circle. The main results in
\cite{l,m} are lower and upper bounds of $|\omega_{N}(z)|$ in terms of the star-discrepancy $D_{N}^*$ of the sequence $(\arg\frac{1}{2\pi}\xi_{k}), k=1,\ldots , N.$\footnote{The second paper was published in a seminar proceedings volume called ``Zahlentheoretische Analysis''. Hlawka introduced this term for applications of number-theoretic methods in real or complex analysis. In particular, he often applied uniformly distributed sequences to give discrete versions of continuous models.} It should also be mentioned that Wagner \cite{wagner} proved the general lower bound
$$\sup_{| z|= 1}|\omega_{N}(z)|\geq(\log N)^{c}$$ for infinitely $N$, where $c> 0$ is some explicitly given constant. This solved a problem stated by Erd\H{o}s.\\
In the sequel we will give a second, essentially optimal theorem which estimates products $\prod^{N}_{k=1} 2 \sin (\pi x_{k})$ in terms of the star-discrepancy of the sequence $(x_{k})_{k \geq 1}$. Let $\omega=\left\{x_{1}, \ldots, x_{N}\right\}$ be numbers in $[0,1]$ and let $P_N(\omega)=\prod^{N}_{k=1} 2 \sin(\pi x_k)$. Let $D_{N}^{*} (\omega)$ denote the star-discrepancy of $\omega$. Furthermore, let $d_N$ be a real number from the interval $[1/(2N),1]$, which is the possible range of the star-discrepancy of $N$-element point sets. We are interested in $$ P_N^{(d_{N})} := \sup_{\omega} P_N(\omega)= \sup_{\omega} \prod^{N}_{k=1}2 \sin(\pi x_k), $$ where the supremum is taken over all $\omega$ with $D_{N}^{*} (\omega) \leq d_{N}$. We will show \begin{theorem} \label{th_a_fritz} Let $\left(d_{N}\right)_{N \geq 1}$ be an arbitrary sequence of reals satisfying $1/(2N) \le d_N \le 1,~N \geq 1,$ and $\lim_{N \rightarrow \infty} d_N = 0$. Then we have: \begin{itemize} \item [a)] For all $\varepsilon > 0$ there exist $c(\varepsilon)$ and $N(\varepsilon)$ such that for all $N > N(\varepsilon)$ we have $$ P_N^{(d_{N})} \leq c(\varepsilon) \frac{1}{N} \left(\left(\frac{{\rm e}}{\pi}+\varepsilon\right) \frac{1}{d_{N}}\right)^{2N d_{N}}. $$ \item [b)] For all sufficiently large $N$ we have $$ P_N^{(d_{N})} \geq \frac{2 \pi^2}{{\rm e}^6} \frac{1}{N} \left(\frac{{\rm e}}{\pi} \frac{1}{d_N}\right)^{2 N d_N}. $$ \end{itemize} \end{theorem}
To check the quality of Theorem \ref{th_a_fritz}, consider the case $d_N=1/(N+1)$ which includes the point sets $x_k=k/(N+1)$ for $k=1,2,\ldots,N$ mentioned before. Then the upper estimate in Theorem~\ref{th_a_fritz} gives the correct order of magnitude $P_N =\mathcal{O}(N)$.\\
Let us now focus on products of the form $$ \prod^{N}_{n=1} 2 \sin (\pi \{n \alpha\}) =\prod^{N}_{n=1} 2 \sin (\pi n \alpha), $$ where $\alpha$ is a given irrational number, i.e., we consider the special case when $(x_n)_{n \geq 1}$ is the Kronecker sequence $(\{n \alpha \})_{n \geq 1}$. Such products play an essential role in many fields and are the best studied such Weyl products in the literature. See for example \cite{h,f,j,i,Knill+Les,Lub,g,VerMes}. Before discussing these products in detail, let us recall some historical facts. By Kronecker's approximation theorem, the sequence $(n\alpha)_{n \ge 1}$ is everywhere dense modulo 1; i.e., the sequence of fractional parts $(\{n\alpha\})_{n \geq 1}$ is dense in $[0,1]$. At the beginning of the 20th century various authors considered this sequence (and generalizations such as $(\{\alpha n^{d}\})_{n \geq 1}$, etc.) from different points of view; see for instance Bohl \cite{bohl}, Weyl \cite{weyl2} and Sierpi\'nksi \cite{sierp}. An important impetus came from celestial mechanics. It was Hermann Weyl in his seminal paper \cite{weyl} who opened new and much more general features of this subject by introducing the concept of uniform distribution for arbitrary sequences $(x_{k})_{k\geq 1}$ in the unit interval (as well as in the unit cube $[0,1]^{s}$). This paper heavily influenced the development of uniform distribution theory, discrepancy theory and the theory of quasi-Monte Carlo integration throughout the last 100 years. For the early history of the subject we refer to Hlawka and Binder \cite{hla2}.\\
Numerical experiments suggest that for integers $N$ with $q_{l} \leq N < q_{l+1}$, where $\left(q_{l}\right)_{l \geq 0}$ is the sequence of best approximation denominators of $\alpha$, \begin{equation} \label{equ_stern_fritz} \text{the product attains its maximal value for}~N = q_{l+1}-1. \end{equation} Moreover we conjecture that always \begin{equation} \label{equ_6neu_fritz}
\limsup_{q \rightarrow \infty} \frac{1}{q} \prod^{q-1}_{n=1} |2 \sin (\pi n \alpha) | < \infty. \end{equation}
Compare these considerations also with the conjectures stated in \cite{Lub}. To illustrate these two assertions see Figures~\ref{fig_b} and \ref{fig_c}, where for $\alpha = \sqrt{2}$ we plot $\prod^{N}_{n=1} |2 \sin (\pi n \alpha)|$ for $N=1,\ldots, 500$ (Figure~\ref{fig_b}) and the normalized version $\tfrac{1}{N} \prod^{N}_{n=1} |2 \sin (\pi n \alpha) |$ for $N=1, \ldots, 500$ (Figure~\ref{fig_c}). Note that the first best approximation denominators of $\sqrt{2}$ are given by $1,2,5,12,29,70,169,408,\ldots.$\\ \begin{figure}
\caption{$\prod^{N}_{n=1} \left|2 \sin (\pi n \alpha) \right|$ for $N=1,\ldots, 500$ and $\alpha=\sqrt{2}$ }
\label{fig_b}
\end{figure}
\begin{figure}
\caption{$\tfrac{1}{N}\prod^{N}_{n=1} \left|2 \sin (\pi n \alpha) \right|$ for $N=1,\ldots, 500$ and $\alpha=\sqrt{2}$ }
\label{fig_c}
\end{figure}
For the case $N= q-1$ for some best approximation denominator $q$ the product $\prod^{q-1}_{n=1} |2 \sin (\pi n \alpha)|$ already was considered in \cite{f,g}. In particular, it was shown there that \begin{equation} \label{equ_delta_fritz}
\lim_{q \rightarrow \infty}\log \prod^{q-1}_{n=1} |2 \sin (\pi n \alpha)| = \lim_{q \rightarrow \infty} \frac{1}{q} \sum^{q-1}_{n=1} \log|2 \sin (\pi n \alpha)| = 0, \end{equation} when $q$ runs through the sequence of best approximation denominators. Indeed, we are neither able to prove assertion \eqref{equ_stern_fritz} nor assertion \eqref{equ_6neu_fritz}. Nevertheless we want to give a quantitative estimate for the case $N=q-1$, i.e., also a quantitative version of \eqref{equ_delta_fritz}, before we will deal with the general case.
\begin{theorem} \label{th_proda} Let $q$ be a best approximation denominator for $\alpha$. Then $$
1 \leq \prod^{q-1}_{n=1} |2 \sin (\pi n \alpha)| \leq \frac{q^{2}}{2}. $$ \end{theorem}
Next we consider general $N\in \mathbb{N}$:
\begin{theorem} \label{th_prodb} Let $\alpha := [0; a_1, a_2,a_3, \ldots]$ be the continued fraction expansion of the irrational number $\alpha \in [0,1]$. Let $N \in \mathbb{N}$ be given, and denote its Ostrowski expansion by $$ N = b_{l}q_{l} + b_{l-1} q_{l-1} + \cdots + b_{1}q_{1} + b_{0} $$ where $l=l(N)$ is the unique integer such that $q_l \le N < q_{l+1}$, where $b_{i} \in \{0,1,\ldots,a_{i+1}\}$, and where $q_1,q_2,\ldots$ are the best approximation denominators for $\alpha$. Then we have $$
\prod^{N}_{n=1} \left|2 \sin (\pi n \alpha) \right| \leq \prod^{l}_{i=0} 2^{b_{i}} q^{3}_{i}. $$ \iffalse Furthermore, \begin{align*}
\lefteqn{\frac{1}{N} \sum^{N}_{n=1} \log |2 \sin (\pi n \alpha)|}\\ & \leq (\log 2) \left(\frac{a_1+a_2+\cdots+a_{l+1}}{N}\right) + 3 \frac{\log N}{N} \left(\frac{\log N}{\log \phi}+1\right) \end{align*} \fi \end{theorem}
\begin{corollary} \label{co_prodb} For all $N$ with $q_{l} \leq N < q_{l+1}$ we have $$
\frac{1}{N} \sum^{N}_{n=1} \log |2 \sin (\pi n \alpha)| \leq (\log 2) \left(\frac{1}{q_{l}} + \frac{l}{2^{(l-3)/2}}\right) + 3 \, \frac{\log q_{l}}{q_{l}} \left(\frac{\log q_l}{\log \phi}+1 \right), $$ where $\phi=(1+\sqrt{5})/2$ and hence $$
\limsup_{N \rightarrow \infty} \frac{1}{N} \sum^{N}_{n=1} \log |2 \sin (\pi n \alpha)| = 0 =\int_0^1 \log(2 \sin(\pi x)) \rd x . $$ \end{corollary}
The second part of Corollary~\ref{co_prodb} can also be obtained from \cite[Lemma~4]{h}.\\
In the following we say that a real $\alpha$ is of type $ t \ge 1$ if there is a constant $c > 0$ such that $$\left|\alpha - \frac{p}{q}\right| > c \frac{1}{q^{1+t}}$$ for all $p,q \in \mathbb{Z}$ with $\gcd(p,q)=1$.
The next result essentially improves a result given in \cite{Knill+Les}. There a bound on $\prod^{N}_{n=1} \left|2 \sin \left(\pi n \alpha \right) \right|$ for $\alpha$ of type $t$ of the form $N^{c N^{1-1/t} \log N}$ instead of our much sharper bound $2^{C N^{1-1/t}}$ was given. Note that our result only holds for $t > 1$, so we cannot obtain the sharp result of Lubinsky \cite{Lub} in the case of $\alpha$ with bounded continued fraction coefficients.
\begin{corollary} \label{co_b}
Assume that $\alpha$ is of type $ t > 1$. Then for some constant $C$ and all $N$ large enough $\prod^{N}_{n=1} \left|2 \sin \left(\pi n \alpha \right) \right|\leq 2^{C N^{1-1/t}}$. \end{corollary}
Now we will deal with $\prod^{N}_{n=1} |2 \sin (\pi x_n)|$, where $(x_n)_{n \geq 1}$ is the van der Corput-sequence. The van der Corput sequence (in base 2) is defined as follows: for $n \in \mathbb{N}$ with binary expansion $n=a_0+a_1 2+a_2 2^3+\cdots$ with digits $a_0,a_1,a_2,\ldots \in \{0,1\}$ (of course the expansion is finite) the $n^{{\rm th}}$ element is given as $$x_n=\frac{a_0}{2}+\frac{a_1}{2^2}+\frac{a_2}{2^3}+\cdots$$ (see the recent survey \cite{FKP} for detailed information about the van der Corput sequence). For this sequence, in contrast to the Kronecker sequence, we can give very precise results. We show:
\begin{theorem} \label{th_a} Let $(x_{n})_{n \geq 1}$ be the van der Corput sequence in base 2. Then $$
\limsup_{N \rightarrow \infty} \frac{1}{N^{2}} \prod^{N}_{n=1} |2 \sin (\pi x_{n})| = \frac{1}{2 \pi} $$ and $$
\liminf_{N \rightarrow \infty} \prod^{N}_{n=1} |2 \sin (\pi x_{n})| = \pi. $$ \end{theorem}
Finally, we study probabilistic analogues of Weyl products, in order to be able to quantify the typical order of such products for ``random'' sequences and to have a basis for comparison for the results obtained for deterministic sequences in Theorems \ref{th_proda}, \ref{th_prodb} and \ref{th_a}. We will consider two probabilistic models. First we study $$\prod^{N}_{k=1}2\sin(\pi X_{k}),$$ where $(X_{k})_{k \ge 1}$ is a sequence of independent, identically distributed (i.i.d.) random variables in $[0,1]$. The second probabilistic model are random subsequences $(n_{k}\alpha)_{k \ge 1}$ of the Kronecker sequences $(n\alpha)$, where the elements of $n_{k}$ are selected from $\mathbb{N}$ independently and with probability $\frac{1}{2}$ for each number. This model is frequently used in the theory of random series (see for example the monograph of Kahane \cite{kah}) and was introduced to the theory of uniform distribution by Petersen and McGregor \cite{peter} and later extensively studied by Tichy \cite{tichy}, Losert \cite{losert}, and Losert and Tichy \cite{lt}.
\begin{theorem} \label{th51} Let $(X_{k})_{k \ge 1}$ be a sequence of i.i.d random variables having uniform distribution on $[0,1]$, and let $$P_{N}= \prod^{N}_{k=1}2\sin (\pi X_{k}).$$ Then for all $\varepsilon > 0$ we have, almost surely, $$ P_N \leq \exp\left(\left(\frac{\pi}{\sqrt{6}} + \varepsilon\right) \sqrt{N \log \log N}\right) $$ for all sufficiently large $N$, and $$ P_N \geq \exp \left(\left(\frac{\pi}{\sqrt{6}}-\varepsilon \right) \sqrt{N \log \log N}\right) $$ for infinitely many $N$.\\
\end{theorem}
\begin{theorem} \label{th52} Let $\alpha$ be an irrational number with bounded continued fraction coefficients. Let $(\xi_n)_{n \geq 1}=(\xi_n(\omega))_{n \geq 1}$ be a sequence of i.i.d.\ $\{0,1\}$-valued random variables with mean $1/2$, defined on some probability space $(\Omega,\mathcal{A},\mathbb{P})$, which induce a random sequence $(n_k)_{k \geq 1}=(n_k(\omega))_{k \geq 1}$ as the sequence of all numbers $\left\{n \geq 1:~ \xi_n = 1 \right\}$, sorted in increasing order. Set $$ P_N = \prod^{N}_{k=1}2\sin (\pi n_k \alpha). $$ Then for all $\ve > 0$ we have, $\mathbb{P}$-almost surely, $$ P_N \leq \exp\left(\left(\frac{\pi}{\sqrt{12}} + \varepsilon\right) \sqrt{N \log \log N}\right) $$ for all sufficiently large $N$, and $$ P_N \geq \exp \left(\left(\frac{\pi}{\sqrt{12}}-\varepsilon \right) \sqrt{N \log \log N}\right) $$ for infinitely many $N$ \end{theorem}
\begin{remark}\rm The conclusion of Theorem \ref{th52} remains valid if $\alpha$ is only assumed to be of finite approximation type (see \cite[Chapter 2, Section 3]{e} for details on this notion). \end{remark}
\begin{remark}\rm It is interesting to compare the conclusions of Theorems \ref{th51} (for purely random sequences) and \ref{th52} (for randomized subsequences of linear sequences) to the results in equations \eqref{tm1} and \eqref{tm2}, which hold for lacunary trigonometric products. The results coincide almost excactly, except for the constants in the exponential term (which can be seen as the standard deviations in a related random system; see the proofs). The larger constant in the lacunary setting comes from an interference phenomenon, which appears frequently in the theory of lacunary functions systems (see for example Kac \cite{kac} and Maruyama \cite{maruyama}). On the other hand, the smaller constant in Theorem \ref{th52} represents a ``loss of mass'' phenomenon, which can be observed in the theory of slowly growing (randomized) trigonometric systems; it appears in a very similar form for example in Berkes \cite{berkes} and Bobkov--G\"otze \cite{bobg}. \end{remark}
The outline of the remaining part of this paper is as follows. In Section \ref{sect_b_end} we will prove Theorems \ref{th_kh} and \ref{th_a_fritz}, which give estimates of Weyl products in terms of the discrepancy of the numbers $(x_k)_{1 \leq k \leq N}$. In Section \ref{sect_c} we prove the results for Kronecker sequences (Theorems \ref{th_proda} and \ref{th_prodb}), and in Section \ref{sect_d} the results for the van der Corput sequence (Theorem \ref{th_a}). Finally, in Section \ref{sect_prob} we prove the results about probabilistic sequences (Theorems \ref{th51} and \ref{th52}).
\section{Proofs of Theorems \ref{th_kh} and \ref{th_a_fritz}} \label{sect_b_end}
\begin{proof}[Proof of Theorem \ref{th_kh}] The Koksma-Hlawka-inequality (see e.g. \cite{e}) states that for any function $g:[0,1] \rightarrow \mathbb{R}$ of bounded variation $V(g)$, any $N$ and numbers $x_1, \dots, x_N \in [0,1]$ we have $$
\left|\int^{1}_{0} g(x) \rd x - \frac{1}{N} \sum^{N}_{k=1} g\left(x_{k}\right)\right| \leq V(g) \, D^{*}_{N} (x_1, \dots, x_N), $$ where $D_{N}^{*}$ is the star-discrepancy of $x_{1}, \ldots, x_{N}$. Let $P_{N} := \prod^{N}_{k=1} 2 \sin( \pi x_{k})$ and $$\Sigma_{N} := \log P_{N} = N \log 2 + \sum^{N}_{k=1} \log \sin(\pi x_{k}).$$ For $0 < \varepsilon < \frac{1}{2}$ let \begin{align*} f_{\varepsilon} (x) := \left\{ \begin{array}{ll}
\log \sin (\pi \varepsilon) & \mbox{if}~\left\|x\right\| \leq \varepsilon \\ \log \sin (\pi x) & \mbox{otherwise.}\\ \end{array} \right. \end{align*} Note, that $\int^{1}_{0} \log \sin (\pi x) \rd x = - \log 2$, hence \begin{align*} \int^{1}_{0} f_{\varepsilon} (x) ~\rd x = & 2 \varepsilon \log \sin (\pi \varepsilon) + \int^{1}_{0} \log \sin (\pi x) \rd x - 2 \int^{\varepsilon}_{0} \log \sin (\pi x) \rd x \\
= & 2 \varepsilon \log \sin (\pi \varepsilon) - \log 2 - 2 \int^{\varepsilon}_{0} \log \sin (\pi x) \rd x. \end{align*} By partial integration we obtain \begin{align*} \int^{\varepsilon}_{0} \log \sin (\pi x) \rd x = & \varepsilon \log \sin (\pi \varepsilon) - \int^{\varepsilon}_{0} x \pi \cot (\pi x) \rd x \\
= & \varepsilon \log \sin (\pi \varepsilon) - \varepsilon - \mathcal{O}(\varepsilon^{3}) \end{align*} (with a positive $\mathcal{O}$-constant for $\varepsilon$ small enough). Furthermore, we have \begin{equation*}
V(f_{\varepsilon}) = \int_0^1 |f_{\varepsilon}'(x)| \rd x = 2 \pi \int_{\varepsilon}^{1/2} \cot(\pi x) \rd x = -2 \log \sin(\pi \varepsilon). \end{equation*}
Altogether we have, using the Koksma-Hlawka inequality and since $\log \sin (\pi \varepsilon) = \log (\pi \varepsilon) - \frac{\pi \varepsilon^{2}}{6} - \mathcal{O}(\varepsilon^{4}),$ \begin{align*} \Sigma_{N} \leq & N \log 2 + \sum^{N}_{k=1} f_{\varepsilon} \left(x_{k}\right)\\
\leq & N \log 2 + N \int^{1}_{0} f_{\varepsilon} (x) \rd x + N D_{N}^{*} V(f_{\varepsilon})\\
=& N\left(2 \varepsilon \log \sin \varepsilon - 2 \int^{\varepsilon}_{0} \log \sin (\pi x) \rd x \right) - 2 N D_{N}^{*} \log \sin (\pi \varepsilon) \\
= & 2N \int^{\varepsilon}_{0} x \pi \cot (\pi x) \rd x - 2ND_{N}^{*} \log \sin (\pi \varepsilon) \\
= & 2 N \varepsilon + N \mathcal{O}(\varepsilon^{3}) + 2 N D_{N}^{*} \, (-\log (\pi \varepsilon) + \mathcal{O} (\varepsilon^{2})) \\
= & 2 N \varepsilon - 2 N D_{N}^{*} \log \pi \varepsilon + N \mathcal{O}(\varepsilon^{2}). \end{align*} Hence $$ P_{N}={\rm e}^{\Sigma_N} \leq {\rm e}^{2 N \varepsilon} \left(\frac{1}{\pi \varepsilon}\right)^{2 N D_{N}^{*}} {\rm e}^{c \varepsilon^{2} N} $$ for some constant $c>0$. We choose $\varepsilon = D_{N}^{*}$ and obtain $$ P_{N} \leq \left(c' \, \frac{N}{N D_{N}^{*}}\right)^{2 ND_{N}^{*}} $$ For some $c'>0$. Note that $c'$ can be chosen such that $c' < 1$ if $\varepsilon = D_{N}^{*} =o(1)$ for $N \rightarrow \infty$. \end{proof}
Next we come to the proof of Theorem \ref{th_a_fritz}. We will need several auxiliary lemmas, before proving the theorem.
\begin{lemma} \label{lem_a_fritz} Let $D \in \mathbb{Q}$, let $N$ be even such that $N D =: M$ for some integer $M$. Then the $N$-element point set $\widetilde{\omega}$ consisting of the points $$\frac{M}{N}, \frac{M+1}{N}, \ldots, \frac{\frac{N}{2}-1}{N}, \frac{\frac{N}{2}+1}{N}, \frac{\frac{N}{2}+2}{N}, \ldots, \frac{N-M}{N}$$ together with $2 M$ times the point $\frac{1}{2}$ has star-discrepancy \begin{equation*} D_{N}^{*} (\omega) = D. \end{equation*} If any of these points is moved nearer to $1/2$, then the star-discrepancy of the new point set is larger than $D$. Furthermore, $\widetilde{\omega}$ is the only sequence with these two properties. \end{lemma}
\begin{proof} See Figure~\ref{bd78}, where the discrepancy function $a \mapsto \frac{A_{N}(a)}{N} -a$ for $a \in [0,1]$ of $\widetilde{\omega}$ is plotted. \begin{figure}\label{bd78}
\end{figure} \end{proof}
\begin{lemma} \label{lem_b_fritz} For $\widetilde{\omega}$ as in Lemma~\ref{lem_a_fritz} we have $P_N^{(d_{N})} = P_N(\widetilde{\omega})$. \end{lemma}
\begin{proof} Let $x_{1}, \ldots, x_{N}$ be any $N$-element point set in $[0,1]$. If one of these points is moved nearer to $1/2$ then this move increases the value $\prod^{N}_{k=1} 2 \sin (\pi x_{k})$. Hence the result immediately follows from Lemma~\ref{lem_a_fritz}. \end{proof}
\begin{lemma} \label{lem_c_fritz} For all $N\in \mathbb{N}$ and all $x\in [0,1]$ we have \begin{enumerate}[(i)] \item \quad $\prod^{N-1}_{k=1} 2 \sin(\pi k/N) = N$, and \item \quad $\prod^{N-1}_{k=0} 2 \sin(\pi (k+x)/N) = 2 \sin(\pi x).$ \end{enumerate} \end{lemma}
\begin{proof} Equation (i) is well known. A nice proof can be found for example in \cite{mu}. Equation (ii) is \cite[Formula 1.392]{GR}. \end{proof}
\begin{lemma} \label{lem_d_fritz} There is an $\varepsilon_{0} > 0$ such that for all $\varepsilon < \varepsilon_{0}$ we have $$ \varepsilon \log (\pi \varepsilon) - \varepsilon - \varepsilon^{2} \leq \int^{\varepsilon}_{0} \log \sin(\pi x) \rd x \leq \varepsilon \log (\pi \varepsilon) - \varepsilon. $$ \end{lemma}
\begin{proof} This follows immediately from the Taylor expansion $$ \int^{\varepsilon}_{0} \log \sin(\pi x) \rd x - \varepsilon \log (\pi \varepsilon) = - \varepsilon - \frac{\pi^{2}}{18} \varepsilon^{3} + \mathcal{O}(\varepsilon^{5}). $$ \end{proof}
\begin{lemma} \label{lem_e_fritz} There is an $\varepsilon_{0} > 0$ such that for all $\varepsilon < \varepsilon_{0}$ we have $$ \log (\pi \varepsilon) - \varepsilon \leq \log \sin (\pi \varepsilon) \leq \log (\pi \varepsilon). $$ \end{lemma}
\begin{proof} This follows from $$ \log \sin (\pi x) - \log (\pi x) = - \frac{\pi^{2}x^{2}}{6} + \mathcal{O}(x^{4}). $$ \end{proof}
\begin{proof}[Proof of Theorem~\ref{th_a_fritz}] Let $N d_N = M$ with $M \geq 2$ (for $M=1$ the result is easily checked by following the considerations below) and $\widetilde{\omega}$ as in Lemmas~\ref{lem_a_fritz} and \ref{lem_b_fritz}. Note that $M=M(N)$ depends on $N$. We have, using also equation (i) of Lemma~\ref{lem_c_fritz}, \begin{align*} P_N(\widetilde{\omega})
= & \left(\prod^{N-1}_{k=1} 2 \sin \left(\pi \frac{k}{N}\right)\right) \ 2^{2 M-1} \ \left(\prod^{M-1}_{k=1} 2 \sin \left(\pi \frac{k}{N}\right)\right)^{-2}\\
= & 2 N\left(\prod^{M-1}_{k=1} \sin \left(\pi \frac{k}{N}\right)\right)^{-2}. \end{align*} Note that the function $x \mapsto \log \sin (\pi x)$ is of the form as presented in Figure~\ref{fig_a}. \begin{figure}
\caption{The function $\log \sin (\pi x)$}
\label{fig_a}
\end{figure} Hence for $M < \frac{N}{2}$ we have \begin{align*} \log \sin \left(\frac{\pi}{N}\right) + & N \int^{\frac{M-1}{N}}_{\frac{1}{N}} \log \sin (\pi x) \rd x\\
\leq & \sum^{M-1}_{k=1} \log \sin \left(\pi \frac{k}{N}\right)\\
\leq & N \int^{\frac{M-1}{N}}_{\frac{1}{N}} \log \sin (\pi x) \rd x + \log \sin \left(\pi \frac{M-1}{N}\right). \end{align*} By Lemma~\ref{lem_d_fritz} for all $M$ with $\frac{M}{N} < \varepsilon_{0}$ for the integral above we have \begin{align*} \lefteqn{N \int^{\frac{M-1}{N}}_{\frac{1}{N}} \log \sin (\pi x) \rd x}\\ &\leq \left(M-1\right) \log \left(\pi \frac{M-1}{N}\right) - (M-1) - \log \left(\frac{\pi}{N}\right) + 1 + \frac{1}{N}, \end{align*} and hence, using also Lemma~\ref{lem_e_fritz}, \begin{align*} \sum^{M-1}_{k=1} \log \sin \left(\pi \frac{k}{N} \right) \leq & (M-1) \log \left(\frac{\pi}{{\rm e}} \frac{M-1}{N}\right) - \log \pi +\log N + 1\\ & + \frac{1}{N} + \log (M-1) - \log N + \log \pi \\
\le & (M-1) \log \left(\frac{\pi}{{\rm e}} \frac{M-1}{N}\right) + \log (M-1) + 2. \end{align*} This gives $$ \left(\prod^{M-1}_{k=1} \sin \left(\pi \frac{k}{N}\right)\right)^{2} = {\rm e}^{2 \sum^{M-1}_{k=1} \log \sin (\pi k/N)} \leq {\rm e}^{4} (M-1)^{2} \left(\frac{\pi}{{\rm e}} \frac{M-1}{N}\right)^{2 (M-1)}, $$ and consequently \begin{align*} P_N(\widetilde{\omega}) \geq & 2N \frac{1}{{\rm e}^{4}} \frac{1}{(M-1)^2} \left(\frac{{\rm e}}{\pi} \frac{N}{M-1}\right)^{2(M-1)} = \frac{2 \pi^2}{{\rm e}^6} \frac{1}{N} \left(\frac{{\rm e}}{\pi} \frac{N}{M-1}\right)^{2 M}\nonumber\\
\ge & \frac{2 \pi^2}{{\rm e}^6} \frac{1}{N} \left(\frac{{\rm e}}{\pi} \frac{1}{d_N}\right)^{2 N d_N}. \end{align*} This proves assertion b) of Theorem~\ref{th_a_fritz}.\\
On the other hand we have \begin{align*} \lefteqn{N \int^{\frac{M-1}{N}}_{\frac{1}{N}} \log \sin (\pi x) \rd x}\\ &\geq (M-1) \log \left(\pi \frac{M-1}{N}\right) -(M-1) - \frac{(M-1)^{2}}{N} - \log \left(\frac{\pi}{N}\right)+1, \end{align*} and hence \begin{align*} \sum^{M-1}_{k=1} \log \sin \left(\pi \frac{k}{N}\right) \geq & (M-1) \log \left(\frac{\pi}{{\rm e}} \frac{M-1}{N}\right) \\ &- \frac{(M-1)^{2}}{N} - \log \pi + \log N + 1 + \log \pi - \log N - \frac{1}{N}\\
= & (M-1) \log \left(\frac{\pi}{{\rm e}} \frac{M-1}{N}\right) + 1-\frac{1}{N} - \frac{(M-1)^{2}}{N}. \end{align*} This gives $$ \left(\prod^{M-1}_{k=1} \sin \left(\pi \frac{k}{N}\right)\right)^{2} = {\rm e}^{2 \sum^{M-1}_{k=1} \log \sin (\pi k/N)} \geq \frac{1}{{\rm e}^{\frac{2 (M-1)^{2}}{N}}} \left(\frac{\pi}{{\rm e}} \frac{M-1}{N}\right)^{2(M-1)}, $$ and consequently \begin{equation} \label{equ_d_fritz} P_N \left(\widetilde{\omega}\right) \leq 2 N {\rm e}^{2 \frac{(M-1)^{2}}{N}} \left(\frac{{\rm e}}{\pi} \frac{N}{M-1}\right)^{2 (M-1)}. \end{equation}
It remains to show that for all $\varepsilon > 0$ there are $c(\varepsilon)$ and $N(\varepsilon)$ such that for all $N \geq N(\varepsilon)$ the right hand side of \eqref{equ_d_fritz} is at most $c(\varepsilon) \frac{1}{N} ((\tfrac{{\rm e}}{\pi} + \varepsilon ) \frac{N}{M})^{2 M}$.
To this end let $B(\varepsilon)$ be large enough such that for all $M > B(\varepsilon)$ we have $(M-1)^{1/M} \frac{M}{M-1} < 1 + \frac{\pi}{2 {\rm e}} \varepsilon$. Furthermore, let $N(\varepsilon)$ be large enough such that for all $N \geq N(\varepsilon)$ the value $\frac{M}{N} = d_{N}$ is so small such that $${\rm e}^{\frac{M-1}{N}} < \frac{1+\frac{\pi}{{\rm e}} \varepsilon}{1 + \frac{\pi}{2 {\rm e}} \varepsilon}.$$ Then for all $M > B(\varepsilon)$ and all $N > N(\varepsilon)$ we have \begin{align*} \lefteqn{2 N {\rm e}^{2 \frac{\left(M-1\right)^{2}}{N}} \left(\frac{{\rm e}}{\pi} \frac{N}{M-1}\right)^{2 (M-1)}}\\ & \le \frac{2 \pi^2}{{\rm e}^2} \frac{\left(M-1\right)^{2}}{N} \left(\frac{{\rm e}}{\pi} \, {\rm e}^{\frac{M-1}{N}} \frac{M}{M-1} \frac{N}{M} \right)^{2 M}\\ & = \frac{2 \pi^2}{{\rm e}^2 N} \left(\frac{{\rm e}}{\pi} \, {\rm e}^{\frac{M-1}{N}} \, (M-1)^{\frac{1}{M}} \, \frac{M}{M-1} \, \frac{N}{M}\right)^{2 M}\\ & \leq \frac{2 \pi^2}{{\rm e}^2 N} \left(\left(\frac{{\rm e}}{\pi} + \varepsilon \right) \frac{N}{M}\right)^{2 M}. \end{align*} If $M \leq B(\varepsilon)$, then the penultimate expression can be estimated by \begin{align*} \lefteqn{\frac{2 \pi^2}{{\rm e}^2N} \left(\frac{{\rm e}}{\pi} \, {\rm e}^{\frac{M-1}{N}} \, (M-1)^{\frac{1}{M}} \, \frac{M}{M-1} \, \frac{N}{M}\right)^{2 M}}\\ & \leq \left(\max_{M\leq B(\varepsilon)}\left(\frac{2 \pi^2}{{\rm e}^2} \, {\rm e}^{2 M (M-1)} (M-1)^2 \left(\frac{M}{M-1}\right)^{2 M}\right)\right) \frac{1}{N} \left(\frac{{\rm e}}{\pi} \frac{N}{M}\right)^{2 M}\\ &= c(\varepsilon) \frac{1}{N} \left(\frac{{\rm e}}{\pi} \frac{N}{M}\right)^{2 M}, \end{align*} where $$c(\varepsilon):=\max_{M\leq B(\varepsilon)}\left(\frac{2 \pi^2}{{\rm e}^2} \, {\rm e}^{2 M \left(M-1\right)} \left(M-1\right)^{2} \left(\frac{M}{M-1}\right)^{2 M}\right).$$ This implies the desired result. \end{proof}
\section{Proofs of the results for Kronecker sequences} \label{sect_c}
\begin{proof}[Proof of Theorem~\ref{th_proda}] Let $\alpha = \frac{p}{q} + \theta$ with $0 < \theta < \frac{1}{q \cdot q^{+}}$, where $q^{+}$ is the best approximation denominator following $q$. The case of negative $\theta$ can be handled quite analogously. There is exactly one of the points $\{k \alpha\}$ for $k=1, \dots, q-1$ in each interval $[\frac{m}{q}, \frac{m+1}{q})$ for $m=1, \ldots, q-1$. Note that the point in the interval $[\frac{q-1}{q}, 1)$ is the point $\left\{q^{-} \alpha\right\}$, where $q^{-}$ is the best approximation denominator preceding $q$. We have $$ \left\{q^{-} \alpha \right\} = \frac{q-1}{q} + q^{-} \theta \leq \frac{q-1}{q} + \frac{q^{-}}{q \cdot q^{+}} < \frac{q-1}{q} + \frac{1}{2 q} = \frac{q- \frac{1}{2}}{q}. $$ Hence, on the one hand (by equation (i) of Lemma~\ref{lem_c_fritz}), $$
\prod^{q-1}_{n=1} |2 \sin( \pi n \alpha)| \leq \left(\prod^{q-1}_{n=2} 2 \sin\left( \pi \frac{n}{q} \right)\right) \, 2 \sin\frac{\pi}{2} = \frac{2 q}{2 \sin (\pi/q)} \leq \frac{q^2}{2}. $$ On the other hand \begin{align*}
\prod^{q-1}_{n=1} |2 \sin (\pi n \alpha)| \geq & \left(\prod^{q-1}_{n=1} 2 \sin \left(\pi \frac{n}{q}\right)\right) \ \frac{1}{2 \sin(\pi \tfrac{\lfloor q/2\rfloor}{q})} \ 2 \sin (\pi q^{-} \alpha) \\ \geq & q \, \sin\left(\pi \frac{q-1/2}{q}\right) = q \, \sin \frac{\pi}{2 q} \geq 1. \end{align*} \end{proof}
\begin{proof}[Proof of Theorem~\ref{th_prodb}] Let $N_{i} := b_{l}q_{l} + b_{l-1}q_{l-1} + \cdots + b_{i+1} q_{i+1}$ for $i=0, \ldots, l-1$ and $N_{l} := 0$. Then $$
\prod^{N}_{n=1} \left|2 \sin( \pi n \alpha) \right| = \prod^{l}_{i=0} \prod^{N_{i} + b_{i}q_{i}}_{n=N_{i}+1} \left|2 \sin (\pi n \alpha) \right|. $$ We consider $$
\Pi_{i} := \prod^{N_{i}+b_{i}q_{i}}_{n=N_{i}+1} \left|2 \sin (\pi n \alpha) \right|. $$ Let $\alpha := \frac{p_{i}}{q_{i}} + \theta_{i}$ with, say, $\frac{1}{2 q_{i} q_{i+1}} < \theta_{i} < \frac{1}{q_{i} q_{i+1}}$. (The case of negative $\theta_{i}$ is handled quite analogously.)
Let $n=N_{i} + dq_{i} + k$ for some $0 \leq d < b_{i}$ and $1 \leq k \leq q_{i}$, then, with $\kappa:= \kappa_{i} := \{N_{i} \alpha \} \pmod{\frac{1}{q_{i}}}$ and $\tilde{\theta}_i:=q_i \theta_i$ we have \begin{equation} \label{equ_proda} \left\{n \alpha \right\} = \left\{N_{i} \alpha + k \frac{p_{i}}{q_{i}} + (dq_{i}+k) \theta_{i} \right\}= \left\{\kappa + \frac{l(k)}{q_{i}} + d \tilde{\theta}_i +k \theta_{i}\right\} \end{equation} for some $l(k)\in \{0,1,\ldots, q_i -1\}$. Since $0 < k \theta_{i} + dq_{i} \theta_{i} \leq \frac{a_{i+1}q_{i}}{q_{i+1}q_{i}} < \frac{1}{q_{i}}$, for given $d$ there is always exactly one point $\left\{n \alpha\right\}$ in the interval $[\kappa + \frac{l}{q_{i}} , \kappa + \frac{l+1}{q_{i}}) =: I_{l}$ for each $l=0,\ldots,q_{i}-1$ (the interval taken modulo one).
We replace now the points $\left\{n \alpha\right\}$ by new points, namely:
\begin{itemize} \item if $\left\{n \alpha \right\} \in I_{l}$ with $\kappa + \frac{l}{q_{i}} \geq\frac{1}{2}$ then in the representation \eqref{equ_proda} of $\left\{n \alpha \right\}$ we replace $k \theta_{i}$ by $0$, unless $l=q_{i}-1$.
\item if $\left\{n \alpha\right\} \in I_{l}$ with $\kappa + \frac{l+1}{q_{i}} < \frac{1}{2}$ then in the representation \eqref{equ_proda} of $ \left\{n \alpha \right\}$ we replace $k \theta_{i}$ by $\tilde{\theta}_{i}.$
\item if $\left\{ n \alpha \right\} \in I_{l_{0}}$, where $l_{0}$ is such that $\kappa + \frac{l_{0}}{q_{i}} < \frac{1}{2} \leq \kappa + \frac{l_{0}+1}{q_{i}}$ then \begin{itemize} \item for the $d$ such that $\kappa + \frac{l_{0}}{q_{i}} + d \tilde{\theta}_{i} \geq \frac{1}{2}$ in the representation \eqref{equ_proda} of $\left\{n \alpha\right\}$ we replace $k \theta_{i}$ by $0$, \item for the $d$ such that $\kappa + \frac{l_{0}}{q_{i}} + (d+1) \tilde{\theta}_{i} < \frac{1}{2}$ in the representation \eqref{equ_proda} of $\left\{n \alpha \right\}$ we replace $k \theta_{i}$ by $\tilde{\theta}_{i}$, \item for the single $d_{0}$ such that $\kappa + \frac{l_{0}}{q_{i}} + d_{0} \tilde{\theta}_{i} < \frac{1}{2} \leq \kappa + \frac{l_{0}}{q_{i}} + \left(d_{0}+1\right)\tilde{\theta}_i$ we replace $\left\{n \alpha\right\}$ by~$\frac{1}{2}$. \end{itemize}
\item if $\left\{n \alpha \right\} \in I_{l}$ with $l=q_{i}-1$, then \begin{itemize} \item for the $h$ such that $\kappa + \frac{q_{i}-1}{q_{i}} + h \tilde{\theta}_{i} \geq 1$ in the representation \eqref{equ_proda} of $\left\{n\alpha \right\}$ we replace~$k \theta_{i}$~by~$\tilde{\theta}_{i}$, \item for the $h$ such that $\kappa + \frac{q_{i}-1}{q_{i}} + (h+1) \tilde{\theta}_{i} \leq 1$ in the representation \eqref{equ_proda} of $\left\{n \alpha \right\}$ we replace $k \theta_{i}$ by $0$,
\item for the single $h_{0}$ such that $\kappa + \frac{q_{i}-1}{q_{i}} + h_{0} \tilde{\theta}_{i} < 1 < \kappa + \frac{q_{i}-1}{q_{i}} + \left(h_{0}+1\right) \tilde{\theta}_{i}$ we replace in the representation \eqref{equ_proda} of $\left\{ n \alpha \right\}$ the $k \theta_{i}$ by $0$ if $g(\kappa + \frac{q_{i}-1}{q_{i}} + h_{0} \tilde{\theta}_{i}) \geq g(\kappa + \frac{q_{i}-1}{q} + (h_{0}+1) \tilde{\theta}_{i})$ and by $\tilde{\theta}_{i}$ otherwise, where here and in the following we use the notation $g(x) := \left|2 \sin \pi x \right|$. Let the second be the case, the other case is handled quite analogously. \end{itemize} \end{itemize}
Using the new points instead of the $\left\{n \alpha\right\}$ by construction we obtain an upper bound $\widetilde{\Pi}_{i}$ for $\Pi_{i}$. Then \begin{align*} \widetilde{\Pi}_{i} = & g(\kappa + \tilde{\theta}_{i} ) g(\kappa+ 2 \tilde{\theta}_{i}) \cdots g(\kappa+b_{i} \tilde{\theta}_{i})\\ & \times g(\kappa + \tfrac{1}{q_{i}}+ \tilde{\theta}_{i}) g(\kappa+\tfrac{1}{q_{i}} + 2 \tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{1}{q_{i}}+b_{i}\tilde{\theta}_{i}) \\ & \vdots \\ & \times g(\kappa + \tfrac{l_{0}-1}{q_{i}} + \tilde{\theta}_{i}) g(\kappa + \tfrac{l_{0}-1}{q_{i}} + 2 \tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{l_{0}-1}{q_{i}} + b_{i} \tilde{\theta}_{i}) \\ & \times g(\kappa+\tfrac{l_{0}}{q_{i}} + \tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{l_{0}}{q_{i}} + d_{0} \tilde{\theta}_{i}) g(\tfrac{1}{2}) \\ & \hspace{1cm}\times g(\kappa + \tfrac{l_{0}}{q_{i}} + (d_{0}+1)\tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{l_{0}}{q_{i}} + (b_{i}-1)\tilde{\theta}_{i}) \\ & \times g(\kappa + \tfrac{l_{0}+1}{q_{i}}) g(\kappa + \tfrac{l_{0}+1}{q_{i}} + \tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{l_{0}+1}{q_{i}} + (b_{i}-1)\tilde{\theta}_{i})\\ & \vdots \\ & \times g(\kappa + \tfrac{q_{i}-2}{q_{i}}) g(\kappa + \tfrac{q_{i}-2}{q_{i}} + \tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{q_{i}-2}{q_{i}} + (b_{i}-1)\tilde{\theta}_{i}) \\ & \times g(\kappa + \tfrac{q_{i}-1}{q_{i}}) \cdots g(\kappa + \tfrac{q_{i}-1}{q_{i}} + (h_{0}-1) \tilde{\theta}_{i}) g(\kappa + \tfrac{q_{i}-1}{q_{i}} + (h_{0} + 1) \tilde{\theta}_{i}) \\ & \hspace{1cm}\times g(\kappa + \tfrac{q_{i}-1}{q_{i}} + (h_{0}+2) \tilde{\theta}_{i}) \cdots g(\kappa + \tfrac{q_{i}-1}{q_{i}} + b_{i}\tilde{\theta}_{i}). \end{align*} Hence \begin{align*} \widetilde{\Pi}_{i} = & \left(\prod^{b_{i}-1}_{d=1} \prod^{q_{i}-1}_{l=0} g \left(\kappa + \frac{l}{q_{i}} + d \tilde{\theta}_{i}\right)\right) \frac{g(\frac{1}{2})}{g(\kappa + \frac{q_{i}-1}{q_{i}} + h_{0} \tilde{\theta}_{i})} \\ & \times \left(\prod^{l_{0}-1}_{l=0} g \left(\kappa + \frac{l}{q_{i}} + b_{i}\tilde{\theta}_{i}\right)\right) \prod^{q_{i}-1}_{l=l_{0}+1} g \left(\kappa + \frac{l}{q_{i}}\right). \end{align*} By equation (ii) of Lemma~\ref{lem_c_fritz} we have $$
\prod^{q_{i}-1}_{l=0} g \left(\kappa + \frac{l}{q_{i}} + d \tilde{\theta}_{i} \right) = 2 |\sin (\pi q_{i} (\kappa + d \tilde{\theta}_{i}))| \leq 2 $$
and hence $$\prod^{b_{i}-1}_{d=1} \prod^{q_{i}-1}_{l=0} g \left(\kappa + \frac{l}{q_{i}} + d \tilde{\theta}_{i}\right) \le 2^{b_i-1} |\sin (\pi q_{i} (\kappa + h_{0} \tilde{\theta}_{i}))|.$$ Note that $b_{i} \tilde{\theta}_i < \frac{a_{i+1}}{q_{i+1}} < \frac{1}{q_{i}}$ and therefore also $\kappa + d \tilde{\theta}_{i} < \frac{2}{q_{i}}$ always. Hence \begin{align*} \lefteqn{\left(\prod^{l_{0}-1}_{l=0} g \left(\kappa + \frac{l}{q_{i}} + b_{i}\tilde{\theta}_{i}\right)\right) \prod^{q_{i}-1}_{l=l_{0}+1} g \left(\kappa + \frac{l}{q_{i}}\right)}\\ & \leq g \left(\frac{2}{q_{i}}\right) g \left(\frac{3}{q_{i}}\right) \cdots g \left(\frac{\lfloor q_{i}/2\rfloor}{q_{i}}\right) g \left(\frac{1}{2}\right)^{2} g \left(\frac{\lfloor q_{i}/2\rfloor+1}{q_{i}}\right) \cdots g \left(\frac{q_{i}-1}{q_{i}}\right)\\ &= \left(\prod^{q_{i}-1}_{l=1} 2 \sin\left( \pi \frac{l}{q_{i}}\right)\right) \frac{4}{\sin(\pi/q_{i})}= \frac{4q_{i}}{\sin( \pi/q_{i})} \leq 2 q_{i}^{2}. \end{align*} Hence \begin{eqnarray*}
\widetilde{\Pi}_{i} \leq 2^{b_{i}-1} \, \frac{2 |\sin (\pi q_{i} (\kappa + h_{0} \tilde{\theta}_{i}))|
}{2 |\sin(\pi (\kappa + \tfrac{q_{i}-1}{q_{i}} + h_{0}\tilde{\theta}_{i}))|} \, 2 q_{i}^{2}. \end{eqnarray*} We have
$$\frac{|\sin (\pi q_{i} (\kappa + h_{0} \tilde{\theta}_{i}))|
}{|\sin(\pi (\kappa + \tfrac{q_{i}-1}{q_{i}} + h_{0}\tilde{\theta}_{i}))|}=\frac{|\sin (\pi q_{i} (\kappa +\tfrac{q_i -1}{q_i}+ h_{0} \tilde{\theta}_{i}))|
}{|\sin(\pi (\kappa + \tfrac{q_{i}-1}{q_{i}} + h_{0}\tilde{\theta}_{i}))|} \le q_i,$$ since $|\sin(nx)/\sin x| \le n$ for $n \in \mathbb{N}$. Hence $$\widetilde{\Pi}_{i} \leq 2^{b_{i}} q_{i}^{3}$$ and therefore $$
\prod^{N}_{n=1} |2 \sin (\pi n \alpha)| \leq \prod^{l}_{i=0} 2^{b_{i}} q_{i}^{3}, $$ as desired. \end{proof}
\begin{proof}[Proof of Corollary~\ref{co_prodb}] By Theorem~\ref{th_prodb} we have \begin{align*}
\frac{1}{N} \sum^{N}_{n=1} \log |2 \sin (\pi n \alpha)| \leq & (\log 2) \frac{b_{0} + \cdots + b_{l}}{b_{0} q_{0} + \cdots + b_{l} q_{l}} +3 \frac{\log q_{1} + \cdots + \log q_{l}}{b_{0} + b_{1}q_{1} + \cdots + b_{l}q_{l}} \\ \leq & (\log 2) \left(\frac{1}{q_{l}} + \frac{l \max_{0 \leq i < l}b_{i}}{q_{l}}\right) + 3 \, \frac{l \log q_{l}}{q_{l}}. \end{align*} We have \begin{align*} q_l \ge b_{l-1} q_{l-1}+q_{l-2}
\ge b_{l-1} b_{l-2} q_{l-2}+b_{l-1} q_{l-3}+q_{l-2}
\ge (b_{l-1}b_{l-2}+1) q_{l-2}. \end{align*} By iteration we obtain \begin{eqnarray*} q_l \ge (b_{l-1}b_{l-2}+1)(b_{l-3}b_{l-4}+1) \cdots (b_1 b_0+1)\ge 2^{\frac{l}{2}-1} \max_{0 \leq i < l}b_{i} \end{eqnarray*} if $l$ is even and $$q_l \ge (b_{l-1}b_{l-2}+1)(b_{l-3}b_{l-4}+1) \cdots (b_2 b_1+1) q_1 \ge 2^{\frac{l-3}{2}} \max_{0 \leq i < l}b_{i}$$ if $l$ is odd. With these estimates we get \begin{eqnarray*}
\frac{1}{N} \sum^{N}_{n=1} \log |2 \sin (\pi n \alpha)| \leq (\log 2) \left(\frac{1}{q_{l}} + \frac{l}{2^{(l-3)/2}}\right) + 3 \, \frac{l \log q_{l}}{q_{l}}. \end{eqnarray*} Note that $q_l \ge \phi^{l-1}$ and hence $l\le \frac{\log q_l}{\log \phi}+1$, where $\phi=(1+\sqrt{5})/2$. Hence \begin{eqnarray*}
\frac{1}{N} \sum^{N}_{n=1} \log |2 \sin (\pi n \alpha)| \leq (\log 2) \left(\frac{1}{q_{l}} + \frac{l}{2^{(l-3)/2}}\right) + 3 \, \frac{\log q_{l}}{q_{l}} \left(\frac{\log q_l}{\log \phi}+1 \right). \end{eqnarray*} \end{proof}
\begin{proof}[Proof of Corollary~\ref{co_b}] Since $\alpha$ is of type $t >1$ we have $$
\frac{c}{q_{i}^{1+t}} < \left| \alpha - \frac{p_{i}}{q_{i}}\right| < \frac{1}{a_{i+1}q_{i}^{2}} $$ and hence $b_{i} \leq a_{i+1} < q_{i}^{t-1}/c$. Especially we have the following: Let $b_{l} := q_{l}^{\gamma}$, then, because of $$ q_{l}^{\gamma+1} = b_{l} q_{l} \leq N < \left(b_{l}+1\right)q_{l} \leq 2 q_{l}^{\gamma+1}, $$ we have $$ b_{l} = q_{l}^{\gamma} \leq N^{\frac{\gamma}{\gamma+1}} \leq c_{1} N^{1-1/t}. $$ Hence the bound from Theorem~\ref{th_prodb} can be estimated by \begin{eqnarray*} \prod^{l}_{i=0} 2^{b_{i}} q_{i}^{3} & \leq & 2^{b_{l}} \left(\prod^{l}_{i=0} q_{i}^{3} \right) \prod^{l-1}_{i=0} 2^{b_{i}}\\ & \leq & 2^{c_{1} N^{1-1/t}} N^{3 \left(l+1\right)} \prod^{l-1}_{i=0} 2^{c_{1} N^{\left(1-1/t\right) \left(1/t\right)^{i}}} \\ & \leq & 2^{c_{2} N^{1-1/t}} N^{c_{3} \log N} \leq 2^{C N^{1-1/t}} \end{eqnarray*} for $N$ large enough. \end{proof}
\section{Proof of the result on the van der Corput sequence} \label{sect_d}
Let $$ P_N:= \prod^{N}_{k=1} 2 \sin (\pi x_k)~\mbox{and}~f(k) := 2 \sin (\pi x_k), $$ where $x_k$ is the $k^{{\rm th}}$ element of the van der Corput sequence.
\begin{lemma} \label{lem_a} Let (in dyadic representation) $$n:= a_{s} a_{s-1} \ldots a_{k+1}\underbrace{ 0 1 1 \ldots 1 1}_{a_ka_{k-1}\ldots a_{l+1}} \underbrace{0 1 1 \ldots 1}_{a_l a_{l-1} \ldots a_0}$$ and $$\overline{n} := a_{s} a_{s-1} \ldots a_{k+1} 111 \ldots 11011 \ldots1.$$ Then $P_{\overline{n}} > 2 P_{n}$. \end{lemma}
\begin{proof} We have $$ P_{\overline{n}}=P_n \, \frac{f(n+1) \cdots f(n+2^{l}) f(n+2^{l}+1)\cdots f(n+2^{l}+2^{k})}{f(n+2^{k}+1)\cdots f(n+2^{k}+2^{l})}. $$ Since $\{x_{n+1}, \ldots, x_{n+2^{l}}\} = \{\xi, \xi + \frac{1}{2^{l}}, \ldots, \xi + \frac{2^{l}-1}{2^{l}}\}$ with $$ \xi =\frac{1}{2^{l+1}} + \cdots + \frac{1}{2^{k}} + \frac{a_{k+1}}{2^{k+2}} + \cdots + \frac{a_{s}}{2^{s+1}}, $$ we obtain from equation (ii) of Lemma~\ref{lem_c_fritz} $$ f(n+1) \cdots f(n+2^{l}) = 2 \sin( \pi 2^l \xi). $$ Furthermore, $\{x_{n+2^{l}+1}, \ldots, x_{n+2^{l}+2^{k}}\}= \{y, y+\frac{1}{2^{k}}, \ldots, y + \frac{2^{k}-1}{2^{k}}\}$ with $$y=\frac{1}{2^{k+1}} + \frac{a_{k+1}}{2^{k+2}} + \cdots + \frac{a_{s}}{2^{s+1}}$$ and hence, again by equation (ii) of Lemma~\ref{lem_c_fritz}, $$ f(n+2^{l}+1) \cdots f(n+2^{l}+2^{k}+1) = 2 \sin (\pi 2^k y). $$ Note that $\frac{1}{2^{k+1}} < y < \frac{1}{2^{k}}$.
In the same way we have $\{x_{n+2^{k}+1}, \ldots, x_{n+2^{k}+2^{l}}\} = \{\tau, \tau + \frac{1}{2^{l}}, \ldots, \tau + \frac{2^{l}-1}{2^{l}}\}$ with $$ \tau = \frac{1}{2^{l+1}} + \cdots + \frac{1}{2^{k+1}} + \frac{a_{k+1}}{2^{k+2}} + \cdots + \frac{a_{s}}{2^{s+1}} $$ and hence by equation (ii) of Lemma~\ref{lem_c_fritz} $$ f(n + 2^{k}+1) \cdots f(n+2^{k}+2^{l}) = 2 \sin( \pi 2^l \tau). $$ So $$ P_{\overline{n}}=P_n \frac{2 \sin(\pi 2^l \xi) \sin(\pi 2^k y)}{\sin(\pi 2^l \tau)}. $$ We have to show that $$ \Gamma:= \frac{2 \sin(\pi 2^l \xi) \sin(\pi 2^k y)}{\sin(\pi 2^l \tau)} > 1. $$
Since $\tau = y + \frac{1}{2^{l}}-\frac{1}{2^{k}}$ and $\xi = y + \frac{1}{2^{l}}-\frac{1}{2^{k}}-\frac{1}{2^{k+1}}$ it follows that $$ \Gamma = \frac{2\sin( \pi (2^{l} y+1-\frac{1}{2^{k-l}} - \frac{1}{2^{k+1-l}})) \sin(\pi 2^{k}y)}{\sin(\pi (2^{l} y+1-\frac{1}{2^{k-l}}))}. $$ Let $k-l =: m$ and $2^{l} y =: \eta$. Then we have $\frac{1}{2^{m+1}}<\eta<\frac{1}{2^{m}}$ and $$ \Gamma = \frac{2 \sin (\pi (\eta +1-\frac{1}{2^{m}} - \frac{1}{2^{m+1}})) \sin(\pi 2^{m} \eta)}{\sin (\pi (\eta +1-\frac{1}{2^{m}}))}. $$ Let $z:= \frac{1}{2^{m}}-\eta$. Then we have $0 < z < \frac{1}{2^{m+1}}$ and \begin{align*} \Gamma = & \frac{2 \sin(\pi(1-z-\frac{1}{2^{m+1}})) \sin(\pi(1-2^{m}z))}{\sin (\pi(1-z))} \\
= & \frac{2 \sin(\pi (z+\frac{1}{2^{m+1}})) \sin(\pi 2^m z)}{\sin (\pi z)} \\
> & \frac{2 \sin(\pi \frac{1}{2^{m+1}}) \sin (\pi 2^{m} \frac{1}{2^{m+1}})}{\sin(\pi \frac{1}{2^{m+1}})} \\
= & 2. \end{align*} Here we used that $\sin (\pi (z + \frac{1}{2^{m+1}}))$ for $0 < z< \frac{1}{2^{m+1}}$ is minimal for $z \rightarrow 0$ and $\frac{\sin (\pi 2^{m} z)}{\sin( \pi z)}$ for $0 < z < \frac{1}{2^{m+1}}$ is minimal for $z\rightarrow \frac{1}{2^{m+1}}$. \end{proof}
\begin{lemma} \label{lem_b} We have: \begin{enumerate}[(i)] \item \begin{tabular}{llll} Let & $n$ & $=$ & $\overset{s}{\overset{\downarrow}{1}}111 \ldots 111\hspace{-0,172cm}\overset{k+1}{\overset{\downarrow}{0}}\hspace{-0,172cm}111 \ldots 1110$ \\ and & $\overline{n}$ & $=$ & $1111 \ldots 1111011 \ldots 1110$ \\ \end{tabular}
\hspace{-0,27cm}then $P_{\overline{n}} \geq P_n$. \item \begin{tabular}{llll} Let & $n$ & $=$ & $1\hspace{-0,165cm}\overset{s-1}{\overset{\downarrow}{0}}\hspace{-0,165cm}11 \ldots111\hspace{-0,165cm}\overset{k+1}{\overset{\downarrow}{0}}\hspace{-0,165cm}111 \ldots1110$ \\ and & $\overline{n}$ & $=$ & $1011\ldots 1111011 \ldots 1110$ \\ \end{tabular}
\hspace{-0,27cm}then $P_{\overline{n}} \geq P_n$. \item \begin{tabular}{llll} Let & $n$ & $=$ & $1111\ldots111\hspace{-0,172cm}\overset{k+1}{\overset{\downarrow}{0}}\hspace{-0,172cm}111 \ldots1111$ \\ and & $\overline{n}$ & $=$ & $1111\ldots1111011\ldots1111$ \\ \end{tabular}
\hspace{-0,27cm}then $P_{\overline{n}} \geq P_n.$\\
\item \begin{tabular}{llll} Let & $n$ & $=$ & $1011 \ldots1110111 \ldots1111$ \\ and & $\overline{n}$ & $=$ & $1011\ldots1111011\ldots1111$ \\ \end{tabular}
\hspace{-0,27cm}then $P_{\overline{n}} \geq P_n$. \end{enumerate} \end{lemma}
\begin{proof} We only prove (ii), which is the most elaborate part of the lemma. The other assertions can be handled in the same way but even simpler. In (ii) we have \begin{align*} P_{\overline{n}} = & P_n f(10111\ldots110111 \ldots111) \prod^{2^{k}-2}_{i=0} f(1011\ldots100\ldots0+i) \\ = & P_n 2 \sin \left(\pi \left(1-\frac{1}{2^{k+2}}- \frac{3}{2^{s+1}}\right)\right) \frac{\sin (\pi x)}{\sin(\pi \frac{1-x}{2^{k}})} \end{align*} with $x=2^k(\frac{1}{2^{k+1}} - \frac{3}{2^{s+1}})$. Hence $$ P_{\overline{n}} = P_n \frac{2 \sin (\pi (\frac{1}{2^{k+2}} + \frac{3}{2^{s+1}}) \cos(\pi \frac{3}{2^{s-k+1}})}{\sin(\pi (\frac{1}{2^{k+1}} + \frac{3}{2^{s+1}}))}. $$ Here $s \geq 4$ and $1 \leq k\leq s-3$. Some tedious but elementary analysis of the function $$ g(x,y) := \frac{2\sin(\pi(\frac{x}{4} + \frac{3}{2} y)) \cos(\pi \frac{3}{2} \frac{y}{x})}{\sin (\pi (\frac{x}{2} + \frac{3}{2} y))} $$ for $0 < y \leq \frac{1}{16}$ and $8 y \leq x\leq \frac{1}{2}$ shows that $g (x,y) > 1$ in this region. Hence $P_{\overline{n}} > P_n$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th_a}] Consider $n$ with $2^{s} \leq n < 2^{s+1}$. From Lemma~\ref{lem_a} and Lemma~\ref{lem_b} it follows that for $2^{s}+2^{s-1} \leq n < 2^{s+1}$ the product $P_n$ has its largest values for $$ n_{1}=111\ldots11110=2^{s+1}-2 $$ $$ n_{2}=111\ldots11101=2^{s+1}-3 $$ $$ n_{3}=111 \ldots11100=2^{s+1}-4 $$ and for $2^{s} \leq n < 2^{s}+2^{s-1}$ the product $P_n$ has its largest values for $$ n_{4}=101\ldots11110=2^{s+1}-2^{s-1}-2 $$ $$ n_{5}=101 \ldots11101=2^{s+1}-2^{s-1}-3 $$ $$ n_{6}=101\ldots11100=2^{s+1}-2^{s-1}-4. $$ By equation (i) of Lemma~\ref{lem_c_fritz} we have $$ P_{n_{1}} = \frac{2^s}{\sin(\pi/2^{s+1})} $$ hence $\frac{1}{n_1^2} P_{n_{1}} \rightarrow \frac{1}{2 \pi}$ for $s$ to infinity. Furthermore, \begin{align*} P_{n_2} = & \frac{2^s}{\sin(\pi/2^{s+1}) f(2^{s+1}-2)} = \frac{2^s}{\sin(\pi/2^{s+1}) 2 \sin(\pi (\frac{1}{2} - \frac{1}{2^{s+1}}))} \\ = & \frac{2^{s-1}}{\sin(\pi/2^{s+1}) \cos(\pi/2^{s+1})}, \end{align*} and hence $\frac{1}{n_2^2} P_{n_{2}} \rightarrow \frac{1}{4 \pi}$ for $s$ to infinity. Finally \begin{align*} P_{n_3} = & \frac{2^{s-1}}{\sin(\pi/2^{s+1}) \cos(\pi/2^{s+1}) f(2^{s+1}-3)} \\ = & \frac{2^{s-1}}{\sin(\pi/2^{s+1}) \cos(\pi/2^{s+1})2\sin(\pi (1-\frac{1}{4}-\frac{1}{2^{s+1}}))} \\ = & \frac{2^{s-2}}{\sin(\pi/2^{s+1}) \cos(\pi/2^{s+1}) \sin(\pi (\frac{1}{4} + \frac{1}{2^{s+1}}))}. \end{align*} Let now $2^{s}+2^{s-1}\leq n \leq n_{3}$ be arbitrary. Then $$ \frac{1}{n^{2}} P_n \leq \frac{1}{(2^{s}+2^{s-1})^2} P_{n_3}, $$ and the last term tends to $$ \frac{2}{9 \pi \sin \frac{\pi}{4}} < \frac{1}{2 \pi}. $$ Hence for all $s$ large enough we have $\frac{1}{n^{2}} P_n < \frac{1}{2 \pi}$ for all $2^{s} + 2^{s-1} \leq n < n_{3}$.
We still have to consider $n$ with $2^{s} \leq n < 2^{s}+2^{s-1}$. With equation (ii) of Lemma~\ref{lem_c_fritz} we have \begin{align*} P_{n_4} = & P_{n_1} \frac{1}{f(1011\ldots111) \prod^{2^{s-1}-2}_{i=0} f(11000\ldots00+i)} \\ = & P_{n_1} \frac{1}{2 \sin( \frac{3\pi}{2} \frac{1}{2^{s}})} \frac{\sin(\frac{\pi}{2^{s+1}})}{\sin \frac{3 \pi}{4}}. \end{align*} The product $\kappa_{s}$ of the last two factors tends to $\frac{1}{3 \sqrt{2}}$ for $s$ to infinity.
Furthermore, it is easily checked that $P_{n_5}$ and $P_{n_6}$ are smaller than $P_{n_4}$. Hence for all $n$ with $2^s \leq n < 2^s + 2^{s-1}$ we have $$ \frac{P_n}{n^2} \leq \frac{P_{n_4}}{2^{2s}} = \frac{P_{n_1}}{n_1^2} \frac{(2^{s+1}-2)^2}{2^{2s}} \kappa_s $$ which tends to $\frac{1}{2 \pi} \frac{4}{3 \sqrt{2}} < \frac{1}{2 \pi}$ for $s$ to infinity. So altogether we have shown that $$ \limsup_{n \rightarrow \infty} \frac{1}{n^{2}} \prod^{n}_{i=1} 2 \sin(\pi x_i) = \frac{1}{2 \pi}. $$ From Lemma~\ref{lem_a} and from equation (i) of Lemma~\ref{lem_c_fritz} it also follows that for all $s$ we have $$ \min_{2^s \leq n < 2^{s+1}} P_{n} = P_{2^s} = 2^{s+1} \sin\left(\frac{\pi}{2^{s+1}}\right) $$ which tends to $\pi$ for $s$ to infinity. This gives the lower bound in Theorem~\ref{th_a}. \end{proof}
\section{Proof of the probabilistic results} \label{sect_prob}
In the first part of this section we consider products
\begin{align} P_{N}=\prod^{N}_{k=1}2\sin (\pi X_{k}), \label{8} \end{align}
where $(X_{k})_{k\geq 1}$ is a sequence of i.i.d.\ random variables on $[0,1]$. We want to determine the almost sure asymptotic behavior of \eqref{8}. We take logarithms and define
\begin{align}S_{N}=\log P_{N}=\sum^{N}_{k=1}\log (2\sin (\pi X_{k}))= \sum^{N}_{k=1}Y_{k}, \label{9} \end {align}
where $Y_{k}= \log (2\sin (\pi X_{k}))$ is again an i.i.d.\ sequence. Thus we can apply Kolmogorov's law of the iterated logarithm \cite{kolm} (see also Feller \cite{feller}) in the i.i.d.\ case. However, for later use we state this LIL in a more general form below.
\begin{lemma} \label{lemma7} Let $(Z_{k})_{k\geq 1}$ be a sequence of independent random variables with expectations $\mathbb{E}Z_{k}= 0$ and finite variances $\mathbb{E}Z^{2}_{k}<\infty$, and let $B_{N}=\sum^{N}_{k=1}\mathbb{E}Z_{k}^{2}$. Assume there are positive numbers $M_{N}$ such that
\begin{align}| Z_{N}|\leq M_{N} \qquad \text{and} \qquad M_{N}=o\left(\sqrt{\frac{B_{N}}{\log\log B_{N}}}\right). \label{10} \end{align} Then $S_{N}=\sum^{N}_{k=1}Z_{k}$ satisfies a law of the iterated logarithm \begin{align}\limsup_{N\rightarrow\infty}\frac{S_{N}}{\sqrt{B_{N}\log\log B_{N}}}= \sqrt{2} \qquad \text{almost surely.} \end{align} \end{lemma}
In the case of centered i.i.d random variables ${Z}_{k}$ with finite variance, we have $B_{N}=bN$ with $b=\mathbb{E}Z^{2}_{1}$. Thus in this case
\begin{align}\limsup_{N\rightarrow\infty}\frac{S_{N}}{\sqrt{N \log\log N}}=\sqrt{2 b}\quad\quad\quad\text{almost surely.} \end{align}
In order to apply Lemma~\ref{lemma7} to the sum \eqref{9}, we note that $$\mathbb{E} Y_k = \mathbb{E} (\log(2\sin (\pi X_{k}))) = \int^{1}_{0}\log (2 \sin (\pi x)) \rd x= 0,$$ and compute the variance $$\mathbb{E} Y_k^ 2 = \mathbb{E} (\log^{2}(2\sin (\pi X_{k}))) =\int^{1}_{0}\log^{2}(2\sin(\pi x)) \rd x=\frac{\pi^2}{12}.$$ This proves Theorem \ref{th51}.\\
For the proof of Theorem \ref{th52} we split the corresponding logarithmic sum into two parts \begin{eqnarray} \lefteqn{\sum_{1 \leq n_k \leq N} \log (2\sin (\pi n_{k}\alpha))} \nonumber\\ & = & \frac{1}{2} \left(\sum^{N}_{n=1}\log (2\sin (\pi n \alpha)) + \sum^{N}_{n=1} R_{n}\log (2\sin (\pi n\alpha)) \right), \label{13} \end{eqnarray} where $R_{n}=R_{n}(t)$ denotes the $n^{{\rm th}}$ Rademacher function on $[0,1]$ and the space of subsequences of the positive integers corresponds to $[0,1]$ equipped with the Lebesgue measure. For irrationals $\alpha$ with bounded continued fraction expansion, by Corollary~\ref{co_prodb} we have
\begin{align}\sum^{N}_{n=1}\log (2\sin (\pi n \alpha)) =O(\log^{2}N). \label{14} \end{align}
For the second sum in \eqref{13} we set $Z_{n}=R_{n}\log (2 \sin (\pi n \alpha))$ and apply Lemma~\ref{lemma7}. The random variables $Z_{n}$ are clearly independent and thus we have to compute the quantities $B_{N}$ and check condition \eqref{10}. Obviously, $\mathbb{E}Z_{n}=0$ and $\mathbb{E}Z^{2}_{n}=\log^{2} (2\sin (\pi n \alpha))$. Using the fact that $$
|\sin (\pi n \alpha)|\geq 2 \| n \alpha \| \geq\frac{{c}_{0}}{n}, $$ with some positive constant $c_0$, we obtain
$$| Z_{N}| \leq c_{1} \log N$$
with some $c_{1}> 0$. Using Koksma's inequality and discrepancy estimates for $(n \alpha)_{n \geq 1}$ it can easily been shown that \begin{eqnarray*} \frac{B_{N}}{N} = \frac{1}{N}\sum_{n=1}^N \log^{2} (2\sin (\pi n \alpha))
\to \int_0^1 \log^{2} (2\sin (\pi n \alpha)) \rd \alpha = \frac{\pi^2}{12}. \end{eqnarray*} Thus, the conditions of Lemma~\ref{lemma7} are satisfied and we have $$ \limsup_{N \to \infty} \frac{\sum_{n=1}^N Y_n}{\sqrt{N \log \log N}} = \frac{\pi}{\sqrt{6}},\qquad \text{$\mathbb{P}$-almost surely.} $$ Consequently, from \eqref{13} and \eqref{14} we obtain \begin{equation} \label{equfin} \limsup_{N \to \infty} \frac{\sum_{1 \leq n_k \leq N} \log (2\sin (\pi n_{k}\alpha))}{\sqrt{N \log \log N}} = \frac{\pi}{2\sqrt{6}}, \qquad \text{$\mathbb{P}$-almost surely.} \end{equation} Finally, note that by the strong law of large numbers we have, $\mathbb{P}$-almost surely, that $$ \# \left\{k:~1 \leq n_k \leq N\right\} \sim \frac{N}{2}. $$ Consequently, from \eqref{equfin} we can deduce that $$ \limsup_{N \to \infty} \frac{\sum_{k=1}^N \log (2\sin (\pi n_{k}\alpha))}{\sqrt{N \log \log N}} = \frac{\pi}{\sqrt{12}}, \qquad \text{$\mathbb{P}$-almost surely.} $$ This proves Theorem \ref{th52}.\\
\textbf{Acknowledgment.} We thank Dmitriy Bilyk who drew our attention to the results given in \cite{Knill+Les, Lub}, and \cite{VerMes}.
\end{document} | arXiv |
An extensive assessment of network alignment algorithms for comparison of brain connectomes
Marianna Milano1,
Pietro Hiram Guzzi1,
Olga Tymofieva2,
Duan Xu2,
Christofer Hess2,
Pierangelo Veltri1 &
Mario Cannataro1
BMC Bioinformatics volume 18, Article number: 235 (2017) Cite this article
Recently the study of the complex system of connections in neural systems, i.e. the connectome, has gained a central role in neurosciences. The modeling and analysis of connectomes are therefore a growing area. Here we focus on the representation of connectomes by using graph theory formalisms. Macroscopic human brain connectomes are usually derived from neuroimages; the analyzed brains are co-registered in the image domain and brought to a common anatomical space. An atlas is then applied in order to define anatomically meaningful regions that will serve as the nodes of the network - this process is referred to as parcellation. The atlas-based parcellations present some known limitations in cases of early brain development and abnormal anatomy. Consequently, it has been recently proposed to perform atlas-free random brain parcellation into nodes and align brains in the network space instead of the anatomical image space, as a way to deal with the unknown correspondences of the parcels. Such process requires modeling of the brain using graph theory and the subsequent comparison of the structure of graphs. The latter step may be modeled as a network alignment (NA) problem.
In this work, we first define the problem formally, then we test six existing state of the art of network aligners on diffusion MRI-derived brain networks. We compare the performances of algorithms by assessing six topological measures. We also evaluated the robustness of algorithms to alterations of the dataset.
The results confirm that NA algorithms may be applied in cases of atlas-free parcellation for a fully network-driven comparison of connectomes. The analysis shows MAGNA++ is the best global alignment algorithm. The paper presented a new analysis methodology that uses network alignment for validating atlas-free parcellation brain connectomes. The methodology has been experimented on several brain datasets.
The brain is a complex organ of vertebrates and it is composed of single specialized cells called neurons. Neurons are connected among them by synapses forming a complex network of connections. Connections among neurons carry signal pulses that carry information [1]. The activity of the brain is mostly due to this set of connections.
Recent studies have demonstrated in an independent way a strict relation among the set of connections, the functions of the brains and the relations among the insurgence of neurological diseases and the variations of mechanims of connections with respect to healthy people [2]. For example, in the Alzheimer Disease a decreased connectivity, and hippocampus changes are detected [3], the Parkinson disease is associated to altered connectivity [3], or in anxiety disorder an increased connectivity and amygdala changes is found [4].
Consequently, the interest for the modeling and the analysis of the whole system of the brain elements and their relations has lead to the introduction of the so-called connectomics, i.e the study of connectome referred to as the set of elements and interactions [5]. Connectomics is based on modern technologies of investigation of the brain that are able to take a sort of picture of the brain connections of patients [6]. Connectome may be analyzed using different zoom, e.g. by focusing on single components, i.e. neurons and axons, or grouping them into regions. Usually the analysis of single components is defined to as anatomic connectivity, while the analysis of regions is called functional connectivity because regions are in general perfoming different functions.
Among the others, one of the main sources for deriving information about connectomes is Magnetic Resonance Imaging (MRI) [7]. A typical MRI experiment produces a set of images providing both anatomical and functional information. The first one is constituted by axonal fibers between cortical regions, the second one provides information about the functional connectivity, i.e. the activation of region of interest (ROI). Such analysis is often conducted by using diffusion tensor imaging (DTI) that is a specialised version of Diffusion-weighted magnetic resonance imaging (DWI or DW-MRI), and a DTI has been used extensively to map white matter tractography in the brain through the analysis of patterns of diffusion of molecules through bundles of neural axons. The anatomical connectivity structures are primarily derived through applying tractography algorithms to DTI data. Functional connectivity data are derived from functional magnetic resonance imaging (fMRI). The fMRI images show active regions of the brain at a given instance, based on the blood oxygen consumption level. The obtained networks are called functional networks. The combined use of these two techniques is used to determine the structure of human brain connectome as depicted in Fig. 1.
Building a representative network from experimental data: example of a workflow. Diffusion or functional MRI images are acquired for a subject according to the study to be conducted. The MRIs are used to perform whole-brain parcellation by selecting a suitable method. Starting from the parcelled whole brain the computation of connections is performed and a weighted adjacency matrix is constructed. Then, the weights of adjacency matrix are binarized. Finally, the resulting brain network is obtained
Once obtained, connectome data needs to be integrated into a suitable model. One of the most used representation of such data is given by the graph theory, whose models have been used by different approaches to extract clinically relevant information [8, 9]. Graph theory ensures the possibility of modeling such data into a single network model and then the possibility to summarize all the characteristics into few measures, giving the understanding of the organization of the entire network as well as individual network elements [10].
Differently to other kind of networks, the modeling of connectomes using graphs is a open research area since there are many possibility for defining the nodes, the edges, that corresponds to different scale of views. For instance, nodes may represent neurons and edges their axons [11]. Here we focus on the representation of region of interest (ROI) as nodes, and the representation of functional or anatomical connections as edges. There exist three main categories of research applied on such networks: (i) the improvement of the reconstruction of graphs starting from MRI images, (ii) the identification of the structure of networks (i.e. which is the theoretical model underlying the brain network organization), (iii) the identification of relevant modules that may be used to understand brain functions and their modifications in case of disease (e.g. for early detection of diseases). The first and the third problem strictly rely on the definition of a framework for the comparison of graphs.
Considering, for instance, the first problem it should be noted that each MRI experiment produces a series of images (either from intra-subject or inter-subject) that need to be aligned into a spatial domain. When using both functional and structural images, coregistration is the process of the alignment of functional and structural images to map functional information into anatomical space. In such a way each region will correspond to a node of a network using an atlas to define anatomically meaningful regions [12].
Nevertheless, such an approach may lead to substantial inaccuracies in cases of abnormal anatomy (e.g. in presence of diseases) and early brain development (e.g. in brain of child). To address this problem, it has been recently proposed to use atlas-free parcellation and to construct and compare individual connectomes only in the network space [13]. In [13] the authors perform the atlas-free parcellation as the finest parcellation that still interconnects the whole brain, leaving no nodes isolated. Then, they group subjects into homogeneous groups and the NA is performed within each group. The sum network is obtained and mapped to the anatomy of a "reference brain."
Such work, demonstrates the possibility to use NA into the atlas-free parcellation workflow and it poses to the research community the challenge to systematically explore the performance of different NA algorithms since different NA approaches are widely applied in molecular biology analysis, but they have not been explored yet in relation to MRI connectomics.
The techniques for the alignment of biological networks fall into two categories: (i) the local network alignment searches relatively small similar subnetworks that are likely to represent conserved functional structures, (ii) the global network alignment looks for the best superimposition of the whole input networks. However, these approaches can not be easily applied in the connectome alignment problem. The reason is related to the strategy underlying methodology of alignment. For example, the local network aligners, widely used to build the alignment of protein interaction networks (PINs) [14], take as input two networks and a list of seed nodes used to build the initial alignment graph (see [15] for complete details about the construction of the alignment graph). These initial nodes are selected based on biological consideration, such as homology relationships between nodes of PINs. Since the nodes of the brain networks represent ROIs, the homology information cannot be obtained in the case of connectome networks and then, the local alignment cannot be applied.
In this paper we selected six existing state of the art global alignment algorithms and we tested these aligners on diffusion MRI-derived brain networks. The algorithms tested here are MAGNA++ [16], NETAL [17], GHOST [18], GEDEVO [19], WAVE [20], Natalie2.0 [21]. The algorithms are applied to build the alignments among the diffusion MRI-derived brain networks. After the alignments were built, we compared the performance of these algorithms and evaluated this robustness.
Brain parcellation
An essential step in the analysis and macroscopic mapping of brain network is the subdivision of the brain into large-scale regions, also known as "parcellation process". The brain parcellation consists of dividing the brain into a set of macroscopic, homogeneous and non-overlapping regions with respect to information provided, generally, by techniques based on magnetic resonance imaging (MRI) [22]. Especially, MRI has allowed to obtain information about anatomical connectivity, functional connectivity, or task-related activation. Different pieces of evidence demonstrate that parcellation of the brain into the homogeneous region is far from being defined, as well as the edges definition and their placement. In the graph representation of a parcellation-based connectome, the nodes of the graph correspond to a brain region and the edges correspond to structural or functional connections between these regions. Despite its relative simplicity, the application of graph theory to the study of connectomes presents some particular challenges related to the meaningful definition of nodes and edges. An ideal model should represent the true subsystems (as nodes) and the true relations (as edges). However, as deeply investigated in [23], there is no clear evidence for the optimal definition of both nodes and edges. For example, an ideal node definition should group a set of neurons to maximize the functional homogeneity within and to maximize the functional heterogeneity among different nodes. Moreover, it should take into account the spatial (and hopefully temporal) relationship among nodes. Besides the definition, the edges representation is also currently an open challenge and this task is related to the type of measured connectivity, and the method used to quantify it. As mentioned above, brain connectivity can refer to different aspects of brain organization including (i) anatomical connectivity consisting of axonal fibers connecting cortical and subcortical regions inferred from diffusion imaging (see Fig. 2 (1)), and (ii) functional connectivity defined as the observed statistical correlations of the Blood oxygenation level dependent (BOLD) signal between brain regions.
Definition of (1) edges and (2) nodes using an atlas-free random parcellation and using diffusion MRI and tractography. In the first box the edges reconstruction is reported, whereas in the second box the two kind of whole brain parcellations in newborns, 6 month-old subjects and adults are shown. The first cortical parcellation is performed by setting the number of equal-area nodes equal to 95. The second cortical parcellation is performed by setting the number of equal-area nodes equal to 1000. In this last one it is possible to note disconnected regions highlighted in green
That is, the choice of parcellation scheme has a significant impact on the subsequent analysis. There currently exist three parcellation-based connectome approaches:
Parcellation of the brain by using predefined anatomical templates. This approach consists of the registration of the structural images from MRI to anatomical atlas based on the Brodmann areas [24]. In this way, the whole brain is subdivided into labeled regions according to the different labels regions of the templates;
Parcellation of the brain by using randomly generated templates [25]. For the random parcellation different algorithms are applied to produce parcels of roughly equal size. Thus, the generated templates are characterized by approximately uniformly sized brain regions to avoid anatomical biases;
Connectivity-based parcellations that aim to delineate brain regions by analyzing the similarities in structural or functional connectivity patterns. Based on the notion that regions with a similar connectivity profile are involved in the same analogous functional roles, the connectivity-based parcellation partitions small seed regions into a largest collection of functionally homogeneous brain regions by clustering seeds with similar connectivity profiles.
However, each method presents some pitfalls. For example, the registration of brain of the studied subject to a generic brain with defined Brodmann areas raises the question of the accuracy of mapping. In fact, in the most of the cases, the borders of the Brodmann areas, originally defined using cytoarchitectural differences between brain regions, do not match with the cortical surface analyzed.
This approach is limited by inter-subject variability and can be especially problematic in the context of brain maturation. Furthermore, it has been demonstrated that parcellation of brain with predefined anatomical templates may impact negatively all the subsequent analysis by introducing evident biases [13]. In this paper we focus on the random, atlas-free definition of nodes in individual subjects (see Fig. 2 (2)), which can allow for a fully network-driven study of the brain and for comparing brains of different subjects and, potentially, species [13].
Global network alignment algorithms
The identification of an accurate node mapping between atlas-free networks may offer significant details on the comparison of brains or structure of groups of subjects, such as healthy versus diseased subjects. Many different network alignment methods have been proposed in biological fields [26].
Formally, a graph G is defined as G={V,E}, where V is a finite set of nodes and E is a finite set of edges. Let G 1={V 1,E 1} and G 2={V 2,E 2} be two graphs, where V 1,2 are sets of nodes and E 1,2 are sets of edges, a graph alignment is the mapping between the nodes of the input networks that maximizes the similarity between mapped entities. From a theoretical point of view, the graph alignment problem consists of finding an alignment function (or a mapping) f:V 1→V 2 that maximizes a cost function Q. The similarity between the graphs is defined by a cost function, Q(G 1,G 2,f), also known as the quality of the alignment.
Let f be an alignment between two graphs G 1 and G 2, given a node u from G 1, f(u) is the set of nodes from G 2 that are aligned under f to u. Q expresses the similarity among two input graphs with respect to a specific alignment f and the formulation of Q strongly influences the mapping strategy.
There exist different formulations of Q that fall into following the classes:
Topological Similarity: Graphs are aligned by considering only edge topology, so that the perfect alignment is reached when input graphs are isomorphic.
Usually, the cost function is defined as the number of edges conserved by f with respect to the total number of edges in the source network (G 1), also referred to as edge correctness (EC) [27]. Therefore, the EC does not take into account the target network (G 2).
$$ EC= \frac{(v_{1},v_{2})\in E_{1}| f(v_{1},v_{2})|\in E_{2} }{|E_{1}|} $$
Another typical measure is the Induced Conserved Structure, ICS [27]. Let D be the number of edges in a subnetwork of G 2 induced on the nodes in G 2 aligned to the nodes in G 1, ICS of f is the ratio of the number of edges conserved by f to D.
$$ ICS= \frac{|f(E_{1})| }{|E(G_{2}[f(V_{1})])|} $$
where D is |E(G 2[f(V 1)])|.
However, ICS fails in the penalization of misaligning edges in the smaller network because it takes into account the target network.
Finally, the Symmetric Substructure Score, S 3[27], takes into account the unique edges in the composite graph created by the overlap of two networks.
$$ S^{3}= \frac{|f(E_{1})| }{|E_{1}|+|E(G_{2}[f(V_{1})])|-|f(E_{1})|} $$
S 3 has been shown to be superior to existing measures since it penalizes both alignments from sparse graph regions to dense graph regions and alignments from dense graph regions to sparse graph regions.
Node Similarity: Such function considers the similarity among mapped nodes. Nodes of the aligned graphs can be more or less similar to each other. Thus the alignment should align each node of one graph to the most similar node of the other one given a node similarity functions, s(v 1,v 2)→R, v 1∈V 1, v 2∈V 2. The overall objective is to maximize the sum of scores considering aligned nodes.
$$ NC=max {sum}_{v_{1},v_{2}}=f(v_{1})s(v_{1},v_{2}) $$
Hybrid approaches: Some recent formulations of Q take into account of both of the approaches by linear combination.
The network alignment problem can be formulated in various ways. In general, the network alignment can be classified as local alignment or global alignment.
The local alignment aims to find multiple and unrelated regions of isomorphism, i.e. same graph structure, between the input networks, where each region implies a mapping independently of other regions. The strategy consists of the mapping or set of mappings between subsets of nodes such that their similarity is maximal over all possible subsets. These subnetworks correspond to conserved patterns of interaction that can represent a conserved motif or pattern of activities (a synopsis is available in [15]). The global alignment aims to find a mapping that should cover all of the nodes of the input networks, associating each node of a network with one node of the other networks or marking the node as a gap when no possible match exists. This strategy does not consider small regions of similarity, i.e. conserved motifs, but tries to find a consistent mapping between the whole set of nodes of the networks.
In this work, six global alignment algorithms were chosen to built the global alignment of brain networks. We give hereafter a short conceptual description.
A popular existing method of global alignment is MAGNA [16]. MAGNA is a global network aligner that simulates a population of alignments that evolves over time by applying a genetic algorithm and a function for the crossover of two alignments into a superior alignment. Since the genetic algorithm simulates the evolutionary process guided by the survival of the fittest principle, only alignments, i.e. those that conserve the most edges, survive. Thus, MAGNA proceeds to the next generation, until the alignment accuracy cannot be optimized further. Recently, an extension of MAGNA algorithm called MAGNA++ was developed.
The second aligner is NETAL [17], an algorithm for the global alignment widely used to protein-protein interaction networks. NETAL builds the best global network alignment by applying a greedy method, based on the alignment scoring matrix, which is derived from both biological and topological information of input networks.
The third algorithm, GHOST [18], is a global pairwise network aligner that uses a novel spectral signature based on the local neighborhood's topology to measure topological similarity between subnetworks. The idea behind GHOST consists of the combination of the novel novel spectral signature with seed-and-extend procedure to build the alignment.
The fourth global aligner is GEDEVO [19], a novel tool for efficient graph alignment.
Underlying the GEDEVO method is the Graph Edit Distance model (GED), where a graph is transferred into another one with a minimal number of edge insertions and deletions. Thus, GEDEVO uses the GED as optimization model for finding the best alignments.
The fifth algorithm is WAVE [20] a general and novel alignment strategy which aim is to optimize both node and edge conservation while constructing an alignment. WAVE is used on top of an established node cost function and it leads to a new superior method for global network alignment, by favoring conserved edges among nodes with node cost function similar over those with node cost function dissimilar.
The last algorithm is Natalie2.0 [21], a network alignment method, which looks at the network alignment problem as a generalization of the quadratic assignment problem and solves it using techniques from integer linear programming.
The dataset consisted of 24 diffusion MRI-derived structural networks of human brain: 12 networks with a number of nodes equal to 95 and the 12 networks with a number of nodes equal to 1000. The brain networks are related to three different stages of development by including newborns (NE), six-month-old infants (6M), and adults (AD). See Methods Section for a complete description.
Building of brain network alignment
We built the alignment of all networks with 95 and 1000 nodes (for convenience we call the two dataset n e t w o r k s 95 and n e t w o r k s 1000) related to same growth stages (NE, 6M, AD) by applying MAGNA++, NETAL, GHOST, GEDEVO, WAVE and Natalie2.0 algorithms. Initially, we aligned each network with itself. We executed this stage in order to test if the algorithm is able to build the alignment (see [28] for more details). Then, we aligned the brain network related to the same growth stage, NE, 6M, AD. We run all NA methods on the same Linux machine with Intel Core i5 and 4GB of RAM. We also generated the same alignments using the six NA algorithm selected. We selected the following MAGNA++ parameters: S 3 as measure of Edge Conservation, the α parameter equal to 0, in order to consider only topology, whereas the population size, number of generation, fraction of elite members were set to default values. We tested different parameters and obtained best results with the default parameters for NETAL, GHOST, GEDEVO, Natalie2.0. WAVE did not require to set specific parameters. The NETAL parameters were: a that controls the weight of similarity and interaction scores, b that controls the weight of biological and topological similarities, c that controls the contribution of neighbors of two nodes in calculating the similarity between them, i that defines the number of iterations for computing similarities. In GEDEVO, pop parameter that controls the number of new individuals per iteration set equal to 1000 and maxsame that controls the stop after N iterations without any score improvement were equal to 300. In Natalie2.0, beta set equal to 1, in order to consider only topology, whereas, maxJsonNodes that controls maximum number of nodes to be generated and verbosity that specifies the verbosity level parameters were set to default values. To build the alignment using GHOST, nneighbors was set to all, serchiter that controls the number of local search iterations that should be performed after the initial global alignment is complete, set equal to 10, beta that controls the edges alignment in the initial seed-and-extend phase of the algorithm, set equal to 1, ratio that controls ratio of bad-moves allowed during the local-search phase of the alignment algorithm, set equal to 8.0.
The global alignments were built among the n e t w o r k s 95 and then among the n e t w o r k s 1000.
At the end of this alignment step, we built 48 global alignments for each selected aligner by using the dataset n e t w o r k s 95. Table 1 presents all the obtained alignments.
Table 1 List of the alignments built among the networks with 95 nodes
About the n e t w o r k s 1000, we built 48 alignments with NETAL, GHOST, GEDEVO, WAVE according to Table 1. Since MAGNA++ requires that network 1 has fewer nodes than network 2 to build the global alignment, we aligned each smaller network, in term number of nodes, to larger networks. Finally we obtained 30 alignments built with MAGNA++. We do not have alignments by using Natalie2.0 because the algorithm was not able to build the alignment among networks with high nodes number.
Table 2 reports the execution time to build the alignment on the networks with 95 nodes and on the networks with 1000 nodes for all global alignment algorithms.
Table 2 Execution Time to build the global alignment with MAGNA++, NETAL, GHOST, GEDEVO, WAVE, Natalie2.0 for the networks with 95 nodes and the networks with 1000 nodes
Topological evaluation
Here, we aim to evaluate the quality of the alignments built with MAGNA++, NETAL, GHOST, GEDEVO, WAVE, Natalie2.0 NA algorithms. The topological quality is related to two alignment algorithm capability as the reconstruction of the true node mapping and the conservation of as much as possible edges. Typically, the Node Correctness (NC) is the measure widely used to evaluate how an alignment reconstructs the true node mapping correctly. Instead, different measures are used to evaluate how well the edges are conserved on an alignment, such as EC, ICS or S 3 (see the previous Section). However, among the selected algorithm, MAGNA++ is the unique tool that enables to compute all quality measures, NC, EC, ICS and S 3. For this reason, we computed the quality of built alignments by using the software for NA evaluation proposed in [26]. The software ensures the computation of six topological measures: Precision Node Correctness (P-NC), Recall Node Correctness (R-NC), F-score of Node Correctness (F-NC), High Node coverage (NCV), Generalized S 3 (G S 3), and NCV combined with G S 3 (NCV-G S 3). P-NC evaluates the the precision of the alignment i.e. the percentage of the aligned node pairs that are also present in the true node mapping. P-NC is defined as:
$$ P-NC= \frac{(M \cap N)}{(M)} $$
were M is the set of node pairs that are mapped under the true node mapping and N be the set of node pairs that are aligned under f.
R-NC evaluates the percentage of all node pairs from the true node mapping that are aligned under f and it is defined as:
$$ R-NC= \frac{(M \cap N)}{(N)} $$
G S 3 is the percentage of conserved edges N c out of the total of both conserved and non-conserved edges N n :
$$ GS^{3}= \frac{N_{c}}{N_{c}+N_{n}} $$
NCV is the percentage of nodes from G1 and G2 that are also in G'1 and G'2 subgraphs:
$$ NCV= \frac{V'_{1}+V'_{2}}{V_{1}+V_{2}} $$
Finally, NCV-G S 3 is the geometric mean of the NCV and G S 3 measures. These six measures evaluate alignment quality from different aspects and they can be divided in two groups, the first one composed by P-NC, R-NC and F-N measures that estimate how well the alignment captures the true node mapping, and the second one formed by NCV, G S 3 and NCV-G S 3 measures that capture the size of the alignment. We computed P-NC, R-NC, F-NC, NCV, G S 3 and NCV-G S 3 for each alignment built with MAGNA++, NETAL, GHOST, GEDEVO, WAVE, Natalie2.0. Then, we compared these measures in order to analyze which algorithm ensures a higher alignment quality. However, we focus on F-NC and NCV-GS3 as the most representative non-redundant measures because these are both a combination of two individual measures.
Figures 3 and 4 show an overview of topological measures comparison on n e t w o r k s 95 whereas Figs. 5 and 6 show an overview of topological measures comparison on n e t w o r k s 1000.
The topological evaluation of alignments built with MAGNA++ (blue marker), NETAL (red marker), GHOST (green marker), GEDEVO (purple marker), WAVE (light blue marker), Natalie2.0 (black marker). The Figure shows the F-NC scores of each alignment built among the networks with 95 nodes by applying the selected six aligners
The topological evaluation of alignments built with MAGNA++ (blue marker), NETAL (red marker), GHOST (green marker), GEDEVO (purple marker), WAVE (light blue marker), Natalie2.0 (black marker). The Figure shows the NCV-G S 3 scores of each alignment built among the networks with 95 nodes by applying the selected six aligners
The topological evaluation of alignments built with MAGNA++ (blue marker), NETAL (red marker), GHOST (green marker), GEDEVO (purple marker), WAVE (light blue marker), Natalie2.0 (black marker). The Figure shows the F-NC scores of each alignment built among the networks with 1000 nodes by applying the selected six aligners
The topological evaluation of alignments built with MAGNA++ (blue marker), NETAL (red marker), GHOST (green marker), GEDEVO (purple marker), WAVE (light blue marker), Natalie2.0 (black marker). The Figure shows the NCV-G S 3 scores of each alignment built among the networks with 1000 nodes by applying the selected six aligners
We note that the best results in terms of edge conservation were obtained when applying MAGNA++ as global aligner both on n e t w o r k s 95 and n e t w o r k s 1000. We also note that values of NCV-G S 3 for n e t w o r k s 95 are higher than NCV-G S 3 for n e t w o r k s 1000.
Regarding the reconstruction of the true node mapping we note that the quality of alignment is homogeneous among n e t w o r k s 95, with exception of the quality of 12 alignments built with MAGNA++ that was better than other algorithms. For the n e t w o r k s 1000, the F-NC values are higher for the alignment built with WAVE with exception of the alignments built with NETAL.
Robustness analysis
We analyzed the robustness of the different algorithms to various levels of graph alteration (edge removal). We generated a series of altered networks derived from the high-confidence brain network. We built the synthetic counterparts with 5, 10, 15, 20 and 25% of added noise. We obtained 60 synthetic networks with 95 nodes and 60 synthetic networks with 1000 nodes. To measure the performance of MAGNA++, NETAL, GHOST, GEDEVO, WAVE, Natalie2.0, we aligned the high-confidence brain network with its noisy counterparts obtained by random removal of edges from the network. Since all networks contain the same nodes, we know the true node mapping. The high-confidence network is an exact subgraph of each noisy network. Exploiting randomness, we ran each experiment 30 times and averaged results over the 30 runs [26]. The aim of the test was to demonstrate that the alignment algorithms are capable of producing high-quality alignments with edge conservation of about 90%. This evaluation test has been widely adopted in different NA studies (see [18, 29]). We performed this test on the brain networks built with each selected global aligner and with NETAL. The results show that, given the high topological similarity of the aligned network with its noisy counterpart, MAGNA++, NETAL, GHOST, GEDEVO, WAVE, Natalie2.0 are capable of discovering alignments with high edge conservation. The better performance was achieved with MAGNA++. Figures 7 and 8 show the validation of the edge conservation when introducing increasing noise level from 5 to 25% into the high-confidence brain networks.
The robustness evaluation of alignments to various alteration levels of networks with 95 nodes. The figure shows the trend of the edge conservation related to alignment of the high-confidence brain network with the synthetic counterparts at 5, 10, 15, 20 and 25% of added noise. The alignments are built with MAGNA++ (blue marker), NETAL (red marker), GHOST (green marker), GEDEVO (purple marker), WAVE (light blue marker), Natalie2.0 (black marker)
The robustness evaluation of alignments to various alteration levels of networks with 1000 nodes. The figure shows the trend of the edge conservation related to alignment of the high-confidence brain network with the synthetic counterparts at 5, 10, 15, 20 and 25% of added noise. The alignments are built with MAGNA++ (blue marker), NETAL (red marker), GHOST (green marker), GEDEVO (purple marker), WAVE (light blue marker), Natalie2.0 (black marker)
Understanding brain connectivity can shed light on the brain cognitive functioning that occurs via the connections and interaction between neurons. The term brain connectivity refers to different aspects of brain organization including anatomical connectivity consisting of axonal fibers across cortical regions and functional connectivity defined as the observed statistical correlations of the BOLD signal between regions of interest. A powerful formalism to represent the brain connectivity derives from graph theory. The graph theoretical modeling of the human connectome has already enabled important discoveries and will most likely continue to do this in the future. In this study we proposed to apply classical global alignment algorithms such as MAGNA++, NETAL, GHOST, GEDEVO, WAVE, Natalie2.0, to align atlas-free human brain networks at three developmental stages. We analyzed the alignment results in term of topological quality measures and performance. According to these analyses, MAGNA++ resulted the best alignment algorithm. Our ongoing study is focused on the implementation of an ad hoc algorithm for connectome alignment. Since there are many conditions in which the classical parcellation is not useful, we retain that this seminal work may open the way for the use of network alignment in atlas-free parcellation.
The dataset consisted of diffusion MRI-derived structural networks of human brain at different stages of development, starting with newborns [13]. Acquisition of the MRI data was compliant with the Health Insurance Portability and Accountability Act (HIPAA) and the study was approved by the Committee on Human Research (CHR) of the University of California, San Francisco. Three age groups were included: 4 newborns imaged in the first 4-5 days of life (NE), 4 six-month-old infants (6M), and 4 adults (age 24-31 years) (AD). The two pediatric groups had transient encephalopathy at birth, but none of the patients had clinical or imaging evidence of brain injury. The subjects were scanned on a 3T GE MR scanner using a spin echo (SE) echo planar imaging (EPI) diffusion tensor imaging DTI sequence with parameter described in [13]. Tensor calculation, tractography, cortical parcellation into 95 equal-area nodes and then in 1000 equal-area nodes (Fig. 2), and construction of the connectivity matrices was performed as described previously [13]. All networks were binarized with a threshold of 1 streamline. Starting form the images we obtained two different datasets. The first dataset consist of 12 networks with number of nodes equal to 95 depending on parcellation step. For convenience we call this dataset n e t w o r k s 95. Table 3 shows the networks parameters. About the second dataset, the 12 networks were constructed by setting the number of equal-area nodes for the cortical parcellation equal to 1000. Since all cortical areas of the brain are connected, a fine parcellation should ensure the interconnectedness of the whole brain, leaving no nodes isolated. In [13] the authors demonstrated that the highest number of nodes at which this condition is fulfilled in equal to 95. For this reason, the networks of the second dataset showed the isolated nodes that were not computed in the construction of the connectivity matrices. For convenience we call this dataset n e t w o r k s 1000 even though the nodes number is different from 1000. Table 4 shows the network parameters.
Table 3 Details of brain networks with 95 nodes used for experiments
Table 4 Details of brain networks with 1000 nodes used for experiments
Alignment algorithms
In this section we describe in detail the global alignment algorithm selected to align the diffusion brain networks.
MAGNA [27] is a global network aligner that uses a genetic algorithm to build an improved alignment starting from existing ones (generated randomly or by using other aligners). While the alignment is constructed, MAGNA optimizes the edge conservation, without decreasing the quality of node mapping. MAGNA is the first algorithm that uses genetic algorithms to build global alignment. In specific, MAGNA simulates a population of alignments that evolves over time by applying the genetic algorithm and a function for crossover of two alignments into a superior alignment. The genetic algorithm simulates the evolutionary process, guided by the survival of the fittest principle. The genetic algorithm input consist of a initial population of a given number of members. In MAGNA, the members of a population are alignments. Members of a population crossover with each other to produce new members. Only the fittest members are more likely to crossover. Thus, the child resulting from a crossover function reflects each parent. To avoid the size of the population to grow without bound, the size is kept constant across all generations, with only the fittest members surviving from one generation to the next. Thus, as the algorithm progresses, only fittest alignments, i.e. those that conserve the most edges, survive and MAGNA proceeds to the next generation, until the alignment accuracy cannot be optimized further. The fittest alignment from the last generation is reported as the final alignment. Practically, MAGNA takes as input two networks with different nodes number (|V 1|<|V2|) and builds the final global alignment. Moreover, to build the alignment MAGNA requires several parameters such as the type of initial population, population size, maximum number of generations (i.e. iterations of the genetic algorithm), and optimization function (i.e., alignment quality measure). Furthermore, MAGNA introduces new and superior alignment quality measure that takes the best from each existing measure, The Symmetric Substructure Score (S 3) [27]. that takes into account the unique edges in the composite graph created by the overlap of two networks:
$$ S^{3}= \frac{|f(E_{1})| }{|E(G_{2}[f(V_{1})-|f(E_{1})| |} $$
S 3 has been shown to be superior to existing measure, see [27] for more details.
There exists a MAGNA extension, named MAGNA++ [30], that introduces important improvements.
While MAGNA maximizes edge conservation during the alignment, MAGNA++ enables both the maximization of any different measures of edge conservation (EC, ICS, S 3) and any desired node conservation measure. Let us define S N and S E , as node and edge conservation measures, then MAGNA++ maximizes the following measure:
$$ \alpha S_{E} + (1-\alpha)S_{N} $$
where α controls the contribution of each node and edge conservation measures and takes the values between 0 and 1. In this way the alignment quality results are improved when MAGNA++ is compared with only node conservation or only edge conservation. Moreover, MAGNA++ provides a graphical user interface for easy use and offers source code for easy extensibility.
NETAL [17] is a global aligner tool that applies a greedy method, based on the alignment scoring matrix derived from biological and topological information of input networks to find the best global network alignment. The alignment building consists of two phases. In the first phase, the Alignment Score Matrix is constructed, exploiting two matrices called Similarity Score Matrix and Interaction Score Matrix. The Similarity Score Matrix is generated from the weighted sum of topological and biological similarities between every two nodes of input networks. The topological and biological similarities are respectively based on the structure of the networks and biological properties (for example in proteins networks). During the building of the alignment the score of the matrix remain always fixed. The Interaction score is based on the estimate of expected value of the number of conserved interactions that involve the node of network 1 aligned with the node of network 2. Since the expected value of the number of conserved interactions changes after the alignment, the interaction scores should be updated iteratively, just as the alignment score matrix should change. In the second phase, a greedy search is used to find the global alignment based on the values of the alignment score matrix. At first, the node pairs with maximum alignment score are chosen and aligned to each other. Then interaction score matrix is updated, and the alignment score matrix is changed based on the new values. The greedy search proceeds until all the nodes of the first network are aligned with the nodes of the second network. Practically, given two networks, NETAL finds an injective mapping so each node in the smaller network is mapped to one node in the larger network. The output consists of final global alignment. Moreover, to build the alignment NETAL requires several parameters such as a value that controls the weight of similarity and interaction scores, b that controls the weight of biological and topological similarities, c that controls the contribution of neighbors of two nodes in calculating the similarity between them, i value that defines the number of iterations for computing similarities.
GHOST is global pairwise network aligner widely used in PINs analysis. GHOST introduces a novel spectral signature based on the application of spectral graph theory to build a topological similarity measure. Initially, a set of different spectra and signatures are extracted by considering the induced subgraphs for a range of different radii centered about each node of the network. Then, the structural distance, defined on the spectral densities of two graphs, is computed between the signatures of two nodes of different networks for a sequence of radii. This signature is used to compute the similarity of nodes between different networks and to guide the building of the final global alignment. The alignment is built in two phases. In the fist phase, GHOST applies a seed-and-extend strategy to detect seed regions of an alignment. The seed regions consist of the pairs of nodes from the different networks which the structural distance is minimal. Then, GHOST expands the alignment around the neighborhoods of these pairs of nodes until all nodes of the smaller network are aligned with the nodes of the larger network. In the second phase, GHOST applies a local search strategy to improve the alignment. GHOST explores the pair of nodes aligned and realigns the nodes to obtain an alignment similar to the initial one but with superior topological quality.
GEDEVO is a global network aligner based on an evolutionary algorithm that uses the Graph Edit Distance (GED) as optimization model for finding the best alignments. The GED is a general model for the Graph Matching problem and it is defined as the minimal modifications required to transfer a graph into another graph. So, lets one-to-one mapping f among two networks, the GED model counts the inserted or deleted edges induced by the mapping f. According to this, the best alignment shows the lowest graph editing cost. The GEDEVO builds the alignment by generating an initial mapping f with random permutations and then each pair of nodes is evaluated by using the pairScore. The pairScore reflects how well two nodes correspond in a given mapping f. The pairScore depend on: the GED that computes the number of deleted and inserted edges induced by mapping of two nodes given the mapping f, and the graphlet signature distance (GSD) that computes the difference in neighboring topologies of two nodes within a distance equal to 4. Afterward, the mapping f is partitioned into two sets of pairs with low and hight scores. The high scoring pairs are swapped randomly, whereas a number of randomly chosen bad pairs in the mapping are swap with directed mutations. After each swap, the scores among the new pairs are recomputed. This operation enables to keep good pairs and to swap a bad pair more often with another bad pair. At the end, the one swap that induces the best score is kept. In this way, the final score of the mapping results improved.
Weighted Alignment VotEr (WAVE) is a novel algorithm which builds an alignment by maximizing both node and edge conservation. When WAVE is used on top of well-established node cost functions, the alignment results improved with respect to different methods that optimize only node or edge conservation or treat each conserved edge the same. The reason consists of the capability of WAVE to favor conserved edges with similar NCF end nodes over those with dissimilar NCF end nodes. Furthermore, WAVE introduces a novel measure of edge conservation denoted as weighted edge conservation (WEC). WEC measure counts the number of conserved edges and weights each conserved edge by the node cost function based similarity of its end nodes. Thus, the edges with highly similar end nodes are preferred to be aligned over the edges with dissimilar end nodes. Starting from an empty alignment, WAVE calculates the marginal gain of adding an available node pair to the alignment. The marginal gain depends on the Alignment Quality that is based on a combination of weighted edge conservation and weighted node conservation measures. Thus, the pair with the highest marginal gain is aligned. Furthermore, when a pair of nodes is aligned, this node pair has a chance to give a weighted vote to their neighbors, where weighted vote for the initial vote of each node pair derives from a node conservation measures. At each step WAVE aligns each node pair with the highest vote and votes for all the pairs of neighbors. Finally, the vote that a node pair gets from its aligned neighbors is the marginal gain to the objective function of aligning them.
Natalie 2.0 is an open source software for global network alignment which supports different scoring schemes taking into account both node-to-node correspondences and network topologies. By formulating the global network alignment as a mathematical program, this can be considered as a special case of the well-studied quadratic assignment problem. Natalie 2.0 focuses on sparse network alignment, where each node can be mapped only to a typically small subset of nodes in the other network. This corresponds to a quadratic assignment problem instance with a symmetric and sparse weight matrix. Thus, Natalie 2.0 obtains strong upper and lower bounds for the problem by improving a Lagrangian relaxation approach.
Assessment of alignment algorithms
To compute the quality of built alignments we used the software for fairly evaluating a NA method proposed by [26]. The software provides a GUI and python source code for any platform. The software aims to analyze an alignment allowing both topological and biological evaluation. In this work we focused only on a topological evaluation. The software requires an input alignment built with any NA method. This input alignment must be provided in the form of aligned node pairs. Once the alignment is supplied, six topological measures, Precision NC (P-NC), Recall NC (R-NC), F-score (F-NC), High Node coverage (NCV), Generalized S 3 (G S 3), and NCV combined with G S 3 (NCV-G S 3) can be selected. To compute topological evaluation with P-NC, R-NC, and F-NC, the true node mapping between the aligned networks as additional input must be provided. For the topological measures NCV, GS3 and NCV-GS3, the two aligned networks are required as additional input. The biological evaluation is allowed by selecting four measures, GO correctness (GC), Precision of known protein function prediction (P-PF), Recall of known protein function prediction (R-PF) and F-score of known protein function prediction (F-PF). For the calculation of these biological measures, GO data of both aligned networks are required as input. The run time to compute the evaluation measures is few seconds.
BOLD:
Blood oxygenation level dependent
CHR:
Committee on human research
DTI:
Diffusion tensor imaging
DWI:
Diffusion weighted imaging
DW-MRI:
Diffusion weighted magnetic resonance imaging
EC:
Edge correctness
F-NC:
F-score
F-PF:
F-score of known protein function prediction
fMRI:
Functional magnetic resonance imaging
GC:
GO correctness
GED:
Graph edit distance
GSD:
Graphlet signature distance
G S 3 :
Generalized S 3
HIPAA:
ICS:
Induced conserved structure
Network alignment
NCV:
High node coverage
NCV-G S 3 :
NCV combined with G S 3
Protein interaction networks
P-NC:
Precision NC
P-PF:
Precision of known protein function prediction
ROIs:
Region of interest
R-NC:
Recall NC
R-PF:
Recall of known protein function prediction
S 3 :
Symmetric substructure score
6M:
Six-month-old
WEC:
Weighted edge conservation
WAVE:
Weighted alignment voter
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PHG, MM, PV and MC have been partially supported by the following research project funded by the Italian Ministry of Education and Research (MIUR): BA2Know-Business Analytics to Know (PON03PE_00001_1).
The cost of this research and publication were founded by the Italian Ministry of Education and Research (MIUR), project BA2Know-Business Analytics to Know (PON03PE_00001_1).
Softwares used in this article are available on their own websites.
PHG and MM conceived the main idea of the algorithm and designed the tests. MC and PV supervised the design of the algorithm. PHG and MM designed the functional requirements of the software tool. OT performed medical experiments and participated in the design of the algorithm. CH and DX supervisioned medical experiments and performed data interpretation. All authors read and approved the final manuscript.
About this supplement
This article has been published as part of BMC Bioinformatics Volume 18 Supplement 6, 2017: Proceedings of the 3rd International Workshop on Data Mining and Visualization for Brain Science in conjunction with 7th ACM Conference on Bioinformatics, Computational Biology, and Health Informatics (ACM BCB'16). The full contents of the supplement are available online at http://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-18-supplement-6.
Department of Surgical and Medical Sciences, University of Catanzaro, Catanzaro, Italy
Marianna Milano, Pietro Hiram Guzzi, Pierangelo Veltri & Mario Cannataro
Department of Radiology University of California, San Francisco, USA
Olga Tymofieva, Duan Xu & Christofer Hess
Marianna Milano
Pietro Hiram Guzzi
Olga Tymofieva
Duan Xu
Christofer Hess
Pierangelo Veltri
Mario Cannataro
Correspondence to Pietro Hiram Guzzi.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Milano, M., Guzzi, P., Tymofieva, O. et al. An extensive assessment of network alignment algorithms for comparison of brain connectomes. BMC Bioinformatics 18, 235 (2017). https://doi.org/10.1186/s12859-017-1635-7
Human connectome
Alignment network algorithms | CommonCrawl |
\begin{document}
\centerline{}
\centerline{}
\centerline {\Large{\bf Representation of vector fields}}
\centerline{}
\centerline{\bf {A. G. Ramm$\dag$\footnotemark[3]}}
\centerline{}
\centerline{$\dag$Mathematics Department, Kansas State University,}
\centerline{Manhattan, KS 66506-2602, USA}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \footnotetext[3]{Corresponding author. Email: [email protected]}
\newtheorem{Theorem}{\quad Theorem}[section]
\newtheorem{Definition}[Theorem]{\quad Definition}
\newtheorem{Corollary}[Theorem]{\quad Corollary}
\newtheorem{Lemma}[Theorem]{\quad Lemma}
\newtheorem{Example}[Theorem]{\quad Example}
\newtheorem{Remark}[Theorem]{\quad Remark}
\newtheorem{thm}{Theorem}[section] \newtheorem{cor}[section]{Corollary} \newtheorem{lem}[section]{Lemma} \newtheorem{dfn}[section]{Definition} \newtheorem{rem}[section]{Remark}
\newcommand{\begin{equation*}}{\begin{equation*}} \newcommand{\end{equation*}}{\end{equation*}} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}} \newcommand{\begin{align}}{\begin{align}} \newcommand{\end{align}}{\end{align}} \newcommand{\par\noindent}{\par\noindent} \newcommand{\mathbb{R}^3}{\mathbb{R}^3}
\begin{abstract} \noindent A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in $\mathbb{R}^3$, decaying at infinity, from the knowledge of its $\nabla \times A$ and $\nabla \cdot A$. The representation of $A$ as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded $C^2-$smooth domain. \end{abstract}
{\bf Mathematics Subject Classification:} MSC 2010, 26B99, 76D99, 78A99 \\
{\bf Keywords:} vector fields; representation of vector fields
\section{Introduction}
In fluid mechanics and electrodynamics one is often interested in the following questions: \begin{itemize} \item[Q1.] Let $A(x), x \in \mathbb{R}^3$, be a twice differentiable in $\mathbb{R}^3$ vector field vanishing at infinity together with its two derivatives. Given $\nabla \times A$ and $\nabla \cdot A$, can one recover $A(x)$ uniquely? Can one give an explicit formula for $A(x)$? \item[Q2.] Can one find a scalar field $u = u(x)$ and a divergence-free vector field $B(x)$, $\nabla \cdot B = 0$, such that \begin{equation}\label{eq1} A = \nabla u + B, \qquad \int_{\mathbb{R}^3}\nabla u \cdot B dx = 0. \end{equation} \end{itemize}
These questions were widely discussed in the literature, for example, in \cite{1} - \cite{3}. Our aim is to give a simple answer to these questions. By $H^m(\mathbb{R}^3)$, $H^m(D)$, the usual Sobolev spaces are denoted, $H^m(D, w(x))$ is the weighted Sobolev space, where $w=w(x)>0$ is the weight function.
\section{Answer to question Q1.} Denote $\nabla \times A := a$, $\nabla \cdot A := f$. Then $\nabla \times \nabla \times A = \nabla \times a$. It is well known that \begin{equation}\label{eq2} -\nabla^2A = \nabla \times \nabla \times A - \nabla \, \nabla \cdot A. \end{equation} Thus, \begin{equation}\label{eq3} -\nabla^2 A = \nabla \times a - \nabla f.
\end{equation} Let $g(x,y) := \frac{1}{4\pi|x - y|}$. Then \begin{equation}\label{eq4} -\Delta g(x,y) = \delta(x - y), \end{equation} where $\delta(x)$ is the delta function. Thus, from \eqref{eq3} one gets \begin{equation}\label{eq5} A(x) = \int_{\mathbb{R}^3}g(x,y) \nabla \times a dy - \int_{\mathbb{R}^3}g(x,y) \nabla f dy. \end{equation}
{\em This formula gives an analytical representation of $A(x)$ in terms of $a = \nabla \times A$ and $f = \nabla \cdot A$.}
To prove {\em the uniqueness of this representation,} assume that there are two different vector fields $A$ and $F$ that have the same $a = \nabla \times A = \nabla \times F$ and $f = \nabla \cdot A = \nabla \cdot F$. Then, by formula \eqref{eq2}, one has \begin{equation}\label{eq6} -\nabla^2(A - F) = 0. \end{equation}
Therefore $A - F$ is a harmonic function in $\mathbb{R}^3$ which vanishes at infinity. By the maximum principle such a function is equal to zero identically.
{\em Thus, $A(x)$ is uniquely determined in $\mathbb{R}^3$ by formula \eqref{eq5} if $\nabla \times A$ and $\nabla \cdot A$ are known and if $A$ vanishes at infinity.}
$\Box$
\section{Answer to question Q2.} Formula \eqref{eq5} can be written as \begin{equation}\label{eq7} A(x) = \nabla \times \int_{\mathbb{R}^3}g(x,y) a(y) dy - \nabla \int_{\mathbb{R}^3}g(x,y) f(y)dy, \end{equation} provided that $a(y)$ and $f(y)$ decay at infinity sufficiently fast, for example, if
$$|A(x)| + |\partial A(x)| + |\partial^2 A(x)| \leq c(1 + |x|)^{-\gamma}, \gamma > 3,$$ where $\partial$ is an arbitrary first order derivative, so that the following integrations by parts can be justified: \begin{equation}\label{eq8} \nabla \times \int_{\mathbb{R}^3}g(x,y) a(y) dy = - \int_{\mathbb{R}^3}\left[ \nabla_y g(x,y), a(y) \right]dy = \int_{\mathbb{R}^3}g(x,y)\nabla \times a(y) dy, \end{equation} \begin{equation}\label{eq9} -\nabla \int_{\mathbb{R}^3}g(x,y) f(y) dy = \int_{\mathbb{R}^3}\nabla_y g(x,y) f(y) dy = - \int_{\mathbb{R}^3} g(x,y) \nabla f(y) dy.
\end{equation} One may also assume that $A \in H^2(\mathbb{R}^3, 1 + |x|^\gamma), \gamma > 2,$ in order to justify formulas \eqref{eq1}, \eqref{eq5}, \eqref{eq7}.
It follows from \eqref{eq1} and \eqref{eq7} that \begin{equation}\label{eq10} u(x) = -\int_{\mathbb{R}^3} g(x,y) f(y) dy, \qquad B(x) = \nabla \times \int_{\mathbb{R}^3} g(x,y) a(y) dy. \end{equation}
To check the second formula \eqref{eq1}, it is sufficient to check that \begin{equation} \label{eq11}
\int_{\mathbb{R}^3} \nabla u \cdot \nabla \times p dx=0, \end{equation}
provided that $u=u(x)$ and $p=p(x)$ decay at infinity sufficiently fast. In our case $u$ is defined by formula \eqref{eq10} and $p(x)=\int_{\mathbb{R}^3} g(x,y) a(y)dy$.
Formula \eqref{eq11} can be verified by a direct calculation. Let $\frac{\partial u}{\partial x_j}:= u_{,j}$ and denote by $e_{jmq}$ the antisymmetric unit tensor: $e_{123}=1, e_{jmq}=\left\{ \begin{array}{ll}
1 & \text{ if $jmq$ is even},\\
-1 & \text{ if $jmq$ is odd}. \end{array} \right.$ The triple $jmq$ is called even if by an even number of transpositions it can be reduced to the triple 123. An odd triple $jmq$ is the one that is not even. A transposition is the change of the order of two neighboring indices.
The vector product can be written with the help of $e_{jmq}$ as follows: $$A \times B = [A,B]=e_{jmq} A_m B_q.$$ Here and below summation is understood over the repeated indices. For example, $(\nabla \times p)_j=e_{jmq} p_{q,m}$. With these notations one has \begin{equation} \label{eq12}
\int_{\mathbb{R}^3} \nabla u\cdot \nabla\times p dx=e_{jmq}\int_{\mathbb{R}^3} u_{,j}p_{q,m}dx=-e_{jmq} \int_{\mathbb{R}^3} u p_{q,mj}dx=0, \end{equation} because $e_{jmq}p_{q,mj}=0$.
Let us summarize the results.
\begin{thm} \label{thm1}
Assume that a vector field $A(x)\in H_{loc}^2(\mathbb{R}^3)$ decays at infinity sufficiently fast, for example, $A(x)\in H^2(\mathbb{R}^3, 1+|x|^\gamma), \gamma >2$. Then, given $a:=\nabla\times A$ and $f:=\nabla\cdot A$ in $\mathbb{R}^3$ one can uniquely recover $A$ by formula \eqref{eq5}.
Moreover, one can uniquely represent $A(x)$ by formula \eqref{eq1}, where $u$ and $B$ are uniquely defined by formula \eqref{eq10}. \end{thm} \begin{thm} \label{thm2}
Assume that $D \subset \mathbb{R}^3$ is a bounded domain with $C^2$-smooth boundary $S$, $A(x) \in H^2(D), a(x):=\nabla\times A(x), f:=\nabla\cdot A(x)$ and $\phi(s)=A|_{s \in S}$ are known. Then $A(x)$ is uniquely recovered by solving the Dirichlet problem
\begin{equation} \label{eq13}
-\nabla^2 A=\nabla\times a(x)-\nabla f(x) \text{ in }D, \qquad A|_S=\phi(s).
\end{equation} \end{thm} \begin{proof}
Theorem \ref{thm1} is already proved.
To prove Theorem \ref{thm2} one reduces it to solving problem \eqref{eq13}. Existence and uniqueness of the solution to the Dirichlet problem \eqref{eq13} are known, so Theorem \ref{thm2} is proved. \end{proof} \begin{Remark}
It follows from formula \eqref{eq7} that if $f=\nabla\cdot A=0$, then $A=B=\nabla\times \int_{\mathbb{R}^3} g(x,y)a(y)dy$, and if $a=\nabla\times A=0$, then $A=\nabla u$, where $u$ is defined in formula \eqref{eq10}. \end{Remark} \begin{Remark}
Under the assumption $\gamma>3$, vector field $A(x)$ decays at infinity so that formulas \eqref{eq1}, \eqref{eq5}, and \eqref{eq10} are valid. \end{Remark}
Let us estimate, for example, an integral of the type \eqref{eq9} assuming that $|\nabla f|\le \frac{c}{(1+|x|)^\gamma}, \gamma > 3$. Let $|x|=r, |y|=\rho$, $\theta$ be the angle between $x$ and $y$, and $x$ is directed along $y_3$ axis. Then one has \begin{align}
I_1: &=\int_{\mathbb{R}^3} \frac{dy}{|x-y|(1+|y|)^\gamma} \\
&=2\pi \int_0^\infty \frac{dr r^2}{(1+r)^\gamma}\int_{-1}^1 \frac{ds}{(r^2-2r\rho s +\rho^2)^{1/2}} \\
&=\frac{\pi}{\rho} \int_0^\infty \frac{dr r}{(1+r)^\gamma} (r+\rho -|r-\rho|) \\
&=\frac{\pi}{\rho} \left(2\int_0^\rho \frac{dr r^2}{(1+r)^\gamma} + \int_\rho^\infty \frac{dr r 2 \rho}{(1+r)^\gamma} \right)\\
&\le \frac{\pi}{\rho}\left( 2\left.\frac{(1+r)^{-\gamma+3}}{-\gamma+3}\right|_0^\rho + 2\rho\left.\frac{(1+r)^{-\gamma+2}}{-\gamma+2}\right|_\rho^\infty \right) \\
&\le \frac{2\pi}{\rho(\gamma-3)}+\frac{2\pi}{\gamma-2}\frac{1}{\rho^{\gamma-2}}. \end{align}
If $A \in H^2(\mathbb{R}^3, (1+|x|)^\gamma), \gamma>2$, let us estimate, for example, the following integral: \begin{align}
I_2^2 &:= \left(\int_{\mathbb{R}^3}\frac{1}{|x-y|}|\nabla\times a|dy \right)^2 \\
&\le \int_{\mathbb{R}^3} \frac{dy}{|x-y|^2(1+|y|)^\gamma} \int_{\mathbb{R}^3}|\nabla\times a|^2(1+|y|)^\gamma dy \\
&\le c\int_{\mathbb{R}^3} \frac{dy}{|x-y|^2(1+|y|)^\gamma} \\
&\le 2\pi c\int_0^\infty \frac{dr r^2}{(1+r)^\gamma}\int_{-1}^1 \frac{ds}{r^2-2r\rho s +\rho^2} \\
&=\frac{\pi c}{\rho} \int_0^\infty \frac{dr r}{(1+r)^\gamma} \ln \frac{c_1 \ln\rho}{\rho}, \quad \rho>1. \end{align} By $c, c_1>0$ estimation constants are denoted.
\end{document} | arXiv |
Demystifying p-value in analysis of variance (ANOVA)
In analysis of variance (ANOVA), p-value is very often used to determine whether an initial hypothesis is accepted or rejected. However, many people still do not know what p-value is. In fact, there is a better parameter to use to accept to reject an initial hypothesis.
Mar 5, 2022 • 8 min read
In analysis of variance (ANOVA), p-value is very often used to determine whether an initial hypothesis ($H_{0}$) is accepted or rejected. However, many people still do not know what p-value is. In fact, there is a better parameter to use to accept to reject an initial hypothesis.
In this post, we will discuss what p-value is and what quantitively a better parameter to use to accept or reject hypothesis.
Let us go into the discussion!
Analysis of variance (ANOVA): A practical introduction
ANOVA analysis was first developed by R. Fischer in 1920. The main goal of Fischer was to analyse and improve the productivity of farm crops.
ANOVA is one of the most used and common tools in statistics and data analysis. This analysis has many applications in all fields, including engineering, medicine, exact science and social science.
With ANOVA analysis, we can perform a simultaneous analysis of effects or factors, with more than two levels or treatments for each effect or factor, of an experiment or observation.
The simultaneous analysis can be done because ANOVA is the results of the generalisation of t-test analysis with two levels or treatments to test the mean equality from two types of population. Hence, the ANOVA can be used to test the mean equality for more than two types of population.
A very common example of ANOVA analysis in measurement science is to analyse, optimise and improve various types of industrial and measurement processes with gauge repeatability and reproducibility (GR&R) test.
Another example of ANOVA analysis to optimise a measurement process is to optimise the value of scanning speed parameter and number of sampled points in a geometric measurement process using coordinate measuring machine (CMM). This optimisation process is performed so that the measurement can have minimal measurement uncertainty.
In this practical introduction, the basic model of ANOVA test, that is one-way ANOVA, will be presented. This basic model of ANOVA is the basis of all others and more complex ANOVA analyses.
The basic ANOVA analysis is presented with example as follow.
Suppose, we have an experiment to optimise the heat treatment process of a material. We want to analysis one factor that affect the process that is temperature with various levels m (100 degree C, 200 degree C, 300 degree C). We want to compare these m levels to find which level gives the best treatment result. Level mcan be called as the number of treatments or levels.
On each level, experiments are repeated r times. Hence, the measurement results of the heat threated materials from the experiment are random variables.
The data observation model of one-way ANOVA analysis shown in table 1. In table 1, the data table contains $y_{ij}$ that represents the $j$-th observation at the $i$-th treatment.
Table 1: The structure of data observation for one-way ANOVA analysis (one factor with more than two levels for each factor).
From the data structure shown in table 1 above, the data can be modelled as:
Where $i=1,2,…,p$ and $j=1,2,…,r$. $\mu$ is the mean and is constant. $\tau _{i}$ is a random variable from the $i$-th treatment or level. $\epsilon _{ij}$ is the error random variable.
The model above is assumed such that $\tau _{i} \sim N(0,\sigma _{\tau}^{2})$ and $\epsilon _{ij} \sim N(0, \sigma _{\epsilon}^{2})$.
In ANOVA analysis, there are two main types of model:
Random-effect model
Fixed-effect model
Random-effect model assumes that a treatment represents a random sample from a population. Hence, random-effect model has a conclusion that can be generalised to the population where the sample drawn from.
Fixed-effect model has a limited conclusion from an analysis result and the conclusion cannot be generalised to other similar treatment. Fixed-effect model has treatments or levels that are selected specifically by a person who does the experiment.
Hence, fixed-effect model (by assuming that $\tau _{i}=\tau=constant$ ) can be formulated as:
Where Where $i=1,2,…,p$ and $j=1,2,…,r$. $\mu _{i}$ is the mean and is constant and $\epsilon _{ij}$ is the error random variable. Also, $\epsilon _{ij} \sim N(0, \sigma _{\epsilon}^{2})$.
The main hypothesis from the fixed-effect model is to compare whether the mean from treatments $\mu _{i}$ is statistically equal or not equal.
Meanwhile, the main hypothesis from the random-effect model is to compare the variation of treatments whether the variation is homogenous or not.
By considering fixed-effect model, the ANOVA analysis is started by partitioning the total variation into several components that are:
Where $SS_{total}$ is the sum-of-squared of total, $SS_{treatment}$ is the sum-of-squared of treatment and $SS_{error}$ sum-of-squared of error in all treatments.
The explanation of $SS_{total}= SS_{treatment}+ SS_{error}$ is as follow. $SS_{total}$ that is the measure of total variation in experiment data (including treatment and error) can be formulated as:
The equations above mean that the total variation from observation or experiment data, $SS_{total}$, can be partitioned into the sum-of-squared of the difference between the treatment mean and total mean plus the sum-of-squared of the difference between an observation in each treatment and the treatment mean.
The total of observations is $pr$. Where $p$ is the number of treatment or level and $r$ is the number of repetition for each treatment or level.
Hence, $SS_{total}$ has $pr-1$ degree of freedom, $SS_{treatment}$ has $p-1$ degree of freedom and $SS_{error}$ has $p(r-1)$ degree of freedom (because, for each repetition, the degree of freedom of the error is $r-1$).
By dividing each sum-of-squared $SS$ with its degree of freedom, we get:
Where $MS_{treatment}$ and $MS_{error}$ are the mean-squared of treatment/level and error, respectively.
The hypothesis test of the fixed-effect model is:
The test statistic for the hypothesis test of the fixed-effect model above is as follow:
$F_{0}$ has F statistical probability distribution. If $F_{0}>F_{\alpha , p-1, pr-p}$, then $H_{0}$ is rejected and a conclusion can be said that there is a significant difference among the treatment/level mean, and otherwise.
The ANOVA table for one factor and many treatment/level fixed-effect model is shown in table 2.
Table 2: the ANOVA table for one factor, multiple treatment/level fixed-effect model.
Meanwhile, figure 1 shows an example of $F$ distribution. From figure 1, if $F_{0}>F_{\alpha, 4,20}$, then $H_{0}$ hypothesis will be rejected. And otherwise, if $F_{0} \leq F_{\alpha, 4,20}$, then $H_{0}$ hypothesis will be accepted.
Figure 1: An example of F distribution.
Hence, the $F_{0}$ is a better parameter to determine whether to accept or reject hypothesis $H_{0}$. Because, in ANOVA calculations and summary, $F_{0}$ values for each factor are shown. Hence we can thoroughly compare this $F_{0}$ value among each other. The $F_{0}$ that is significantly big suggests that we must reject $H_{0}$.
Demystifying p-value in ANOVA analysis
Commonly, ANOVA analysis is performed by using a commercial statistic software such as Minitab or SPSS.
For ANOVA analysis that is performed with these commercial software, very often, these commercial software will convert the $F_{0}$ value (that is the test statistic for an hypothesis test) into $p$-value.
The value of $F_{0}$ is inversely equal to $p$-value. If the $F_{0}$ value is large, then the $p$-value will be small so that we will reject $H_{0}$. And otherwise, if the $F_{0}$ value is low, then the $p$-value will be large so that we will accept $H_{0}$.
The question is, what is actually the $p$-value?
$p$-value is the area under an $F$ distribution curve from $F_{0}$ until the tail of the $F$ distribution. That is $p$-value = $P(F_{0}<X<\infty)$.
That is why, the $F_{0}$ and $p$-value have an inverse relation. Because, the larger the value of $F_{0}$, the smaller (the lower probability) the area under an $F$ distribution curve from $F_{0}$ to the tail.
Figure 2 shows the illustration of $p$-value and its relation to $F_{0}$.
Figure 2: the relation between $F_{0}$ value and $p$-value on a $F$ distribution curve.
ANOVA analysis with Random-effect model
For ANOVA analysis with random-effect model, aspects that are compared are the variations between treatments or levels. That is why, this model is the basic model used for Gauge R&R test.
The hypothesis test for random-effect model is:
It is important to note that, for ANOVA with one factor (one-way ANOVA), the parameters of the statistical test for random-effect model is equal to fixed-effect model. However, the test statistic will be different if the ANOVA analysis has more than one factor.
To estimate the variation for each component in the random-effect model (from the treatment, error and total), the estimation is formulated as follow:
In this post, we have demystified $p$-value in ANOVA analysis.
This $p$-value is always used to accept or reject hypothesis $H_{0}$ in ANOVA analysis.
However, as being explained in this post, $F_{0}$ is a better parameter to use in accepting or rejecting the hypothesis. Because, the values of $F_{0}$ are always available for each factor.
Hence, we can thoroughly compared the $F_{0}$ values for each factor. The one that is significantly big, shows that we must reject $H_{0}$. | CommonCrawl |
\begin{document}
\begin{abstract} We study comparison properties in the category $\mathrm{Cu}$ aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that R\o rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.
\end{abstract}
\title{Comparison properties of the Cuntz semigroup and applications to C*-algebras}
\section*{Introduction} Over the last 25 years, the classification of simple, nuclear C*-algebras has inspired a great wealth of research. Recently, a classification has been carried out in two groundbreaking articles \cite{EGLN16,TWW15} for simple C*-algebras of finite nuclear dimension. Nuclear dimension plays the role of a non-commutative covering dimension for nuclear C*-algebras. Requesting this dimension to be finite is one of the strong regularity conditions occurring in the Toms-Winter conjecture, which predicts that three regularity conditions, each with a different flavour, are in fact equivalent. Counterexamples to the conjecture stating that the same classifying invariant, which works in the case of finite nuclear dimension (the so-called Elliott invariant), should work in the general case, appeared in 2003 due to R\o rdam \cite{R03} and in 2008 due to Toms \cite{T08}. The latter exhibited two non-isomorphic AH-algebras that agreed not only on the Elliott invariant, but also on a whole swathe of topological invariants. However, Toms' examples can be distinguished using the Cuntz semigroup $\mathrm{Cu}(\_)$. In this paper, we focus on studying some comparison properties - such as the corona factorization property and weak comparison conditions for the Cuntz semigroup- to capture the structure of some simple C*-algebras.
There are a number of regularity properties dividing those C*-algebras not handled by the classification theorem from \cite{TWW15} into classes of `regular' and `irregular' C*-algebras. One of them is the corona factorization property (short, CFP), which is a mild regularity property introduced in \cite{EK01} in order to understand the theory of extensions and in particular of when extensions are absorbing \cite{KN06, NgCFP}. Zhang proved that, under the additional assumption of the CFP, there is no simple C*-algebra of real rank zero containing both a finite and an infinite projection. (The same follows from the methods developed in \cite{OPR1}.) In addition to its analytical definition, the CFP has been characterized in \cite{OPR2}, for $\sigma$-unital C*-algebras, as a certain comparison property of the Cuntz semigroup, also called the CFP (for semigroups). Another related comparison property is the $\omega$-comparison, a generalization of the almost unperforation property, which holds in the case of well-behaved C*-algebras (in the sense of the above mentioned classification theorem)(\cite{Robert11}).
The study of comparison and divisibility properties for the Cuntz semigroup was initiated in \cite{BRTTW,OPR1,OPR2, RobertRordam}, where the preceding properties play important roles. In particular, in \cite[Proposition 2.17]{OPR1} it is shown that $\omega$-comparison implies the CFP for Cuntz semigroups; and the converse is left open. In Example \ref{Ex:CFP-noW}, we answer this question negatively providing an abstract Cuntz semigroup that satisfies the CFP, but not $\omega$-comparison. Our abstract semigroup lies in the category $\mathrm{Cu}$ (as defined in \cite{CEI08} and extended by additional axioms from \cite{RorWin10} and \cite{Robert13}) to which the Cuntz semigroup of a C*-algebra naturally belongs.
All of the comparison properties mentioned above have in common (suitably stated) that they are characterized by those elements in the Cuntz semigroup of a unital C*-algebra, that cannot be represented by the Cuntz equivalence class of a positive element in some matrix algebra over the given algebra, but only appear as the equivalence class of a positive element in the stabilization. If there is a largest element in the Cuntz semigroup (which does exist in the simple case), then we are more precisely concerned about properties of this largest element. In particular, we focus on questions such as : {\it if a multiple of some element $x$ in a Cuntz semigroup equals the largest element, is $x$ itself already the largest element?} Or, {\it if all functionals on the Cuntz semigroup are infinite on an element $x$, must $x$ be the largest element?} Or thirdly, {\it if the sum of two elements $x$ and $y$ equals to the largest element, and $y$ is small in a suitable sense, must $x$ already be equal to this largest element?}
To this end, we introduce a new comparison property involving the largest element in a $\mathrm{Cu}$- semigroup, which we call $\beta$-comparison, using the concept of an order unit norm as defined in \cite{Good.} as our motivation. We set this new property in relation to the other comparison properties, and we highlight differences with the help of examples.
In the C*-algebra framework, there are multiple implications which can be deduced from our study of comparison properties. Firstly, Example \ref{Ex:stablyNoFunctionals} suggests that the expected dichotomy (of being either stably finite or purely infinite) in the simple real rank zero case might require an analytical approach: The given examples of abstract $\mathrm{Cu}$-semigroups answer the corresponding question negatively in the algebraic setting, but we don't know whether these examples can be realized as the Cuntz semigroup of a C*-algebra. Secondly, we relate the comparison properties studied in this paper for those $\mathrm{Cu}$-semigroups coming from a C*-algebra. In this setting, we show that for simple C*-algebras $\beta$-comparison and $\omega$-comparison are equivalent properties and that the so-called elementary $\mathrm{Cu}$-semigroups as in \cite{APT14} can not arise as the Cuntz semigroup of a C*-algebra. Finally, in Theorem \ref{RordamsAlgebraNoCFP} we conclude that the only known example of a simple, nuclear C*-algebra with both finite and infinite projections constructed in \cite{R03}, does not have the CFP.
The outline of the paper is as follows. After fixing notation and recalling some basic facts on Cuntz semigroups in Section \ref{Sec:Preliminaries}, we explore the difference between states and functionals on $\mathrm{Cu}$-semigroups in Section \ref{SectionFunctionals}. The results are used subsequently in the definition of the value $\beta(x,y)$ in Section \ref{SectionBeta} and we explore some of its characteristics. Section \ref{SecCompProp} is focused on comparison properties. We recall all properties relevant for this paper and define both the $\beta$-comparison property, associated to the value $\beta(x,y)$, and cancellation of small elements at infinity. We further show some relations between all described comparison properties and give examples. In Section \ref{SectionApplications}, we apply and expand the results obtained in the algebraic framework to the framework of C*-algebras.
\section{Notation and Preliminaries}\label{Sec:Preliminaries} \subsection{Partially ordered Abelian Semigroups} Throughout, $(W,\leq)$ will denote a partially ordered abelian monoid, i.e., a partially ordered abelian semigroup with neutral element $0$. We shall exclusively be interested in positively ordered semigroups, i.e., semigroups where $0\leq x$ for all $x\in W$. In particular, $\leq$ will extend the algebraic order, that is, if $x+z=y$, then $x\leq y$.
In the following we want to remind the reader of some commonly used terminology. An \textbf{order unit} in $W$ is a non-zero element $u$ such that, for all $x\in W$, there is an $n\in\mathbb{N}$ such that $x\leq nu$. We define the i\textbf{deal generated by an element} $y$ as $$W_{y}:=\{x\in W \mid \text{ there exists } n\in \mathbb{N}\text{ such that } x\leq ny\}.$$ Given two elements $x,y\in W$, one writes \textbf{$x\propto y$} if $x$ satisfies $x\leq ny$ for some $n\in\mathbb{N}$.
Given an increasing sequence $(y_n)$ in $W$, an element $y$ is a \textbf{supremum} of $(y_n)$ when it is a least upper bound. When they exist, suprema of increasing sequences are unique, and we will denote them by $\sup (y_n)$. We say that an ordered abelian semigroup $(W,\leq)$ is \textbf{complete} if all the increasing sequences have suprema in $W$.
One writes \textbf{$x\ll y$} if, whenever $\{x_{n}\}$ is an increasing sequence for which the supremum exists and satisfies $y\leq \sup\,x_{n}$, then $x\leq x_{n}$ for some $n$. An element $x$ is called \textbf{compact}, if $x\ll x$. We write \textbf{$y<_{s}x$} if there exists $k\in \mathbb{N}$ such that $(k+1)y\leq kx$.
Finally, an element $x$ in $W$ is said to be \textbf{full} if for any $y',y\in W$ with $y'\ll y$, one has $y'\propto x$, denoted by $y\,\bar{\propto}\, x$. A sequence $\{x_{n}\}$ in $W$ is said to be \textbf{full} if it is increasing and for any $y',y\in W$ with $y'\ll y$, one has $y'\propto x_{n}$ for some (hence all sufficiently large) n. Notice that if $x\in W$ is an order unit, it is also a full element, but the reverse is not true.
We say that $W$ is \textbf{simple} if $x\,\bar{\propto}\, y$ for all nonzero $x,y\in W$. In other words, every nonzero element in a simple semigroup is full.
\subsection{$\mathrm{Cu}$-semigroups} Given a partially ordered abelian monoid $S$, the following axioms were introduced in \cite{CEI08} in order to define a category $\mathrm{Cu}$ of semigroups containing Cuntz semigroup $\mathrm{Cu}(A)$ of any {C*-al\-ge\-bra}\,. \begin{enumerate}[(O1)] \item Every increasing sequence $(a_n)_{n\in\mathbb{N}}$ in $S$ has a supremum in $S$. \item Every element $a\in S$ is the supremum of a sequence $(a_n)_n$ such that $a_n\ll a_{n+1}$ for all $n$. \item If $a,a',b,b'\in S$ satisfy $a'\ll a$ and $b'\ll b$, then $a'+b'\ll a+b$. \item If $(a_n)_n$ and $(b_n)_n$ are increasing sequences in $S$, then $\sup_n(a_n+b_n)=\sup_n(a_n)+\sup_n(b_n)$. \end{enumerate} A sequence as in {\rm (O2)} is called \textbf{rapidly increasing}. Moreover, note that, for semigroups in $\mathrm{Cu}$, the order satisfies $x\leq y$ if and only if $x'\leq y$ for all $x'\ll x$.
\begin{definition} A $\mathrm{Cu}$-semigroup is a partially ordered monoid that satisfies axioms {\rm (O1)-(O4)} from the above paragraph. That is, $S$ is a $\mathrm{Cu}$-semigroup precisely when $S$ lies in the category $\mathrm{Cu}$. \end{definition}
If $A$ denotes a C*-algebra $A$, then its Cuntz semigroup is the ordered semigroup of Cuntz-equivalence classes of positive elements in the stabilization of $A$, with the direct sum as addition and the order is given by Cuntz-subequivalence. We refer the reader to the overview article \cite{APT} for the definition of the Cuntz relation and information on the Cuntz semigroup of a C*-algebra. If $A$ denotes a C*-algebra, then its Cuntz semigroup $\mathrm{Cu}(A)$ is a $\mathrm{Cu}$-semigroup (\cite{CEI08}). If $a$ is a positive element in a the stabilization of a C*-algebra $A$, then we denote by $\langle a \rangle$ its equivalence class in $\mathrm{Cu}(A)$. We will also consider the (original) Cuntz semigroup $W(A)$ of equivalence classes of positive elements in matrix algebras over $A$.
There are two additional axioms (O5) and (O6) that have been shown to hold for any $\mathrm{Cu}$-semigroup $S$ coming from a C*-algebra, i.e., for any $S$ such that there is a C*-algebra $A$ with $S=\mathrm{Cu}(A)$ (see \cite{Robert13} for (O5) and \cite{RorWin10} for (O6)).
\hspace{-0.5cm}{\rm (O5)} (Almost algebraic order) If $x'\ll x \leq y$ in $S$, then there is some $z\in S$ such that $x'+z\leq y\leq x+z$.
\hspace{-0.5cm}{\rm (O6)} (Almost Riesz decomposition) If $x'\ll x\leq y_1+y_2$ in $S$, then there are elements $x_1\leq x,y_1$ and $x_2\leq x,y_2$ such that $x_1+x_2\geq x'$.
Therefore, it would be natural to include (O5) and (O6) into the definition of the category $\mathrm{Cu}$ and into the definition of a $\mathrm{Cu}$-semigroup. Since at times we would like to highlight the usage of these additional axioms, we leave the definition of the category $\mathrm{Cu}$ (and $\mathrm{Cu}$-semigroup) as it is and mention explicitily when we assume a given $\mathrm{Cu}$-semigroup to satisfy the additional axioms.
If $S$ is a $\mathrm{Cu}$-semigroup, let us recall that an element $a\in S$ is {\bf finite} if for every element $b\in S$ such that $a+b\leq a$, one has $b=0$. An element is \textbf{infinite} if it is not finite. An infinite element $a\in S$ is {\bf properly infinite} if $2a\leq a$. We say that $S$ is {\bf stably finite} if an element $a\in S$ is finite whenever there exists $\tilde{a}\in S$ with $a\ll \tilde{a}$. In particular, if $S$ contains a largest element, denoted by $\infty$, then the latter condition is equivalent to $a\ll \infty$ (\cite{APT14}). A largest element, $\infty$, always exists whenever $S$ is simple, and it is unique whenever it exists (\cite[Paragraph 5.2.2.]{APT14}). If $S$ is simple, then we say that $S$ is {\bf purely infinite} if $S=\{0,\infty\}$, i.e., if $S$ only contains of the zero element and the largest element.
We finish our preliminary part by recalling that a $\mathrm{Cu}$-semigroup $S$ is said to be {\bf algebraic} if every element $x\in S$ is supremum of a sequence of compact elements, i.e. of elements such that $a\ll a$. A $\mathrm{Cu}$-semigroup $S=\mathrm{Cu}(A)$ coming from a C*-algebra $A$ is algebraic, whenever the underlying C*-algebra $A$ has real rank zero.
\section{States and Functionals on $\mathrm{Cu}$-semigroups}\label{SectionFunctionals}
In this section, we recall the notions of an (extended valued) state and a functional on $\mathrm{Cu}$-semigroups, and we will study their interplay. While functionals (by definition) better preserve the order structure of the $\mathrm{Cu}$-semigroup, there are better results on states to conclude how two given elements order-relate. For additional information on states and functionals we refer the reader to \cite[Section 5.2]{APT14}.
For an ordered semigroup $S$ with distinguished element $y\in S$, by an (extended valued) state we mean an ordered semigroup map $f:S\rightarrow [0,\infty]$ such that $f(y)=1$. The set of states on $S$ normalized at $y$ is denoted by $\mathcal{S}(S,y)$.
By a functional on a $\mathrm{Cu}$-semigroup $S$ we mean an ordered semigroup map that preserves suprema of increasing sequences (which always exist in a $\mathrm{Cu}$-semigroup), and we denote the set of functionals on $S$ by $F(S)$.
If $S$ is a simple $\mathrm{Cu}$-semigroup, then every (nonzero) functional is faithful, but states don't need to be faithful. To understand this difference better, we consider a helpful subsemigroup. If $S$ is a simple $\mathrm{Cu}$-semigroup, then $S$ contains a largest element $\infty$, and we let $$S_{\ll\infty}:=\{s\in S\ |\ s\ll\infty\}.$$
Note that, whenever $S$ is simple, $x\in S_{\ll\infty}$ and $y\in S$ is nonzero, then there is some $n\in\mathbb{N}$ such that $x\leq n\cdot y$ (because $\infty=\sup_n n\cdot y$). This implies for both states and functionals on $S$ alike that, if a state, or functional, is zero on some nonzero element $x\in S_{\ll \infty}$, then it is zero on all of $S_{\ll \infty}$. If this happens for a functional then, as every $x\in S$ can be written as the supremum of a rapidly increasing sequence (in particular, as the supremum of elements in $S_{\ll\infty}$) and because functionals preserve suprema of increasing sequences, the functional must be zero everywhere. That is, nonzero functionals are faithful on simple $\mathrm{Cu}$-semigroups. But for states there is no such condition on suprema of increasing sequences, so a state may very well be nonzero, but zero on $S_{\ll \infty}$. In fact, faithfulness of functionals on simple $\mathrm{Cu}$-semigroups is a consequence of the more general fact that functionals are uniquely determined (also for nonsimple $S$) on $S_{\ll}=\{s\in S|\ \exists t\in S \mbox{ with } s\ll t\}$ (and states are not), which agrees with $S_{\ll\infty}$ whenever $S$ contains a largest element.
\begin{remark} If $A$ is a unital C*-algebra, then consider its (original) Cuntz semigroup $W(A)$ of equivalence classes of positive elements in matrix algebras over $A$. For every $x\in W(A)\subseteq \mathrm{Cu}(A)$ we have that $x\leq \langle N\cdot 1_A\rangle \ll \infty$ for some $N$. Hence $W(A)\subseteq \mathrm{Cu}(A)_{\ll}$. Whether the converse holds, a problem that appeared in the literature under the name of `hereditariness of the Cuntz semigroup', is an open question in general. Positive answers in quite general settings can be found in, e.g., \cite{ABPP} and \cite{BRTTW}. \end{remark}
Existence of states is connected to infiniteness of the element at which we would like to normalize. This follows from the following result of \cite{Blac.Ror.} on extensions of states on preordered semigroups, which generalizes the well-known corresponding result on extensions of states on ordered groups by Goodearl and Handelman (\cite{GoodHand}):
\begin{theorem}(\cite[Corollary 2.7]{Blac.Ror.})\label{Blac} Let $(W,\leq,u)$ be a preordered semigroup with $u$ an order-unit, and let $W_{0}$ be a subsemigroup containing $u$ (equipped with the relative preordering). Then every state on $W_{0}$ extends to a state on $W$. \end{theorem}
If $u$ is not an order unit, consider $I(u)$, the order ideal generated by $u$. Then a state on a subsemigroup $W_0$ containing $u$ can be extended to a state $f$ in $\mathcal{S}(I(u),u)$. As we consider extended valued states (i.e., we allow states to take the value $\infty$), we can extend $f$ to all of $W$ by setting $$\bar{f}(x):= \left \{\begin{array}{lll} f(x) &,\ x\in I(u) \\ \infty &,\ x\notin I(u) \end{array}\right. $$
Hence, the assumption on $u$ being an order unit can be dropped if one considers extended valued states.
Of course, $\mathcal{S}(S,y)=\emptyset$ whenever some multiple of $y$ is properly infinite. If, on the other hand, no multiple of $y$ is properly infinite, then $f(n\cdot y)=n$ is a well-defined state on $\{0,y,2y,3y,\ldots\}\subseteq S$, which extends to a state on $S$. For later reference, we put this observation into a lemma.
\begin{lemma}\label{NoStates} Let $S$ be an ordered semigroup and let $y\in S$. Then the set $\mathcal{S}(S,y)$ of states normalized at $y$ is empty if and only if some multiple of $y$ is properly infinite. \end{lemma}
For functionals there is no such characterization. Obviously, if $y$ is properly infinite, then $\lambda(y)=\infty$ for all functionals $\lambda$. But it is possible that in a (simple) $\mathrm{Cu}$-semigroup $S$ (with $S=\mathrm{Cu}(A)$ for some C*-algebra $A$) there is some $y\in S$ such that $y$ is infinite on all functionals, while no multiple of $y$ is infinite (see Example \ref{ExampleBetaVsQQ}). Further, we have the following observation:
\begin{lemma}\label{FaithfulStates} Let $S$ be a simple $\mathrm{Cu}$-semigroup and let $y\in S$. Then $\lambda(y)=\infty$ for all functionals $\lambda\in F(S)$ if and only if there is no faithful state in $\mathcal{S}(S,y)$. \end{lemma}
\begin{proof} Suppose $f\in \mathcal{S}(S,y)$ is faithful. Then $\tilde{f}(x):=\sup_{x'\ll x}f(x')$ is a functional (see \cite{Robert13}) with $0<f(y)\leq 1$. Conversely, as functionals on simple $\mathrm{Cu}$-semigroups are faithful, a suitable scaling of a functional with $\lambda(y)<\infty$ would give a faithful state on $S$. \end{proof}
Another, related, useful observation is content of the following lemma. Let us denote by $\lambda_\infty$ the functional assigning the value $\infty$ to all (nonzero) $z\in S$.
\begin{lemma}\label{InfiniteMultiples} Let $S$ be a simple $\mathrm{Cu}$-semigroup. If there is some $x\in S_{\ll\infty}$ such that $\lambda(x)=\infty$ for all $\lambda\in F(S)$, then $\lambda=\lambda_\infty$. In this case, for every (nonzero) $z\in S$, some multiple of $z$ is properly infinite. \end{lemma}
\begin{proof} The first statement is clear as for any two $x,y \in S_{\ll\infty}$ there is $m\in\mathbb{N}$ such that $x\leq m\cdot y$. For the second statement, pick (nonzero) $z\in S$ and suppose that there is a state $f\in \mathcal{S}(S,z)$. Then $\tilde{f}(x):=\sup_{x'\ll x}f(x')$ is a nonzero functional (\cite{Robert13}) with finite values, which contradicts the assumption. By Lemma \ref{NoStates}, some multiple of $z$ is properly infinite. \end{proof}
By \cite[Proposition 2.1]{OPR2} (which also follows from Goodearl and Handelman's paper \cite{GoodHand}), for elements $x,y$ in an ordered semigroup $S$, $x<_s y$ is equivalent to the statement that $f(x)< f(y)=1$ for all states $f$ normalized at $y$. It is not known to the authors whether, in the case that $S$ is a $\mathrm{Cu}$-semigroup, this is equivalent to the statement that $\lambda (x)< \lambda (y)=1$ for all functionals $\lambda$ normalized at $y$. A slightly weaker statement was shown by Robert:
\begin{lemma}(\cite{Robert13})\label{ComparisonFunctionals} Let $S$ be a simple $\mathrm{Cu}$-semigroup, let $x,y\in S$. Suppose that $\lambda(x)<\lambda(y)$ for all functionals $\lambda\in F(S)$ finite on $S_{\ll\infty}$. Then for all $x'\ll x$ we have $x'<_s y$. \end{lemma}
\begin{proof} Let $(y_n)$ be a rapidly increasing nonzero sequence in $S$ with supremum $y$.
Suppose first that there is no functional that is finite on $S_{\ll\infty}$. Then, by Lemma \ref{InfiniteMultiples}, some multiple of each $y_n$ is infinite. In particular, there is $k\in\mathbb{N}$ such that $(k+1) x\leq \infty =ky$, implying that $x'<_sy$.
Hence, from now on we may assume that for every $n\in \mathbb{N}$ there exists at least one functional normalized at $y_n$. As functionals preserve suprema of increasing sequences, and since $F(S)$ is compact (\cite{ERS11}), we find $n\in \mathbb{N}$ so that for all functionals $\lambda$, which are finite on $S_{\ll\infty}$, we have the strict inequality $\lambda(x)<\lambda(y_n)$. Let $f$ be a state normalized at $y_n$. Then $\tilde{f}(x):=\sup_{x'\ll x}f(x')$ is a nonzero functional (\cite{Robert13}). For all $x'\ll x$, we get $$f(x')\leq \sup_{z\ll x} f(z)=\tilde{f}(x)<\tilde{f}(y_n)\leq f(y_n).$$ Hence, $f(x')<f(y_n)$ for all states $f$ normalized at $y_n$, and \cite[Proposition 2.1]{OPR2} shows that $x'<_s y_n\leq y$. \end{proof}
From the above we were not able to deduce that $\lambda (x)< \lambda (y)=1$ for all functionals $\lambda$ normalized at $y$ implies that $x <_s y$, but we get a weaker statement, sufficient for our later application.
\begin{lemma}\label{UglyWayOut} Let $S$ be a simple $\mathrm{Cu}$-semigroup and $x\in S$ such that $x\ll \infty$. Then there is some $m\in \mathbb{R}$ such that whenever $y\in S$ satisfies that $\lambda(x)\cdot m<\lambda(y)$ for all functionals $\lambda\in F(S)$ finite on $S_{\ll\infty}$, then $x<_s y$. \end{lemma}
\begin{proof} Find a rapidly increasing sequence $z_n\ll z_{n+1}$ such that $\sup_{n}z_n=\infty$. Then, as $x\ll \infty$, it follows that $x\ll z_n\ll \infty$ for some $n$ and there exists some $m\in \mathbb{N}$ such that $z_n\leq m\cdot x$. Hence, $x\ll m\cdot x$. Whenever now $\lambda(m\cdot x)=m\cdot \lambda(x)<\lambda(y)$ for all functionals finite on $S_{\ll\infty}$, then we get that $x<_s y$ by Lemma \ref{ComparisonFunctionals}. \end{proof}
\section{The value $\beta(x,y)$}\label{SectionBeta}
In this section we explore a value associated to any two elements $x,y$ in a $\mathrm{Cu}$-semigroup $S$, called $\beta(x,y)$. This value is induced by extending the order-unit norm for partially ordered abelian groups as described in (\cite{GoodHand}) to semigroups. In Section \ref{SecCompProp} we will use it to define a comparison property on $\mathrm{Cu}$-semigroups.
Although we will be mainly concerned about $\mathrm{Cu}$-semigroups, we define $\beta(x,y)$ more generally for $x,y$ in an arbitrary ordered abelian semigroup $(W,\leq)$.
\begin{definition} (cf. \cite[Section 4]{GoodHand}) Let $(W,\leq)$ be an ordered abelian semigroup and $x,y\in W$ such that $x\propto y$.
We define the real number $\beta(x,y)$ as $$\beta(x,y)=\inf\{l/k \mid kx\leq ly\text{ where } k,l\in\mathbb{N}\}.$$ \end{definition}
Recall that we denote by $W_y$ the order ideal of $W$ such that for all $x\in W_y$ there exists $n\in \mathbb{N}\text{ such that } x\leq ny$. We provide an equivalent definition of the value $\beta(x,y)$ in the case that $W_y$ allows a state normalized at $y$..
\begin{proposition}\label{equality}
Let $(W,\leq)$ be an ordered abelian semigroup. If $x\in W_y$ and $\mathcal{S}(W_{y},y)\neq\emptyset$, then $$\beta(x,y)=\inf\{l/k\mid kx\leq ly\}=\sup\{f(x)\mid f\in \mathcal{S}(W_{y},y)\}.$$ \end{proposition}
\begin{proof} Let us start by defining $W_{0}:=\langle x,y\rangle=\{kx+ly\mid k,l\geq 0\}\subseteq W_y$. Note that the existence of a state $f\in \mathcal{S}(W_{y},y)$ implies the following property:
\begin{equation}\label{EqOne} \mbox{Whenever } z\in W_0 \mbox{ and } ky+z \leq ly+z\mbox{, then } k\leq l. \end{equation}
Consider the map \[ \begin{tabular}{c c c c}
$f_{0} : $ & $W_{0}$ &$\longrightarrow$ & $\mathbb{R}^{+}$ \\
& $kx+ly$& $\mapsto$ & $k\beta(x,y)+l$. \\
\end{tabular}\\\, \] We claim that $f_{0}$ is a state in $\mathcal{S}(W_{0},y)$.
As additivity is clear, we subsequently prove that $f_{0}$ is well-defined and that it preserves the order. Namely, given two elements in $W_{0}$, $k_{1}x+l_{1}y$ and $ k_{2}x+l_{2}y$, we must show that $f_{0}(k_{1}x+l_{1}y)\leq f_{0}(k_{2}x+l_{2}y)$ if $k_{1}x+l_{1}y\leq k_{2}x+l_{2}y$. We divide the proof in four cases: \begin{enumerate}[\rm (i)]
\item $k_{1}\leq k_{2}$ and $l_{1}\leq l_{2}$,
\item $k_{1}\leq k_{2}$ and $l_{1}\geq l_{2}$,
\item $k_{1}\geq k_{2}$ and $l_{1}\leq l_{2}$,
\item $k_{1}> k_{2}$ and $l_{1}> l_{2}$. \end{enumerate}
We note that {\rm (i)} is trivial and {\rm (iv)} stands in contradiction to Condition (\ref{EqOne}), so let us start showing {\rm (ii)}. In this case, since $k_{1}, k_{2}, l_{1}, l_{2}$ are integers, we can write $k_{2}=k_{1}+k'$ and $l_{1}=l_{2}+l'$ for some $k',l'\in\mathbb{N}$, obtaining $k_{1}x+(l_{2}+l')y\leq (k_{1}+k')x+l_{2}y$. Setting $z:= k_{1}x+l_{2}y$, we get $l'y+z\leq k'x+z\text{ where } z\in W_{0}$. It follows then that $ml'y+z\leq mk'x+z$ for all $m\in \mathbb{N}$.
We have to show that $k_1\beta(x,y) + l_1 \leq k_2\beta(x,y) + l_2$, equivalently $\beta(x,y)\geq l'/k'$. To end up with a contradiction, suppose that $\beta(x,y) < l'/k'$. Then, there exist $a,b\in \mathbb{N}$ such that $\beta(x,y)\leq b/a < l'/k'$ and $ax\leq by$. Then $y + k'ax\leq y + k'by \leq al'y.$ Taking $m=a$ in the above equation for the second equality, we get $$(al'+1)y+z= aly'+z+y\leq ak'x+z+y\leq al'y+z,$$ which contradicts Condition (\ref{EqOne}).
Finally to prove {\rm (iii)}, we start as before to obtain $mk''x+z'\leq ml''y+z'$ for all $m$, where $z'=k_{2}x+l_{1}y.$ As we can find $l_0$ such that $z\leq l_0y$, it follows that $mk''x\leq (ml''+l_0)y$ for all $m$ Hence, $\beta(x,y)\leq \inf \{\frac{ml''+l_0}{mk''} |\ m\in\mathbb{N}\}=l''/k''$, which implies that $f_{0}(k_{1}x+l_{1}y)\leq f_{0}(k_{2}x+l_{2}y)$ as desired.
By Theorem \ref{Blac}, $f_{0}$ extends to
$\mathcal{S}(W_{y},y)$. As all the states on $\mathcal{S}(W_{y},y)$ satisfy that $f(x)\leq \beta(x,y)$ and $f_{0}(x)=\beta(x,y)$,
we conclude that $\beta(x,y)=\sup\{f(x)\mid f\in \mathcal{S}(W_{y},y)\}$. \end{proof}
\begin{remark} \rm{Let $(W,\leq)$ be an ordered abelian semigroup, $x,y\in W$ and $x\in W_{y}$ and $\mathcal{S}(W_y,y)\neq \emptyset$. Then $$\{f(x)\mid f\in \mathcal{S}(W_{y},y)\} = \{f(x)\mid f\in \mathcal{S}(W,y)\}.$$ Indeed, let $\phi: \mathcal{S}(W_{y},y)\to \mathcal{S}(W,y)$ be the map that sends $f\mapsto \bar{f}$, where $\bar{f}$ is defined by $\bar{f}=f$ on $W_{y}$ and $\infty$ otherwise. Clearly $\bar{f}$ is a state and $\phi$ is well-defined. Notice that $\phi$ is injective. Hence, the map $\varphi: \mathcal{S}(W,y)\to \mathcal{S}(W_{y},y)$ defined by $f\mapsto f_{W_{y}}$ is surjective. It follows that $\{f(x)\mid f\in \mathcal{S}(W_{y},y)\}= \{f(x)\mid f\in \mathcal{S}(W,y)\}$, and that $\beta(x,y)=\inf\{l/k\mid kx\leq ly\}=\sup\{f(x)\mid f\in \mathcal{S}(W,y)\}.$} \end{remark}
The next lemmas show some properties of the value $\beta(x,y)$.
\begin{lemma}\label{properties}
Let $(W,\leq)$ be an ordered abelian semigroup and $x,y,z\in W$. \begin{enumerate}[\rm(i)]
\item If $x \propto y$ and $y\leq z$ then $\beta(x,y)\geq \beta(x,z)$.
\item If $y \propto z$ and $x\leq y$ then $\beta(x,z)\leq \beta(y,z)$. \end{enumerate} \end{lemma} \begin{proof} $\rm(i)$ If $kx\leq ly$, then $kx\leq ly\leq lz$; thus, $$\{l/k\mid kx\leq ly\}\subseteq\{m/n\mid nx\leq mz\}.$$ Therefore, $\inf\{l/k\mid kx\leq ly\}\geq \inf\{m/n\mid nx\leq mz\}.$
$\rm(ii)$ If $ky\leq lz$ then $kx\leq ky\leq lz$; thus, $$\{l/k\mid ky\leq lz\}\subseteq \{m/n\mid nx\leq mz\}.$$ Therefore, $\inf\{l/k\mid ky\leq lz\}\geq \inf\{m/n\mid nx\leq mz\}$. \end{proof}
Note that $\beta(x,y)< 1$ if and only if $x<_s y$. (Recall that $x<_s y$ if $(k+1)\cdot x\leq k\cdot y$ for some $k\in\mathbb{N}$.)
\begin{lemma}\label{properties2} Let $(W,\leq)$ be an ordered abelian semigroup. Then:
\begin{enumerate}[\rm(i)]
\item If $x\in W$ and $\{y_{n}\}$ is a sequence in $W$ satisfying $x\propto y_{j}$ for all $j$,
$\beta(x,y_{1}+\ldots+y_{n})\leq (\sum^{n}_{j=1}\beta(x,y_{j})^{-1})^{-1}$,
\item If $y\in W$ and $\{x_{n}\}$ is a sequence in $W$ satisfying $y\propto x_{j}$ for all $j$,
$\beta(x_{1}+\ldots+x_{n},y)\leq \sum^{n}_{j=1}\beta(x_{j},y)$,
\end{enumerate} \end{lemma}
\begin{proof} We only prove the second statement, since the first one is shown in a similar fashion. We assume $n=2$ and note that the general case then follows easily.
Let $\epsilon>0$. For $i=1,2$ find $l_{i},k_{i}\in\mathbb{N}$ such that $\beta(x_{i},y)\leq \frac{l_{i}}{k_{i}}\leq \beta(x_{i},y)+\epsilon$ and $k_{i}x_{i}\leq l_{i}y$. Then $k_1k_2(x_1+x_2)\leq k_2l_1y + k_1l_2 y$, so $$\beta(x_1+x_2,y)\leq \frac{k_2l_1+k_1l_2}{k_1k_2}=\frac{l_1}{k_1}+\frac{l_2}{k_2}\leq \beta(x_{1},y)+\beta(x_{2},y) +2\epsilon.$$ \end{proof}
\begin{remark}\label{almost_zero} We would like to emphasize that given $x$ and a sequence $\{y_{n}\}$ belonging to $W$ such that $x<_s y_{j}$ for all $j$, we have:
$$\beta(x,y_{1}+y_{2}+\ldots+y_{k})\leq 1/k \text{ for all } k.$$ \end{remark}
\begin{proof} Assume $k=2$ since it is easy to extend the proof to general $k\in\mathbb{N}$. Consider $y_{1}, y_{2}$ such that $(k_{1}+1)x\leq k_{1}y_{1}$ and $(k_{2}+1)x\leq k_{2}y_{2}$. Using \cite[Proposition 2.1]{OPR2}, there exists $k_{0}\in \mathbb{N}$ such that $(k+1)x\leq ky_{1}$ and $(k+1)x\leq ky_{2}$ for all $k\geq k_{0}$. Adding both inequalities, we obtain $ 2kx\leq k(y_{1}+y_{2})$ for all $k\geq k_0$. Thus, $ \beta(x, y_{1}+y_{2})\leq 1/2 .$ \end{proof}
\begin{lemma}\label{prop1} Let $W$ be an abelian ordered semigroup and $x,y\in W$, such that $y\leq mx$ and $x\leq ny$ for some $m,n\in\mathbb{N}$. Then the following statements are equivalent. \begin{itemize} \item[(i)] $\beta(x,y)=0$ \item[(ii)] Some multiple of $x$ is properly infinite. \item[(iii)] Some multiple of $y$ is properly infinite. \item [(iv)] $\mathcal{S}(W,y)=\emptyset.$ \end{itemize} \end{lemma}
\begin{proof} If $\beta(x,y)=0$, then we can find $k,l\in\mathbb{N}$ such that $kx\leq ly$ and $l/k\leq 1/2m$. Then $2kx\leq 2ly \leq 2lm x\leq kx$, hence $kx$ is properly infinite. It then follows further that $2ny\leq 2nmx\leq x\leq ny$, so $y$ is properly infinite. It is easy to see that {\rm (iii)} implies {\rm (i)}. Finally, the equivalence with {\rm (iv)} follows from Lemma \ref{NoStates}.
\end{proof}
\section{Comparison Properties}\label{SecCompProp}
In this section we recall the properties of corona factorization (CFP) and $\omega$-comparison for $\mathrm{Cu}$-semigroups as defined in \cite{OPR2} and show a few equivalent reformulations of each property. We use the value $\beta(x,y)$ from Section 3 to introduce a new comparison notion, called $\beta$-comparison property, which we also prove to be equivalent to $\omega$-comparison for many $\mathrm{Cu}$-semigroups of importance. We demonstrate differences of all mentioned properties with the help of examples. Finally, we discuss the relation between the CFP and the non-existence of stably infinite but finite elements.
Recall that a sequence $(x_n)_n$ in an ordered abelian semigroup $W$ is called full, if it is increasing and for any $z'\ll z$, one has $z'\leq m\cdot x_n$ for some $n,m\in\mathbb{N}.$ For future use note that, if $(x_n)$ is a full sequence in a $\mathrm{Cu}$-semigroup $S$, then $(n\cdot x_n)$ is an increasing sequence in $S$ with supremum equal to the maximal element $\infty$ in $S$. (In particular, a maximal element exists in $S$.) Indeed, for any $t\ll s$ one has $t\leq n \cdot x_n$ for some $n$. Now taking the supremum on both sides (first on the right, then on the left hand side) yields $s\leq \sup_n (n\cdot x_n)$.
We will also want to change a full sequence to a more suitable one in the following way.
\begin{lemma}\label{NewFullSequence} Let $S$ denote a $\mathrm{Cu}$-semigroup, and let $(x_n)_n$ be a full sequence in $S$. Then there is a full sequence $(x_n')_n$ in $S$ such that $x_n'\ll x_n$ for each $n\in \mathbb{N}$. \end{lemma}
\begin{proof} For each $n\in\mathbb{N}$ find a rapidly increasing sequence $(x_n^k)_k$ with supremum $x_n$. Starting with $x_1^1\ll x_1\leq x_2$, we find $k_2$ such that $x_1^1\leq x_2^{k_2}$. Inductively, we find an increasing sequence $(y_n)_n:=(x_n^{k_n})_n$ with $x_i^j\leq x_n^{k_n}\ll x_n$ for each $i,j,n\in \mathbb{N}$ such that $1\leq i,j\leq (n-1)$. Let us show that the sequence $(y_n)_n$ is full.
Let $z'\ll z$, and find $z''$ such that $z'\ll z''\ll z$. By assumptions on $(x_n)_n$, we find $m$ and $k$ such that $z'\ll z''\leq m\cdot x_k$. Therefore, there is some $l$ such that $z'\leq m\cdot x_k^l$. We conclude that $$z'\leq m\cdot x_k^l\leq m\cdot x_M^{k_M}=m\cdot y_M,$$ where $M=\max\{k,l\}+1$. \end{proof}
\begin{definition}(\cite{OPR2})\label{DefCFP} Let $W$ be an ordered abelian semigroup.
\begin{itemize} \item $W$ satisfies the corona factorization property (CFP) if, given any full sequence $(x_{n})_n$ in $W$, any sequence $(y_{n})_n$ in $W$, an element $x'$ in $W$ such that $x'\ll x_{1}$, and a positive integer $m$ satisfying $x_{n}\leq m y_{n}$ for all $n$, then there exists a positive integer $k$ such that $x'\leq y_{1}+\ldots+y_{k}$.
\item $W$ satisfies the strong corona factorization property (StCFP) if, given $x',x\in W$, a sequence $(y_n)$ in $W$, and a positive integer $m$ such that $x'\ll x\leq my_n$ for all $n$, then there exists a positive integer $k$ such that $x'\leq y_1+\ldots+y_k$. \end{itemize} \end{definition}
Note that the CFP and the StCFP are equivalent in the simple case. The terminology comes from the fact that a $\sigma$-unital C*-algebra has the corona factorization property if and only if its Cuntz semigroup $W(A)$ has the CFP as defined above (\cite{OPR2}).
The following proposition was basically shown in \cite{RobertRordam}, but our version differs from theirs in that we reduce to elements in $S_{\ll\infty}$. Recall that we denote by $S_{\ll\infty}$ the set of all $y\in S$ such that $y\ll\infty$.
\begin{proposition}(\cite{RobertRordam})\label{PropCFP} Let $S$ be a $\mathrm{Cu}$-semigroup. Then the following statements are equivalent: \begin{itemize} \item[(i)] $S$ has the CFP. \item[(ii)] Given any full sequence $(x_{n})_n$ in $S$, any sequence $(y_{n})_n$ in $S_{\ll\infty}$, an element $x'$ in $S$ such that $x'\ll x_{1}$, and a positive integer $m$ satisfying $x_{n}\leq m y_{n}$ for all $n$, then there exists a positive integer $N$ such that $x'\leq y_{1}+\ldots+y_{N}$. \item[(iii)] Given any full sequence $(x_{n})_n$ in $S$, any sequence $(y_{n})_n$ in $S_{\ll\infty}$ and any positive integer $m$ satisfying $x_{n}\leq m\cdot y_{n}$ for all $n$, then $\infty = \sum_{n=1}^\infty y_{n}$. \item[(iv)] Given a sequence $(y_n)_n$ in $S_{\ll\infty}$ such that $m\cdot \sum_{n=k}^\infty y_n=\infty$ for some $m$ and all $k\in\mathbb{N}$, then $\sum_{n=1}^\infty y_n=\infty$. \end{itemize} \end{proposition}
\begin{proof} We will show first that, in the definition of the CFP, we can reduce to a sequence $(y_n)_n\in S_{\ll\infty}$, i.e., condition (ii) implies the CFP. So let $(x_{n})_n$ be a full sequence in $S$, $(y_{n})_n$ a sequence in $S$, $x'\ll x_{1}$ in $S$, and $m$ a positive integer $m$ satisfying $x_{n}\leq m y_{n}$ for all $n$. Use Lemma \ref{NewFullSequence} to find a new full sequence $(x'_n)_n$ with $x_n'\ll x_n$, and we may choose the sequence such that $x'\ll x'_1$. Choose for each $n$ a rapidly increasing sequence $(y_n^k)_k$ with supremum $y_n$. Then, for each $n\in\mathbb{N}$, there is $k(n)$ such that $x'_n\leq m\cdot y_n^{k(n)}$. As each $y_n^k\in S_{\ll\infty}$, we can apply (ii) to find $N$ in $\mathbb{N}$ such that $x'\leq y_1^{k(1)}+\ldots + y_N^{k(N)}\leq y_1+\ldots +y_N$.
The converse, that (i) implies (ii), is trivial.
Let us then suppose statement (ii) and that we are given a full sequence $(x_{n})_n$ in $S$, a sequence $(y_{n})_n$ in $S_{\ll\infty}$ and a positive integer $m$ satisfying $x_{n}\leq m y_{n}$ for all $n$. We choose an injective map $\alpha:\mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N}$. Fix some $k,l \in\mathbb{N}$. Then for any $x'\ll x_{k}$, (ii) gives some $N\in\mathbb{N}$ such that $x'\leq \sum_{n=1}^N y_{\alpha(l,n)}$. Since $x'\ll x_{k}$ was arbitrary, we get, taking supremum on both sides, that $x_{k}\leq \sum_{n=1}^\infty y_{\alpha(l,n)}$. Hence, $$\infty\cdot x_{k}\leq \sum_{l=1}^\infty \sum_{n=1}^\infty y_{\alpha(l,n)}\leq \sum_{n=1}^\infty y_n.$$ The latter holds for arbitrary $k$; thus, $\infty=\sup_k \left ( \infty \cdot x_k\right ) \leq \sum_{n=1}^\infty y_n$.
Let us suppose (iii) to hold and that we are given a sequence $(y_n)_n$ in $S_{\ll\infty}$ such that $m\cdot\sum_{n=k}^\infty y_n=\infty$ for all $k\in\mathbb{N}$. Pick an arbitrary full sequence $(x_n)_n$ and, with the help of Lemma \ref{NewFullSequence}, find a new full sequence $(x_n')_n$ such that $x_n'\ll x_n$ for all $n$ in $S$. Then, for any $n,k\in \mathbb{N}$ we have that $x_n'\ll x_n\leq \infty = m\cdot \sum_{j=k}^\infty y_j$. Hence, for each $n$ and $k$ there is $N(n,k)\in\mathbb{N}$ such that $x_n'\leq m\cdot \sum_{j=k}^{N(n,k)} y_j$. We choose $z_1:=\sum_{j=1}^{N(1,1)} y_j$, and then, inductively, for given $z_n=\sum_{j=k}^{M} y_j$, we choose $z_{n+1}:=\sum_{j=M+1}^{N(n+1,M+1)} y_j$. (Note that each $z_n\in S_{\ll\infty}$.) It follows that $x_n'\leq m\cdot z_n$ for all $n$, and, by (iii), we obtain that $\infty= \sum_{n=1}^\infty z_n= \sum_{n=1}^\infty y_n$.
Finally, suppose that (iv) holds and that we are given a full sequence $(x_n)_n$ in $S$, some $x'\ll x_1$, a sequence $(y_n)_n$ in $S_{\ll\infty}$, and a positive integer $m$ such that $ x_n \leq m\cdot y_n$ for each $n$. Then for all $l,k$ such that $l\geq k$, $$m\cdot \sum_{n=k}^\infty y_n\geq \sum_{n=l}^\infty m\cdot y_n \geq \sum_{n=l}^\infty x_n\geq \infty\cdot x_l.$$ Taking supremum over $l$, $m\cdot \sum_{n=k}^\infty y_n=\infty$ for each $k$, which by {\rm(iv)} implies that $\sum_{n=1}^\infty y_n=\infty$. Hence, $x'\ll x \leq \sum_{n=1}^\infty y_n$, which shows the existence of some $k$ such that $x'\leq \sum_{n=1}^k y_n$. Thus, (iv) implies (ii), which completes the proof.
\end{proof}
\begin{definition} An ordered abelian semigroup $W$ has the $\omega$-comparison property if whenever $x',x,$ $ y_{0}, y_{1}, y_{2},\ldots$ are elements in $W$ such that $x<_{s}y_{j}$ for all $j$ and $x'\ll x$, then $x'\leq y_{0}+y_{1}+\ldots+y_{n}$ for some $n$. \end{definition}
The following lemma constitutes a reduction step in the proof of Proposition \ref{Omega}.
\begin{lemma}\label{ReductionStepOmega} Let $S$ be a $\mathrm{Cu}$-semigroup. In the definition of $\omega$-comparison one may, without loss of generality, assume that $y_j\ll\infty$ for all $j$. That is, $\omega$-comparison is equivalent to the following property: \begin{itemize} \item Whenever $x',x,$ $ y_{0}, y_{1}, y_{2},\ldots$ are elements in $S$ such that $x<_{s}y_{j}\ll \infty$ for all $j$ and $x'\ll x$, then $x'\leq y_{0}+y_{1}+\ldots+y_{n}$. \end{itemize} \end{lemma}
\begin{proof} We suppose that we are given a $\mathrm{Cu}$-semigroup $S$ for which we only know the condition of $\omega$-comparison to hold when $y_j\ll\infty$ for all $j$. We will show that then the $\omega$-comparison holds.
Let $x',x,$ $ y_{0}, y_{1}, y_{2},\ldots$ be elements in $S$ such that $x<_{s}y_{j}$ for all $j$ and $x'\ll x$. We find $x''$ such that $x'\ll x''\ll x$. For each fixed $j$ there is some $n\in \mathbb{N}$ such that $(n+1)x''\ll n y_j$. Choosing a rapidly increasing sequence $(y_j^k)_k$ with supremum $y_j$, the sequence $(n\cdot y_j^k)_k$ increases rapidly and has supremum $n\cdot y_j$. It follows that $x''<_s y_j^l\ll \infty$ for some $l=l(j)$. Set $y_j':=y_j^{l(j)}$ for each $j$. Now $x'\ll x'' <_sy_j' \ll\infty$ and our assumption implies that $x'\leq \sum_{j=1}^N y_j'\leq \sum_{j=1}^N y_j$. \end{proof}
For equivalent notions of $\omega$-comparison we only consider simple $\mathrm{Cu}$-semigroups. The reason for restricting ourselves to simple $\mathrm{Cu}$-semigroups is that in the definition of the $\omega$-comparison, as it was introduced in \cite{OPR2}, there is no assumption on fullness of $x$. In this way, $\omega$-comparison is more similar to the strong CFP than the CFP.
\begin{proposition}\label{Omega} Let $S$ be a simple $\mathrm{Cu}$-semigroup. Then the following statements are equivalent. \begin{itemize} \item[(i)] $S$ has $\omega$-comparison. \item[(ii)] Whenever $(y_n)$ is a sequence of nonzero elements in $S_{\ll\infty}$ such that $y_n <_s y_{n+1}$ for all $n$, then $\sum_{n=1}^\infty y_n=\infty$ (in $S$). \item [(iii)] Whenever $(y_n)$ is a sequence in $S_{\ll\infty}$ such that $\lambda \left (\sum_{n=k}^\infty y_n\right )=\infty$ for all $k\in\mathbb{N}$ and all functionals $\lambda\in F(S)$, then $\sum_{n=1}^\infty y_n=\infty$. \item [(iv)]Whenever $(y_n)$ is a sequence of nonzero elements in $S_{\ll\infty}$ such that $\lambda \left (\sum_{n=1}^\infty y_n\right )=\infty$ for all functionals $\lambda\in F(S)$, then $\sum_{n=1}^\infty y_n=\infty$. \end{itemize} \end{proposition}
\begin{proof} By Lemma \ref{ReductionStepOmega} we may assume in the definiton of $\omega$-comparison that $y_j\ll\infty$ for all $j$. Now note, that instead of the condition that $x<_{s}y_{j}\ll \infty$ for all $j$, one may assume that $y_j<_s y_{j+1}\ll\infty$ for all $j$. Indeed, use simplicity to find for given $\sigma:=\sum_{j=k}^l y_j$ some $m\in\mathbb{N}$ such that $\sigma \leq m\cdot x$. Then, $$\sigma=\sum_{j=k}^l y_j\leq m\cdot x<_s\sum_{j=l+1}^{l+m}y_j.$$ Using this and starting with $\sigma:=y_1=\tilde{y}_1$, by iteration, we find the desired sequence $(\tilde{y}_j)_j$ with $\tilde{y}_j<_s \tilde{y}_{j+1}\ll\infty , j\in\mathbb{N}$, and $\sum_{j=1}^\infty y_j = \sum_{j=1}^\infty \tilde{y}_j$. One now shows that (i) implies (ii) in an analogous way to the proof of (i) implies (ii) of Proposition \ref{PropCFP}, and the implication from (ii) to (i) is easy.
To connect the statements (i) and (ii) to statements (iii) and (iv) consider first the case that there is $x\in S_{\ll\infty}$ with $\lambda(x)=\infty$ for all $\lambda\in F(S)$. Then, by Lemma \ref{InfiniteMultiples}, $\lambda=\lambda_\infty$ and every element is stably infinite. In this case, the conditions on the $y_n$ in (ii), (iii), and (iv) all just reduce to the condition that $y_n\neq 0$ for every $n$. Hence (ii) through (iv) are trivially equivalent in this case. We may therefore assume in what follows that there are functionals finite on $S_{\ll\infty}$.
To see that (iii) implies (ii), we use that $y_j<_s y_{j+1}$ implies that $\lambda(y_j)< \lambda(y_{j+1})$ for all functionals $\lambda$ finite on $S_{\ll\infty}$. The converse is shown by induction and by using Lemma \ref{UglyWayOut} as follows: If $y_j'=\sum_{k=a}^{b}y_j$ has been determined, then find $m$ as in Lemma \ref{UglyWayOut} for $y'_j$, and then find $c\in \mathbb{N}$ such that $m\cdot \lambda(y'_j)<\sum_{j=b+1}^c \lambda(y_j)$. Set $y'_{j+1}:=\sum_{j=b+1}^c y_j$.
Finally, to see that (iii) and (iv) are equivalent is easy as we have already reduced to the case that $\lambda(y_j)<\infty$ for all $j$. \end{proof}
\begin{remark} The assumption that all elements $y_j$ in {\rm(iv)} are nonzero is necessary (cf. Example \ref{Ex:Omega-noBeta}). \end{remark}
We will now define a new regularity property on comparison in an ordered abelian semigroup $W$ based on the value $\beta(x,y)$ for $x,y\in W$ as defined in Section \ref{SectionBeta}. Note that $\beta(x,y)=0$ if and only if for every $\epsilon>0$ there is $k\in \mathbb{N}$ such that $kx\leq \lfloor k\epsilon \rfloor y$, where $\lfloor a \rfloor$ denotes the largest smaller integer than $a$.
\begin{definition} ($\beta$-comparison) Let $W$ be an ordered abelian semigroup. We will say that $W$ satisfies the $\beta$-comparison property if whenever $x,y$ are two elements in $W$ with $\beta(x,y)=0$, then $x\leq y$. \end{definition}
\begin{lemma}\label{BetaLemma} Let $S$ be a simple $\mathrm{Cu}$-semigroup. Then the following statements are equivalent. \begin{itemize} \item[(i)] $S$ has $\beta$-comparison. \item[(ii)] Whenever $\beta(x,y)=0$ for some non-zero $x$, then $y=\infty$. \end{itemize} \end{lemma}
\begin{proof} Suppose $\beta(x,y)=0$ for some non-zero $x$, which implies $\beta(n\cdot x,y)=0$ for each $n\in\mathbb{N}$ by a simple computation. Letting $x'\ll x$, we have $\beta(n\cdot x',y)=0$ by Lemma \ref{properties}. Now (i) implies that $n\cdot x\leq y$ for all $n\in \mathbb{N}$, hence $y=\infty$ by simplicity. The other implication is trivial. \end{proof}
The following lemma characterizes $\beta$-comparison using functionals and states.
\begin{lemma}\label{BetaAndFunctionals} Let $S$ be a simple $\mathrm{Cu}$-semigroup. Then the following statements are equivalent for an element $y\in S$: \begin{itemize} \item[(i)] There is some nonzero $x\in S$ such that $\beta(x,y)=0$. \item[(ii)] There is no faithful state $f\in \mathcal{S}(S,y)$. \item[(iii)] $\lambda(y)=\infty$ for all functionals $\lambda\in F(S)$. \end{itemize} \end{lemma}
\begin{proof} By Lemma \ref{FaithfulStates}, (ii) and (iii) are equivalent.
Suppose that the set of states $\mathcal{S}(S,y)$ normalized at $y$ is empty. Then, by Lemma \ref{NoStates}, there is $n\in \mathbb{N}$ such that $n\cdot y$ is properly infinite; hence, $n\cdot y=\infty$. In this case, $\beta(x,y)=0$ and $\lambda(y)=\infty$ for all functionals $\lambda$, so all three statements hold.
Otherwise, there is a state $f \in \mathcal{S}(S,y)$ and $\beta(x,y)=\sup\{f(x)\ |\ f\in \mathcal{S}(S,y)\}.$ Thus, (i) and (ii) are also equivalent in this case. \end{proof}
With respect to Lemma \ref{BetaAndFunctionals}, one may wonder about the existence of a simple $\mathrm{Cu}$-semigroup $S$ with a {\bf stably finite} element $y\in S$ such that $\beta(x,y)=0$ for some nonzero $x$ (equivalently, $\lambda(y)=\infty$ for all $\lambda\in F(S)$). This question on existence has a positive answer, which is explained based on the C*-algebra constructed in \cite{Petzka13}:
\begin{example}\label{ExampleBetaVsQQ} In \cite{Petzka13}, the second author constructed a stably finite projection $Q$ in the multiplier algebra of a separable stable simple C*-algebra $A$, which is of the form $Q=\sum_{j=1}^\infty {p_j}$, where the $p_j$ are pairwise orthogonal projections in $A$ and such that $\lambda(p_j)=\lambda(p_i)$ for all $i,j$ and all functionals $\lambda$.
Considering $a:=\sum_{j=1}^\infty \frac{1}{2^j} p_j$, we find a positive element $a\in A$ such that its Cuntz class, $y=\langle a \rangle$, satisfies that $\lambda(y)=\infty$ for all $\lambda$, hence $\beta(x,y)=0$. We will show that $y$ is stably finite.
Indeed, that the multiplier projection $Q$ is stably finite is shown in \cite{Petzka13} by the existence of projections $g_n$ such that for each $n$ we have $g_n\npreceq n\cdot Q$. Assuming $n\cdot y=\infty$, we get in particular that $g_n\preceq n\cdot a$. By compactness of $\langle g_n\rangle$, there is some $N(n)\in\mathbb{N}$ such that $g_n\preceq n\cdot \sum_{j=1}^{N(n)} \frac{1}{2^j} p_j\sim \sum_{j=1}^{N(n)} p_j$. Now, the Cuntz subequivalence is just Murray-von Neumann subequivalence of projections; hence, $g_n\preceq \sum_{j=1}^{N(n)} p_j<Q$, a contradiction. It follows that $n\cdot y<\infty$ for all $n$ and $y$ is stably finite. \end{example}
Our goal is now to relate $\beta$-comparison to $\omega$-comparison.
\begin{proposition} \label{BetaThenOmega}
Let $S$ be $\mathrm{Cu}$-semigroup. If $S$ has $\beta$-comparison, then $S$ has $\omega$-comparison. \end{proposition}
\begin{proof} Assume that $S$ does not satisfy $\omega$-comparison. Then, there exists a sequence $\{y_{n}\}$, and $x,x'$ in $S$ such that $x'\ll x<_{s}y_{j}$ for all $j$ and $x'\not\leq y_{1}+y_{2}+\ldots y_{k}$ for any $k$. Let $y:=\sup_{n} (\sum^{n}_{j=1} y_{j})$ and notice that $x\not\leq y$.
We have that $x\leq \infty\cdot y_j \leq \infty \cdot y$ for all $j$, so $x\, \overline{\propto}\, y$. By Lemma \ref{properties}, $\beta(x',y)\leq \beta(x',\sum^{k}_{j=1} y_{j})$ for any $k$. Using Remark \ref{almost_zero}, one gets $\beta(x',y)\leq 1/k$ for all $k$. Letting $k$ go to infinity, $\beta(x',y)=0$. Hence $\beta(x',y)=0$ for all $x'\ll x$, but $x\nleq y$, so $S$ does not satisfy $\beta$-comparison. \end{proof}
The converse is not true for general (simple) $\mathrm{Cu}$-semigroups, but it is true for simple $\mathrm{Cu}$-semigroup satisfying the additional axioms (O5) and (O6) and containing no minimal element, which is the content of the following example and proposition.
\begin{example}\label{Ex:Omega-noBeta} There exists a simple $\mathrm{Cu}$-semigroup $S$, such that $S$ has $\omega$-comparison, but no $\beta$-comparison. \end{example}
\begin{proof} Let $$S=\{0\}\cup \{1\}\cup\{\infty\}$$ with $1+1=\infty$ and $1$ compact. If one wants to exclude minimal (necessarily compact) elements in a $\mathrm{Cu}$-semigroup, then one can similarly consider $$\tilde{S}=\{0\}\cup (1,2]\cup\{\infty\}$$ with $x+y=\infty$ for any nonzero $x,y\in\tilde{S}$. Both semigroups are given the order inherited from $\mathbb{R}$. Since the endpoint at 1 is not included in $\tilde{S}$, every element can be written as the supremum of a rapidly (i.e. strictly) increasing sequence.
One checks that $S$ and $\tilde{S}$ are both simple satisfying the axioms (O1)--(O4). Moreover, note that $S$ satisfies (O5) and (O6). However, the order in $\tilde{S}$ is not almost algebraic, so (O5) fails in $\tilde{S}$.
We show that $S$ and $\tilde{S}$ have $\omega$-comparison, but no $\beta$-comparison. For the failure of $\beta$-comparison, note that $2x=\infty$ for every non-zero $x$ in $S$ and $\tilde{S}$. Thus, we have that $\beta(x,y)=0$ for arbitrary nonzero $x,y$. But in $S$ we have $1\neq \infty$, and in $\tilde{S}$ we have $3/2\neq\infty$.
On the other hand, $S$ and $\tilde{S}$ both have $\omega$-comparison. To verify this, fix some nonzero $x$ in $S$ or $\tilde{S}$. Then for any sequence $(y_j)$ so that $x<_s y_j$ for all $j$, the $y_j$'s are necessarily non-zero, so $\sum_{j=1}^\infty y_j=\infty\geq x$. \end{proof}
In the previous example, instead of the semigroup $S$, we could have considered more generally the semigroup $S_n=\{0,1,2,\ldots,n,\infty\}$ for some $n\in\mathbb{N}$, equipped with the natural order and the natural addition except that $x+y=\infty$ whenever the sum of $x$ and $y$ exceeds $n$ in $\mathbb{R}$. These $\mathrm{Cu}$-semigroups were also studied in \cite{APT14}. The following proposition states that, in the class of simple $\mathrm{Cu}$-semigroups satisfying (O1)--(O6), these are the only semigroups that distinguish $\omega$-comparison from $\beta$-comparison.
\begin{proposition}\label{BetaVsOmega}
Let $S$ be a simple $\mathrm{Cu}$-semigroup satisfying axioms (O1)--(O6) and $S\neq S_n$ for any $n\in\mathbb{N}$. If $S$ has $\omega$-comparison, then $S$ has $\beta$-comparison (and hence they are equivalent by Proposition \ref{BetaThenOmega}). \end{proposition}
\begin{proof} Let $y\in S$ such that $\beta(x,y)=0$ for some nonzero $x$. Then, by Lemma \ref{BetaAndFunctionals} we have that $\lambda(y)=\infty$ for all functionals $\lambda\in F(S)$. Pick a rapidly increasing sequence $(y_i)$ of nonzero elements with supremum $y$.
As $\lambda(y)=\infty$ and $\lambda$ preserves suprema, we can (after possibly changing to a subsequence) assume that $\lambda(y_i)+i\leq \lambda(y_{i+1})$ for all $i$ and all $\lambda$. (Here we are using compactness of $F(S)$, which was shown in \cite[Theorem 4.8]{ERS11}). Using (O5), we can find for each $i$ some $z_i\in S$ such that $y_i+z_i\leq y_{i+2}\leq y_{i+1}+z_i$.
We will distinguish between the case that $\lambda(y_i)=\infty$ for all $\lambda\in F(S)$ (for some and hence all (nonzero) $y_i$, c.f. Lemma \ref{InfiniteMultiples}) and the case that there is some functional $\mu$ with $\mu(y_i)<\infty$ for all $i$.
First suppose the latter. Then $\mu(z_i)\geq \mu(y_{i+2})-\mu(y_{i+1})\geq i+1$ for such a functional finite on $S_{\ll\infty}$. It follows in particular that $z_i$ is nonzero. If now $\nu$ is a functional, then either $\nu$ is infinite on $S_{\ll\infty}$ and hence on all of $S\setminus\{0\}$, or $\nu$ is finite on each $y_j$ and $\nu(z_i)\geq i+1$. In particular, we get for all functionals $\lambda$ that $\lambda \left (\sum_{i=k}^\infty z_{2i+1}\right )=\infty$ for all $k$. Now, $\omega$-comparison implies that $\sum_{i=1}^\infty z_{2i+1}=\infty$ (by Proposition \ref{Omega}). But, for any $N\in \mathbb{N}$, we have $$\sum_{i=1}^{2N+1} z_{2i+1}\leq y_1+z_1+z_3+z_5+\ldots+z_{2N+1} \leq y_3+z_3+z_5+\ldots+z_{2N+1}\leq\ldots\leq y_{2N+1}.$$ Hence $$\infty=\sum_{i=1}^\infty z_{2i+1}\leq \sup_i y_i=y.$$
In the other case, in which the only functional on $S$ is the trivial functional $\lambda_\infty$ assigning $\infty$ to all nonzero $z\in S$, we only need to find for each $y\in S$ a sequence of nonzero $z_j$'s such that $\sum_{j=1}^\infty z_j\leq y$. Then, an application of $\omega$-comparison on the sequence $(z_j)_j$ yields the desired conclusion that $y=\infty$. That such a sequence can be found in the simple $\mathrm{Cu}$-semigroup $S\neq S_n$ satisfying axioms (O1)--(O6) follows from the fact that these semigroups have the Glimm halving property (\cite{Robert13}) (i.e., for every nonzero $x\in S$ there is some nonzero $z\in S$ such that $2z\leq x$). \end{proof}
In \cite[Proposition 2.17]{OPR2} it is proved that if a complete abelian ordered semigroup satisfies $\omega$-comparison, then it also satisfies the corona factorization property. In the same paper it was left unanswered whether the converse holds. The example below shows that both properties are not equivalent for $\mathrm{Cu}$-semigroups, and hence neither for general complete abelian ordered semigroups. It remains an open question whether the two notions are equivalent for $\mathrm{Cu}$-semigroups $S$ coming from a C*-algebra, i.e., for $S=\mathrm{Cu}(A)$ for some C*-algebra $A$.
\begin{example}\label{Ex:CFP-noW} There exists a simple $\mathrm{Cu}$-semigroup that satisfies the corona factorization property but does not have the $\omega$-comparison property. \end{example} \begin{proof} Let $$S=[0,1]\cup\{\infty\}$$ equipped with the usual order and addition, except that if $x,y\in S$ are such that $x+y> 1$ in $\mathbb{R}$, we set $x+y=\infty$.
It is easy to check that $S$ is simple, totally ordered and that it satisfies the (strong) corona factorization property.
On the other hand, let us check that $S$ does not satisfy $\omega$-comparison. To do so, firstly note that, for $0\leq x,y\leq 1$ in $S$, $x\ll y$ if and only if $x<y$, and that $\infty\ll\infty$. Now, consider the sequence $\{y_{n}\}=\{1/2^{n+1}\}$ and the elements $x=1\,, x'=3/4$. Clearly $x'\ll x$ and we get $x<_{s} y_{j}$ for all $j$, since in fact $x<_sy$ holds for arbitrary $x,y$ in $S$.
But as $\sum^{\infty}_{j=1}y_{j}=1/2 \ngeq 3/4$, we conclude that $S$ does not satisfy the $\omega$-comparison property. \end{proof}
There is also a stably finite $\mathrm{Cu}$-semigroup distuingishing $\omega$-comparison and the CFP.
\begin{example}\label{Ex:stablyNoFunctionals}
Let $$S=\{ (0,0)\} \cup ((0,1]\cup\{\infty\})\times (0,\infty] ,$$ with addition defined by componentwise addition and with the additional condition that if in the first component we have $x+y>1$ , then $x+y=\infty$. Namely, $(x,r)+(y,s)=(\infty,r+s)$ whenever $x+y>1$ in $\mathbb{R}$. The order is componentwise with the natural ordering in each component. Note that the relation of compact containment is given by componentwise strict inequalities and with $(\infty,r)\ll(\infty,s)$ whenever $r< s$.
It is easy to check that $S$ is simple and satisfies all axioms {\rm (O1)--(O6)}. Note that all elements except those of the form $(x,\infty)$ are stably finite. This makes $S$ a simple stably finite $\mathrm{Cu}$-semigroup.
If $\lambda$ is a nonzero functional on $S$, then $\lambda((x,\infty))=\infty$ for all $\lambda\in F(S)$. Indeed, by construction, there exists $m\in\mathbb{N}$ such that $m(x,\infty)=\infty$, so $m\cdot \lambda((x,\infty))= \lambda((\infty,\infty))=\infty$. Therefore, by Proposition \ref{BetaAndFunctionals}, there is some nonzero $(x,r)\in S$ such that $\beta((x,r),(y,\infty))=0$, while $(y,\infty)$ is stably finite. Hence, $S$ does not have $\beta$-comparison, and neither does $S$ satisfy $\omega$-comparison by Proposition \ref{BetaVsOmega}. On the other hand, it is easy to see with Proposition \ref{PropCFP} that $S$ satisfies the (strong) corona factorization property.
\end{example}
\begin{remark} One can even find an algebraic, simple, stably finite $\mathrm{Cu}$-semigroup with the CFP and failing $\omega$-comparison by a small modification of the previous example. Take $$S=\{ (0,0)\} \cup (((\mathbb{Q}\cap(0,1])\sqcup (0,1])\cup\{\infty\})\times ((\mathbb{Q}\cap(0,\infty))\sqcup (0,\infty])\,$$ with addition and order similar to before, but now addition and order in each component are defined as in the Cuntz semigroup of the universal UHF-algebra (see e.g. \cite{BPT}). By construction, the semigroup $S$ is algebraic, and one shows the required properties of $S$ in the same way as above. \end{remark}
Note that the simple $\mathrm{Cu}$-semigroup in Example \ref{Ex:CFP-noW} is neither stably finite nor purely infinite. This behavior is in general ruled out by the property of $\beta$-comparison.
\begin{proposition}\label{betaImpliesDichotomy} Let $S$ be a simple $\mathrm{Cu}$-semigroup with $\beta$-comparison. Then $S$ is either stably finite or purely infinite. \end{proposition}
\begin{proof} Suppose $S$ is not stably finite, so that there is some $x\ll\infty$ which is not finite. In other words, $\infty$ is compact, and for every nonzero $y\in S$ some finite multiple of it is properly infinite. By Lemma \ref{prop1}, $\beta(x,y)=0$ for every nonzero $x,y$. By $\beta$-comparison, every nonzero element is infinite and $S=\{0,\infty\}$, i.e., $S$ is purely infinite. \end{proof}
Hence, in the simple case, $\beta$-comparison implies the dichotomy of being either stably finite or purely infinite. By Proposition \ref{BetaVsOmega}, $\omega$-comparison implies the same dichotomy of a simple $\mathrm{Cu}$-semigroup $S$ satisfying (O1)--(O6) and different from $S_n$ for any $n$.
One sees from the above example that the CFP allows for the existence of both finite and infinite elements in simple $\mathrm{Cu}$-semigroups. In particular, the CFP is not equivalent to the following stronger statement:
$$\mbox{If }m\cdot \sum_{n=1}^\infty y_n=\infty\mbox{, then }\sum_{n=1}^\infty y_n=\infty.$$
We now turn our attention to $\mathrm{Cu}$-semigroups without the CFP (therefore without $\omega$-comparison and $\beta$-comparison) that are neither stably finite nor purely infinite. The next result provides a characterization of a simple $\mathrm{Cu}$-semigroup not having the CFP, which we subsequently use to find an explicit simple $\mathrm{Cu}$-semigroup without the CFP that is neither stably finite nor purely infinite. However, this semigroup does not satisfy the axiom (O6). (See Theorem \ref{RordamsAlgebraNoCFP} for the existence of a simple $\mathrm{Cu}$-semigroup with (O6) and without the CFP that is neither stably finite nor purely infinite, which is given as the Cuntz semigroup of a C*-algebra. This Cuntz semigroup, however, has not been computed yet.)
\begin{proposition}\label{CharactCFP} Let $S$ be a simple $\mathrm{Cu}$-semigroup, containing a finite compact element, and with $\infty\ll\infty$. Then $S$ does not have the CFP if and only if there is a sequence of elements $(z_n)$ in $S_{\ll\infty}$ such that $2z_n=\infty$ and $\sum_{n=1}^\infty z_n<\infty$. \end{proposition}
\begin{proof} It is clear that the existence of such a sequence implies the lack of CFP. Conversely, suppose $S$ does not have the CFP. Then, by Proposition \ref{PropCFP}, there is $m\in \mathbb{N}$ and a sequence $(y_n)$ in $S_{\ll\infty}$ such that $m\cdot \sum_{n=k}^\infty y_n=\infty$ for all $k$; however, $\sum_{n=1}^\infty y_n\neq \infty$. Choosing $m$ to be minimal, replacing each $y_n$ with a suitable multiple of itself and possibly discarding a finite number of $y_n$'s, we may assume that $m=2$. Now $2\cdot \sum_{n=k}^\infty y_n=\infty$ for all $k$ and $\infty$ is compact. Thus, for each $k$ there is some $N(k)$ such that $2\cdot \sum_{n=k}^{N(k)} y_n=\infty$. Choose $z_1=\sum_{n=1}^{N(1)}y_n$, and then choose $z_{n+1}$ inductively from $z_n=\sum_{n=s}^{t}y_n$ to be $z_{n+1}=\sum_{t+1}^{N(t+1)}y_n$. Then, the sequence $(z_n)$ in $S_{\ll\infty}$ satisfies $2z_n=\infty$, and $\sum_{n=1}^\infty z_n =\sum_{n=1}^\infty y_n<\infty$ as required. \end{proof}
Explicitly, we have the following example. \begin{example}\label{Ex:nonStablynoPurely} We construct an example of a simple $\mathrm{Cu}$-semigroup satisfying (O5), without the CFP, and which is neither stably finite nor purely infinite. However, our example does not satisfy the axiom (O6) of almost Riesz refinement. \end{example}
\begin{proof} Let $$S=[0,1]^\mathbb{N}\cup\{\infty\},$$ with addition given by componentwise addition and with the relation that $x+y=\infty$, whenever any component exceeds 1. One checks that $S$ is simple and that it satisfies all the required axioms, i.e. (O1)--(O5).
Letting $y_n$ denote the element in $S$, which is 1 at position $n$ and zero elsewhere, Proposition \ref{CharactCFP} applies to show that $S$ does not have the CFP.
To see that (O6) does not hold, consider $y_1=(1,0,0,\ldots)$ and $y_2=(0,1,0,0,\ldots)$. We have $y_1\leq y_2+y_2=\infty$, but we cannot find any nonzero elements $x_1,x_2\leq y_1,y_2$. \end{proof}
Alternatively, in the previous example one could have used $S=\{0,1\}^\mathbb{N}\cup\{\infty\}$ instead, but our aim was to show that one can guarantee the non-existence of minimal nonzero elements in $S$. The failure of axiom (O6) in the last example can be generalized as follows.
\begin{proposition}(\cite[Lemma 5.1.18]{APT14}) Let $S$ be a simple $\mathrm{Cu}$-semigroup satisfying (O6). Then for any finite number of elements $y_1,\ldots, y_n$ in $S_{\ll\infty}$ there is some nonzero $z\in S$ such that $z\leq y_j$ for all $j$. \end{proposition}
\begin{remark} Theorem \ref{RordamsAlgebraNoCFP} shows the existence of a simple C*-algebra $A$ such that its Cuntz semigroup $\mathrm{Cu}(A)$ is neither stably finite nor purely infinite and fails to have the CFP. On the other hand, it seems difficult to write down an explicit example of a simple $\mathrm{Cu}$-semigroup $S$, neither stably finite nor purely infinite, satisfying all the axioms (O1)--(O6) and failing to satisfy the CFP. By the previous proposition, in such a semigroup, for any finite number of elements $y_1,\ldots,y_n$ one can find a nonzero element $z\in S$ such that $z\leq y_j$ for all $j=1,2,\ldots,n$. However, if the CFP fails in $S$ and this failure is witnessed by a sequence $(y_j)$, then there is no nonzero $z\in S$ such that $z\leq y_j$ for all $j\in\mathbb{N}$, as otherwise $\sum_{j=1}^\infty y_j\geq \infty\cdot z=\infty$.
\end{remark}
The CFP is closely related to the property {\rm (QQ)}, which was introduced in \cite{OPR2}.
\begin{definition}(\cite{OPR2}) A positively ordered abelian semigroup $W$ satisfies the property {\rm (QQ)} if every element in $W$, for which a multiple is properly infinite, is itself properly infinite. \end{definition}
The following relations are immediate.
\begin{proposition}\label{PropQQ} Let $S$ be a simple $\mathrm{Cu}$-semigroup. \begin{itemize} \item[(i)] If $S$ has $\beta$-comparison, then $S$ has property {\rm (QQ)}. \item[(ii)] If $S$ has {\rm (QQ)}, then $S$ has the CFP. \end{itemize} \end{proposition}
\begin{proof} For (i), let $x\in S$ and $n\in \mathbb{N}$ with $n\cdot x=\infty$. Then $\beta(z,x)=0$ for all $z$, and by $\beta$-comparison, $x=\infty$. Statement (ii) is trivial (with Proposition \ref{PropCFP}). \end{proof}
\begin{remark} Attempting to prove the converse to {\rm (i)} in the most direct fashion, one would hope that $\beta(x,y)=0$ for some nonzero $x$ (equivalently, $\lambda(y)=\infty$ for all functionals $\lambda$) implies that some multiple of $y$ should be infinite. That this does not hold in general has already been noted in Example \ref{ExampleBetaVsQQ}. Hence, to show that the converse to (i) holds, one would need that the existence of some $y\neq \infty$ with $\beta(x,y)=0$ for some nonzero $x$ implies the existence of some $z\neq \infty$ in $S$ (possibly $z\neq y$) such that $n\cdot z=\infty$. \end{remark}
We introduce a new property related to the existence of both finite and infinite elements in $S_{\ll\infty}$:
\begin{definition}
A complete abelian positively ordered semigroup $W$, containing a largest element $\infty$, has cancellation of small elements at infinity, if whenever $x$ and $y$ are elements in $W$ with $x\ll \infty$, $y\neq 0$ and $x+y=\infty$, then $y=\infty$. \end{definition}
It is clear that if $S\neq \{0,\infty\}$ and $\infty$ is compact in $S$, then cancellation of small elements at infinity fails. Example \ref{Ex:stablyNoFunctionals} shows that cancellation of small elements at infinity can also fail in a stably finite $\mathrm{Cu}$-semigroup satisfying all axioms (O1)--(O6). It is not known (but possibly expected) whether cancellation at infinity holds for the Cuntz semigroup $\mathrm{Cu}(A)$ of a simple stably finite C*-algebra $A$.
The next result shows that cancellation of small elements at infinity holds when $S$ satisfies either {\rm (QQ)} or $\omega$-comparison or $\beta$-comparison.
\begin{proposition}\label{QQImpliesCAI} Let $S$ be a simple $\mathrm{Cu}$-semigroup. \begin{enumerate}[\rm(i)]
\item If $S$ has property {\rm (QQ)}, then it has cancellation of small elements at infinity.
\item If $S$ has $\beta$-comparison, then it has cancellation of small elements at infinity. \end{enumerate} \end{proposition}
\begin{proof} Suppose $x\ll \infty$ and $x+y=\infty$. Since $x\ll \infty$ and also using simplicity, there is $n\in \mathbb{N}$ such that $x\leq n\cdot y$. Hence $(n+1)\cdot y=\infty$. By property (QQ), we get $y=\infty$. The second statement easily follows from combining {\rm (i)} with Proposition \ref{PropQQ}.
\end{proof}
As we shall see below, the converse to Proposition \ref{QQImpliesCAI}(i) holds for certain simple $\mathrm{Cu}$-semigroups. Recall that a Cuntz semigroup is called algebraic, if every element can be written as the supremum of an increasing sequence of compact elements.
\begin{proposition}\label{CFPVsQQ} Let $S$ be a simple algebraic $\mathrm{Cu}$-semigroup with {\rm (O5)}. Then $S$ has property {\rm (QQ)} if and only if $S$ has both the CFP and cancellation of small elements at infinity. \end{proposition}
\begin{proof} Proposition \ref{QQImpliesCAI} and Proposition \ref{PropQQ} show the 'only if'-direction.
Using Proposition \ref{PropCFP} one sees that, under the assumption of cancellation of small elements at infinity, the CFP can be rephrased as the statement that if $(y_j)_j$ is a sequence in $S_{\ll\infty}$ such that $m\cdot \sum_{j=1}^\infty y_j=\infty$ for some $m\in \mathbb{N}$, then $\sum_{j=1}^\infty y_j=\infty$. Loosely speaking, the CFP equals property {\rm (QQ)} for elements of the form $y=\sum_{j=1}^\infty y_j$ with $y_j\ll\infty$ for all $j$. Using (O5) and the assumption that $S$ is algebraic, one see that every element in $S$ can be written as $\sum_{j=1}^\infty y_j$ for suitable $y_j\ll\infty$. Hence, property {\rm (QQ)} holds. \end{proof}
Proposition \ref{CFPVsQQ} can be generalized to the simple non-algebraic case with a minor technical limitation. (The `only if'-direction holds for a general simple $\mathrm{Cu}$-semigroup.)
\begin{proposition}\label{CFPVsQQPart2} Let $S$ be a simple algebraic $\mathrm{Cu}$-semigroup with (O5) and with cancellation of small elements at infinity. Suppose that $S$ does not have property {\rm (QQ)} and the failure of {\rm (QQ)} is witnessed by an element $x\in S$ and some $m>2$, such that $m\cdot x=\infty$, but $(m-1)x<\infty$. Then $S$ does not have the CFP. \end{proposition}
We omit the proof as the arguments are similar to the ones in (the first part of) the proof of Theorem \ref{CFPVsQQAlg}, in which we overcome the technical limitation (of needing $m$ to be strictly greater than 2) and prove the conclusion of Proposition \ref{CFPVsQQ} for any $\mathrm{Cu}$-semigroup $S=\mathrm{Cu}(A)$ coming from a simple C*-algebra $A$.
\section{Applications to the Cuntz semigroup of a C*-algebra}\label{SectionApplications}
In this final section we use the results obtained in the previous sections at the level of general $\mathrm{Cu}$-semigroups to $\mathrm{Cu}$-semigroups arising from C*-algebras, i.e., the case where $S=\mathrm{Cu}(A)$. Theorem \ref{CFPVsQQAlg} shows that, for any simple C*-algebra $A$, the CFP in combination with cancellation of small elements at infinity is equivalent to property (QQ). We summarize the relations between all regularity properties studied in this paper in Theorem \ref{thm:end}. Finally, we show in Theorem \ref{RordamsAlgebraNoCFP} that the C*-algebra described in \cite{R03}, containing both a non-zero finite projection and an infinite projection, does not have the CFP.
\begin{definition} Let $W$ be an ordered abelian semigroup. We
say that $W$ has the weak halving property if for every $x \in W$ there are $y_1,y_2 \in W$ such
that $y_1+y_2 \le x$ and $x \propto y_j$ for $j=1,2$. \end{definition}
Note that if an ordered abelian semigroup $W$ has the weak halving property, then inductively one can find a sequence $(y_n)_n$ of elements in $W$ such that for each $n\in \mathbb{N}$ one has $y_1+y_2+ \cdots+y_n\le x$ and $x \propto y_n$ for all $n$. In the case of a complete ordered semigroup, we also get $\sum_{j=1}^\infty y_j\leq x$.
\begin{lemma} \label{lm:halving}
Let $A$ be a unital simple {C*-al\-ge\-bra}{} not of type I. Denoting by $W(A)$ the (original) Cuntz semigroup given by equivalence classes of positive elements in matrix algebras over $A$, it follows that
$W(A)$ has the weak halving property. \end{lemma}
\begin{proof}Let $x \in W(A)$ be given. Upon replacing $A$ by a
matrix algebra over $A$, we may assume that $x = \langle a \rangle$
for some positive element $a$ in $A$. We may also assume that $a$ is
non-zero (as it is trivial to halve the zero-element). Take a
maximal abelian sub-{C*-al\-ge\-bra}{} $D$ of $\overline{aAa}$. Then $D$ is
infinite dimensional (by the assumption that $A$ is not of type I),
and hence contains two non-zero pairwise orthogonal positive
elements $b_1,b_2$. Put $y_j = \langle b_j \rangle$. Then $y_1+y_2 =
\langle b_1 +b_2 \rangle \le \langle a \rangle = x$, and $x \propto
y_j$ for $j=1,2$, because $W(A)$ is algebraically simple, i.e. $x\propto y$ for all $x,y\in W(A)$. \end{proof}
The next example shows that the $\mathrm{Cu}$-semigroups $S_n$ (see the paragraph before Proposition \ref{BetaVsOmega}) can not arise as the Cuntz semigroup of a C*-algebra (cf. \cite[Remark 5.1.17]{APT14}). Hence, this shows that $\omega$-comparison and $\beta$-comparison are equivalent properties for any $\mathrm{Cu}$-semigroup coming from a C*-algebra.
\begin{example}\label{elementaryNon} Let $n$ be a natural number and let $$S_n = \{0,1,2, \dots,n,\infty\}$$ as in Proposition \ref{BetaVsOmega}. $S_n$ is simple and satisfies $\omega$-comparison, but not the property (QQ) (the element $1$ is finite but $\infty = (n+1) \cdot 1$ is properly infinite), therefore neither $\beta$-comparison.
However, note that the semigroup $S_n$ fails to have the weak halving property, hence it can not be the Cuntz semigroup of a simple C*-algebra by Lemma \ref{lm:halving}. (Note that if $\mathrm{Cu}(A)=S_n$ for the completed Cuntz semigroup of a simple C*-algebra $A$, then also $W(A)=S_n$.) \end{example}
\begin{remark} Leonel Robert shows in \cite{Robert13} that a simple Cuntz semigroup $S$ with axioms {\rm (O1)--(O6)} has either Glimm halving (for every nonzero $x\in S$ there is some nonzero $z\in S$ such that $2z\leq x$) or $S=S_n$ for some $n\in\mathbb{N}\cup\{\infty\}$. By the C*-algebraic proof of the weak halving property above, we can rule out the possibility of $S=S_n$ for some $n\in\mathbb{N}$. It follows that every Cuntz semigroup $S=\mathrm{Cu}(A)$, coming from a simple nonelementary C*-algebra $A$, has the Glimm halving property. \end{remark}
We characterize $\omega$-comparison for simple $\mathrm{Cu}$-semigroups $S=\mathrm{Cu}(A)$ coming from a C*-algebra.
\begin{proposition}(cf. \cite{BRTTW})\label{BRTTW} If there is a simple C*-algebra $A$ such that $S=\mathrm{Cu}(A)$, then the $\omega$-comparison is also equivalent to the following statements (see Propostion \ref{Omega}). \begin{itemize} \item[(iv)] If $y\in S$ is such that $\lambda(y)=\infty$ for all functionals $\lambda \in F(S)$, then $y=\infty$. \item[(v)] $A$ is regular, i.e., whenever $D$ is a non-unital hereditary subalgebra of $A\otimes\mathcal{K}$ with no bounded quasitrace, then $D$ is stable. \end{itemize} \end{proposition}
\begin{proof} By Lemma \ref{lm:halving} (and Example \ref{elementaryNon}), $S\neq S_n$ for any $n\in \mathbb{N}$. It follows from Proposition \ref{BetaVsOmega} that $S$ has the $\omega$-comparison if and only if it has the $\beta$-comparison. This shows the equivalence of $\omega$-comparison and {\rm (iv)} with the help of Lemma \ref{BetaLemma} and Lemma \ref{BetaAndFunctionals}.
Since {\rm (iv)} and {\rm (v)} both imply dichotomy (by Proposition \ref{betaImpliesDichotomy} and \cite[Lemma 4.6]{NgCFP} respectively), it suffices to show the equivalence of {\rm (iv)} and {\rm (v)} in the case that all projections in the stabilization of $A$ are finite. In this case, the desired equivalence was shown in \cite[Theorem 4.2.1 (i) \& (iii)]{BRTTW} (see also the last paragraph of \cite[Section 3]{BRTTW}). \end{proof}
\begin{remark}\label{Rm:BRTTW} Notice that it follows from Proposition \ref{BRTTW} that for a simple C*-algebra $r_{A,\infty}$ (radius of comparison with respect to $\infty$ (see \cite{BRTTW} for further details)) is zero if and only if $\mathrm{Cu}(A)$ satisfies $\omega$-comparison. Combining this with Proposition \ref{betaImpliesDichotomy}, ones gets that a simple C*-algebra $A$ with $r_{A,\infty}=0$ is either stably finite or purely infinite. \end{remark}
It was shown in \cite{OPR2} that a $\sigma$-unital C*-algebra $A$ has the corona factorization property (i.e., every full projection in $\mathcal{M}(A\otimes \mathcal{K})$ is properly infinite) if and only if $\mathrm{Cu}(A)$ has the CFP. We discussed in Section 4 that the corona factorization property might allow for the existence of both finite and infinite compact elements in a simple $\mathrm{Cu}$-semigroup (Example \ref{Ex:CFP-noW}). One may therefore ask the question whether the simple nuclear C*-algebra $A$ containing both a non-zero finite and an infinite projection constructed in \cite{R03} has the CFP. That this is not the case is proven in Theorem \ref{RordamsAlgebraNoCFP}. Before proving Theorem \ref{RordamsAlgebraNoCFP}, let us first state the following result, which follows immediately from Proposition \ref{CharactCFP}. (But note that the proof to Theorem \ref{RordamsAlgebraNoCFP} only requires the trivial direction of Propostion \ref{NoCFPCharact}.)
\begin{proposition}\label{NoCFPCharact} Let $A$ be a simple C*-algebra containing both a finite and an infinite projection. Then $A$ does not have the CFP if and only if there is a sequence of elements $(z_n)_n$ in $\mathrm{Cu}(A)$ such that $z_n\ll\infty$, $2z_n=\infty$ and $\sum_{n=1}^\infty z_n<\infty$. \end{proposition}
\begin{theorem}\label{RordamsAlgebraNoCFP} The Cuntz semigroup of the simple nuclear C*-algebra $C$ constructed in \cite{R03}, containing both a non-zero finite and an infinite projection, does not have the CFP. \end{theorem}
\begin{proof} We will remind the reader of some key features of the construction retaining the notation from \cite{R03}. The algebra in question is a crossed product $C=D\rtimes_\alpha \mathbb{Z}$. We will then find the desired elements for the application of Proposition \ref{NoCFPCharact} right from its construction.
At first, let $A:=C(\prod_{j=1}^\infty S^2,\mathcal{K})$. There is an injective map $\varphi$ from $A$ into its multiplier algebra $\mathcal{M}(A)$ with certain properties (see \cite[Proposition 5.2]{R03}), which extends to an injective map $\bar{\varphi}:\mathcal{M}(A)\rightarrow \mathcal{M}(A)$. This extension $\bar{\varphi}$ induces an inductive sequence with limit $B$ given by $$\xymatrix{\mathcal{M}(A) \ar[r]^{\bar{\varphi}} \ar@/_1.5pc/[rrrr]_{\mu_{\infty,0}} & \mathcal{M}(A) \ar[r]^{\bar{\varphi}}&\mathcal{M}(A) \ar[r]^{\bar{\varphi}} & \ldots \ar[r] & B}.$$ Let $\alpha$ denote the natural automorphism on $B$ coming from this inductive limit structure. Now, the algebra $D$ in the crossed product is given by the inductive limit of building blocks $D_n=\text{C*}(A_{-n},\ldots,A_{-1},A_0,A_1,\ldots, A_n)$ with injective connecting maps given by inclusion. Here, $A_0:=\mu_{\infty,0}(A)\cong C(\prod_{j=1}^\infty S^2,\mathcal{K})$, $A_n:=\alpha^n(\mu_{\infty,0}(A))$ for all $n\in\mathbb{Z}$. The properties of $\varphi$ imply that $A_n\cap A_m=\{0\}$ and $A_nA_m=A_{min\{n,m\}}$.
The infinite projection $\mu_{\infty,0}(g)$ in $D\rtimes_\alpha\mathbb{Z}$ is given by the image of the trivial projection $g$ in $C(\prod_{j=1}^\infty S^2,\mathcal{K})\cong A_0=D_0$. (The map is given by the composition of the inclusion of $D_0$ into $D$ and the natural inclusion of $D$ into $D\rtimes_\alpha\mathbb{Z}$.) The finite projection is given by the image of the Bott projection, $Q:=\mu_{\infty,0}(p_1)\in D_0\hookrightarrow D\rtimes_\alpha\mathbb{Z}$, where $p_1$ denotes the Bott projection over the first coordinate of $\prod_{j=1}^\infty S^2$.
We have that $\alpha(\mu_{\infty,0}(p_1))=\mu_{\infty,0}(\varphi(p_1))$. In $\mathcal{M}(A)$, we have that $\varphi(p_1)>q_n, n=1,2,\ldots$ for an infinite sequence of mutually orthogonal projections $q_n$ in $A$. Each $q_n$ is equivalent in $A$ to a Bott projection $p_{\nu(n)}$ with $\nu(n)\in \mathbb{N}$ denoting the coordinate of $\prod_{j=1}^\infty S^2$ over which the Bott projection is taken. (In the notation of \cite{R03} we have $\varphi(p_1)> \sum_{j=-\infty}^0 S_j p_{\nu(j,1)} S_j^*$, so $q_n:=S_{(-n)} p_{\nu(-n,1)} S_{(-n)}^*$.)
Setting $z_n:=\langle \mu_{\infty,0}(q_n) \rangle$ (where $\langle a \rangle$ denotes the Cuntz class of $a$), we have that $$\sum_{n=1}^\infty z_n= \sum_{n=1}^\infty \langle \mu_{\infty,0}(q_n) \rangle<\langle \mu_{\infty,0}(\varphi(p_1)) \rangle=\langle \alpha(\mu_{\infty,0}(p_1)) \rangle =\langle \mu_{\infty,0}(p_1) \rangle=\langle Q\rangle$$ is finite, and $2\cdot z_n=2\cdot \langle \mu_{\infty,0}(q_n)\rangle =\langle \mu_{\infty,0}( p_{\nu(-n,1)}\oplus p_{\nu(-n,1)})\rangle\geq \langle \mu_{\infty,0}(g)\rangle =\infty,\ n\in\mathbb{N}.$ \end{proof}
The next result provides the relation between the corona factorization property and property {\rm (QQ)} for simple C*-algebras. Proposition \ref{PropQQ} shows that property (QQ) implies the CFP. Example \ref{Ex:CFP-noW} and Example \ref{Ex:stablyNoFunctionals} show that the converse does not hold. However, if we rule out examples like the ones in \ref{Ex:CFP-noW} and \ref{Ex:stablyNoFunctionals} by assuming cancellation of small elements at infinity, then we do get the converse.
\begin{theorem}\label{CFPVsQQAlg} Let $A$ be a simple $C^*$-algebra. Then $\mathrm{Cu}(A)$ has property {\rm (QQ)} if and only if $\mathrm{Cu}(A)$ has both the CFP and cancellation of small elements at infinity. \end{theorem}
\begin{proof} Proposition \ref{QQImpliesCAI} and Proposition \ref{PropQQ} show that the 'only if'-direction holds.
For the converse let us assume cancellation of small elements at infinity to hold. As in the proof of Proposition \ref{CFPVsQQ} we note that all we need to show is that, if there is some $x\in Cu(A)$ with $m\cdot x=\infty$ for some $m$, yet $x\neq\infty$, then there is a sequence $(z_n)_n$ such that $N\cdot \sum_{n=1}^\infty z_n=\infty$ for some $N$, yet $\sum_{n=1}^\infty z_n\neq \infty$. (In Proposition \ref{CFPVsQQ} we saw that this is easy with axiom (O5) in the algebraic case, i.e., in the case that every element in $Cu(A)$ can be written as the supremum of compact elements.)
The proof is divided in cases:
At first, suppose that we have $x\in \mathrm{Cu}(A)$ such that $m\cdot x=\infty$ for some $m>2$, but $(m-1)\cdot x\neq \infty$. Find $a\in (A\otimes \mathcal{K})_+$ of norm 1 with $\langle a \rangle=x$. For given $\alpha< \beta \in\mathbb{R}$, let $f_{\alpha,\beta}$ denote the function from $\mathbb{R}_+$ into itself given by $$f_{\alpha, \beta}(t)=\left \{\begin{array}{ll} 0 & ,0\leq t\leq \alpha \mbox{ and }t \geq \beta \\ 1 &, t=(\beta+\alpha)/2 \end{array} \right. \mbox{, and linear elsewhere.} $$ Note that $f_{\alpha,\beta}(a)\ll\infty$ for each $0<\alpha<\beta$. We set $a_n:=f_{1/2^{n},3/ 2^{n}}(a) $ and $z_n:=\langle a_n \rangle$, $n\geq 1$. Then $a_{2n}$ is orthogonal to $a_{2k}$, and $a_{2n+1}$ is orthogonal to $a_{2k+1}$, whenever $k\neq n$. It follows that $\sum_{n=1}^\infty z_{2n}\leq \langle a \rangle =x$, and also $\sum_{n=1}^\infty z_{2n-1}\leq x$, so $\sum_{n=1}^\infty z_{n}\leq 2x$. On the other hand, $\sum_{n=1}^N z_{n}\geq \langle (a-1/2^N)_+ \rangle$ for each $N$, where $(a-\epsilon)_+=g(a)$ for $g(t)=\max\{0,t-\epsilon\}$. Hence, $$ \sum_{n=1}^\infty z_{n}\leq 2x<\infty, \mbox{ and }m\cdot \sum_{n=1}^\infty z_{n}\geq m\cdot x=\infty.$$ We found our desired sequence $(z_n)_n$.
In the case that $m=2$, we may try to proceed as before to find the sequence $(z_n)_n$. In this case there exist two possibilities: If we are lucky, the $z_n$'s satisfy $\sum_{n=1}^\infty z_n <\infty$, in which case we are done, just as before. But possibly $\sum_{n=1}^\infty z_n =\infty$, in which case we need to restart to choose the $z_n$'s more carefully. Let us study this second case.
Suppose $2x=\infty$, $x\neq \infty$, and find $a\in (A\otimes \mathcal{K})_+$ of norm 1 with $\langle a \rangle=x$. Set $a_1:=(a-1/2)_+$, and $z_1:=\langle a_1 \rangle \in \mathrm{Cu}(A)_{\ll\infty}$. Let $y_2:=\langle f_{0,3/4}(a)\rangle$. Then $2y_2+2z_1\geq 2x=\infty$. By cancellation of small elements at infinity we must have that $2y_2=\infty$. We can write $y_2=\sup_n \langle f_{1/n,3/4}(a)\rangle $. Hence, $$z_1\ll \infty = 2\cdot y_2=2\cdot \sup_n \langle f_{1/n,3/4}(a)\rangle ,$$ so we can find $0< \delta_2< 1/2$ such that $z_1\leq 2\cdot \langle f_{\delta_2,3/4}(a)\rangle$. We set $a_2:=f_{\delta_2,3/4}(a)$ and $z_2:=\langle a_2 \rangle$.
Now find $\delta_2<\gamma_2<1/2$, set $a_3:=f_{\delta_2/2,\gamma_2}(a)$ and set $z_3:=\langle a_3 \rangle \in Cu(A)_{\ll\infty}$.
Similar to the previous step, we set $y_4:=\langle f_{0,3\delta_2/4}(a)\rangle$ and get $$z_3\ll \infty = 2\cdot y_4=2\cdot \sup_n \langle f_{1/n,3\delta_2/4}(a)\rangle.$$ Thus, proceeding inductively, we get a sequence $(a_n)_n$ of positive elements in $A\otimes \mathcal{K}$ and a sequence $(z_n)_n$ in $\mathrm{Cu}(A)$, such that: \begin{itemize} \item[(1)] $z_n=\langle a_n\rangle\ll\infty$ for all $n$. \item[(2)] $a_n\leq a$ for all $n$. \item[(3)] For all $n\neq k$, $a_{2n}$ is orthogonal to $a_{2k}$, and $a_{2n+1}$ is orthogonal to $a_{2k+1} $. \item[(4)] $z_{2n-1}\leq 2 z_n$ for all $n$. \item[(5)] $\sum_{n=1}^\infty z_{2n}\leq \langle a \rangle=x$ and $\sum_{n=1}^\infty z_{2n+1}\leq x$. \item[(6)] $\sum_{n=1}^\infty z_{n}\geq \langle a \rangle=x$. \end{itemize}
Recall that by assumption we have $\sum z_n=\infty$, and that by (4) it follows that $\sum_{n=1}^\infty z_{2n-1}\leq 2\cdot \sum_{n=1}^\infty z_{2n}$. Therefore, $$3\cdot \sum_{n=1}^\infty z_{2n}\geq \sum_{n=1}^\infty z_{n}=\infty \mbox{ , while } \sum_{n=1}^\infty z_{2n}\leq x \neq \infty.$$ We found the desired sequence with $(z_{2n})_n$. \end{proof}
We conclude this paper with an overview of our results on comparison properties for the Cuntz semigroup of a C*-algebra, together with a list of interesting open questions that naturally arise from our studies.
\begin{theorem}\label{thm:end} Let $A$ be a simple C*-algebra. Then we have the following diagram of relations for comparison properties of the $\mathrm{Cu}$-semigroup $\mathrm{Cu}(A)$:
$$\xymatrix{ \omega-comparison \ar@{<=>}[d] \ar@{=>}[rd]& \\ \beta-comparison \ar@{<=>}[dd]&{\rm (QQ)} \ar@{<=>}[dd] \\& \\\ \lambda(y)=\infty \mbox{ for all functionals } \lambda\Leftrightarrow y=\infty\ar@{<=>}[d]&\mbox{CFP $\&$ cancel. small elements at }\infty\\ \nexists \mbox{ faithful state }f\in\mathcal{S}(\mathrm{Cu}(A),y)\Leftrightarrow y=\infty \ar@{=>}[ru] & }$$ \end{theorem}
\begin{question}\label{QN} \begin{itemize} \item Is there any simple C*-algebra $A$ such that $\mathrm{Cu}(A)=[0,1]\cup\{\infty\}$?\newline Or any
stably finite C*-algebra such that $\mathrm{Cu}(A)=\{(0,0)\} \cup ((0,1]\cup\{\infty\})\times(0, \infty]?$ \item Does $\mathrm{Cu}(A)$ have cancellation of small elements at infinity for any simple stably finite C*-algebra? \item Is CFP plus cancellation of small elements at infinity equivalent to $\omega$-comparison for any $\mathrm{Cu}$-semigroup? What about for the $\mathrm{Cu}$-semigroup arising from a C*-algebra? \item Is CFP is equivalent to $\omega$-comparison for the Cuntz semigroup arising from a C*-algebra? \end{itemize} \end{question}
\end{document} | arXiv |
Serial relation
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.
Bertrand Russell used serial relations in The Principles of Mathematics[1] (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.[2]
A serial relation R is an endorelation on a set U. As stated by Russell, $\forall x\exists y\ xRy,$ where the universal and existential quantifiers refer to U. In contemporary language of relations, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest.
For a relation R, let {y: xRy } denote the "successor neighborhood" of x. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood".[3]
In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.[4]
Russell's series
Relations are used to develop series in The Principles of Mathematics. The prototype is Peano's successor function as a one-one relation on the natural numbers. Russell's series may be finite or generated by a relation giving cyclic order. In that case, the point-pair separation relation is used for description. To define a progression, he requires the generating relation to be a connected relation. Then ordinal numbers are derived from progressions, the finite ones are finite ordinals.[1]: Chapter 28: Progressions and ordinal numbers Distinguishing open and closed series[1]: 234 results in four total orders: finite, one end, no end and open, and no end and closed.[1]: 202
Contrary to other writers, Russell admits negative ordinals. For motivation, consider the scales of measurement using scientific notation, where a power of ten represents a decade of measure. Informally, this parameter corresponds to orders of magnitude used to quantify physical units. The parameter takes on negative as well as positive values.
Stretch
Russell adopted the term stretch from Meinong, who had contributed to the theory of distance.[5] Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole."[1]: 181 To explain Meinong, Russell refers to the Cayley-Klein metric, which uses stretch coordinates in anharmonic ratios which determine distance by using logarithm.[1]: 255 [6]
References
1. Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858.
2. B. A. Bernstein (1926) "On the Serial Relations in Boolean Algebras", Bulletin of the American Mathematical Society 32(5): 523,4
3. Yao, Y. (2004). "Semantics of Fuzzy Sets in Rough Set Theory". Transactions on Rough Sets II. Lecture Notes in Computer Science. Vol. 3135. p. 309. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.
4. James Garson (2013) Modal Logic for Philosophers, chapter 11: Relationships between modal logics, figure 11.1 page 220, Cambridge University Press doi:10.1017/CBO97811393421117.014
5. Alexius Meinong (1896) Uber die Bedeutung der Weberische Gesetze
6. Russell (1897) An Essay on the Foundations of Geometry
External links
• Jing Tao Yao and Davide Ciucci and Yan Zhang (2015). "Generalized Rough Sets". In Janusz Kacprzyk and Witold Pedrycz (ed.). Handbook of Computational Intelligence. Springer. pp. 413–424. ISBN 9783662435052. Here: page 416.
• Yao, Y.Y.; Wong, S.K.M. (1995). "Generalization of rough sets using relationships between attribute values" (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..
| Wikipedia |
Experimental Physics Mechanics Pdf
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Philipp Habegger
Philipp Habegger (born 23 July 1978) is a Swiss[1] mathematician and a professor of mathematics at the University of Basel who works in Diophantine geometry.
Philipp Habegger
Born (1978-07-23) July 23, 1978
NationalitySwiss
Alma materUniversity of Basel
Scientific career
FieldsMathematics
InstitutionsUniversity of Basel
University of Zurich
ETH Zurich
ThesisHeights and Multiplicative Relations on Algebraic Varieties (2007)
Doctoral advisorDavid Masser
Early life and education
Habegger was born on 23 July 1978.[1] He received his Ph.D. under the supervision of David Masser at the University of Basel in 2007.[2]
Career
From 2008 to 2010, Habegger was a ETH Fellow at ETH Zurich.[1] He moved to the University of Zurich for a lectureship position in 2010.[1] In 2013, he was a von Neumann Fellow at the Institute for Advanced Study.[3] As of 2021, Habegger is a professor of mathematics at the University of Basel.[4]
Research
Habegger's research focuses on height functions and their applications to unlikely intersections.[3]
Selected publications
• Gao, Ziyang; Habegger, Philipp (2019). "Heights in families of abelian varieties and the Geometric Bogomolov Conjecture". Annals of Mathematics. 189 (2): 527. arXiv:1801.05762. doi:10.4007/annals.2019.189.2.3. JSTOR 10.4007/annals.2019.189.2.3. S2CID 56339965.
• Habegger, Philipp (2009). "Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension". Inventiones Mathematicae. 176 (2): 405–447. doi:10.1007/s00222-008-0170-6. hdl:20.500.11850/156804. ISSN 0020-9910. S2CID 122167169.
• Habegger, Philipp (2013). "Small height and infinite nonabelian extensions". Duke Mathematical Journal. 162 (11): 2027–2076. arXiv:1109.5859. doi:10.1215/00127094-2331342. ISSN 0012-7094. S2CID 119435006.
• Habegger, Philipp (2013). "Special points on fibered powers of elliptic surfaces". Journal für die reine und angewandte Mathematik. 2013 (685). arXiv:1110.1908. doi:10.1515/crelle-2012-0007. ISSN 0075-4102. S2CID 119138713.
• Habegger, P.; Pila, J. (2012). "Some unlikely intersections beyond André–Oort". Compositio Mathematica. 148 (1): 1–27. doi:10.1112/S0010437X11005604. ISSN 0010-437X.
References
1. "Curriculum Vitae" (PDF). Philipp Habegger. November 2010. Retrieved 5 March 2021.
2. Philipp Habegger at the Mathematics Genealogy Project
3. "Philipp Habegger". Institute for Advanced Study. 9 December 2019. Retrieved 5 March 2021.
4. "Philipp Habegger". University of Basel (in German). Retrieved 5 March 2021.
External links
• Personal website
Authority control
International
• ISNI
• VIAF
National
• Catalonia
• Israel
• Belgium
• United States
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• MathSciNet
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Other
• IdRef
| Wikipedia |
\begin{document}
\begin{abstract} We provide an expression for the conductor $c(\Delta)$ of a semimodule $\Delta$ of a numerical semigroup $\Gamma$ with two generators in terms of the syzygy module of $\Delta$ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of $\Gamma$, as well as a formula for $c(\Delta)$ in terms of the dual semimodule of $\Delta$. \keywords{Numerical semigroup \and Frobenius problem \and $\Gamma$-semimodule \and syzygy}
\end{abstract} \maketitle \section{Introduction} \label{intro} A classical problem in the combinatorics of natural numbers is to find a closed expression for the largest natural number that is not representable as a nonnegative linear combination of some relatively prime numbers, called the Frobenius number. This problem can be encoded in terms of getting a formula for the conductor of a numerical semigroup; it is known under the name ``Frobenius problem''.
Consider $\mathbb{N}=\{x\in \mathbb{Z}: x\geq 0\}$. A numerical semigroup $\Gamma$ is an additive sub-monoid of the monoid $(\mathbb{N},+)$ such that the greatest common divisor of all its elements is equal to $1$. The complement $\mathbb{N}\setminus \Gamma$ is therefore finite, and its elements are called gaps of $\Gamma$. Moreover, $\Gamma$ is finitely generated and it is not difficult to find a minimal system of generators of $\Gamma$, see. e.g. Rosales and Garc\'ia S\'anchez \cite{RosalesGarciaSanchez}.
The number $c(\Gamma)=\max (\mathbb{N}\setminus \Gamma)+1$ is called the conductor of $\Gamma$; in particular $c(\Gamma)-1$ is the Frobenius number of $\Gamma$. The computation of $c(\Gamma)$ for an arbitrary number of minimal generators of $\Gamma$ is NP-hard (see Ram\'irez Alfons\'in \cite{ram} for a good account of this), but there are some special cases in which a closed formula is available. For example, if $\Gamma=\alpha \mathbb{N}+\beta\mathbb{N}:=\langle \alpha , \beta \rangle$, then $c(\Gamma)=\alpha \beta -\alpha -\beta +1$. However, for a numerical semigroup with more than two generators it is not possible in general to obtain a closed polynomial formula for its conductor in terms of the minimal set of generators (see Curtis \cite{curtis}).
We are interested in subsets of $\mathbb{N}$ which have an additive structure over $\Gamma$ (in analogy with the structure of module over a ring): a $\Gamma$-semimodule is a non-empty subset $\Delta$ of $\mathbb{N}$ with $\Delta+\Gamma\subseteq \Delta$. A system of generators of $\Delta$ is a subset $\mathcal{E}$ of $\Delta$ such that $\Delta=\bigcup_{x\in \mathcal{E}} (x+\Gamma)$; it is called minimal if no proper subset of $\mathcal{E}$ generates $\Delta$. Notice that, since $\Delta\setminus \Gamma$ is finite, every $\Gamma$-semimodule is finitely generated and has a conductor $$ c(\Delta)=\max (\mathbb{N}\setminus \Delta)+1. $$
Motivated by the Frobenius problem, it is natural to ask for a closed formula for the conductor of a \(\Gamma\)-semimodule. The purpose of this note is to give a formula for $c(\Delta)$ in the case $\Gamma=\langle \alpha, \beta \rangle$ in terms of the generators of the semimodule of syzygies of $\Delta$, see \cite{MU1}, as well as in terms of the generators of the dual of this semimodule, see \cite{MU2}. These are the contents of our two main results, namely Theorem \ref{formula} resp.~Corollary \ref{cor:main}.
\section{Semimodules over a numerical semigroup}
Let $\Gamma$ be a numerical semigroup. This section is devoted to collect the main properties concerning $\Gamma$-semimodules. The reader is referred to \cite{RosalesGarciaSanchez} or \cite{ram} for specific material about numerical semigroups.
Every $\Gamma$-semimodule $\Delta$ has a unique minimal system of generators (see e.g. \cite[Lemma 2.1]{MU1}). Two $\Gamma$-semimodules $\Delta$ and $\Delta'$ are called isomorphic if there is an integer $n$ such that $x\mapsto x+n$ is a bijection from $\Delta$ to $\Delta'$; we write then $\Delta\cong \Delta'$. For every $\Gamma$-semimodule $\Delta$ there is a unique semimodule $\Delta' \cong \Delta$ containing $0$; such a semimodule is called normalized.
Moreover, the minimal system of generators $\{x_0=0,\ldots , x_n\}$ of a normalized $\Gamma$-semimodule is a $\Gamma$-lean set, i.e. it satisfies that $$
|x_i-x_j| \notin \Gamma \ \ \mbox{for~any} \ \ 0\leq i <j \leq n, $$ and conversely, every $\Gamma$-lean set of $\mathbb{N}$ minimally generates a normalized $\Gamma$-semimo\-dule. Hence there is a bijection between the set of isomorphism classes of $\Gamma$-semimodules and the set of $\Gamma$-lean sets of $\mathbb{N}$. See Sect. 2 in \cite{MU1} for the proofs of those statements.
There is another kind of system of generators---not minimal---for a semimodule $\Delta$ of $\Gamma$ relative to $s\in \Gamma\setminus \{0\}$: this is the set of the $s$ smallest elements in $\Delta$ in each of the $s$ classes modulo $s$, namely the set $\Delta \setminus (s+\Delta)$, and is called the Ap\'ery set of $\Delta$ with respect to $s$; we write $\mathrm{Ap}(\Delta,s)$.
A formula for the conductor in terms of $\mathrm{Ap}(\Delta,s)$ for $s\in \Gamma\setminus \{0\}$ is easily deduced.
\begin{proposition}\label{prop:cond_Apery}
Let $\Delta$ be a $\Gamma$-semimodule. For any $s\in \Gamma\setminus\{0\}$ we have that
$$
c(\Delta)-1=\mathrm{max}_{\leq_\mathbb{N}} \mathrm{Ap}(\Delta,s)-s.
$$ \end{proposition}
\begin{proof}
The equality follows as in the case $\Delta=\Gamma$, see e.g.~Lemma 3 in Brauer and Shockley \cite{bs}. \end{proof}
In this paper we will consider numerical semigroups with two generators, say $\Gamma=\langle \alpha, \beta \rangle$, with $\alpha,\beta \in \mathbb{N}$ with $\alpha < \beta $ and $\mathrm{gcd}(\alpha, \beta) = 1$. As mentioned above, the conductor of $\Gamma$ can be expressed as $c=c(\langle \alpha,\beta \rangle)=(\alpha-1)(\beta-1)$. The gaps of $\langle \alpha, \beta \rangle$ are also easy to describe: they admit a unique representation $\alpha \beta -a\alpha -b \beta$, where $a\in \ ]0,\beta-1]\cap \mathbb{N}$ and $b\in \ ]0,\alpha-1]\cap \mathbb{N}$. This writing yields a map from the set of gaps of $\langle \alpha, \beta \rangle$ to $\mathbb{N}^2$ given by $$\alpha \beta -a\alpha -b \beta \mapsto (a,b),$$ which allows us to identify a gap with a lattice point in the lattice $\mathcal{L}=\mathbb{N}^2$; since the gaps are positive numbers, the point lies inside the triangle with vertices $(0,0),(0,\alpha),(\beta, 0)$.
In the following we will use the notation $$ e=\alpha\beta - a(e)\alpha-b(e)\beta $$ for a gap $e$ of the semigroup $\langle \alpha,\beta \rangle$; if the gap is subscripted as $e_i$ then we write $a_i=a(e_i)$ and $b_i=b(e_i)$.
Let us denote by \(\leq\) the total ordering in $\mathbb{N}$; sometimes we will write \(\leq_{\mathbb{N}}\) to emphasize that it is the natural ordering. In addition, we define the following partial ordering $\preceq$ on the set of gaps:
\begin{definition}\label{jorder}
Given two gaps $e_1,e_2$ of $\langle \alpha , \beta \rangle$, we define
$$
e_1 \preceq e_2 \ \ :\Longleftrightarrow \ a_1\leq a_2 \ \ \wedge \ \ b_1 \geq b_2
$$
and
$$
e_1 \prec e_2 \ \ :\Longleftrightarrow \ a_1 < a_2 \ \ \wedge \ \ b_1 >b_2.
$$ \end{definition}
Observe that the ordering $\preceq$ differs from the one used by the second author and Uliczka in \cite{MU,MU1,MU2}: there the gaps $e_i$ are ordered by decreasing sequence of the corresponding $a_i$.
Let $\mathcal{E}=\{0,e_1,\ldots , e_n\} \subseteq \mathbb{N}$ with gaps $e_i=\alpha\beta -a_i \alpha -b_i\beta$ of $\langle \alpha, \beta \rangle$ for every $i=1,\ldots, n$ such that $a_1<a_2<\cdots < a_n$. Corollary 3.3 in \cite{MU1} ensures that $\mathcal{E}$ is $\langle \alpha , \beta \rangle$-lean if and only if $b_1>b_2>\cdots > b_n$.
This simple fact leads to an identification (cf. \cite[Lemma 3.4]{MU1}) between an $\langle \alpha, \beta \rangle$-lean set and a lattice path with steps downwards and to the right from $(0,\alpha)$ to $(\beta,0)$ not crossing the line joining these two points, where the lattice points identified with the gaps in $\mathcal{E}$ mark the turns from the $x$-direction to the $y$-direction; these turns will be called ES-turns for abbreviation. Figure \ref{fig1} shows the lattice path corresponding to the $\langle 5,7 \rangle$-lean set $\{0,9,11,8\}$.
\begin{center}
\begin{figure}\label{fig1}
\end{figure} \end{center}
Let $g_0=0,g_1,\ldots , g_n$ be the minimal system of generators of a $\langle \alpha, \beta \rangle$-semi\-module $\Delta$. From now on, we will assume that the indexing in the minimal set of generators of $\Delta$ is such that $g_0=0\preceq g_1\preceq\cdots\preceq g_n$; accordingly we will use the notation $[g_0,\ldots, g_n]$ rather than $\{g_0,\ldots , g_n\}$. In \cite{MU1} it was introduced the notion of syzygy of $\Delta$ as the $\langle \alpha, \beta \rangle$-semimodule $$ \mathrm{Syz}(\Delta):=\bigcup_{i,j\in \{0,\ldots n\}, i\neq j} \Big ((\Gamma + g_i)\cap (\Gamma + g_j) \Big ). $$
The semimodule of syzygies of the semimodule $\Delta$ minimally generated by $[g_0=0,g_1,\dots,g_n]$ can be characterized as follows (see \cite[Theorem 4.2]{MU1}; since Definition \ref{jorder} differs from the corresponding in \cite{MU,MU1,MU2}---as mentioned above, Definition \ref{defin:syz} must be conveniently adapted here): \begin{definition}\label{defin:syz}
\begin{equation*}
\mathrm{Syz}(\Delta) =\bigcup_{0\leq k<j\leq n} \Big ((\Gamma + g_k)\cap (\Gamma + g_j) \Big )= \bigcup_{k=0}^{n} (\Gamma + h_k),
\end{equation*}
where $h_1,\ldots , h_{n-1}$ are gaps of $\Gamma$, $h_0,h_n \leq \alpha \beta$, and
\begin{align*}
&h_k \equiv g_k ~\mathrm{ mod }~ \beta, ~h_ k > g_{k} \ \mbox{for } k=0,\ldots , n \\
&h_k \equiv g_{k+1} ~ \mathrm{mod } ~\alpha,~ h_ k > g_{k+1} \ \mbox{for } k=0,\ldots , n-1\\
&h_n \equiv 0~\mathrm{mod } ~\alpha, \ \mathrm{and } ~h_n \geq 0
\end{align*} \end{definition}
In particular, \(J=[h_0,\dots,h_n]\) is a minimal system of generators of the semimodule \(\mathrm{Syz}(\Delta)\), hence \(h_0\preceq h_1\preceq\cdots\preceq h_n.\) Therefore it is easily seen that the SE-turns of the lattice path associated to \(\Delta\) can be identified with the minimal set of generators of the syzygy module (we call SE-turns to the turns from the \(y\)--direction to the \(x\)--direction). After that, we can associate to any \(\Gamma\)-semimodule \(\Delta\) a lean set \([I,J]\), where \(I\) is a minimal set of generators of \(\Delta\) and \(J\) a minimal set of generators of \(\mathrm{Syz}(\Delta);\) or, equivalently, a lattice path. An easy consequence of this fact is the following lemma.
\begin{lemma}\label{lem:aux}
Let \(\Delta\) be a \(\Gamma\)-semimodule with associated $\Gamma$-lean set \([I,J]\) for \(I=[g_0=0,g_1,\dots,g_n]\) and \(J=[h_0,\dots,h_n]\). Then, for any \(h\in J\) we have \(h-\alpha-\beta\notin\Delta.\) \end{lemma}
\begin{proof}
Consider \(h\in J\) such that that \(g_i\prec h\prec g_{i+1}\). Let us denote $(a_j,b_j)$ resp. $(a_{j+1},b_{j+1})$ the coordinates of $g_j$ resp. $g_{j+1}$ in the lattice $\mathcal{L}$; then the element \(h\) is represented in the lattice path as $(a_j,b_{j+1})$, see Definition \ref{defin:syz}. By contradiction, assume that $h-\alpha-\beta\in \Delta$; then there exists a gap \(g\in I\) together with two integers $\nu_1,\nu_2\in\mathbb{N}$ such that
\[
h-\alpha-\beta=\nu_1\alpha+\nu_2\beta+g.
\]
Since $h-\alpha-\beta\notin\Gamma$, we may write
\[
h-\alpha-\beta=\alpha\beta-(a_j+1)\alpha-(b_{j+1}+1)\beta.
\]
The writing of $g$ as $g=\alpha\beta-a\alpha-b\beta$ is unique whenever $(a,b)\in \mathcal{L}$, therefore
\[
a_j+1=a-\nu_1,\quad b_{j+1}+1=b-\nu_2.
\]
These equalities yield the conditions $a_j<a$ and \(b_{j+1}<b\). But the unique minimal generator which fulfills these conditions is \(g_{j+1}\); however, \(h\) cannot be expressed as \(h=g_{j+1}+\nu+\alpha+\beta\) since \(h\) is represented in the lattice path as $(a_j,b_{j+1})$, a contradiction. \end{proof} \begin{example}
For \(\Gamma=\langle 5,7 \rangle\) and the \(\Gamma\)--semimodule \(\Delta_I\) minimally generated by $I=[0,9,11,8]$, it is easily deduced that the syzygy module $\mathrm{Syz}(\Delta_I)$ is minimally generated by $J=[14,16,18,15]$, cf.~Figure \ref{fig1}; there we have extended the labelling beyond the axis in the natural way in order to have also an interpretation of $J$ in terms of the lattice path.
Observe that by Lemma \ref{lem:aux} we have \(14-7-5=2\notin \Delta,\) \(16-7-5=4\notin \Delta,\) \(18-7-5=6\notin \Delta\) and \(15-7-5=3\notin \Delta\); this can be read off from Figure \ref{fig1} as well. \end{example}
\section{A formula for the conductor of an $\langle \alpha, \beta \rangle$-semimodule}
In this section we are going to provide a formula for the conductor of a \(\Gamma\)-semimodule with any number of generators in terms of the generators of $\Gamma$ and a special syzygy of the $\Gamma$-semimodule. In particular, we will obtain some relations between the conductor of $\Gamma$ and the conductor of the $\Gamma$-semimodule. Finally, we will provide a formula for the conductor of the \(\Gamma\)-semimodule in terms of its dual.
\begin{theorem}\label{formula}
Let \(\Delta\) be a \(\Gamma\)-semimodule with associated lean set \([I,J]\) as above, and let \(M:=\max_{\leq_\mathbb{N}}\{h\in J\}\) denote the biggest (with respect to the total ordering of the natural numbers) minimal generator of $\mathrm{Syz}(\Delta)$. Then
\[c(\Delta)=M-\alpha-\beta+1.\]
In particular, if $(m_1,m_2)$ are the coordinates of the point representing $M$ in the lattice $\mathcal{L}$, then we have
\[c(\Delta)=c(\Gamma)-m_1\alpha-m_2\beta.\] \end{theorem}
\begin{proof}
Since \(c(\Delta)-1\) is the Frobenius number of the \(\Gamma\)--semimodule \(\Delta\), it is enough to check that (i) \(M-\alpha-\beta\notin\Delta\), and (ii) if \(\ell\notin\Delta\), then \(\ell\leq M-\alpha-\beta.\)
The statement (i) is clear by Lemma \ref{lem:aux}, since \(M\in J\). To see (ii), consider an element \(\ell\notin\Delta\), which in particular means \(\ell\notin\Gamma\). So we can associate to $\ell$ a point $(a,b)$ in the lattice $\mathcal{L}$. Moreover, \(\ell\) is upon and not contained in the lattice path associated to $I$. This means that there exists some \(j\in J\) with coordinates \((j_1,j_2)\) in the lattice path such that \(a> j_1\) and \(b> j_2\), otherwise \(\ell\) would be an element of \(\Delta\), since the elements represented by lattice points on and under the lattice path belong to \(\Delta\). Therefore, \(a\geq j_1+1\) and \(b\geq j_2+1\). Thus, from the representation of \(\ell\) and \(j\) as gaps we can check that
\[\ell=\alpha\beta-a\alpha-b\beta\leq_{\mathbb{N}}\alpha\beta-(j_1+1)\alpha-(j_2+1)\beta=j-\alpha-\beta.\]
Hence, since \(M=\max_{\leq_\mathbb{N}}\{h\in J\}\) and \(M\in J\), we have that \(M-\alpha-\beta\geq_{\mathbb{N}}\ell\) for any \(\ell\notin\Delta\), which proves (ii).
Finally, since $M$ can be represented as a lattice point $(m_1,m_2)\in \mathcal{L}$, we have
\[c(\Delta)=M-\alpha-\beta+1=\alpha\beta-m_1\alpha-m_2\beta-\alpha-\beta+1=c(\Gamma)-m_1\alpha-m_2\beta.\] \end{proof}
\begin{example}\label{example}
Again in the case of \(\Gamma=\langle 5,7 \rangle\) and the \(\Gamma\)--semimodule minimally generated by $[0,9,11,8]$, Figure \ref{fig1} illustrates that the maximal syzygy is \(M=18\), and so the conductor of the semimodule is $c(\Gamma)-5 m_1-7 m_2 = 24-5\cdot 2 - 7 \cdot 1=7$. \end{example}
Notice that for the particular case of $\Delta=\Gamma$ we have $M=\alpha\beta$, and we recover the well-known formula $c(\Gamma)=\alpha\beta-\alpha-\beta +1$. The value $M$ can be easily characterized in terms of the Ap\'ery set of $\Delta$ with respect to $\alpha+\beta$:
\begin{proposition}
Let \(M:=\max_{\leq_\mathbb{N}}\{h\in J\}\) be the biggest minimal generator of the syzygy module with respect to the natural ordering of $\mathbb{N}$ as above, then
$$
M=\max_{\leq_\mathbb{N}} \mathrm{Ap}(\Delta,\alpha+\beta).
$$ \end{proposition}
\begin{proof}
This is a consequence of Proposition \ref{prop:cond_Apery} for $s=\alpha+\beta \in \langle \alpha, \beta \rangle$. \end{proof}
A straightforward consequence of Theorem \ref{formula} is the following.
\begin{corollary}
Let \(\Delta\) be a \(\Gamma\) semimodule. Then
\[c(\Gamma)-c(\Delta)\in\Gamma.\] \end{corollary}
We conclude this paper rewriting the formula of Theorem \ref{formula} in terms of the dual \(\Gamma\)--semimodule of \(\Delta\), \[
\Delta^{\ast}:=\{z\in\mathbb{Z}\;|\;z+\Delta\subset\Gamma\}, \] see \cite{MU2}. An important fact about the dual semimodule is that the minimal set of generators of \(\mathrm{Syz}(\Delta)\) is in bijection with the minimal set of generators of \(\Delta^{\ast}\): \begin{lemma}[\cite{MU2}, Lemma 6.1]\label{lemma:61}
The minimal sets of generators of \(\Delta^{\ast}\) and \(\mathrm{Syz}(\Delta)\) are in correspondence via the map \(x\mapsto\alpha\beta-x.\) \end{lemma} In particular, this bijection together with Theorem \ref{formula} allows us to compute the conductor of the semimodule \(\Delta\) in terms of the minimal generators of $\Delta^{\ast}$ in a natural way: \begin{corollary}\label{cor:main}
Let $\Delta$ be a $\Gamma$-semimodule, and let $\Delta^{\ast}$ be its dual, minimally generated by $x_0,\dots,x_n$. Then
\[c(\Delta)=\alpha\beta-\min_{\leq\mathbb{N}}\{x_0,\ldots, x_n\}-\alpha-\beta+1.\] \end{corollary} \begin{proof}
By Theorem \ref{formula} we have that \(c(\Delta)=\max_{\leq_\mathbb{N}}\{h\in J\}-\alpha-\beta+1, \) where \(J\) is a minimal set of generators of \(\mathrm{Syz}(\Delta).\) Lemma \ref{lemma:61} yields the equality
$$
\min_{\leq\mathbb{N}}\{x_0, x_1,\ldots, x_n\}=\alpha\beta-\max_{\leq_\mathbb{N}}\{h\in J\},
$$ which allows us to conclude. \end{proof}
\begin{example}
By \cite[Theorem 2.5]{MU2}, the minimal generators of the dual of the $\langle 5,7 \rangle$-semimodule $\Delta_I$ are given by $[20,17,19,21]$; notice that, for the explicit calculation, the mentioned theorem requires the reverse ordering $\succeq$ instead of the ordering $\preceq$ we use here. The minimum of this set is $17$, therefore by Corollary \ref{cor:main} we have $c(\Delta)=35-17-12+1=7$, as computed in Example \ref{example}. \end{example}
\end{document} | arXiv |
Squares are constructed on each of the sides of triangle $\triangle ABC$, as shown. If the perimeter of $\triangle ABC$ is 17, then what is the perimeter of the nine-sided figure that is composed of the remaining three sides of each of the squares? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
picture a;
draw(a,(0,0)--(3,0)--(0,4)--cycle);
label("$A$",(-2.1,-2.2),SW);
label("$B$",(3.1,-2.2),SE);
label("$C$",(0.05,0.3),N);
draw(a,(3,0)--(3,-3)--(0,-3)--(0,0));
draw(a,(0,0)--(-4,0)--(-4,4)--(0,4));
draw(a,shift(-2.4,2.2)*rotate(90 - aTan(4/3))*((3,0)--(8,0)--(8,-5)--(3,-5)));
add(currentpicture,rotate(-130)*a);
[/asy]
Since all sides of a square have the same length, the perimeter of the nine-sided figure is equal to \[ AB + AB + AB + AC + AC + AC + BC + BC + BC. \]But we know that $AB+AC+BC=17$, the perimeter of $\triangle ABC$. Therefore the perimeter of the nine-sided figure is $3(17)=\boxed{51}$. | Math Dataset |
Group penalized generalized estimating equation for correlated event-related potentials and biomarker selection
Ye Lin1,
Jianhui Zhou1,
Swapna Kumar2,3,
Wanze Xie2,3,
Sarah K. G. Jensen2,3,
Rashidul Haque4,
Charles A. Nelson2,
William A. Petri Jr1 &
Jennie Z. Ma ORCID: orcid.org/0000-0002-5839-28821
Event-related potentials (ERP) data are widely used in brain studies that measure brain responses to specific stimuli using electroencephalogram (EEG) with multiple electrodes. Previous ERP data analyses haven't accounted for the structured correlation among observations in ERP data from multiple electrodes, and therefore ignored the electrode-specific information and variation among the electrodes on the scalp. Our objective was to evaluate the impact of early adversity on brain connectivity by identifying risk factors and early-stage biomarkers associated with the ERP responses while properly accounting for structured correlation.
In this study, we extend a penalized generalized estimating equation (PGEE) method to accommodate structured correlation of ERPs that accounts for electrode-specific data and to enable group selection, such that grouped covariates can be evaluated together for their association with brain development in a birth cohort of urban-dwelling Bangladeshi children. The primary ERP responses of interest in our study are N290 amplitude and the difference in N290 amplitude.
The selected early-stage biomarkers associated with the N290 responses are representatives of enteric inflammation (days of diarrhea, MIP1b, retinol binding protein (RBP), Zinc, myeloperoxidase (MPO), calprotectin, and neopterin), systemic inflammation (IL-5, IL-10, ferritin, C Reactive Protein (CRP)), socioeconomic status (household expenditure), maternal health (mother height) and sanitation (water treatment).
Our proposed group penalized GEE estimator with structured correlation matrix can properly model the complex ERP data and simultaneously identify informative biomarkers associated with such brain connectivity. The selected early-stage biomarkers offer a potential explanation for the adversity of neurocognitive development in low-income countries and facilitate early identification of infants at risk, as well as potential pathways for intervention.
The related clinical study was retrospectively registered with https://doi.org/ClinicalTrials.gov, identifier NCT01375647, on June 3, 2011.
Event-related potentials (ERPs) have been widely used in studies of perceptual and cognitive development. ERPs represent the volume-conducted electrical signals generated by large populations of synchronously activated neurons activated in response to stimuli. Specifically, with multiple electrodes on the scalp, ERPs are small parts of electroencephalogram (EEG) recording of the brain response elicited to specific stimuli such as viewing pictures or words on the computer screen [1]. As the brain response to a single stimulus is usually weak or noisy in the EEG recording of a single trial, an ERP waveform is actually generated from the aggregated EEG recordings over many trials for better brain response measuring [2, 3]. In general, an ERP waveform consists of a series of positive and negative voltage deflections, characterized by the amplitudes of negative- or positive-going peaks or the latencies to these peaks in milliseconds (ms). For example, the N290 component surfaces as a negative deflection in voltage and with a peak latency between 250 and 350 ms, while the P400 component appears as a positive-going waveform that peaks between 350 and 450 ms depending on the age of the child [4–6]. Consequently, ERP data (amplitudes or latencies) are hierarchical in that there are multiple ERP measurements for each subject corresponding to multiple treatment or stimulus conditions and multiple channels (i.e., electrodes), while channels are further clustered in different regions of the brain. Comparisons of brain activities between different treatment conditions for different channels in different brain regions are of research interest [7].
In the previous literature, there are a few approaches to compare ERPs between different stimulus conditions from multiple channels. One approach is to compare ERPs between conditions for each channel individually, which is often subjected to multiple comparison problem. Lage -Castellanos et al. [8] applied false discovery rate method and performed a permutation test for comparisons within each channel and at each time point. Causeur et al. [9] introduced a dynamic factor model for multiple testing to account for the dependence among hypotheses. The second approach is to analyze the data from all channels simultaneously. One popular tactic is to group the channels by the brain regions such as frontal, central and parietal, and then perform Analysis of Variance (ANOVA) separately for each region, or include region as a factor in Multivariate Analysis of Variance (MANOVA) for all ERPs together [10]. Yet another approach is to average the ERPs over the multiple channels of interest and then compare conditions using one-way ANOVA. Either way, the channels within a brain region would be treated the same and the variations or the correlation structure between individual channels would not be accounted for. In fact, ERP measures do not only vary but also are highly correlated among channels. Vossen et al. [11] showed the correlated structure among ERP data and applied mixed regression approach. However, they only considered the correlation among repeated measurements from different conditions while channels are still modeled separately. To improve estimation efficiency, a model accounting for both the individual channel effects and the correlation structure is highly desired in ERP data analysis.
In addition to evaluating the effect of treatment conditions on the brain response of interest in ERP data, motivated by our clinical study, we are also interested in that whether such brain response is attributable to a set of important clinical risk factors and biomarkers. Since a large number of risk factors and biomarkers are available in the clinical study, variable selection using penalized methods would be preferred for such high-dimensional data to select the important predictors and estimate their impacts on the brain response. Many penalized methods have been developed based on different penalties for high-dimensional data, such as Least Absolute Shrinkage and Selection Operator (LASSO) [12], Smoothly Clipped Absolute Deviation (SCAD) [13], Elastic Net [14] and Adaptive LASSO [15]. Penalized methods for correlated data have also been proposed for marginal models [16] and for mixed effects models [17]. In addition, Wang et al. [18] proposed penalized generalized estimating equations (PGEE) for high-dimensional correlated data based on SCAD penalty. However, these available methods are not readily applicable to ERP data mainly due to the lack of consideration of the specific structured correlation among different channels in ERP data, especially when both conditions and channels are included. Second, the SCAD-based PGEE does not allow group variable selection, which is pivotal in the clinical studies as many risk factors or biomarkers are clustered or potentially correlated.
In this paper, we extend the PGEE method to a Group Penalized Generalized Estimating Equations (GPGEE) that can accommodate a multi-level structured correlation and achieve group-wise variable selection. Thus our proposed method can be readily applied to test the condition difference in ERP measures and simultaneously perform group variable selection to identify important predictors associated with ERP for brain response. To our knowledge, hierarchical models with complex correlation structure are rarely used for ERP response analysis in the ERP research, nor are the regularized regression methods with penalty. Our modeling development was motivated by the ERP data from a birth cohort of Bangladeshi children, the Performance of Rotavirus and Oral Polio Vaccines in Developing Countries (PROVIDE) study. A large and comprehensive set of non-invasive biomarkers were developed in the PROVIDE study from fecal and blood samples [19]. Children in low-resource communities such as those in the PROVIDE cohort are exposed to numerous adversities, including malnutrition, infectious disease exposure, and extreme poverty. In turn, exposure to early adversity can limit their cognitive developmental potentials with long lasting effects. Using EEG as a neuro-imaging tool for cognitive and neural development assessment, a subset of children in the PROVIDE birth cohort were measured at 3 years of age for ERP response. The primary objective of our clinical ERP study was to evaluate the impact of early adversity on brain connectivity and identify risk factors and biomarkers associated with the brain response. With the challenges and limitations in ERP research described earlier, the GPGEE model is developed to achieve the clinical objective.
Our method addresses the following major challenges in analyzing the ERP data from the PROVIDE study. First, due to the design of experiment, ERP data are hierarchical or multilevel by nature with multiple conditions and multiple channels for each study subject. Second, ERP data are highly correlated across channels under each condition, and across conditions for each channel. Lots of information would be lost by simply averaging ERPs over these channels to compare ERPs between conditions. Third, although variable selection methods for high-dimensional data have been intensively studied [12, 13] and applied in clinical and genetic studies [19–21], to our knowledge, no variable selection technique has been applied to ERP data. Further, since many predictors in the high dimensional data are categorized with multiple levels or potentially correlated, group penalty needs to be imposed in the variable selection process to ensure informative predictors and groups can be correctly selected.
The rest of the paper is organized as follows. In "Methods" section, we present the models for high-dimensional correlated data, propose the model specifically for the structured correlation matrix in ERP data, expand the PGEE method to allow group penalty, and derive the algorithm for solving group-penalized estimating equations. In "Simulations" section, we conduct a simulation study to compare the relative performance of our proposed GPGEE with the existing model under several scenarios, without and with group penalty, and with different correlation structures. In "Results" section, we apply our proposed method to ERP data from the PROVIDE study. Compared to the existing methods such as regularized regression or PGEE, our proposed method doesn't only model the ERP multi-level structure appropriately, but also promotes group-wise variable selection. The simulation results show that our proposed method outperforms the existing modeling approaches in variable selection and parameter estimation. Our work would be one of the pioneering efforts in ERP research to test the difference in ERPs between conditions while identifying important biomarkers associated with ERPs simultaneously.
Clinical data and ERP measurements
The PROVIDE (Performance of Rotavirus and Oral Polio Vaccines in Developing Countries) study was a randomized controlled clinical trial with a 2-by-2 factorial design to investigate the efficacy of Rotavirus and Oral Polio Vaccines in Bangladeshi children, conducted between May 2011 and August 2018 in Dhaka, Bangladesh. The cohort consisted of 700 children enrolled within 72 hours of birth after written parental consent and were followed through twice weekly household visits and regularly scheduled clinical visits during the first 5 years of life. Details about the study design, enrollment, surveillance and biomarker development were described previously [19, 20, 22, 23]. Children were 36 months old at the time of neuro-imaging test for cognitive assessment. The study was approved by the Ethical Review Committee of the International Centre for Diarrhoeal Disease Research, Bangladesh (icddr,b), and the Institutional Review Board at Boston Childrens Hospital and the University of Virginia. This study is reported in line with the Consolidated Standards of Reporting Trials (CONSORT) Statement, and the CONSORT Checklist can be found in Additional file 2.
ERPs were measured in a subset of children at 36 months of age. After data processing and quality checking, 70 children out of 130 had valid data for the final ERP analysis. Each child was tested with a face oddball paradigm in which standard (70% of chance) and oddball (30% of chance) faces were presented in a random order. This paradigm has been widely employed to examine the neural correlates of social attention and recognition memory of faces in children [24–26]. The current study focused on one ERP component that can be elicited using this paradigm- the N290 component as the neurocognitive response. The N290 component is regarded as the precursor of the adult N170 face-sensitive component and potentially be generated by the fusiform face and occipital face areas in children [5, 27, 28]. The N290 amplitude in response to the two conditions (standard and oddball) in different electrode channels reflects the averaged synchronous brain activation of large number of neurons occurring around 290 ms following stimulus onset.
Figure 1 shows that N290 peak amplitudes are different among 13 channels (see Additional file 1 for the 13 electrode locations) under either condition, suggesting that modelling the variations among channels would capture more accurate information than simply taking average of all channels. Also, the N290 amplitudes are highly correlated among the 13 channels (Fig. 2 for the correlation plot). Furthermore, ERP response data are more structured with respect to multiple conditions by multiple channels for each subject, thus a structured correlation matrix will be needed to appropriately characterize the ERP data structure.
Boxplot of N290 amplitude under oddball/standard condition. X-axis represents 13 electrodes on the occipital regions of interest (see the supplemental figure for exact electrode locations). Y-axis shows the amplitude in uV
Correlation plot of N290 amplitude under oddball/standard condition
The clinical factors included maternal information (maternal height, weight, and education), socioeconomic status, and sanitation and environmental factors such as water source and water treatment. Biomarker data were obtained from the fecal and blood samples collected at early age of life to measure inflammation [19]. We hypothesized that only a small subset of these clinical factors and biomarkers are associated with the ERP response, thus the variable selection methods would be suitable in this investigation.
The PGEE model proposed by Wang et al. [18] for correlated data is limited in that it can only handle a simple correlation structure. While the estimator obtained by PGEE [18] is consistent with any working correlation matrix, the efficiency of the estimator can be improved when the specified correlation matrix is closer to the true matrix. To characterize the ERP correlations between conditions and among channels, we specify the within-subject correlation matrix as the Kronecker product of the channel correlation matrix and condition correlation matrix. In addition, to enable group variable selection of the categorized channel variable with 13 levels in our study, the penalty for individual variable selection in PGEE is adapted for group selection. Therefore, our method extends PGEE and prompts an integrated model for ERP responses such that we can evaluate the differences between conditions and identify informative clinical factors and biomarkers simultaneously, while accounting for the complex correlations among ERPs and allowing group variable selection.
Proposed model for ERPs
Suppose that there are I subjects, each subject is placed under J treatments, and K repeated measurements are recorded under each treatment. We use Yijk to denote the kth repeated measures under the jth treatment for the ith subject. By vectorizing Yi=(Yi11,...,Yi1K,Yi21,...,Yi2K,...,YiJ1,...,YiJK)T, we consider group variable selection for a generalized linear model for the correlated data in Yi:
$$E\left(\mathbf Y_{i}\right)={\boldsymbol{\mu}_{i}},$$
$$g\left(\mu_{ijk}\right)=\mathbf X_{i}^{T} \mathbf \beta + condition_{j} + channel_{k},$$
$$Var\left(Y_{ijk}\right)=\phi v\left(\mu_{ijk}\right),$$
$$Cov\left(\mathbf Y_{i}\right) = \mathbf V_{i},$$
where Vi denotes the covariance structure and ϕ is an overdispersion parameter. Without loss of generality, we assume ϕ=1 in the rest of the paper.
Structured correlation matrix
For correlated data, a working correlation matrix needs to be pre-specified in many estimation methods, and its appropriate specification improves the estimation efficiency considerably for regression parameters. Some commonly used correlations, such as unstructured, AR1, exchangeable, etc., are often adopted in the practice. However, none of the commonly used correlations can appropriately account for the structured correlation for ERP data. Given that how ERP data were collected, it is natural to assume that ERP measurements from different conditions are correlated, and under each of treatment conditions the correlation structures among the channels are the same. Thus for the structured covariance matrix, we adopt a separate correlation for treatment condition and channel. Letting Bi be the covariance matrix for conditions and Σi be the covariance matrix for channels, the structured covariance matrix for each subject is the Kronecker product of the two matrices, Vi=Bi⊗Σi.
Group selection for GEE
Variable selection for correlated data has been studied in Wang et al. [18], where penalized generalized estimating equations is adopted for simultaneous model estimation and variable selection, the SCAD penalty is used for individual variable selection. However, for many biomarker studies, predictors are highly correlated and/or pre-classified into different groups, and variables need to be selected in groups, as shown in Yuan and Lin [29]. Here we adopt SCAD with group selection and extend the PGEE to Group Penalized Generalized Estimating Equations (GPGEE) to select variables for the ERP data. Suppose covariates {X1,X2,...,Xp} are classified into d groups
$$\left\{ X_{11},...,X_{1p_{1}} \right\},...,\left\{ X_{d1},...,X_{dp_{d}} \right\}.$$
The corresponding coefficients are \(\boldsymbol {\beta }=\left (\boldsymbol {\beta ^{G}_{1}}^{T},...,\boldsymbol {\beta ^{G}_{d}}^{T}\right)\), where \(\boldsymbol {\beta ^{G}_{i}}\) is the coefficient vector for group i. For the group variable selection, we will either select the whole group of variables or remove the whole group from the model.
We define the estimating functions as
$$U\left(\boldsymbol{\beta}\right)=S\left(\boldsymbol{\beta}\right)-n \mathbf q_{\lambda}^{G}\left(\boldsymbol{\beta}\right){sign}\left(\boldsymbol{\beta}\right),$$
$$S\left(\boldsymbol{\beta}\right)=\sum_{i=1}^{n}\mathbf X_{i}^{T} \mathbf A_{i}^{1/2}\left(\boldsymbol{\beta}\right)\hat{\mathbf R}^{-1}\mathbf A_{i}^{-1/2}\left(\boldsymbol{\beta}\right)\left(\mathbf Y_{i}-\mu_{i}\left(\boldsymbol{\beta}\right)\right)$$
is a vector of estimating functions defining the GEE [18], \(\hat {\mathbf R}\) is the estimated working correlation matrix \(\left (\mathbf V_{i} = \mathbf A_{i}^{1/2}\left (\boldsymbol {\beta }\right)\hat {\mathbf R}^{-1}\mathbf A_{i}^{-1/2}\left (\boldsymbol {\beta }\right)\right)\), and \(\mathbf q_{\lambda }^{G}\left (\boldsymbol {\beta }\right)\mathbf sign\left (\boldsymbol {\beta }\right)\) denotes the component wise product with
$$sign\left(\boldsymbol{\beta}\right)=\left(sign\left(\beta_{1}\right),...,sign\left(\beta_{p}\right)\right)$$
$$\mathbf{q}_{\lambda}^{G}\left(\boldsymbol{\beta}\right)= \left(\boldsymbol{q}_{\lambda}\left(\left|\left|\boldsymbol{\beta}^{G}_{1}\right|\right|_{1}\right)^{T},...,\boldsymbol{q}_{\lambda}\left(\left|\left|\boldsymbol{\beta}^{G}_{d}\right|\right|_{1}\right)^{T}\right).$$
Here, \(\boldsymbol {q}_{\lambda }\left (\left |\left |\boldsymbol {\beta }^{G}_{i}\right |\right |_{1}\right)= q_{\lambda }\left (\left |\left |\boldsymbol {\beta }^{G}_{i}\right |\right |_{1}\right) * \boldsymbol {1}_{p_{i}}\) denotes the group penalty vector for group i, and qλ(θ) is the derivative of the SCAD penalty function imposed on the L1- norm of the group vector \(\boldsymbol {\beta }_{i}^{G}\).
The notation qλ(θ) is the derivative of the SCAD penalty,
$$q_{\lambda}(\theta)=\lambda \left\{I\left(\theta < \lambda \right) + \frac{\left(a\lambda-\theta \right)_{+}}{(a-1)\lambda} I\left(\theta > \lambda\right) \right\}$$
for θ≥0 and some a>2. As suggested in Fan and Li [13], we let a=3.7.
Algorithm for GPGEE
Similar to the algorithm proposed in Wang et al. [18], we apply the Newton-Raphson algorithm combined with the minorization-maximization to solve the penalized estimating equations.
By the minorization-maximization algorithm, for a small ε>0, the penalized estimator βn approximately satisfies
$$S_{nj} - nq_{\lambda_{n}}\left(\hat{\boldsymbol{\beta}}_{nj}^{G}\right) sign\left(\hat{\beta}_{nj}\right) \frac{\left|\hat{\beta}_{nj}\right|}{\epsilon + \left|\hat{\beta}_{nj}\right|} =0, j=1,..., p.$$
To solve the above equations, we apply the Newton-Raphson algorithm as follows,
$$\begin{array}{@{}rcl@{}} \boldsymbol{\beta}_{n}^{k}& = & \boldsymbol{\beta}_{n}^{k-1} + \left[\boldsymbol{H}_{n}\left(\boldsymbol{\beta}_{n}^{k-1}\right) + n\boldsymbol{E}_{n}\left(\boldsymbol{\beta}_{n}^{k-1}\right)\right]^{-1}\\ && \times \left[\boldsymbol{S}_{n}\left(\boldsymbol{\beta}_{n}^{k-1}\right) - n\boldsymbol{E}_{n}\left(\boldsymbol{\beta}_{n}^{k-1}\right) \boldsymbol{\beta}_{n}^{k-1}\right], \end{array} $$
$$\boldsymbol{H}_{n}\left(\boldsymbol{\beta}_{n}^{k-1}\right) = \sum_{i=1}^{n} \mathbf X_{i}^{T} \mathbf A_{i}^{1/2}\left(\boldsymbol{\beta}_{n}^{k-1}\right)\mathbf R^{-1}\mathbf A_{i}^{1/2}\left(\boldsymbol{\beta}_{n}^{k-1}\right) \mathbf X_{i},$$
$$\boldsymbol{E}_{n}\left(\boldsymbol{\beta}_{n}^{k-1}\right) = diag \left\{ \frac{q_{\lambda_{n}}\left(\boldsymbol{\hat{\beta}_{n1}^{G}}\right)}{\epsilon + \left|\hat{\beta}_{n1}\right|},..., \frac{q_{\lambda_{n}}\left(\boldsymbol{\hat{\beta}_{np}^{G}}\right)}{\epsilon + \left|\hat{\beta}_{np}\right|} \right\}.$$
In practice, we set ε=10−6 and take \(\hat {\boldsymbol {\beta }}\), the GEE estimator with independence working correlation matrix, as the initial value of β. The stopping criterion for the iterative algorithm is \(\sum _{j=1}^{p}\left |\hat {\boldsymbol {\beta }}_{j}^{k+1}-\hat {\boldsymbol {\beta }}_{j}^{k}\right | < 10^{-5}\). In our study, we use Bayesian information criterion (BIC) developed for correlated data [30] for selecting the tuning parameter λ.
In this section, we illustrate the numerical strength of our developed method by comparing it with existing methods through a simulation study. In our simulation study, the sample sizes are set at 50 and 100. There are 20 correlated measurements and 40 covariates for each subject. The correlated normal responses are generated from the model
$$Y_{ij}=\boldsymbol{X}^{T}_{ij} \boldsymbol{\beta} + \epsilon_{ij},$$
where \(\boldsymbol {X}^{T}_{ij}=\left (x_{ij,1},..., x_{ij, 40}\right)^{T}\) is a vector of 40 covariates for i=1,..,50 and j=1,...,20, and
$${} \boldsymbol{\beta}\,=\,\left(2, 1, 1, 1, 1, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0.1, 0.1, 1, 1, 1, 0,..., 0\right)^{T}$$
containing 8 groups with every 5 covariates in each group. For the covariates, we generate xij,1 from Bernoulli(0.5) distribution and the rest from the multivariate normal distribution with mean 0 and an AR1 covariance matrix with marginal variance 1 and auto-correlation coefficient 0.5. The covariance matrix of random errors for each subject is Vi=Bi⊗Σi, where Bi is a 2-by-2 identity matrix and Σi is a 10-by-10 AR1 matrix with marginal variance 10 and auto-correlation coefficient 0.9. We compare our GPGEE model with PGEE model to illustrate the importance of incorporating group penalty and using structured correlations. Five models are evaluated for comparison in the simulations: original PGEE with AR1 working correlation (Model 1), a modified PGEE incorporated with our structured correlation (Model 2), our GPGEE with AR1 working correlation (which is unstructured correlation, Model 3), our GPGEE with structured correlation but with misspecified working correlation (Model 4) and our proposed model with both group penalty and correct structured correlation (Model 5). We assume that the true group memberships of the covariates to impose the group-SCAD penalty are known, and divide the covariates into 8 groups. For the structured correlation that is correctly specified, we use AR1 as working correlation structure for Σi and assume Bi to be unstructured. For the misspecified structured correlation, we use CS (compound symmetry) as working correlation structure for Σi instead. The selection results are defined as exact-selection when the selected model is the true model, under-selection when at least one true covariate is not selected, and all other cases are defined as over-selection. We also report the mean squared error (MSE) which is defined as the average of \(\left |\left | \hat {\boldsymbol {\beta }} - \boldsymbol {\beta } \right |\right |^{2}_{2}\) and corresponding standard error (SE) defined as the standard deviation of \(\left |\left | \hat {\boldsymbol {\beta }} - \boldsymbol {\beta } \right |\right |^{2}_{2}\) from the simulated datasets.
We conduct the simulation by generating 200 datasets for each sample size, and summarize the percentages of over-selection, under-selection, exact-selection, and MSEs (SE) in Table 1, top panel for sample size 50. Our GPGEE model (Model 5) has the smallest MSE and SE among the 5 models, and selects the true model for 94.5% of the simulated datasets, compared to the existing PGEE model (Model 1) with only 2.5% exact-selection and much higher MSE. The results further show that without the pre-specified structured correlation (Model 3), the model selection is less accurate, and it is more likely to have higher under-selection and larger MSE (SE), which suggests it is crucial to incorporate the structured correlation when the data is multi-level by nature. The results also show the importance of adopting group penalty, especially to deal with under-selection problem when there are covariates with smaller coefficients. In addition, if we capture the multi-level correlation structure correctly but mis-specify the true correlation for one layer (Model 4), the method can still be helpful to select the true model (96% selection rate) though the MSE is larger due to the mis-specification. Thus, as long as we specify this multi-level correlation structure, our method is relatively robust for variable selection against misspecification of the working correlation structure. When the sample size is increased 100, the comparsion results remain to be similar, as shown in Table 1, bottom panel.
Table 1 Comparison of model selection performance. O for over-selection, U for under-selection and Exact for exact-selection
In the PROVIDE cohort, there were 47 clinical factors and early-stage biomarkers available for the analysis, including children's enteric and systemic inflammatory biomarkers, nutritional measures, maternal health and socioeconomic status (SES), and sanitation conditions [19]. Of them, 14 biomarkers are categorical measures, and the rest are continuous variables. The CRP index is a cumulative number of times that children experienced elevated CRP level over the first two years (i.e., being on the top 50% at 6, 18, 40, 53 and 104 weeks), thus measuring the sustained inflammation burden. For 70 children with ERP measurements, the descriptive statistics of these clinical factors and biomarkers are summarized in Table 2.
Table 2 Descriptive summary of risk factors and biomarkers in ERP Study (N=70)
As described earlier, in the ERP study, the children were shown with the face pictures, 70% of time for the same face (standard condition) and 30% of time with new different faces (oddball condition) over 150 trials. Brain activities were recorded for all electrodes during the observation of each picture, and ERP components were derived from multiple trials to measure the electrical activity of the brain immediately in response to a direct stimulus event [6]. In this clinical application, the mean peak amplitude of N290 component was used as a clinical example, which measures the brain response with face processing around 290 ms, obtained under each treatment condition from 13 electrodes placed on different locations of occipital region. The N290 amplitude reflects the synchronous activation of large number of neurons, and large amplitude is generally deemed to have greater underlying neuronal activity. It is hypothesized that the N290 amplitude response originates in areas of the brain dedicated to face processing, such as the occipital face areas and the inferior temporal cortex (such as the fusiform). In addition to evaluate the difference in N290 amplitude for neural activity of face processing between the two conditions, we aimed to study the association of biomarkers in infancy with the ERP response in early childhood. Ultimately, we hope to gain insights on how infant's health and nutrition markers affects the development of the brain.
For each child, there are 26 N290 amplitude responses corresponding to 13 channels under 2 conditions. Those 26 ERP responses are highly correlated with multilevel correlation structure due to the nature of this experiment, that is, N290 measurements are not only correlated across channels, but also vary under different treatment conditions. As shown in Fig. 2, the N290 measurements among 13 channels (aligned by their locations on the brain) are highly correlated, and the correlations appear to be autoregressive in that channels closer to each other in the brain yield higher correlations than that further apart. Also, the correlation patterns appear to be different between oddball and standard conditions. In addition, N290 measurements vary considerably across the 13 channels and across conditions as depicted in Fig. 1. For the special data features, our proposed GPGEE model described in "Methods" section can properly evaluate the relationship between biomarkers and N290 response while accounting for the hierarchical correlation structures and variations across channels/conditions. To apply our proposed model, the correlation matrix between conditions for the same channel was assumed to be unstructured, and that among channels for the same condition to be autoregressive with order 1 (AR1). The group penalty was applied to electrode or channel which is a multi-level categorical covariate with 13 levels. By using 12 dummy variables and grouping them together, we are able to conduct variable selection for this covariate. The N290 responses were assumed to be normally distributed with identity link. For biomarkers and clinical predictors, prescreening was performed based on their correlations, and representative predictors were selected for those with corrections >0.7. Thus 6 biomarkers were removed, including IL-4 at week18, IL-6 at week 18, TNFa at week 18, WAZ at birth, WHZ at birth and monthly household income.
The results of variable selection with our proposed GPGEE for N290 response were presented in Table 3. A total of 10 biomarkers were selected using BIC after adjusting for condition and channel differences. Among those selected biomarkers, IL-10, RBP, Zinc, Calprotectin, Neopterin and water treatment were positively associated with N290 amplitude, while IL-5, MIP1b, MPO and maternal height have negative effects on the N290. These results provide some supporting evidence that children's health conditions in early childhood indeed are associated with brain development at 3 years of age. While N290 amplitude measures the strength of the signal of brain activity for brain connectivity, some researchers have also focused on studying the change of N290 amplitude between conditions. The differences in N290 between oddball and standard conditions reflects how the brain behaves differently when seeing a new face vs. a familiar face, and therefore measures the child's ability to discriminate between a novel and a familiar face. In particular, A differential response in these ERP components between the two experimental conditions indicates the detection or discrimination of the infrequent from the frequent faces by the brain and reflects some aspect of memory updating and the efficiency of stimulus processing [26, 31, 32].
Table 3 Risk factors and biomarkers selected for N290 amplitude
When considering the difference in N290 as the outcome variable, the analysis would be performed similarly under our GPGEE framework, where the correction structure is reduced at channel level only. For the difference in N290 response, 13 biomarkers were selected (Table 4), of which 8 biomarkers have positive effects and 5 have negative effects on the N290 difference. Obviously, RBP at week 6, Zinc, mother height, and water treatment were associated with both N290 amplitude and the difference in N290, while some biomarkers (Days of diarrhea in the first 18 weeks, RBP at week 18, Mannitol in urines at week 24, LPS at week 18, CRP index, Monthly household expenditure, Mother weight, Reg1B at week 6, Gender) were only informative to the difference of N290 between the conditions, indicating that these biomarkers contributing to a stronger overall brain EEG signal don't necessarily contribute to a better EEG power, the ability to identify new faces.
Table 4 Risk factors and biomarkers selected for N290 difference
The primary objective of our clinical study was to identify biomarkers in early childhood that could affect children's brain development measured by ERP data at 3 years of age. To our best knowledge, no previous study has analyzed ERP data under correlated hierarchical data framework where the correlation structure among both channels and conditions are accounted for. Many available statistical methods couldn't be directly applied here because of the nature of ERP's correlation structure. Further, group penalty needs to be incorporated in variable selection for ERP data so the clustered clinical risk factors and biomarkers can be selected together. Therefore our proposed group penalized GEE estimator with structured correlation matrix for ERP data can properly model the complex ERP response and simultaneously identify informative biomarkers associated with ERP amplitude and ERP difference, respectively. Our proposed method outperforms the existing modeling approaches in the simulation study. Further, our work would be one of the pioneering efforts in ERP research to test the condition difference in ERPs and, simultaneously, to identify important covariates associated with ERPs.
In our study, N290 measure was analyzed in the clinical application, but the developed method can be applied to any other ERP measurements with tasks focusing on different brain functions. Our clinical findings were limited by the small sample size, missing data in biomarkers, and time lag between collection of biomarkers and ERP measurement. Nevertheless, our proposed method emphasizes on the correlation structure among channels based on their physical locations on the brain, thus improves the model estimation efficiency for ERP data analysis. For future work, if data is normally distributed with identity link function, our proposed method can be extended further with choices of penalty, such as elastic net, and computing algorithms, such as Fast Iterative Shrinkage-Thresholding Algorithm [33] or Alternating Direction Method of Multipliers [34], which would improve the computational time for larger datasets. In addition, although the ERP responses were considered as the continuous outcomes, our model is also applicable to other types of response such as categorical or count response. In addition, the systemic and enteric inflammation biomarkers identified in this study for their association with ERPs are similar and consistent with the previous findings in the cognitive development research [35].
Using the proposed group penalized GEE, we modeled the complex ERP data with structured correlation and identified informative early-stage biomarkers associated with such brain connectivity. Our findings are clinically important in understanding early childhood neurocognitive development in low-income countries. Particularly, the selected early-stage biomarkers offer a potential explanation for the adversity of brain connectivity, which will facilitate early identification of infants at risk and potential pathways for effective intervention in the malnourished children. Our proposed method is not only applicable to the ERP studies but also to other biomedical studies for biomarker selection with highly correlated responses.
The datasets supporting the conclusions of this article are available at icddr,b, and at Boston Children's Hospital and University of Virginia. According to the data protection regulation and informed consent form, the authors are not permitted to deposit the individual participant data elsewhere.
ERP:
Event-Related Potentials
GEE:
Generalized Estimating Equation
PGEE:
Penalized Generalized Estimating Equation
GPGEE:
Group Penalized Generalized Estimating Equation
Electroencephalogram
LASSO:
Least Absolute Shrinkage and Selection Operator
SCAD:
Smoothly Clipped Absolute Deviation
PROVIDE:
Performance of Rotavirus and Oral Polio Vaccines in Developing Countries
Bayesian Information Criterion
AR1:
Autoregressive with Order 1
MPO:
Myeloperoxidase
Alpha-1 Anti-trypsin
Reg1B:
Lithostathine-1-Beta
C Reactive Protein
Endocab Lipopolysaccharide
L1b:
Interleukin 1 beta
IL4:
Interleukin 4
IL10:
Interleukin 10
MIP1b:
Macrophage Infammatory Protein 1 Beta
TNFa:
Tumor Necrosis Factor alpha
RBP:
Retinol Binding Protein
HAZ:
Height for Age Z score
WAZ:
Weight for Age Z score
WHZ:
Weight for Height Z score
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We sincerely thank the children and parents of Mirpur for their participation in the study. We gratefully acknowledge the clinical investigators and professional personnel at icddr,b, Boston Children's Hospital and University of Virginia, who made this study possible.
This study was supported by the Bill & Melinda Gates Foundation and by the National Center For Advancing Translational Sciences of the National Institutes of Health (Award Number UL1TR003015). The funders had no role in study design, data analysis, and interpretation of the data.
University of Virginia, Charlottesville, US
Ye Lin, Jianhui Zhou, William A. Petri Jr & Jennie Z. Ma
Harvard University, Cambridge, US
Swapna Kumar, Wanze Xie, Sarah K. G. Jensen & Charles A. Nelson
Boston Children's Hospital, Boston, US
Swapna Kumar, Wanze Xie & Sarah K. G. Jensen
International Centre for Diarrhoeal Disease Research, Dhaka, Bangladesh
Rashidul Haque
Ye Lin
Jianhui Zhou
Swapna Kumar
Wanze Xie
Sarah K. G. Jensen
Charles A. Nelson
William A. Petri Jr
Jennie Z. Ma
All authors contributed significantly to the work of this manuscript. RH and WAP led the original PROVIDE study of the birth cohort, and CAN, RH and WAP led the neurocognitive study. SK, WZ and SKJ participated and/or contributed to the neurocognitive data. YL, JZ and JZM contributed to the statistical method development and conducted the data analysis. All authors interpreted the clinical or statistical results, wrote and revised the manuscript, and approved the final version. None of the authors reported any conflicts of interest.
Correspondence to Jennie Z. Ma.
The study was approved by the Ethical Review Committee of the International Centre for Diarrhoeal Disease Research, Bangladesh (icddr,b), and the Institutional Review Board at Boston Children's Hospital and the University of Virginia. Informed written consent was obtained from the parents or guardians for the participation of their child in the study. Study staff were trained in the best ethical practices of clinical research, including approaches to the consenting of illiterate individuals, assessing comprehension of the study by the parents or guardians prior to their signing consent, and prevention of coercion in any step of the study process including recruitment, retention, study practices and consenting in the PROVIDE study (www.clinicaltrials.gov, identifier: NCT01375647).
Not applicable as no individual data are published.
Supplemental figure and table. Electrode locations, corresponding electrode numbers and channel numbers used in the paper.
CONSORT checklist for the related clinical study.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
Lin, Y., Zhou, J., Kumar, S. et al. Group penalized generalized estimating equation for correlated event-related potentials and biomarker selection. BMC Med Res Methodol 20, 221 (2020). https://doi.org/10.1186/s12874-020-01103-x
Correlated data
Penalized generalized estimating equations (GEE) | CommonCrawl |
Here are examples from Chapter 6 to help you understand these concepts better. These were taken from the real world and supplied by FSDE students in Summer 2021. If you'd like to submit your own examples, please send them to the author [email protected].
1. Problem
The setup that holds the solar panels at the UPEI FSDE is modeled below. Considering beam S (1.9 m length), find the internal forces at point C. Assume the intensity of the solar panel on the beam is 200 N/m.
Sketch:
Free-body diagram:
3. Knowns and Unknowns
Knowns
w = 220 N/m
OA = 0.5 m
AC = 0.2 m
AB = 0.4 m
L = 1.9 m
Unknowns: Nc, Vc, Mc
4. Approach
Use equilibrium equations. First solve for reaction forces, then make a cut at C and solve for the internal forces.
$$w=\frac{F}{L}\\F=wL\\F_R=220N/m\cdot 1.9m\\F_R=418N\\\sum F_X=0=B_X$$
Find reaction forces:
$$\sum M_A=0=B_y(0.4m)-F_R(0.55m)\\(0.4m)B_y=418N(0.55m)\\B_y=\frac{229.9 N\cdot m}{0.4m}\\B_y=574.75N$$
$$\sum F_y=0=-F_R+A_y+B_y\\A_y=F_R-B_y\\A_y=418N-574.74N\\A_y=-156.75N$$
The answer we got for Ay is negative, which means that the arrow should be drawn in the other direction. We will change it for our next sketch.
Make a cut at C:
Now solve for the internal forces:
$$\sum F_x=0\:\:;\:\:N_c=0\\\sum F_y=0=-A_y-V_c-(w\cdot L)\\V_c=-156.75N-(220N/m\cdot0.95m)\\V_c=-346.75 N\\\sum M_c=A_y(0.2m)+M_c+(F_{Rc}\cdot 0.475m)\\M_c=-156.75 N (0.2m)-(220N/m\cdot 0.95m\cdot 0.475m)\\M_c=-130.625N\cdot m$$
Final FBD, showing the arrows in the correct directions:
It makes sense that Ay and By are in different directions, because the resultant force Fr of the solar panel on the beam is not between A and B. It also makes sense that the moment at C is in the clockwise direction rather than the counterclockwise directions, when you think about the direction of the forces applied to the beam.
A beam that is simply supported has two point loads acting on it. One acts 2 m from point A and the other acts at 2.5 m from C. Point B is in the middle of the beam. The first point load is 500 N and the second is 300 N. What are the internal forces at point B? Solve for reaction forces and include a shear/moment diagram.
Knowns:
F1 = 500 N
Unknowns: Ay, Ax, Cy, VB, MB, NB
Shear/moment equations, EOM equations
Solve for reaction forces:
(Ax, Cy)
\sum F_{x}=0=A_{x}=0 \\
\sum M_{A}=0 &=-F_{1} \cdot 2 m-F_{2} \cdot 5.5 m+C_{y} \cdot 8 m \\
C_{y}=&+F_{1} \cdot 2 m+F_{2} \cdot 5.5 m \\
C_{y} &=\frac{500 N \cdot 2 m+300 N \cdot 5.5 m}{8 m} \\
C_{y} &=331.25 \mathrm{~N}
(Ay)
\sum F_{y}=0 &=A_{y}+C_{y}-F_{1}-F_{2} \\
A_{y} &=F_{1}+F_{2}-C_{y} \\
A_{y} &=500 \mathrm{~N}+300 \mathrm{~N} – 331.25 \mathrm{~N} \\
A_{y} &=468.75 \mathrm{~N}
Cut 1: at B
\sum F_{X}=0=A_{X} &+N_{B}=0 \\
& N_{B}=0 \\
\sum F_{y}=0 &=A_{y}-V_{B}-F_{1} \\
V_{B} &=A_{y}-F_{1} \\
V_{B} &=468.75 N – 500 N \\
V_{B}=-31.25 N
\sum M_{B}=& 0=-A_{y}(4 m)+F_{1}(2 m)+M_{B} \\
& M_{B}=A_{y}(4 m)-F_{1}(2 m) \\
& M_{B}=468.75 N(4 m)-500 N(2 m) \\
M_{B} &=875 \mathrm{~N} \cdot \mathrm{m}
Cut 2: At the point where F1 is applied
\sum M_{1}=0 &=-A_{y}(2 m)+M_{1}=0 \\
M_{1} &=A_{y}(2 m) \\
M_{1} &=468.75 N(2 m) \\
M_{1} &=937.5 \mathrm{~N} \cdot m
\sum M_{2}=0 =-A_{y}(5.5 \mathrm{~m})+F_{1}(3.5 \mathrm{~m})+M_{2} \\
M_{2} &=A_{y}(5.5 \mathrm{~m})-F_{1}(3.5 \mathrm{~m}) \\
M_{2} &=468.75 \mathrm{~N}(5.5 \mathrm{~m})-500 \mathrm{~N}(3.5 \mathrm{~m}) \\
M_{2} &=828.125 \mathrm{~N} \cdot \mathrm{m}
Answer: NB = 0, VB = -31.25 N, MB = 875 Nm
The reaction forces make sense as they offset the applied forces. The shear/moment diagrams returned to zero so they are correct too. The moment found at B is in the moment diagram, it is smaller than the maximum.
Previous: 6.2 Shear/Moment Diagrams
Next: Chapter 7: Inertia | CommonCrawl |
\begin{document}
\title[Vortex solutions in the Ginzburg-Landau-Painlev\'e theory of phase transition]{Vortex solutions in the Ginzburg-Landau-Painlev\'e theory of phase transition}
\author{Panayotis Smyrnelis} \address[P.~ Smyrnelis]{Institute of Mathematics, Polish Academy of Sciences, ul. \'{S}niadeckich 8, 00-656 Warsaw, Poland} \email[P. ~Smyrnelis]{[email protected]}
\subjclass{Primary 35J47; 35J50 35Q56; Secondary 35B40; 35B07.} \keywords{Painlev\'e equation, Ginzburg-Landau system, vortex, minimizer, liquid crystals.}
\begin{abstract}
The extended Painlev\'e P.D.E. system $\Delta y -x_1 y - 2 |y|^2y=0$, $(x_1,\ldots,x_n)\in \mathbb{R}^n$, $y:\mathbb{R}^n\to\mathbb{R}^m$, is obtained by multiplying by $-x_1$ the linear term of the Ginzburg-Landau equation $\Delta \eta=|\eta|^2\eta-\eta$, $\eta:\mathbb{R}^{n}\to\mathbb{R}^{m}$. The two dimensional model $n=m=2$ describes in the theory of light-matter interaction in liquid crystals, the orientation of the molecules at the boundary of the illuminated region. On the other hand, the one dimensional model reduces to the second Painlev\'e O.D.E. $y''-xy-2y^3=0$, $x\in \mathbb{R},$ which has been extensively studied, due to its importance for applications. The solutions of the extended Painlev\'e P.D.E. share some characteristics both with the Ginzburg-Landau equation and the second Painlev\'e O.D.E. The scope of this paper is to construct standard vortex solutions $y:\mathbb{R}^{n}\to\mathbb{R}^{n-1}$ ($\forall n\geq 3$) of the extended Painlev\'e equation. These solutions have in every hyperplane $x_1=\mathrm{Const.}$, a profile similar to the standard vortices $\eta:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$ of the Ginzburg-Landau equation, but their amplitude is determined by the Hastings-McLeod solution $h$ of the second Painlev\'e O.D.E. evaluated at $x_1$.
\end{abstract}
\maketitle
\section{The extended Painlev\'e P.D.E.}\label{sec:sec2} \subsection{Origin of the model}
In the physical context of light-matter interaction in liquid crystals (cf. \cite{clerc2}), we recently discovered that the extended Painlev\'e equation: \begin{equation}\label{pain 1}
\Delta y-x_1 y-2|y|^2y=0 , \qquad \forall x=(x_1,\ldots,x_n)\in \R^n, y=(y_1,\ldots,y_m):\R^n\to\R^m, \end{equation} is relevant to describe the orientation of the molecules at the boundary of the illuminated region (cf. \cite{Clerc2017} when $n=m=1$, \cite{Clerc2018} when $n=m=2$, and \cite{panayotis_4} when $n=2$, $m=1$). It can alternatively be written as \begin{equation} \label{pain 1db} \Delta y(x)= H_y(x_1,y(x)), \qquad x\in \R^n, \end{equation}
with a non autonomous potential $H(x_1,y):=\frac{1}{2} x_1 |y|^2 +\frac{1}{2} |y|^4$, $(x_1,y)\in\R\times\R^m$, and $H_y=(\frac{\partial H}{\partial y_1},\ldots,\frac{\partial H}{\partial y_m})\in\R^m$. Also note that equation \eqref{pain 1} has variational structure. Let \begin{equation}\label{funcp}
E_{\mathrm{P_{II}}}(u, \Omega)=\int_\Omega \left[ \frac{1}{2} |\nabla u|^2 +\frac{1}{2} x_1 |u|^2 +\frac{1}{2}| u|^4\right],\ u \in H^1(\Omega;\R^m), \ \Omega\subset \R^n, \end{equation} be its associated functional. In the physical models studied in the aforementioned works, the extended Painlev\'e equation is obtained as the singular limit (after appropriate rescaling) of the system \begin{equation}\label{sing}
\epsilon^2\Delta u_\epsilon-\mu(x)u_\epsilon-|u_\epsilon|^2u_\epsilon+\epsilon a(\epsilon) f(x)=0 , \ \forall x=(x_1,\ldots,x_n)\in \R^n, u_\epsilon:\R^n\to\R^m, \ \epsilon>0, \end{equation}
where $\mu(x)=e^{\,-|x|^2}-\chi$, $\chi\in (0,1)$, $f:\R^n\to\R^m$ is a specific map related to $\mu$, and $a(\epsilon)$ is a nonnegative parameter. More precisely, the function $\mu$ describes light intensity and is sign changing due to the fact that the light is applied to the sample locally, and areas where $\mu< 0$ are interpreted as shadow zones, while areas where $\mu > 0$ correspond to illuminated zones. On the other hand the map $f$ describes the electric field induced by the light. Finally, the intensity of the applied laser light is represented by the parameter $a$. The relevant solution of \eqref{sing} to model the orientation of the molecules in the liquid crystal sample, is a minimizer $v_\epsilon\in H^1(\R^n;\R^m)$ of the energy functional associated to equation \eqref{sing}: \begin{equation} \label{funct00}
E(u)=\int_{\R^n}\left[\frac{\epsilon}{2}|\nabla u|^2-\frac{1}{2\epsilon}\mu(x)|u|^2+\frac{1}{4\epsilon}|u|^4-a(\epsilon)f(x) \cdot u\right], \end{equation} where $\cdot$ stands for the inner product in $\R^m$. Assuming that $\lim_{\epsilon\to 0}a(\epsilon)=0$ (cf. \cite[Theorems 1.2 and 1.3 (i)]{Clerc2017}, and \cite[Theorem 1.1 (ii)]{Clerc2018}), we discovered that the minimizers $v_\epsilon$ appropriately rescaled in a neighbourhood of a point where $\mu$ vanishes, converge as $\epsilon \to 0$, to a solution $y$ of \eqref{pain 1}. In addition, $y$ is by construction bounded in the half-spaces $[s_0,\infty)\times \R^{n-1}$, $\forall s_0\in\R$, and \emph{minimal} in the sense that \begin{equation}\label{minnn} E_{\mathrm{P_{II}}}(y, \mathrm{supp}\, \phi)\leq E_{\mathrm{P_{II}}}(y+\phi, \mathrm{supp}\, \phi) \end{equation}
for all $\phi\in C^\infty_0(\R^n;\R^m)$. To explain formally the relation between (\ref{pain 1}) and the energy $E$, one can see from the expression of $E$, that as $\epsilon\to 0$, the modulus of the minimizer $v_{\epsilon}$ should approach a nonnegative root of the polynomial $-\mu(x)z+z^3=0$, or in other words, $|v_{\epsilon}|\to \sqrt{\mu^+}$ as $\epsilon\to 0$, in some perhaps weak sense. This function called the Thomas-Fermi limit of the minimizer is nonsmooth, so the transition near the set $\mu(x)=0$ has to be mediated somehow via a solution of \eqref{pain 1}. \subsection{Topological defects}
One of the most interesting phenomenon occuring in the two dimensional model ($n=m=2$) considered in \cite{Clerc2018} is the presence in the liquid crystal sample of a new type of topological defect that we named the \emph{shadow vortex}\footnote{The shadow vortex was discovered experimentally in \cite{clerc2}. Then, its existence was confirmed mathematically in \cite[Theorem 1.2 (ii)]{Clerc2018}. Manipulating light vortices has applications in quantic computation, telecommunications, and astronomy (improvementof images, detection of exoplanets).}. It appears at the boundary of the illuminated region when the intensity of the applied laser light is of order $a=o(\epsilon |\ln \epsilon|)$. The computation of the distance of the shadow vortex from the boundary of the illuminated region is a difficult open problem. In view of \cite[Theorem 1.1 (ii)]{Clerc2018}, the shadow vortex is located at a distance of order $O(\epsilon^{\frac{2}{3}})$ from the boundary, if and only if its local profile is given by a \emph{minimal} solution $y:\R^2\to\R^2$ of \eqref{pain 1} having at least one zero. This explains why the investigation of equation \eqref{pain 1} is crucial to understand the formation mechanism, and the properties of shadow vortices in liquid crystals. We expect that equation \eqref{pain 1} may also be relevant to describe the singularities of other Ginzburg-Landau type models similar to \eqref{funct00}, for instance in the context of Bose-Einstein condensates (cf. \cite{ignat,MR2062641,MR3355003,sourdis0}). On the other hand, in the one dimensional model ($n=m=1$) topological defects appear at the boundary of the illuminated region, only when $\lim_{\epsilon\to 0}a(\epsilon)\in (0,\sqrt{2})$. In that case (cf. \cite{sourdis2}) the local profile of the \emph{shadow kink} is given by a sign changing minimal solution of the \emph{nonhomogeneous} O.D.E. $y''-x y-2y^3-\alpha=0$. We also refer to \cite{troy1} for an alternative constuction of this sign changing solution, and to \cite{Clerc2017} for the existence of a positive minimal solution of the nonhomogeneous Painlev\'e O.D.E.
\subsection{One dimensional solutions}\label{subsec:1dd} In the one dimensional case ($n=m=1$), \eqref{pain 1} reduces to the second Painlev\'e O.D.E.: \begin{equation}\label{pain 0} y''-x y-2y^3=0 , \qquad \forall x\in \R, \end{equation} which is known to play an important role in the theory of integrable systems \cite{MR1149378}, random matrices \cite{2006math.ph...3038D, Flaschka1980,2005math.ph...8062C}, Bose-Einstein condensates \cite{MR2062641, MR2772375, MR3355003,sourdis0} and other problems \cite{alikoakos_1,helffer1998,KUDRYASHOV1997397}. It has been extensively studied by Painlev\'e and others since the early 1900's. In particular, the solutions of \eqref{pain 0} satisfying the boundary condition $\lim_{+\infty}y=0$ have been classified in \cite{MR555581}. In view of this result, we could establish \cite[Theorem 1.3 (i)]{Clerc2017} that the Hastings-McLeod solution, denoted in this paper by $h$, is up to sign change, the only minimal solution of \eqref{pain 0} which is bounded at $+\infty$. We recall (cf. \cite{MR555581}) that $h:\R\to\R$ is positive, strictly decreasing ($h' <0$) and such that \begin{align}\label{asy0} h(x)&\sim \mathop{Ai}(x), \qquad x\to \infty, \nonumber \\
h(x)&\sim \sqrt{|x|/2}, \qquad x\to -\infty, \end{align} where $\mathop{Ai}$ is the Airy function.
Having a closer look at the potential $H(x,y)=\frac{1}{2}xy^2+\frac{1}{2}y^4$, one can explain formally the asymptotic behaviour of $h$. Indeed, for $x$ fixed, $H$ attains its global minimum equal to $0$ when $y=0$ and $x\geq 0$, and equal to $-\frac{x^2}{8}$ when $y=\pm \sqrt{|x|/2}$ and $x<0$. Thus, the global minima of $H$ bifurcate from the origin, and the two minimal solutions $\pm h$ of \eqref{pain 0} interpolate these two branches of minima.
\subsection{Scalar solutions} In higher dimensions (cf. \eqref{pain 1} with $n\geq 2$, $m=1$), the scalar P.D.E. \begin{equation}\label{scalarpde} \Delta y -x_1y-2y^3=0,\ x=(x_1,\ldots,x_n)\in\R^n,\ y:\R^n\to\R, \end{equation} involves the non autonomous potential $H(x_1,y)=\frac{1}{2} x_1 y^2 +\frac{1}{2} y^4$ which is bistable for every fixed $x_1<0$. We have shown in \cite{panos405}, that \eqref{scalarpde} describes a phase transition model, as the Allen-Cahn equation below: \begin{equation}\label{ac} \Delta u=W'(u)=u^3-u, \ u:\R^n\to\R, \ W:\R\to [0,\infty), \ W(u)=\frac{1}{4}(u^2-1)^2. \end{equation} For the latter the phase transition connects the two minima $\pm1$ of $W$, while for the former the phase transition connects the two branches $\pm \sqrt{(-x_1)^+/2}$ of minima of $H$ parametrized by $x_1$.
More precisely, there exists (cf. \cite{panos405}) a solution $y:\R^2\to\R$ of (\ref{scalarpde})\footnote{This solution was constructed by taking the limit of odd minimizers $v_\epsilon:\R^2\to\R$ of $E$, in a neighbourhood of an appropriate point of the circle $\{x\in\R^2:\mu(x)=0\}$ (cf. \cite{panos405}, and \cite{panayotis_4} for further relation between shadow domain walls and the extended Painlev\'e equation).} converging as $x_2\to \pm \infty$ and $x_1$ is fixed, to the two minimal solutions $\pm h(x_1)$ of \eqref{pain 0}. We detail below its main properties:
\begin{itemize} \item[(i)] $y$ is positive in the upper half-plane and odd with respect to $x_2$ i.e. $y(x_1,x_2)=-y(x_1,-x_2)$. \item[(ii)] $y_{x_2}(x_1,x_2)>0$, $\forall x_1, x_2\in\R$, and $\lim_{l\to\pm\infty} y(x_1,x_2+l)=\pm h(x_1)$ in $C^2_{\mathrm{loc}}(\R^2)$. \item[(iii)] $y$ is minimal (cf. definition \eqref{minnn})\footnote{The minimality of $y$ for general perturbations was pointed out to us by Sourdis \cite{sour}.}. \item[(iv)] For every $x_2\in\R$ fixed, let $\tilde y(t_1,t_2):=\frac{\sqrt{2}}{(-\frac{3}{2} t_1)^{\frac{1}{3}}}\, y\big(-(-\frac{3}{2} t_1)^{\frac{2}{3}}, x_2+t_2(-\frac{3}{2} t_1)^{-\frac{1}{3}}\big)$. Then \begin{equation}\label{scale1} \lim_{l\to -\infty} \tilde y(t_1+l,t_2)= \begin{cases} \tanh(t_2/\sqrt{2}) &\text{when } x_2=0, \\ 1 &\text{when } x_2>0, \\ -1 &\text{when } x_2<0, \end{cases} \end{equation} for the $C^1_{\mathrm{ loc}}(\R^2)$ convergence. \item[(v)] $y_{x_1}(x_1,x_2)<0$, $\forall x_1\in\R$, $\forall x_2>0$.
\end{itemize}
In view of (i), (ii) and (iii) above, the solution $y$ plays a similar role that the heteroclinic orbit $\gamma( x)= \tan (x/\sqrt{2})$, of the Allen-Cahn O.D.E. $\gamma''=\gamma^3-\gamma$. First of all, both solutions $y$ and $\gamma$ are minimal and odd. Next, $y$ connects monotonically along the vertical direction $x_2$, the two minimal solutions $\pm h(x_1)$, in the same way that $\gamma$ connects monotonically the two global minimizers $\pm 1$ of the potential $W$. What's more, the two global minimizers $\pm 1$ of the Allen-Cahn functional $E_{\mathrm{AC}}=\int_\R\big(\frac{1}{2}|u'|^2+W(u)\big)$ have their counterparts in the two minimal solutions $\pm h$ of the Painlev\'{e} equation. While $\gamma$ is a one dimensional object, the solution $y(x_1,x_2)$ is two dimensional, since $x_1$ parametrizes the branches of minima of the potential $H$, and only $x_2$ is involved in the phase transition. The analogy between equations \eqref{scalarpde} and \eqref{ac} also appears in property (iv). Indeed, after rescaling, the solution $y$ converges as $x_1\to-\infty$, to a minimal solution of the Allen-Cahn O.D.E., which is depending on the case either $\gamma$ or $\pm 1$. This is not so surprising because the scalar Painlev\'e P.D.E. \eqref{scalarpde} is obtained by multiplying by $-x_1$, the linear term $u$ in the Allen-Cahn P.D.E. \eqref{ac}, and after rescaling as in \eqref{scale1}, the dependence on $x_1$ disappears as $x_1\to-\infty$.
\subsection{The vector Painlev\'e P.D.E. and the Ginzburg-Landau system}
The scope of the present paper is to investigate the vector equation \eqref{pain 1} (with $m\geq 2$), and construct the first to our knowledge nontrivial examples of solutions. First of all, we shall point out the deep connection of \eqref{pain 1} with the Ginzburg-Landau system \begin{equation}\label{gl}
\Delta u=\nabla W(u)=|u|^2 u-u, \ u:\R^n\to\R^m, \ W:\R^m\to [0,\infty), \ W(u)=\frac{1}{4}(|u|^2-1)^2, \end{equation} which has been extensively studied (cf. in particular \cite{bethuel1} and \cite{SS}) due to its application in the theory of superconductors and superfluids. Actually, the vector Painlev\'e P.D.E. \eqref{pain 1} only differs from \eqref{gl} by the factor $-x_1$ multiplying its linear term.
On the one hand, the non autonomous potential $H(x_1,y)=\frac{1}{2}x_1|y|^2+\frac{1}{2}|y|^4$ associated to \eqref{pain 1} attains its global minimum equal to $0$ when $y=0$ and $x_1\geq 0$, and equal to $-\frac{x^2_1}{8}$ when $|y|=\sqrt{|x_1|/2}$ and $x_1<0$. Thus, for every fixed $x_1<0$, the set of minima of $H$ is the sphere $\{y\in\R^m: |y|= \sqrt{|x_1|/2}\}$. On the other hand, the Ginzburg-Landau potential $W$ is nonnegative, radial, and vanishes only on the unit sphere. These properties of $W$ imply that \eqref{gl} admits when $n=m\geq 2$, a unique \emph{standard vortex} solution $\eta \in C^\infty(\R^n;\R^n)$ such that \begin{itemize}
\item[(a)] $\eta$ is $O(n)$-equivariant (i.e. $\eta(gx)=g \eta (x)$, $\forall x\in \R^n$, $\forall g\in O(n)$), or equivalently $\eta(x)=\eta_{\mathrm{rad}}(|x|)\frac{x}{|x|}$, $\forall x\neq 0$, where $\eta_{\mathrm{rad}}$ is a function having an odd extension in $C^\infty(\R)$. \item[(b)] $\eta_{\mathrm{rad}}$ is increasing, and converges to $1$ at $+\infty$. \end{itemize} In addition the solution $\eta$ is \emph{minimal} in the sense that $E_{\mathrm{GL}}(\eta, \mathrm{supp}\, \phi)\leq E_{\mathrm{GL}}(\eta+\phi, \mathrm{supp}\, \phi)$, for all $\phi\in C^\infty_0(\R^n;\R^n)$, where \begin{equation}\label{funcgl}
E_{\mathrm{GL}}(u, \Omega)=\int_\Omega \left[ \frac{1}{2} |\nabla u|^2 +\frac{1}{4} (|u|^2 -1)^2\right],\ \Omega\subset \R^n, \end{equation} is the Ginzburg-Landau energy functional. Actually in dimension $n=2$, Mironescu \cite{mironescu} established (cf. also \cite{MR1267609}) that any minimal solution of (\ref{gl}) is either constant of modulus $1$ or (up to orthogonal transformation in the range and translation in the domain) the standard vortex $\eta$. In higher dimensions $n=m\geq 3$, the minimality of $\eta$ was proved by Pisante \cite{pisante}, however it is not clear if there exist other nontrivial minimal solutions.
Our purpose in this paper is to construct the analog of the standard vortex solution $\eta$ for the vector Painlev\'e P.D.E. \eqref{pain 1}. Since for every fixed $x_1<0$, the potential $H(x_1,y)$ attains its global minimum on the sphere $\{y\in\R^m: |y|= \sqrt{|x_1|/2}\}$, we should have $m=n-1$ in order to allow the formation of vortices in the hyperplanes $x_1=\mathrm{Const}$. On the other hand, the amplitude of these vortices will depend on the radius $\sqrt{|x_1|/2}$. Thus, the standard vortex solution of \eqref{pain 1} should be a solution $y:\R^n\to\R^{n-1}$, $n\geq 3$, such that \begin{itemize}
\item[(a)] $y$ is $O(n-1)$-equivariant with respect to $(x_2,\ldots,x_n)=:z$ (i.e. $y(x_1,gz)=g y(x_1,z)$, $\forall x_1\in \R$, $\forall z \in \R^{n-1}$, $\forall g\in O(n-1)$).
\item[(b)] $|y(x_1,z)|\approx \sqrt{|x_1|/2}$, as $|z|\to\infty$, and $x_1<0$ is fixed. \end{itemize}
More precisely, we have:
\begin{theorem}\label{corpain2} There exists a solution $y\in C^\infty(\R^n;\R^{n-1})$ (with $ n \geq 3$) to \begin{equation}\label{painhom}
\Delta y-x_1 y-2|y|^2y=0, \text{ with } x=(x_1,\ldots,x_n)\in \R^n,\ y=(y_2,\ldots,y_n):\R^n\to\R^{n-1}, \end{equation} such that \begin{itemize}
\item[(i)] Setting $z:=(x_2,\ldots,x_n)$, $e_z:=\frac{z}{|z|}$, and $\sigma:=|z|$, we have $y(x)= y_{\mathrm{rad}}(x_1,\sigma)e_z$, where $y_{\mathrm{rad}}(x_1,\sigma)$ is a function having an odd with respect to $\sigma$ extension in $C^\infty(\R^2;\R)$. \item[(ii)] In addition, $y_{\mathrm{rad}}(x_1,\sigma)>0$, $\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,\sigma)<0$, and $\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,\sigma)>0$, $\forall x_1\in\R$, $\forall \sigma>0$.
\item[(iii)] $|y(x)|<h(x_1)$, $\forall x \in \R^{n}$, and $\lim_{l\to\infty} y_{\mathrm{rad}}(x_1,\sigma+l)=h(x_1)$ in $C^1_{\mathrm{loc}}(\R^2;\R)$, where $h$ is the Hastings-McLeod solution of (\ref{pain 0}). \item[(iv)] For every $z=(x_2,\ldots,x_n)\in\R^{n-1}$ fixed, let \begin{equation}\label{ytilde} \tilde y(t_1,\ldots,t_n):=\frac{\sqrt{2}}{(-\frac{3}{2} t_1)^{\frac{1}{3}}}\, y\big(-(-\frac{3}{2} t_1)^{\frac{2}{3}}, x_2+t_2(-\frac{3}{2} t_1)^{-\frac{1}{3}}, \ldots, x_n+t_n(-\frac{3}{2} t_1)^{-\frac{1}{3}} \big). \end{equation} Then \begin{equation}\label{scale2} \lim_{l\to -\infty} \tilde y(t_1+l,t_2,\ldots, t_n)= \begin{cases} \eta(t_2,\ldots,t_n) &\text{when } z=0, \\ e_z &\text{when } z\neq 0, \end{cases} \end{equation} for the $C^1_{\mathrm{ loc}}(\R^n;\R^{n-1})$ convergence, where $\eta\in C^\infty(\R^{n-1};\R^{n-1})$ is the standard vortex solution of the Ginzburg-Landau system \eqref{gl}. \end{itemize} \end{theorem}
Theorem \ref{corpain2} provides a solution $y:\R^n\to\R^{n-1}$ having in every hyperplane $x_1=\mathrm{Const.}$, a profile similar to $\eta$. Indeed, for fixed $x_1$, the $O(n-1)$-equivariant map $z\mapsto y(x_1,z)$ only vanishes at $z=0$, and its modulus increases as $|z|$ increases. The amplitude of these vortices is determined by the Hastings-McLeod solution $h$ evaluated at $x_1$. As we mentioned in subsection \ref{subsec:1dd}, $h$ interpolates smoothly the function $x_1\mapsto \sqrt{(-x_1)^+/2}$ describing the radius of the sphere where the potential $H(x_1,y)$ attains its global minimum. On the other hand, for every fixed $z\neq 0$, the map $x_1\mapsto |y(x_1,z)|$ is decreasing, like $h$. Finally, property (iv) shows that after rescaling, the solution $y$ converges as $x_1\to-\infty$, to a solution of the Ginzburg-Landau system \eqref{gl}, which is depending on the case either $\eta$ or a constant of modulus $1$ (compare with \eqref{scale1}). Actually, it is proved in Lemma \ref{ass}, that the rescaling \eqref{scale2} applied to any solution of \eqref{pain 1} satisfying the bound \eqref{boundass}, provides after passing to limit, a solution of \eqref{gl}.
\subsection{Minimal solutions of the vector Painlev\'e P.D.E} Despite the deep connection of the vector Painlev\'{e} P.D.E. with the Ginzburg-Landau system, we are not aware if the structure of minimal solutions of \eqref{pain 1} exactly mirrors that of \eqref{gl} at least in low dimensions. Although by construction the solution $y:\R^n\to\R^{n-1}$ provided by Theorem \ref{corpain2} is only minimal for $O(n-1)$-equivariant perturbations, we expect that $y$ is actually minimal for general perturbations, as the standard vortex $\eta:\R^{n-1}\to\R^{n-1}$ of the Ginzburg-Landau system. While $\eta$ is defined in $\R^{n-1}$, the solution $y$ depends on $n$ variables, since $x_1$ parametrizes the set of minima of the potential $H$, and only $x_2,\ldots,x_n$ are involved in the vortex formation mechanism.
In the one dimensional case, one can easily see (cf. Lemma \ref{cara}) that the only minimal solutions $y:\R\to\R^m$ of \eqref{pain 1} bounded at $+\infty$, are the maps $\R\ni x_1\mapsto h(x_1)\textbf{\em n}$, with $\textbf{\em n}\in\R^m$ a unit vector. These maps have their counterparts in the constant solutions of modulus $1$ of \eqref{gl} (which are also the global minimizers of $E_{\mathrm{GL}}$). On the other hand, in dimension two, the existence of nontrivial minimal solutions $y:\R^2\to\R^m$ of \eqref{pain 1} is not clear, since the Painlev\'e system with $n=2$ and $m\geq 2$, is related to the O.D.E. system $u''=|u|^2-u$, $u:\R\to\R^m$, which has only constant minimal solutions (cf. \cite[Remark 3.5.]{antonop}, and also \cite{farina} for nonexistence results of minimal solutions of the Ginzburg-Landau system in higher dimensions).
As far as the liquid crystal model \eqref{sing} is concerned, with $n=m=2$ as in \cite{Clerc2018}, the nonexistence of a minimal solution $y:\R^2\to\R^2$ of \eqref{pain 1} having an isolated zero, would imply the following two important results. Firstly, that the profile of the shadow vortex (appearing when the intensity of the applied laser light is of order $o(\epsilon |\ln \epsilon|)$, cf. \cite[Theorem 1.2 (ii)]{Clerc2018}), is given by the Ginzburg-Landau system \eqref{gl}. Secondly, that it is located at a distance $d \gg \epsilon^{2/3}$ from the boundary of the illuminated region.
\section{General results for solutions $y:\R^n\to\R^m$ of \eqref{pain 1}}\label{sec:sec3}
In this section we collect some general results holding for every solution $y:\R^n\to\R^m$ of \eqref{pain 1}. These results may be useful to construct different types of solutions of \eqref{pain 1}. They will be used and particularized in the proof of Theorem \ref{corpain2}, in Section \ref{sec:sec4}.
In the next two lemmas, we shall first examine the asymptotic behavior of solutions of \eqref{pain 1}, as $x_1\to+\infty$, and $x_1\to-\infty$.
\begin{lemma}\label{expcvv} Let $y:\R^n\to\R^m$ be a solution of \eqref{pain 1} which is bounded in the half-space $\{x_1\geq 1\}$. Then, we have
$|y(x)|=O(e^{-\frac{2}{3}x_1^{3/2}})$, as $x_1\to\infty$ (uniformly in $z=(x_2,\ldots,x_n)$). \end{lemma} \begin{proof}
Let $M=e^{\frac{2}{3}}\sup_{x_1\geq 1} |y(x)|$, let $y=(y_1,\ldots,y_m)\in\R^m$, and let $\Omega=\{ x_1>1\}$. We shall compare $\pm y_i$, $\forall i=1,\ldots,m$, with the function $\psi(x):=M e^{-\frac{2}{3} x_1^{3/2}}$. It is clear that $\Delta \psi \leq x_1\psi\leq (x_1+2|y|^2)\psi$ on $\Omega$, and $\Delta (y_i-\psi) \geq (x_1+2|y|^2) (y_i-\psi)$ on $\Omega$. Since we have $y_i-\psi\leq 0$ on $\partial \Omega$, it follows from the maximum principle (cf. \cite[Lemma 2.1]{beres}) that $y_i-\psi\leq 0$ holds on $\Omega$. Similarly, we obtain that $-y_i-\psi\leq 0$ holds on $\Omega$. \end{proof}
\begin{lemma}\label{ass} Let $y:\R^n\to\R^m$ be a solution of \eqref{pain 1} such that the function \begin{equation}\label{boundass}
x\mapsto \frac{|y(x)|}{\sqrt{|x_1|}} \text{ is bounded in the half-space $\{x_1\leq -1\}$}, \end{equation} and let $$t=(t_1,t_2,\ldots,t_n):=\big(-\frac{2}{3} (-x_1)^{\frac{3}{2}}, (-x_1)^{\frac{1}{2}}r_2, \ldots, (-x_1)^{\frac{1}{2}}r_n,\big), \ \forall x_1\leq -1, \ \forall r:=(r_2,\ldots, r_n)\in\R^{n-1}.$$ Equivalently, setting $\tau:=(t_2,\ldots, t_n)\in\R^{n-1}$, we have $(x_1,r)=\big(-(-\frac{3}{2} t_1)^{\frac{2}{3}}, (-\frac{3}{2} t_1)^{-\frac{1}{3}}\tau\big)$. Next, define $\tilde y(t_1,\tau):=\frac{\sqrt{2}}{(-\frac{3}{2} t_1)^{\frac{1}{3}}}\, y(x_1, r+z)$, for every $z:=(x_2,\ldots,x_n)\in\R^{n-1}$ \emph{fixed}, or equivalently \begin{equation}\label{eqder} y(x_1, r+z)=\frac{(-x_1)^{\frac{1}{2}}}{\sqrt{2}}\tilde y(t_1,\tau). \end{equation} Then, up to subsequence, \begin{equation}\label{scale2p} \lim_{l\to -\infty} \tilde y(t_1+l,t_2,\ldots, t_n)=u(t_1,t_2,\ldots,t_n), \end{equation}
for the $C^1_{\mathrm{ loc}}(\R^n;\R^{m})$ convergence, where $u\in C^\infty(\R^n;\R^m)$ solves $\Delta u=|u|^2u-u$ in $\R^n$. In addition, if $y$ is a minimal solution of \eqref{pain 1}, then $u$ is also minimal. \end{lemma} \begin{proof}
We are going to show that $\tilde y(t_1,\ldots,t_n)$ is uniformly bounded up to the second derivatives, when $\tau=(t_2,\ldots,t_n)$ belongs to a compact set and $t_1\to-\infty$. By differentiating \eqref{eqder} with respect to $x_1$ and $r$ we obtain \begin{subequations}\label{eqderder} \begin{equation}\label{eqder1} \sqrt{2}y_{x_i}(x_1, r+z)=(-x_1)\tilde y_{t_i}(t_1,\tau), \ \forall i=2,\ldots,n. \end{equation} \begin{equation}\label{eqder2} \sqrt{2}y_{x_ix_j}(x_1, r+z)=(-x_1)^{\frac{3}{2}}\tilde y_{t_it_j}(t_1,\tau),\ \forall i,j=2,\ldots,n. \end{equation} \begin{equation}\label{eqder3}
\sqrt{2}y_{x_1}(x_1, r+z)=-\frac{(-x_1)^{-\frac{1}{2}}}{2}\tilde y(t_1,\tau)+(-x_1) \tilde y_{t_1}(t_1,\tau)-\sum_{i=2}^n\frac{r_i}{2}\tilde y_{t_i}(t_1,\tau). \end{equation} \begin{equation}\label{eqder4}
\sqrt{2}y_{x_1x_j}=-\tilde y_{t_j}+(-x_1)^{\frac{3}{2}}\tilde y_{t_1t_j}-(-x_1)^{\frac{1}{2}}\sum_{i=2}^n\frac{r_i}{2}\tilde y_{t_it_j}, \ \forall j=2,\ldots,n. \end{equation} \begin{equation}\label{eqder5}
\sqrt{2}y_{x_1x_1}=-\frac{(-x_1)^{-\frac{3}{2}}}{4}\tilde y-\frac{3}{2}\tilde y_{t_1} +\frac{(-x_1)^{-1}}{4} \sum_{i=2}^n r_i \tilde y_{t_i} +(-x_1)^{\frac{3}{2}}\tilde y_{t_1t_1}-(-x_1)^{\frac{1}{2}}\sum_{i=2}^n r_i \tilde y_{t_1t_i}+\frac{(-x_1)^{-\frac{1}{2}}}{4}\sum_{i,j=2}^nr_ir_j \tilde y_{t_i,t_j}. \end{equation} \end{subequations}
Since by assumption $y$ satisfies $|y(x)|=O(|-x_1|^{\frac{1}{2}})$ as $x_1\to-\infty$ (i.e. $\tilde y$ is bounded), we obtain by \eqref{pain 1} and standard elliptic estimates \cite[\S 3.4 p. 37 ]{1987130} that \begin{equation}\label{eqder6}
\text{$|D y(x)|=O(|-x_1|^{\frac{3}{2}})$ and $|D^2 y(x)|=O(|-x_1|^{\frac{5}{2}})$, as $x_1\to-\infty$.} \end{equation} From \eqref{eqder6} and \eqref{eqderder} it follows that \begin{equation}\label{eqder7}
\text{$|\nabla \tilde y(t)|=O(|-x_1|^{\frac{1}{2}})$ and $|D^2 \tilde y(t)|=O(|-x_1|)$, as $x_1\to-\infty$,} \end{equation}
provided that $t=(t_1,\tau)\in \Sigma_{\alpha,\beta}:=\{ t\in \R^n: t_1\leq \alpha, \, |\tau|\leq \beta (-\frac{3}{2} t_1)^{\frac{1}{3}}\}$, where $\alpha<0$ and $\beta>0$ are arbitrary constants. In particular, we have $\sqrt{2}\Delta y(x_1,r+z)=(-x_1)^{\frac{3}{2}}\Delta \tilde y(t)+O(|-x_1|^{\frac{3}{2}})$, for $t\in \Sigma_{\alpha,\beta}$. On the other hand it is clear by \eqref{pain 1} that
$\sqrt{2}\Delta y(x_1,r+z)=(-x_1)^{\frac{3}{2}}(|\tilde y(t)|^2-1)\tilde y(t)$, thus \begin{equation}\label{eqder8}
\text{$|\Delta \tilde y(t)|$ and $|\nabla\tilde y(t)|$ are bounded, $\forall t\in \Sigma_{\alpha,\beta}$.} \end{equation} Similarly, by differentiating once more equations \eqref{eqderder} with respect to $x_1$ and $r$, one can show that \begin{equation}\label{eqder9}
\text{$|D^2\tilde y(t)|$ is bounded, $\forall t\in \Sigma_{\alpha,\beta}$.} \end{equation} Next, we apply the theorem of Ascoli to the sequence $\tilde y(t_1+ l,t_2,\ldots t_n)$ as $ l\to-\infty$. Up to a subsequence $l_k\to -\infty$, we obtain via a diagonal argument, the convergence in $C^1_{\mathrm{ loc}}(\R^n;\R^m)$ of $\tilde y_k(t_1,t_2,\ldots,t_n):=\tilde y(t_1+l_k,t_2,\ldots,t_n)$ to a bounded function $ u(t_1,t_2,\ldots,t_n)$ that we are going to determine.
Let $(e_1,\ldots,e_n)$ be the canonical basis of $\R^n$, and let $\tilde \phi(t_1,\ldots,t_n)\in C^\infty_0(\R^n;\R^m)$ be a test function such that $\tilde S:=\supp \tilde \phi \subset\{(t_1,\ldots,t_n): c-d\leq t_1\leq c\} $, for some constants $c \in\R$ and $d>0$. Given $l\in\R$, we consider the translated functions $\tilde \phi^{-l}(t_1,\ldots,t_n):=\tilde \phi(t_1-l,t_2,\ldots,t_n)$, and $\tilde y^l(t_1,\ldots,t_n):=\tilde y(t_1+l,t_2,\ldots,t_n)$. Note that $\tilde S^{l}:=\supp \tilde \phi^{-l}=\tilde S+l e_1$, and $\supp\tilde \phi^{-l} \subset\{(t_1,\ldots,t_n): t_1<-1\} $ when $l<-1-c$. Thus, for $l<1-c$, we can define $\phi^{-l}\in C^\infty_0(\R^n;\R^m)$ by $\phi^{-l}(x_1, r+z)=\frac{(-x_1)^{\frac{1}{2}}}{\sqrt{2}}\tilde \phi^{-l}(t_1,\tau)$ as in \eqref{eqder}, and we set $S^{l}:=\{(x_1(t_1),r(t_1,\ldots,t_n)+z): \, (t_1,\ldots,t_n)\in \tilde S^{l}\}$.
As a consequence, we have \begin{equation}\label{weak}
\int_{\R^n}\nabla y\cdot\nabla \phi^{-l}+x_1 y\cdot\phi^{-l}+2|y|^2y\cdot\phi^{-l}=0, \ \forall l<-1-c, \end{equation}
that becomes after changing variables as in (\ref{eqder}): \begin{equation}\label{weak2}
\int_{\R^n}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[\nabla \tilde y\cdot\nabla \tilde \phi^{-l}+( |\tilde y|^2-1)\tilde y\cdot\tilde \phi^{-l}+A(\tilde y,\tilde \phi^{-l})]=0, \ \forall l<-1-c, \end{equation} with \begin{multline*} A(\psi,\xi):=-\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{-1}[\psi\cdot\xi_{t_1}+\psi_{t_1}\cdot \xi+\sum_{i=2}^nt_i (\psi_{t_i}\cdot\xi_{t_1}+\psi_{t_1}\cdot \xi_{t_i})]\\ +\frac{1}{4}\big(-\frac{3}{2}t_1\big)^{-2}[\psi+\sum_{i=2}^nt_i \psi_{t_i}][\xi+\sum_{j=2}^nt_j \xi_{t_j}]. \end{multline*} On the other hand, in view of \eqref{eqder8}, one can see that \begin{equation}\label{ll1} (-t_1)^\beta=(-c-l_k)^\beta+O((-c-l_k)^{\beta-1}), \ \forall \beta<1, \ \forall (t_1,\ldots,t_n)\in \tilde S^{l_k}, \end{equation} \begin{equation*}\label{ll2} \int_{\R^n}(-t_1)^{\frac{4-n}{3}}A(\tilde y,\tilde \phi^{-l_k})=O((-c-l_k)^{\frac{1-n}{3}}), \end{equation*} and \begin{equation*}\label{ll3}
\int_{\R^n}(-t_1)^{\frac{4-n}{3}}[\nabla \tilde y\cdot\nabla \tilde \phi^{-l_k}+( |\tilde y|^2-1)\tilde y\cdot\tilde \phi^{-l_k}]=(-c-l_k)^{\frac{4-n}{3}}\int_{\R^n}[\nabla \tilde y\cdot\nabla \tilde \phi^{-l_k}+( |\tilde y|^2-1)\tilde y\cdot\tilde \phi^{-l_k}]+O((-c-l_k)^{\frac{1-n}{3}}). \end{equation*} Gathering the previous results, it follows from \eqref{weak2} that \begin{equation*}\label{ll4}
(-c-l_k)^{\frac{4-n}{3}}\int_{\R^n}[\nabla \tilde y\cdot\nabla \tilde \phi^{-l_k}+( |\tilde y|^2-1)\tilde y\cdot\tilde \phi^{-l_k}]+O((-c-l_k)^{\frac{1-n}{3}})=0. \end{equation*} Finally, since \begin{align*}\label{ll4}
\lim_{k\to\infty}\int_{\R^n}[\nabla \tilde y\cdot\nabla \tilde \phi^{-l_k}+( |\tilde y|^2-1)\tilde y\cdot\tilde \phi^{-l_k}]&=
\lim_{k\to\infty}\int_{\R^n}[\nabla \tilde y_k \cdot\nabla \tilde \phi+( |\tilde y_k|^2-1)\tilde y\cdot\tilde \phi]\\
&=\int_{\R^n}[\nabla u\cdot\nabla \tilde \phi+( |\tilde u|^2-1)u\cdot\tilde \phi], \end{align*}
we deduce that $\int_{\R^n}[\nabla u\cdot\nabla \tilde \phi+( |\tilde u|^2-1)u\cdot\tilde \phi]$, $\forall \tilde \phi \in C^\infty_0(\R^n;\R^m)$, i.e. $u$ solves $\Delta u=|u|^2u-u$ in $\R^n$.
Similarly, if $y$ is a minimal solution of \eqref{pain 1}, we have \begin{equation}\label{weakb} E_{\mathrm{P_{II}}}(y, S^l)\leq E_{\mathrm{P_{II}}}(y+ \phi^{-l},S^l), \ \forall l<-1-c, \end{equation}
that becomes after changing variables as in (\ref{eqder}): \begin{equation}\label{weak2b} \int_{\tilde S^l}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[B( \tilde y)+C(\tilde y)]\leq \int_{\tilde S^l}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[B(\tilde y+\tilde \phi^{-l})+C(\tilde y+\tilde \phi^{-l})], \ \forall l<-1-c, \end{equation} with \begin{equation}\label{bdef}
B(\psi)=[\frac{1}{2}|\nabla \psi|^2-\frac{|\psi|^2}{2}+\frac{|\psi|^4}{4}], \end{equation} and \begin{equation}\label{cdef}
C(\psi):=-\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{-1}[\psi\cdot\psi_{t_1}+\sum_{i=2}^n t_i (\psi_{t_i}\cdot\psi_{t_1})]+\frac{1}{8}\big(-\frac{3}{2}t_1\big)^{-2}|\psi +\sum_{i=2}^nt_i \psi_{t_i}|^2. \end{equation} As previously \eqref{ll1} holds, and one can see that \begin{equation*}\label{ll2} \int_{\tilde S^{l_k}}(-t_1)^{\frac{4-n}{3}}C(\tilde y)=O((-c-l_k)^{\frac{1-n}{3}}),\ \int_{\tilde S^{l_k}}(-t_1)^{\frac{4-n}{3}}C(\tilde y+\tilde \phi^{-l_k})=O((-c-l_k)^{\frac{1-n}{3}}), \end{equation*} \begin{equation*}\label{ll3} \int_{\tilde S^{l_k}}(-t_1)^{\frac{4-n}{3}}B(\tilde y)=(-c-l_k)^{\frac{4-n}{3}}\int_{\tilde S^{l_k}}B(\tilde y)+O((-c-l_k)^{\frac{1-n}{3}}), \end{equation*} and \begin{equation*}\label{ll3} \int_{\tilde S^{l_k}}(-t_1)^{\frac{4-n}{3}}B(\tilde y+\tilde \phi^{-l_k})=(-c-l_k)^{\frac{4-n}{3}}\int_{\tilde S^{l_k}}B(\tilde y+\tilde \phi^{-l_k})+O((-c-l_k)^{\frac{1-n}{3}}). \end{equation*} Gathering the previous results, it follows from \eqref{weak2b} that \begin{equation*}\label{ll4} (-c-l_k)^{\frac{4-n}{3}}E_{\mathrm{GL}}(\tilde y, \mathrm{supp}\, \tilde \phi^{-l_k})\leq (-c-l_k)^{\frac{4-n}{3}}E_{\mathrm{GL}}(\tilde y+\tilde \phi^{-l_k}, \mathrm{supp}\, \tilde \phi^{-l_k})+O((-c-l_k)^{\frac{1-n}{3}}), \end{equation*} or equivalently \begin{equation*}\label{ll5} E_{\mathrm{GL}}(\tilde y_k, \mathrm{supp}\, \tilde \phi) \leq E_{\mathrm{GL}}(\tilde y_k+\tilde\phi, \mathrm{supp}\, \tilde \phi) +O((-c-l_k)^{-1}), \end{equation*} Finally, passing to the limit, we obtain \begin{equation*}\label{ll4} E_{\mathrm{GL}}(u, \mathrm{supp}\, \tilde \phi)\leq E_{\mathrm{GL}}(u+\tilde \phi, \mathrm{supp}\, \tilde \phi), \end{equation*} i.e. $u$ is a minimal solution of \eqref{gl}.
\end{proof}
Next, we establish some properties of minimal solutions of \eqref{pain 1}. More precisely, we prove in Lemma \ref{lemzero} that a minimal solution is not identically zero, while in Lemma \ref{cara} (resp. Lemma \ref{lema}) we characterize the one dimensional minimal solutions of \eqref{pain 1} (resp. the nonnegative minimal solution of the scalar equation \eqref{scalarpde}).
\begin{lemma}\label{lemzero} Let $y:\R^n\to\R^m$ be a minimal solution of \eqref{pain 1}. Then, $y$ is not identically $0$. \end{lemma} \begin{proof} The minimality of $y$ implies that the second variation of the energy $E_{\mathrm{P_{II}}}$ is nonnegative: \begin{equation}\label{secondvabis}
\int_{\R^n} (|\nabla \phi(x)|^2+(2|y(x)|^2+4(y\cdot\phi)^2+x_1)|\phi(x)|^2)\mathrm{d} x \geq 0,\quad \forall \phi\in C^1_0(\R^n;\R^m). \end{equation} Clearly \eqref{secondvabis} does not hold when $y\equiv 0$, if we take $\phi(x_1,z)=\phi_0(x_1+l,z)$, with $l\to\infty$, and $\phi_0\in C^1_0(\R^n;\R^m)$, such that $\phi_0 \not\equiv 0$. \end{proof}
\begin{lemma}\label{cara} Let $y:\R\to\R^m$ be a solution of \eqref{pain 1} which is bounded at $+\infty$. Then, $y$ is minimal iff $y(x)= h(x)\textbf{\em n}$, for some unit vector $\textbf{\em n}\in \R^m$. \end{lemma} \begin{proof}
Assume that $y$ is minimal. Since $y\not\equiv 0$ (cf. Lemma \ref{lemzero}), let $x_0\in\R$ be such that $y(x_0)\neq 0$, and let $\textbf{\em n}_0:=\frac{y(x_0)}{|y(x_0)|}$. Next, we consider the competitor map \begin{equation} \xi(x)=\begin{cases} y(x) &\text{when } x\leq x_0, \\
|y(x)|\textbf{\em n}_0 &\text{when } x\geq x_0. \end{cases} \end{equation}
It is clear that $|\xi'|\leq |y'|$ holds on $\R$, and that $E_{\mathrm{P_{II}}}(\xi,[a,b])\leq E_{\mathrm{P_{II}}}(y,[a,b])$, $\forall a<b$. Assume by contradiction that $E_{\mathrm{P_{II}}}(\xi,[x_0,\infty))+2\epsilon< E_{\mathrm{P_{II}}}(y,[x_0,\infty))$, for some $\epsilon>0$. This implies that for $b>b_0$ large enough we have $E_{\mathrm{P_{II}}}(\xi,[x_0,b])+\epsilon< E_{\mathrm{P_{II}}}(y,[x_0,b])$. Setting \begin{equation} \psi(x)=\begin{cases} \xi(x) &\text{when } x\in [x_0,b] \\
(y(b+1)-|y(b)|\textbf{\em n}_0 )(x-b)+|y(b)|\textbf{\em n}_0 &\text{when } x\in [b,b+1], \end{cases} \end{equation}
it follows from Lemma \ref{expcvv} that $E_{\mathrm{P_{II}}}(\psi,[b,b+1])<\epsilon$ provided that $b\geq b_1\geq b_0$ is large enough. Thus we deduce that $E_{\mathrm{P_{II}}}(\psi,[x_0,b_1+1])< E_{\mathrm{P_{II}}}(y,[x_0,b_1+1])$, with $y(x_0)=\psi(x_0)$, and $y(b_1+1)=\psi(b_1+1)$, in contradiction with the minimality of $y$. This proves that $E_{\mathrm{P_{II}}}(\xi,[x_0,\infty))= E_{\mathrm{P_{II}}}(y,[x_0,\infty))$. In particular, we have $E_{\mathrm{P_{II}}}(\xi,[x_0,b])=E_{\mathrm{P_{II}}}(y,[x_0,b])$, on an interval $[x_0,b]$ where $y\neq 0$, and setting $\textbf{\em n}(x):=\frac{y(x)}{|y(x)|}$ on $[x_0,b]$, we obtain $|\xi'|^2=|y'|^2=|\xi'|^2+|\xi|^2|\textbf{\em n}'|^2$ on $[x_0,b]$. As a consequence, $\textbf{\em n}'\equiv 0$, and $y(x)=|y(x)|\textbf{\em n}_0$ hold on $[x_0,b]$. Finally, for any unit vector $\nu$ perpendicular to $\textbf{\em n}_0$, let $\chi(x)=y(x)\cdot \nu$. In view of \eqref{pain 1}, $\chi$ solves $\chi''=x\chi+2|y|^2\chi$ on $\R$, and since $\chi\equiv 0$ on $[x_0,b]$, we conclude by the uniqueness result for O.D.E. that $\chi\equiv 0$ on $\R$. This proves that $y(x)=(y(x)\cdot\textbf{\em n}_0)\textbf{\em n}_0$ holds on $\R$. Therefore, it follows from the characterization of minimal solutions of \eqref{pain 0} established in \cite[Theorem 1.3 (i)]{Clerc2017}, that $y(x)=h(x)\textbf{\em n}_0$, $\forall x\in\R$.
Conversely, let $y(x)=h(x)\textbf{\em n}$, for some unit vector $\textbf{\em n}\in\R^m$, and let us check that $y$ is minimal.
For every test function $\phi\in H^1_0([a,b];\R^m)$, let $\xi(x):=|y(x)+\phi(x)|\textbf{\em n}$. Since $|\xi'|\leq |y'+\phi'|$ holds on $\R$, we have $E_{\mathrm{P_{II}}}(\xi,[a,b])\leq E_{\mathrm{P_{II}}}(y+\phi,[a,b])$. On the other hand, it follows from the minimality of $h:\R\to\R$ that $E_{\mathrm{P_{II}}}(y,[a,b])\leq E_{\mathrm{P_{II}}}(\xi,[a,b])$. This establishes that $y$ is minimal.
\end{proof} \begin{lemma}\label{lema}\footnote{A similar result holds for the nonnegative minimal solutions of the Allen-Cahn equation \eqref{ac} (cf. for instance \cite[Corollary 5.2]{book} which also applies in the vector case).} Let $y:\R^n\to\R$ be a nonnegative minimal solution of \eqref{scalarpde} such that $y(x)\leq h(x_1)$, $\forall x\in\R^n$. Then $y(x_1,\ldots,x_n)=h(x_1)$, $\forall x\in \R^n$. \end{lemma} \begin{proof}
First of all we show that $y>0$. Indeed, if $y(p)=0$ for some $p\in\R^n$, then the maximum principle implies that $y \equiv 0$, and this is ruled out by Lemma \ref{lemzero}. Let $B_R\subset \R^n$ be the open ball of radius $R$, centered at the origin, and let $y_R$ be a minimizer of $E_{\mathrm{P_{II}}}$ in $H_0^{1}(B_R;\R)$ (cf. Lemma \ref{lem1} for the existence of minimizers). We notice that $|y_R|$ is also a minimizer of $E_{\mathrm{P_{II}}}$ in $H_0^{1}(B_R;\R)$, thus we can choose $y_R$ such that $y_R\geq 0$. It is clear that $y_R\in C^\infty(\overline B_R;\R)$ is a smooth solution of \eqref{scalarpde} in $B_R$. To establish Lemma \ref{lema}, we shall compare $y$ with $y_R$. Our claim is that $y_R\leq y$, $\forall R$. Indeed, assume by contradiction that $S:=\{x\in B_R: y_R>y \}\neq \emptyset$, and let $\phi:=\max(y_R-y,0)\in H_0^{1}(B_R;\R)$. On the one hand, we have by the minimality of $y$ that \begin{equation*}\label{mm1}
E_{\mathrm{P_{II}}}(y,B_R)\leq E_{\mathrm{P_{II}}}(y+\phi,B_R)=E_{\mathrm{P_{II}}}(y_R,S) +E_{\mathrm{P_{II}}}(y,B_R\setminus S), \end{equation*} thus $ E_{\mathrm{P_{II}}}(y,S)\leq E_{\mathrm{P_{II}}}(y_R,S)$. On the other hand, by definition of the minimizer $y_R$, we obtain \begin{equation*}\label{mm2} E_{\mathrm{P_{II}}}(y_R,B_R)\leq E_{\mathrm{P_{II}}}(y_R-\phi,B_R)=E_{\mathrm{P_{II}}}(y,S) +E_{\mathrm{P_{II}}}(y_R,B_R\setminus S), \end{equation*} that is, $E_{\mathrm{P_{II}}}(y_R,S)\leq E_{\mathrm{P_{II}}}(y,S)$. As a consequence, $E_{\mathrm{P_{II}}}(y_R,S)= E_{\mathrm{P_{II}}}(y,S)$, and the function $ \tilde y_R(x):=\min (y_R,y)$ is another minimizer of $E_{\mathrm{P_{II}}}$ in $H_0^{1}(B_R;\R)$. In particular $\tilde y_R$ is a $C^\infty(\overline B_R;\R)$ smooth solution of \eqref{scalarpde} in $B_R$. Finally, since $y_R$ and $\tilde y_R$ coincide on an open ball $B\subset \{x\in B_R:y_R(x)<y(x)\}\neq \emptyset$, it follows by unique continuation that $y_R\equiv \tilde y_R$, which is a contradiction.
At this stage, we are going to prove that $y(x_1,\ldots,x_n)=h(x_1)$, $\forall x\in \R^n$ by induction on the dimension $n$. For $n=1$ the statement is true, since $h$ is the only nonnegative minimal solution of O.D.E. \eqref{pain 0}, which is bounded at $+\infty$ (cf. \cite[Theorem 1.3 (i)]{Clerc2017}). Now, assume that $n\geq 2$. We will first establish that $y_R>0$ on $B_R$, provided that $R$ is large enough. As we mentioned before, the existence of $p\in B_R$ such that $y_R(p)=0$ implies that $y_R\equiv 0$. On the other hand, one can see that $0$ is not a minimizer of $E_{\mathrm{P_{II}}}$ in $H_0^{1}(B_R;\R)$, when $R$ is large enough. Indeed, by taking $\phi(x_1,z)=\phi_0(x_1+l,z)$, with $z=(x_2,\ldots,x_n)$, $l>0$, and $\phi_0\in C^ \infty_0(B_1;\R)$ such that $\phi_0 \not\equiv 0$, we have $\phi \in H_0^{1}(B_{l+1};\R)$, and $E_{\mathrm{P_{II}}}(\phi,B_{l+1})<0$, provided that $l$ is large enough. Our claim, is that $\frac{\partial y_R}{\partial x_n}(x)<0$ (resp. $\frac{\partial y_R}{\partial x_n}(x)>0$) provided that $x\in B_R$ (with $R$ large enough) and $x_n>0$ (resp. $x\in B_R$ and $x_n>0$). This follows from the moving plane method applied to $\psi_\lambda(x)=y_R(x)-y_R(x_1, x_2, \ldots,x_{n-1},2\lambda-x_n)$ in the domain $B_{R,\lambda}:=\{x\in B_R:x_n>\lambda\}$, for every $\lambda\in (0,R)$, since $\psi_\lambda$ satisfies $\Delta\psi_\lambda(x)=c(x)\psi_\lambda(x)$ in $B_{R,\lambda}$ with $c(x)=x_1+2(y_R^2(x)+y_R^2(x_1,\ldots, x_{n-1},2\lambda-x_n)+y_R(x)y_R(x_1,\ldots,x_{n-1},2\lambda-x_n))$. We refer to \cite[section 9.5.2.]{evans} for more details. The bound $y_R(x)\leq y(x)\leq h(x_1)$ implies that for every $L>0$ fixed, the functions $y_R$, with $R>L+1$, are uniformly bounded in $B_L$, up to the second derivatives. Therefore, by applying the theorem of Ascoli to $y_R$, via a diagonal argument, we can see that (up to subsequence) $y_R$ converges in $C^2_{\mathrm{loc}}(\R^n;\R)$ to a solution $y_\infty\in C^\infty(\R^n;\R)$ of \eqref{painhom}. Furthermore, $y_\infty$ is by construction minimal, and satisfies $0\leq y_\infty\leq y$ in $\R^n$, as well as $\frac{\partial y_\infty}{\partial x_n}(x)\leq 0$ (resp. $\frac{\partial y_\infty}{\partial x_n}(x)\geq 0$) provided that $x_n\geq 0$ (resp. $x_n\leq 0$). In view of this monotonicity property, a second application of the theorem of Ascoli to the sequence $\tilde y_l(x):=y_\infty(x_1,\ldots,x_{n-1},x_n+l)$, shows that $\lim_{l\to\pm\infty}\tilde y_l(x)=Y^\pm(x_1,\ldots,x_{n-1})$, where $Y^\pm:\R^{n-1}\to\R$ is a nonnegative minimal solution of \eqref{scalarpde}. It is clear that $Y^\pm(x)\leq h(x_1)$, thus our induction hypothesis implies that $Y^\pm(x_1,\ldots,x_{n-1})=h(x_1)$. Finally, since $\lim_{l\to\pm\infty} y_\infty(x_1,\ldots,x_{n-1},x_n+l)=h(x_1)$, $\forall x\in\R^n$, and the function $x_n\mapsto y_\infty(x_1,\ldots,x_n)$ is decreasing on $(0,\infty)$ (resp. increasing on $(-\infty,0)$) for every $(x_1,\ldots,x_{n-1})$ fixed, we deduce that $y_\infty(x_1,\ldots,x_n)=h(x_1)\leq y(x)$, and $y(x_1,\ldots,x_n)=h(x_1)$. \end{proof}
\section{Proof of Theorem \ref{corpain2}}\label{sec:sec4}
In the next lemma we show the existence of an $O(n-1)$-equivariant solution $y$ of \eqref{painhom}. We first construct in every ball $B_R\subset \R^n$ of radius $R$, an $O(n-1)$-equivariant minimizer $y_R$ of $E_{\mathrm{P_{II}}}$. Then, by passing to the limit as $R\to\infty$, we obtain the solution $y$. By construction $y$ vanishes only on the $x_1$ coordinate axis, and $|y(x)|$ is bounded by $h(x_1)$. \begin{lemma}\label{lem1} There exists a solution $y\in C^\infty(\R^n;\R^{n-1})$ (with $ n \geq 3$) to \eqref{painhom} such that \begin{itemize} \item[(i)] $y$ is $O(n-1)$-equivariant with respect to $z:=(x_2,\ldots,x_n)$, i.e. \begin{equation}\label{equivv} y(x_1,gz)=g y(x_1,z),\ \forall x_1\in \R, \ \forall z \in \R^{n-1}, \ \forall g\in O(n-1). \end{equation}
Consequently, $y$ can be written as $y(x)= y_{\mathrm{rad}}(x_1,|z|)\frac{z}{|z|}$, where $y_{\mathrm{rad}}(x_1,\sigma)$ is a function having and odd with respect to $\sigma$ extension in $C^\infty(\R^2;\R)$. \item[(ii)] $y$ is minimal with respect to $O(n-1)$-equivariant perturbations, i.e. \begin{equation}\label{minnn1} E_{\mathrm{P_{II}}}(y, \mathrm{supp}\, \phi)\leq E_{\mathrm{P_{II}}}(y+\phi, \mathrm{supp}\, \phi) \text{ for all $\phi\in C^\infty_0(\R^n;\R^{n-1})$ satisfying \eqref{equivv}.} \end{equation}
\item[(iii)] $y(x_1,z)\cdot z>0$, $\forall x_1\in\R$, $\forall z\neq 0$, and $|y(x)|\leq h(x_1)$, $\forall x \in \R^{n}$.
\item[(iv)] \eqref{minnn1} also holds for maps $\psi$ that can be written as $\psi(x_1,z)=\chi(x_1,z)\frac{z}{|z|}$, with $\chi\in C^\infty_0(\R^n;\R)$ and $\supp \chi\subset\R\times (\R^{n-1}\setminus\{0\})$.
\end{itemize} \end{lemma} \begin{proof} Let $B_R\subset \R^n$ (resp. $D_R\subset \R^2$) be the open ball (resp. the open disc) of radius $R$, centered at the origin. The existence of a minimizer $y_R$ of $E_{\mathrm{P_{II}}}$ in the class $$\mathcal A=\{u\in H_0^{1}(B_R;\R^{n-1}): u(x_1,gz)=g u(x_1,z),\text{ for a.e. } x=(x_1,z)\in B_R,\ \forall g\in O(n-1)\}$$ follows from the direct method. We first show that $\inf\{\,E_{\mathrm{P_{II}}}(u):\ u \in \mathcal A \}>-\infty$.
To see this, we notice that $ \frac{x_1}{2}|u|^2+\frac{1}{4}|u|^4< 0 \Longleftrightarrow \frac{1}{2}|u|^2<-x_1$, thus $\frac{x_1}{2}|u|^2+\frac{1}{4}|u|^4\geq -x_1^2$. Now, let $m:=\inf_{\mathcal A} E_{\mathrm{P_{II}}}>-\infty$, and let $u_n$ be a sequence such that $E_{\mathrm{P_{II}}}(u_n) \to m$. According to what precedes, we obtain the bound $\int_{\R^2}\big[\frac{1}{2}| \nabla u_n|^2+\frac{1}{4}|u_n|^4\big]\leq E_{\mathrm{P_{II}}}(u_n)+\int_{B_R} x_1^2 $, hence $\left\|u_n \right\|_{H^{1}(B_R,\R^m)}$ is bounded. As a consequence, for a subsequence still called $u_n$, we have $u_n\rightharpoonup y_R$ weakly in $H^1$, as well as $u_n\to y_R$ strongly in $L^4$ and $L^2$. In particular,
$\int_{B_R} |\nabla y_R|^2\leq \liminf_{n \to \infty} \int_{\R^2} |\nabla u_n|^2$ holds by lower semicontinuity, while $\int_{B_R} |y_R|^4=\lim_{n \to \infty} \int_{B_R} |u_n|^4$, and $\int_{B_R}x_1|v|^2 = \lim_{n \to \infty} \int_{B_R}x_1 |u_n|^2$ hold due to the strong convergence. Gathering the previous results it is clear that $m= E_{\mathrm{P_{II}}}(y_R,B_R)$. Finally, since $\mathcal A$ is a closed subspace of $ H_0^{1}(B_R;\R^{n-1})$, we deduce that $y_R$ is a minimizer of $ E_{\mathrm{P_{II}}}$ in $\mathcal A$, and a critical point of $ E_{\mathrm{P_{II}}}$ for $O(n-1)$-equivariant perturbations.
Next, since the potential $H$ is invariant with respect to the group $O(n-1)$ (i.e. $H(x_1,gy)=H(x_1,y)$, $\forall x_1\in \R$, $\forall y\in \R^{n-1}$, $\forall g \in O(n-1)$), we obtain in view of \cite{palais}, that $y_R$ is a critical point of $E_{\mathrm{P_{II}}}$ for general $H_0^{1}(B_R;\R^{n-1})$ perturbations. Thus, $y_R\in C^\infty(\overline{B_R};\R^m)$ is a classical solution of \eqref{painhom} in $B_R$, and moreover $y_R$ can be written as $y_R(x)= y_{\mathrm{rad},R}(x_1,|z|)\frac{z}{|z|}$, where $y_{\mathrm{rad},R}(x_1,\sigma)$ is a function having and odd with respect to $\sigma$ extension in $C^\infty(D_R;\R)$. Computing the energy in cylindrical coordinates, we obtain:
\begin{equation}\label{cyl} E_{\mathrm{P_{II}}}(y_R, B_R)=A(\mathbb{S}^{n-2})E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad},R}, D_R\cap\{\sigma>0\}) \end{equation} with $A(\mathbb{S}^{n-2})$ the area of the $(n-2)$-dimensional sphere $\mathbb{S}^{n-2}$, and \begin{equation}\label{cyl2}
E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad},R},D_R\cap\{\sigma>0\})=\int_{D_R\cap\{\sigma>0\} } \Big[\frac{1}{2}|\nabla y_{\mathrm{rad},R}|^2 +\frac{n-2}{2}\frac{y_{\mathrm{rad},R}^2}{\sigma^2}+\frac{x_1y_{\mathrm{rad},R}^2}{2}+\frac{y_{\mathrm{rad},R}^4}{2}\Big]\mathrm{d} x_1\mathrm{d} \sigma. \end{equation}
From this expression of $E_{\mathrm{P_{II}}}$, it follows that $\tilde y_R(x):=|y_R(x)|\frac{z}{|z|}$ is another minimizer of $E_{\mathrm{P_{II}}}$ in $\mathcal A$, and also a solution of \eqref{painhom} in $B_R$. Thus, one can choose $y_R$ such that \begin{equation}\label{psca} y_R(x_1,z)\cdot z\geq 0, \ \forall x\in B_R. \end{equation}
Finally, we are going to show that $|y_R(x)|\leq h(x_1)$ holds on $B_R$. Indeed, assuming by contradiction that $|y_{\mathrm{rad},R}(\tilde x_1,\tilde \sigma)|>h(\tilde x_1)$ holds for some $(\tilde x_1,\tilde \sigma)\in D_R\cap\{\sigma>0\}$, we consider the competitor $\tilde y_{\mathrm{rad},R}(x_1, \sigma):=\min( y_{\mathrm{rad},R}( x_1,\sigma),h(x_1))$, for $(x_1,\sigma)\in D_R\cap\{\sigma>0\} $, and we notice that the function $$\xi(x_1,\sigma):=\max( y_{\mathrm{rad},R}(x_1, \sigma),h(x_1))-h(x_1)$$ is such that $\supp\xi\subset D_R\cap\{\sigma>0\} $. Thus, since $\R^2\ni (x_1,\sigma)\mapsto h(x_1,\sigma)$ is a minimal solution of \eqref{scalarpde}, we have \begin{equation}\label{pp11}
\int_{\supp\xi} \Big[\frac{1}{2}|\nabla \tilde y_{\mathrm{rad},R}|^2 +\frac{x_1\tilde y_{\mathrm{rad},R}^2}{2}+\frac{\tilde y_{\mathrm{rad},R}^4}{2}\Big]\mathrm{d} x_1\mathrm{d} \sigma \leq \int_{\supp\xi } \Big[\frac{1}{2}|\nabla y_{\mathrm{rad},R}|^2 +\frac{x_1y_{\mathrm{rad},R}^2}{2}+\frac{y_{\mathrm{rad},R}^4}{2}\Big]\mathrm{d} x_1\mathrm{d} \sigma. \end{equation} On the other hand, it is obvious that \begin{equation}\label{pp12}
\int_{\supp\xi} \frac{n-2}{2}\frac{\tilde y_{\mathrm{rad},R}^2}{\sigma^2}\mathrm{d} x_1\mathrm{d} \sigma < \int_{\supp\xi }\frac{n-2}{2}\frac{y_{\mathrm{rad},R}^2}{\sigma^2}\mathrm{d} x_1\mathrm{d} \sigma, \end{equation} and \begin{equation}\label{pp33} E_{\mathrm{P_{II},rad}}(\tilde y_{\mathrm{rad},R},(D_R\cap\{\sigma>0\})\setminus\supp\xi\})= E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad},R},(D_R\cap\{\sigma>0\})\setminus\supp\xi\}). \end{equation}
Combining, \eqref{pp11}, \eqref{pp12}, and \eqref{pp33}, we deduce that the map $\tilde y_R(x):= \tilde y_{\mathrm{rad},R}(x_1,|z|)\frac{z}{|z|}$ belonging to $\mathcal A$, satisfies $E_{\mathrm{P_{II}}}(\tilde y_R, B_R)<E_{\mathrm{P_{II}}}(y_R, B_R)$, which is a contradiction. Thus, we have established that $|y_R(x)|\leq h(x_1)$ holds in $B_R$, for every $R>0$.
The previous bounds imply that for every $L>0$ fixed, the maps $y_R$, with $R>L+1$, are uniformly bounded in $B_L$, up to the second derivatives. Therefore, by applying the theorem of Ascoli to $y_R$, via a diagonal argument, we can see that (up to subsequence) $y_R$ converges in $C^2_{\mathrm{loc}}(\R^n;\R^m)$ to a solution $y\in C^\infty(\R^n;\R^{n-1})$ of \eqref{painhom} satisfying \eqref{equivv} and \eqref{minnn1}. Again, we point out that $y$ can be written as $y(x)= y_{\mathrm{rad}}(x_1,|z|)\frac{z}{|z|}$, where $y_{\mathrm{rad}}(x_1,\sigma)$ is a function having and odd with respect to $\sigma$ extension in $C^\infty(\R^2;\R)$.
Our next claim is that $y$ cannot be identically zero. Indeed, the minimality of $y$ implies that the second variation of the energy $E_{\mathrm{P_{II}}}$ is nonnegative in the class of $O(n-1)$-equivariant maps: \begin{equation}\label{secondva}
\int_{\R^n} (|\nabla \phi(x)|^2+(6 |y(x)|^2+x_1)|\phi(x)|^2)\mathrm{d} x \geq 0,\quad \forall \phi\in C^1_0(\R^n;\R^m) \text{ satisfying \eqref{equivv}. } \end{equation}
Clearly \eqref{secondva} does not hold when $y\equiv 0$, if we take $\phi(x_1,z)=\phi_0(x_1+l,z)$, with $l\to\infty$, and $\phi_0$ a smooth, $O(n-1)$-equivariant map, such that $\phi_0 \not\equiv 0$. Now, let us check that $y(x_1,z)\cdot z>0$ holds for every $ x_1\in\R$ and $ z\neq 0$, or equivalently $x_2>0 \Rightarrow y_2(x)>0$. By construction, we have $y(x_1,z)\cdot z\geq 0$, $\forall x\in \R^n$, and in particular, $y_2\geq 0$ holds in the half-space $\Omega=\{x_2>0\}$. Since $y_2$ satisfies $\Delta y_2=(x_1+2|y|^2)y_2$, the existence of a point $p\in \Omega$ such that $y_2(p)=0$ would imply by the maximum principle that $y_2\equiv 0$, and $y\equiv 0$. This is a contradiction. Thus, we have checked that $y(x_1,z)\cdot z>0$ holds for every $ x_1\in\R$ and $ z\neq 0$, and since it is obvious that $|y(x)|\leq h(x_1)$, $\forall x\in\R^n$, the proof of (iii) is complete.
To prove (iv), we write $y(x)= y_{\mathrm{rad}}(x_1,|z|)\frac{z}{|z|}$, $\phi(x)= \phi_{\mathrm{rad}}(x_1,|z|)\frac{z}{|z|}$, and utilize \eqref{cyl}. One can see that \eqref{minnn1} is equivalent to \begin{equation}\label{cyl3}
E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad}}, \{|x_1|\leq \alpha, \sigma\in[0, \beta]\})\leq
E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad}}+\phi_{\mathrm{rad}}, \{|x_1|\leq \alpha, \sigma\in[0, \beta]\}), \end{equation}
provided that $\supp \phi_{\mathrm{rad}}\subset [-\alpha,\alpha]\times[-\beta,\beta]$. Now, given $\psi(x_1,z)=\chi(x_1,z)\frac{z}{|z|}$, with $\chi\in C^\infty_0(\R^n;\R)$ and $\supp \chi\subset[-\alpha,\alpha]\times \{z: 0<|z|\leq\beta\}$, we define for every $\nu\in\mathbb{S}^{n-2}$, the $O(n-1)$-equivariant map $\psi_\nu(x_1,z):=\chi(x_1,|z|\nu)\frac{z}{|z|}$. Setting $\chi_\nu(x_1,\sigma):=\chi(x_1,\sigma\nu)$, we have in view of \eqref{cyl3}: \begin{equation}\label{cyl4}
E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad}},\{|x_1|\leq \alpha, \sigma\in[0, \beta]\})\leq
E_{\mathrm{P_{II},rad}}(y_{\mathrm{rad}}+\chi_\nu, \{|x_1|\leq \alpha, \sigma\in[0, \beta]\}), \forall \nu\in\mathbb{S}^{n-2}. \end{equation} We can also check that \begin{equation}\label{cyl5}
\frac{1}{2}|\nabla \chi_\nu(x_1,\sigma)|^2 +\frac{n-2}{2}\frac{|\chi_{\nu}(x_1,\sigma)|^2}{\sigma^2}\leq \frac{1}{2}|\nabla \psi(x_1,\sigma\nu)|^2 \end{equation}
holds for every $|x_1|\leq \alpha$, $\sigma\in[0, \beta]$, $\nu\in\mathbb{S}^{n-2}$. As a consequence, an integration of \eqref{cyl4} over $\nu\in\mathbb{S}^{n-2}$ gives: \begin{equation}\label{cyl6} E_{\mathrm{P_{II}}}(y, \supp \psi)\leq E_{\mathrm{P_{II}}}(y+\psi, \supp \psi). \end{equation}
\end{proof} The uniqueness of the solution $y$ provided by Lemma \ref{lem1} is an open question. In what follows we establish for \emph{any} satisfying the assumptions of Lemma \ref{lem1}, the statements of Theorem \ref{corpain2} hold. We will proceed in few steps. By particularizing Lemma \ref{ass}, we obtain the limit in \eqref{scale2} in the case where $z\neq 0$:
\begin{lemma}\label{lem44}
Let $y$ be the solution provided by Lemma \ref{lem1}, and consider the rescaled map $\tilde y(t_1,\ldots,t_n)$ as in \eqref{ytilde}, with $z\neq 0$ fixed. Then, $\lim_{l\to\-\infty}\tilde y(t_1+l,t_2,\ldots,t_n)=e_z:=\frac{z}{|z|}$ for the $C^1_{\mathrm{loc}}(\R^n;\R^{n-1})$ convergence. \end{lemma} \begin{proof} We proceed as in the proof of Lemma \ref{ass}. Let $(e_1,\ldots,e_n)$ be the canonical basis of $\R^n$, and let $\tilde \chi(t_1,\ldots,t_n)\in C^\infty_0(\R^n;\R)$ be a test function such that
$\tilde S:=\supp \tilde \phi \subset\{(t_1,\ldots,t_n): c-d\leq t_1\leq c, |(t_2, \ldots ,t_n)|\leq d\}$, for some constants $c \in\R$ and $d>0$. Given $l\in\R$, we consider the maps $\tilde \chi^{-l}(t_1,\ldots,t_n):=\tilde \chi(t_1-l,t_2,\ldots,t_n)$, and $\tilde y^l(t_1,\ldots,t_n):=\tilde y(t_1+l,t_2,\ldots,t_n)$. Note that $\tilde S^{l}:=\supp \tilde \chi^{-l}=\tilde S+l e_1$, and
$\supp\tilde \chi^{-l} \subset\{(t_1,\ldots,t_n): t_1<-1\} $ when $l<-1-c$. Furthermore, for $l<l_0:=\min\big(-1-c, -\frac{2}{3}\big(\frac{d}{|z|}\big)^3-c\big)$, we can define $\phi^{-l}\in C^\infty_0(\R^n;\R^{n-1})$ as in \eqref{eqder}, by
$$\phi^{-l}(x_1, r+z)=\frac{(-x_1)^{\frac{1}{2}}}{\sqrt{2}}\tilde \chi^{-l}(t_1,\tau)\frac{r+z}{|r+z|},$$ since we have $S^{l}:=\{(x_1(t_1),r(t_1,\ldots,t_n)+z): \, (t_1,\ldots,t_n)\in \tilde S^{l}\}\subset (-\infty,0)\times (\R^{n-1}\setminus\{0\})$ for $l<l_0$.
As a consequence of Lemma \ref{lem1} (iv), it follows that \begin{equation}\label{weakaa} E_{\mathrm{P_{II}}}( y, S^l)\leq E_{\mathrm{P_{II}}}( y+ \phi^{-l},S^l), \ \forall l<l_0. \end{equation} Now, we compute \begin{subequations}\label{eqderder1} \begin{equation}\label{eqder1b1}
\sqrt{2}\phi^{-l}_{x_i}(x_1, r+z)=(-x_1)\tilde \chi^{-l}_{t_i}(t_1,\tau)\frac{r+z}{|r+z|}+(-x_1)^{\frac{1}{2}}\tilde \chi^{-l}(t_1,\tau)\Big(\frac{e_i}{|r+z|}-\frac{r_i(r+z)}{|r+z|^3}\Big), \ \forall i=2,\ldots,n, \end{equation} \begin{equation}\label{eqder3b1}
\sqrt{2}\phi^{-l}_{x_1}(x_1, r+z)=-\frac{(-x_1)^{-\frac{1}{2}}}{2}\tilde \chi^{-l}(t_1,\tau)\frac{r+z}{|r+z|}+(-x_1) \tilde \chi^{-l}_{t_1}(t_1,\tau)\frac{r+z}{|r+z|}-\sum_{i=2}^n\frac{r_i}{2}\tilde \chi^{-l}_{t_i}(t_1,\tau)\frac{r+z}{|r+z|}, \end{equation} \end{subequations} and we set \begin{equation}\label{ksi}
\xi(t):=\frac{r+z}{|r+z|}=\frac{(-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z}{|(-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z|}=e_z+O((-t_1)^{-\frac{1}{3}}), \text{ provided that $|\tau|$ remains bounded}. \end{equation} After changing variables (cf. also \eqref{eqderder}), \eqref{weakaa} becomes: \begin{equation}\label{weakaab} \int_{\tilde S^l}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[B( \tilde y)+C(\tilde y)]\leq \int_{\tilde S^l}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[G(\tilde y,\tilde \chi^{-l})+R(\tilde y,\tilde \chi^{-l})], \ \forall l<l_0, \end{equation} with $B$ (resp. $C$) as in \eqref{bdef} (resp. \eqref{cdef}), and
$$G(\tilde y,\tilde \chi^{-l})=\Big[\frac{1}{2}\sum_{i=1}^n| \tilde y_{t_i}+ \tilde \chi^{-l}_{t_i}\xi|^2-\frac{|\tilde y +\tilde \chi^{-l}\xi |^2}{2}+\frac{|\tilde y +\tilde \chi^{-l}\xi |^4}{4}\Big],$$ \begin{align*}
R(\tilde y,\tilde \chi^{-l})&=\big(-\frac{3}{2}t_1\big)^{-\frac{1}{3}}\tilde \chi^{-l}\sum_{i=2}^n (\tilde y_{t_i}+\tilde\chi^{-l}_{t_i}\xi)\cdot\Big(\frac{e_i}{|(-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z|}-\frac{t_i(-\frac{3}{2}t_1)^{-\frac{1}{3}}((-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z)}{|(-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z|^3}\Big)\\
&+\big(-\frac{3}{2}t_1\big)^{-\frac{2}{3}}\frac{|\tilde \chi^{-l}|^2}{2}\sum_{i=2}^n\Big|\frac{e_i}{|(-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z|}-\frac{t_i(-\frac{3}{2}t_1)^{-\frac{1}{3}}((-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z)}{|(-\frac{3}{2}t_1)^{-\frac{1}{3}}\tau+z|^3}\Big|^2\\ &-\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{-1}(\tilde y_{t_1}+\tilde\chi^{-l}_{t_1}\xi)\cdot\big(\tilde y+\tilde\chi^{-l}\xi+\sum_{i=2}^nt_i(\tilde y_{t_i}+\tilde \chi^{-l}_{t_i}\xi)\big)\\
&+\frac{1}{8}\big(-\frac{3}{2}t_1\big)^{-2}\Big|\tilde y+\tilde\chi^{-l}\xi-\sum_{i=2}^nt_i(\tilde y_{t_i}+\tilde \chi^{-l}_{t_i}\xi)\Big|^2 \end{align*}
Since estimates \eqref{eqder8} hold in view of the bound $|y(x)|\leq h(x_1)$ provided by Lemma \ref{lem1} (iii), we have $R(\tilde y,\tilde \chi^{-l})=O((-t_1)^{-\frac{1}{3}})$. On the other hand, \eqref{ksi} and \eqref{eqder8} imply that \begin{equation}\label{eqq1}
G(\tilde y,\tilde \chi^{-l})=\Big[\frac{1}{2}\sum_{i=1}^n| \tilde y_{t_i}+ \tilde \chi^{-l}_{t_i}e_z|^2-\frac{|\tilde y +\tilde \chi^{-l}e_z |^2}{2}+\frac{|\tilde y +\tilde \chi^{-l}e_z |^4}{4}\Big]+O((-t_1)^{-\frac{1}{3}}), \end{equation} or equivalently \begin{equation}\label{eqq2} G(\tilde y,\tilde \chi^{-l})=B(\tilde y+\tilde \chi^{-l}e_z)+O((-t_1)^{-\frac{1}{3}}). \end{equation}
Finally, given a sequence $l_k\to -\infty$, one can show as in the proof of Lemma \ref{ass}, that up to subsequence, $\tilde y_k(t_1,t_2,\ldots,t_n):=\tilde y(t_1+l_k,t_2,\ldots,t_n)$ converges in $C^1_{\mathrm{ loc}}(\R^n;\R^{n-1})$ to a solution $u(t_1,t_2,\ldots,t_n)$, $u:\R^n\to\R^{n-1}$, of $\Delta u=|u|^2-u$. Moreover, in view of Lemma \ref{lem1}, we have $u(t)=v(t)e_z$, with $v\geq 0$, since the vectors $\tilde y_k(t)$ and $f_k(t):=z+\big(-\frac{3}{2}(t_1+l_k)\big)^{-\frac{1}{3}}\tau\in\R^{n-1}$ have the same direction, and $\lim_{k\to\infty}f_k(t)=z$.
To conclude, we reproduce the argument at the end of the proof of Lemma \ref{ass}, and obtain that \begin{align*} \int_{\tilde S^{l_k}}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[B( \tilde y)+C(\tilde y)]&= (-c-l_k)^{\frac{4-n}{3}}\int_{\tilde S^{l_k}}B( \tilde y)+O((-c-l_k)^{\frac{1-n}{3}})\\ &=(-c-l_k)^{\frac{4-n}{3}}\int_{\tilde S}B( \tilde y_k)+O((-c-l_k)^{\frac{1-n}{3}}), \end{align*} while \begin{align*} \int_{\tilde{ S}^{l_k}}\frac{1}{2}\big(-\frac{3}{2}t_1\big)^{\frac{4-n}{3}}[G(\tilde{ y},\tilde{ \chi}^{-l_k})+ R(\tilde{ y},\tilde{ \chi}^{-l_k})]&=(-c-l_k)^{\frac{4-n}{3}}\int_{\tilde{ S}^{l_k}}B( \tilde{ y}+\tilde{ \psi}^{-l_k})+O((-c-l_k)^{\frac{3-n}{3}})\\ &=(-c-l_k)^{\frac{4-n}{3}}\int_{\tilde{ S}}B( \tilde{ y}_k+\tilde{ \chi} e_z)+O((-c-l_k)^{\frac{3-n}{3}}), \end{align*} As a consequence, it follows from \eqref{weakaab} that \begin{equation}\label{weq3} \int_{\tilde S}B( \tilde y_k)+O((-c-l_k)^{-1})\leq\int_{\tilde S}B( \tilde y_k+\tilde \chi e_z)+O((-c-l_k)^{-\frac{1}{3}}), \end{equation} holds for $k$ large enough. Next, passing to the limit, we get \begin{equation}\label{weq3} E_{\mathrm{GL}}(v e_z, \tilde S)\leq E_{\mathrm{GL}}(v e_z+\tilde \chi e_z, \tilde S), \end{equation} i.e. $v:\R^n\to\R$ is a nonnegative minimal solution of $\Delta v=v^3-v$. Therefore, in view of \cite[Corollary 5.2]{book}, we deduce that $v \equiv 1$, and since the limit of $\tilde y_k$ is independent of the sequence $l_k\to-\infty$, we have established that $\lim_{l\to\-\infty}\tilde y(t_1+l,t_2,\ldots,t_n)=e_z$. This completes the proof of Lemma \ref{lem44}. \end{proof}
Next, we examine the asymptotic convergence of $y(x_1,z)$, as $|z|\to\infty$.
\begin{lemma}\label{ass22}
Let $y(x_1,z)=y_{\mathrm{rad}}(x_1,|z|)\frac{z}{|z|}$ be the solution provided by Lemma \ref{lem1}, and let $\{z_k\}\subset\R^{n-1}$ be a sequence such that $\lim_{k\to\infty}|z_k|=\infty$, and $\lim_{k\to\infty}\frac{z_k}{|z_k|}=\textbf{\em n}_0$. Then, $y_k(x_1,z):=y(x_1,z+z_k)$ converges as $k\to\infty$, to $h(x_1)\textbf{\em n}_0$ in $C^1_{\mathrm{loc}}(\R^n;\R^{n-1})$. In particular, $\lim_{l\to \infty} y_{\mathrm{rad}}(x_1,\sigma+l)=h(x_1)$ for the $C^1_{\mathrm{loc}}(\R^2;\R)$ convergence. \end{lemma} \begin{proof}
In view of the bound $|y(x_1,z)|\leq h(x_1)$ provided by Lemma \ref{lem1} (iii), we obtain by the theorem of Ascoli that (up to subsequence) $ y_k$ converges in $C^1_{\mathrm{loc}}( \R^n;\R^{n-1})$ to a solution $y_\infty:\R^n\to\R^{n-1}$ of \eqref{painhom}. In addition, since $\ y_k(x_1,z)=y_{\mathrm{rad}}(x_1,|z+z_k|)\frac{z+z_k}{|z+z_k|}$, and $\lim_{k\to\infty} \frac{z+z_k}{|z+z_k|}=\textbf{\em n}_0$, we deduce that $ y_\infty(x_1,z)=\tilde y(x)\textbf{\em n}_0$, with $\tilde y:\R^n\to\R$ a solution of \eqref{scalarpde}.
By construction, $\tilde y$ is nonnegative. We are going to show that $\tilde y$ is also minimal. Indeed, given $\chi\in C^\infty_0(\R^n;\R)$, let $\psi_k(x_1,z):=\chi(x_1,z-z_k)\frac{z}{|z|}$ and $\phi_k(x_1,z):=\chi(x_1,z)\frac{z+z_k}{|z+z_k|}$. In view of Lemma \ref{lem1} (iv), we have $E_{\mathrm{P_{II}}}(y, \supp \psi_k)\leq E_{\mathrm{P_{II}}}(y+\psi_k, \supp \psi_k)$, or equivalently $E_{\mathrm{P_{II}}}(y_k, \supp \chi)\leq E_{\mathrm{P_{II}}}(y_k+\phi_k, \supp \chi)$. Next, passing to the limit, we obtain $E_{\mathrm{P_{II}}}(\tilde y \textbf{\em n}_0, \supp \chi)\leq E_{\mathrm{P_{II}}}((\tilde y +\chi )\textbf{\em n}_0, \supp \chi)$ i.e. $\tilde y$ is a minimal solution of \eqref{scalarpde}. Thus, since $\tilde y$ clearly satisfies $\tilde y(x)\leq h(x_1)$, we deduce from Lemma \ref{lema} that $\tilde y(x)=h(x_1)$, $\forall x\in\R^n$. Moreover, since this limit is uniquely determined, it is independent of the subsequence extracted from $\{y_k\}$. Finally, setting $y=(y_2,\ldots,y_n)\in\R^{n-1}$, we have
$y_{\mathrm{rad}}(x_1,\sigma)=y_2(x_1,\sigma,0,\ldots,0)$,
$\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,\sigma)=\frac{\partial y_2}{\partial x_1}(x_1,\sigma,0,\ldots,0)$, and
$\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,\sigma)=\frac{\partial y_2}{\partial x_2}(x_1,\sigma,0,\ldots,0)$.
Therefore, $\lim_{l\to \infty} y_{\mathrm{rad}}(x_1,\sigma+l)=h(x_1)$ holds in $C^1_{\mathrm{loc}}(\R^2;\R)$, according to what precedes. \end{proof}
To establish the monotonicity properties stated in Theorem \ref{corpain2} (ii), we shall work with the projection $y_2$ of the solution $y=(y_2,\ldots, y_{n})\in\R^{n-1}$ provided by Lemma \ref{lem1}. We shall first compute in Lemmas \ref{lll1} and \ref{lll2}, bounds for $\frac{\partial y_2}{\partial x_1}$ and $\frac{\partial y_2}{\partial x_2}$ when $x_1$ is large enough and $x_2>0$. Our main tool is a version of the maximum principle in unbounded domains \cite[Lemma 2.1]{beres}. We also utilize the asymptotic behaviour of $y$ provided by Lemmas \ref{lem44} and \ref{ass22}. Next, these bounds are extended to the whole space by applying the moving plane method (cf. Lemma \ref{movingplane}).
\begin{lemma}\label{lll1}
Let $y$ be the solution provided by Lemma \ref{lem1}. Then, we have $\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,|z|)<0$, $\forall x_1\geq 0$, $\forall z\neq 0$. In addition, for every $d>0$, there holds $\sup_{|z|\geq d} \frac{\partial y_{\mathrm{rad}}}{\partial x_1}(1,|z|)<0$, and $\inf_{|z|\geq d} y_{\mathrm{rad}}(1,|z|)>0$. \end{lemma}
\begin{proof} Given $\lambda\geq 0$, we define the function $\psi_\lambda(x_1,z):=y_2(x_1,z)-y_2(-x_1+2\lambda,z)$ for $x\in D_\lambda:=\{ (x_1,\ldots,x_n): x_1 > \lambda , x_2 >0\}$. One can check that
$\psi_\lambda=0$ on $\partial D_\lambda$, and $$\Delta \psi_\lambda-c(x)\psi_\lambda= 2(x_1-\lambda)y_2(-x_1+2\lambda,z)\geq 0 \text{ on } D_\lambda,$$ with $c(x)=x_1+2(|y(x)|^2+|y(x)||y(-x_1+2\lambda,z)|+|y(-x_1+2\lambda,z)|^2)\geq 0$. Furthermore, $\psi_\lambda$ is bounded above and not identically zero (cf. Lemma \ref{lem1} (iii) and Lemma \ref{lem44}). As a consequence of the maximum principle (cf. \cite[Lemma 2.1]{beres}), it follows that
$\psi_{\lambda}(x)< 0$, $\forall x_1> \lambda$, $\forall x_2> 0$, $\forall (x_3,\ldots,x_n)\in\R^{n-2}$, and thus by Hopf's Lemma we have $\frac{\partial \psi_\lambda}{\partial x_1}(\lambda,z)=2\frac{\partial y_2}{\partial x_1}(\lambda,x_2,\ldots,x_n)<0$, provided that $ x_2>0$. This proves that $\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,|z|)<0$, $\forall x_1\geq 0$, $\forall z\neq 0$. Finally, Lemma \ref{ass22} implies that $\lim_{\sigma\to\infty}y_{\mathrm{rad}}(1,\sigma)=h(1)$, and $\lim_{\sigma\to\infty}\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(1,\sigma)=h'(1)$. Therefore, it is clear that $\sup_{|z|\geq d} \frac{\partial y_{\mathrm{rad}}}{\partial x_1}(1,|z|)<0$, as well as $\inf_{|z|\geq d} y_{\mathrm{rad}}(1,|z|)>0$ hold. \end{proof}
\begin{lemma}\label{lll2} Let $y=(y_2,\ldots,y_m)\in\R^{n-1}$ be the solution provided by Lemma \ref{lem1}. Then, for every vector $\textbf{\em n}=(\cos(\theta+\frac{\pi}{2}), \sin(\theta+\frac{\pi}{2}),0,\ldots,0)\in\R^n$, with $\theta \in (0,\frac{\pi}{2})$, there exists $s_\textbf{\em n}>0$ such that $\nabla y_2(x)\cdot \textbf{\em n}>0$ holds, provided that $ x_1 >s_\textbf{\em n}$, and $x_2 >0$. \end{lemma}
\begin{proof} Let $(e_2,\ldots,e_n)$ be the canonical basis of $\R^{n-1}$. Our first claim is that there is a constant $k_1>0$, such that $k_1 \frac{\partial y_2}{\partial x_1} (x)\leq -\sqrt{x_1} y_2(x)$, provided that $x_1\geq 1$, and $x_2\geq 0$. Indeed, let \begin{equation}\label{psi9}
\psi(x)=k_1 \frac{\partial y_2}{\partial x_1} (x)+\sqrt{x_1} y_2( x)=\Big(k_1\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,|z|)+\sqrt{x_1} y_{\mathrm{rad}}(x_1,|z|)\Big)\frac{x_2}{|z|}, \end{equation} for $x\in D:=\{x:\R^n:x_1>1, x_2>0\}$, where the constant $k_1>0$ will be adjusted later. It is clear that $\psi$ vanishes on the hyperplane $x_2=0$. We also notice that $\frac{\partial^2 y_2}{\partial x_1\partial x_2}(1,0,x_3,\ldots,x_n)<0$ by Hopf's Lemma, since the function $\frac{\partial y_2}{\partial x_1}$ vanishes on $\{x_1=1,x_2=0\}$, is negative on $\{ x_1> 0, x_2> 0\}$, and satisfies \begin{equation}\label{eqqq11b}
\Delta \frac{\partial y_2}{\partial x_1}= y_2 +(x_1+6 | y|^2)\frac{\partial y_2}{\partial x_1}\geq(x_1+6|y|^2)\frac{\partial y_2}{\partial x_1} \text{ on } D. \end{equation}
As a consequence, when $k_1$ is large enough, there exists $d>0$ such that $\psi(1,z)\leq 0$, provided that $|z|\leq d$ and $x_2\geq 0$. In addition, \eqref{psi9} and $\sup_{|z|\geq d} \frac{\partial y_{\mathrm{rad}}}{\partial x_1}(1,|z|)<0$, $\forall d>0$, imply that when $k_1$ is large enough, we have $\psi(1,x_2,\ldots,x_n)\leq 0$, $\forall x_2\geq 0$, $\forall (x_3,\ldots,x_n)\in\R^{n-2}$. Next, we compute \begin{align*}
\Delta \psi&=\Big(x_1+6|y|^2+\frac{1}{k_1\sqrt{x_1}}\Big)k_1\frac{\partial y_2}{\partial x_1}+\Big(x_1+2|y|^2+\frac{k_1}{\sqrt{x_1}}-\frac{1}{4x_1^2}\Big) \sqrt{x_1}y_2\\
&=\Big(x_1+2|y|^2+\frac{k_1}{\sqrt{x_1}}-\frac{1}{4x_1^2}\Big) \psi+\Big(4|y|^2+\frac{1}{k_1\sqrt{x_1}}-\frac{k_1}{\sqrt{x_1}}+\frac{1}{4x_1^2}\Big) k_1 \frac{\partial y_2}{\partial x_1}. \end{align*}
By choosing $k_1$ large enough we can ensure that $\big(x_1+2|y|^2+\frac{k_1}{\sqrt{x_1}}-\frac{1}{4x_1^2}\big) \geq 0$ and $\big(4|y|^2+\frac{1}{k_1\sqrt{x_1}}-\frac{k_1}{\sqrt{x_1}}+\frac{1}{4x_1^2}\big)\leq 0$, when $x_1\geq 1$, and $x_2\geq 0$. Thus, our claim follows from the maximum principle (cf. \cite[Lemma 2.1]{beres}).
Similarly, we are going to establish that there is a constant $k_2>0$, such that $\frac{\partial y_2}{\partial x_2} (x)\geq -k_2 y_2(x)$, provided that $x_1\geq 1$, and $x_2\geq 0$. To do this we let \begin{equation}\label{psipsi}
\psi(x)=- \frac{\partial y_2}{\partial x_2}(x)-k_2 y_2(x) =-\frac{x_2}{|z|}\Big(\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,|z|)\frac{x_2}{|z|}+k_2y_{\mathrm{rad}}(x_1,|z|)\Big) -y_{\mathrm{rad}}(x_1,|z|)\Big(\frac{|z|^2-x_2^2}{|z|^3}\Big)\text{ for $x\in D$}, \end{equation} where the constant $k_2$ will again be adjusted later. We first notice that
$\frac{\partial y_2}{\partial x_2}(x_1,0,x_3,\ldots,x_n)>0$ holds on the hyperplane $x_2=0$, since the function $y_2$ vanishes on the hyperplane $x_2=0$, is positive in $\{x_2> 0\}$, and satisfies $\Delta y_2=(x_1+2|y|^2)y_2$.
As a consequence, when $k_2$ is large enough, there exists $d>0$ such that $\psi(1,z)\leq 0$, provided that $|z|\leq d$ and $x_2\geq 0$. In addition, \eqref{psipsi} and $\inf_{|z|\geq d} y_{\mathrm{rad}}(1,|z|)>0$, $\forall d>0$, imply that when $k_1$ is large enough, we have $\psi(1,x_2,\ldots,x_n)\leq 0$, provided that $x_2\geq 0$. On the other hand, it is clear that $\psi(x_1,0,x_3,\ldots,x_n)<0$, $\forall x_1\geq 1$, $\forall (x_3,\ldots,x_n)\in\R^{n-2}$. Next, we compute successively for $x\in D$:
$$\frac{\partial y}{\partial x_2}(x)=\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,|z|)\frac{x_2}{|z|}\frac{z}{|z|}+y_{\mathrm{rad}}(x_1,|z|) \Big(\frac{e_2}{|z|}-\frac{x_2z}{|z|^3}\Big),$$
$$y(x)\cdot \frac{\partial y}{\partial x_2}(x)=\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,|z|) y_{\mathrm{rad}}(x_1,|z|) \frac{x_2}{|z|} ,$$
$$\Big(y(x)\cdot \frac{\partial y}{\partial x_2}(x)\Big)y_2(x)=\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,|z|) y_{\mathrm{rad}}^2(x_1,|z|) \frac{x_2^2}{|z|^2} \leq |y(x)|^2\frac{\partial y_2}{\partial x_2}(x),$$
$$\Delta \frac{\partial y_2}{\partial x_2}(x)=(x_1+2|y(x)|^2) \frac{\partial y_2}{\partial x_2}(x) +4 \Big( y(x)\cdot \frac{\partial y}{\partial x_2}(x)\Big)y_2(x)\leq (x_1+6|y(x)|^2) \frac{\partial y_2}{\partial x_2}(x),$$
$$\Delta y_2(x)=(x_1+2|y(x)|^2)y_2(x)\leq (x_1+6|y(x)|^2)y_2(x),$$ \begin{align*}
\Delta \psi&\geq (x_1+6|y|^2)\psi. \end{align*} Thus, it follows from the maximum principle that $\psi\leq 0$ in $D$.
Finally, given $\theta\in (0,\pi/2)$, we have \[\nabla y_2(x)\cdot \textbf{\em n}=-\frac{\partial y_2}{\partial x_1}(x)\sin\theta+\frac{\partial y_2}{\partial x_2}(x) \cos\theta\geq\Big(\frac{\sqrt{x_1}}{k_1}\sin\theta-k_2 \cos\theta\Big) y_2(x), \ \forall x \in [1,\infty)\times[ 0,\infty)\times\R^{n-2},\] and therefore $\nabla y_2(x)\cdot \textbf{\em n}>0$ provided that $x_1>s_\textbf{\em n}:=\big(\frac{k_1k_2}{\tan \theta}\big)^2$, and $x_2>0$. \end{proof}
\begin{lemma}\label{movingplane} Let $y=(y_2,\ldots,y_m)\in\R^{n-1}$ be the solution provided by Lemma \ref{lem1}, and let $\theta\in (0,\frac{\pi}{2})$ be fixed. For every $\lambda \in \R$, we consider the reflection $\rho_\lambda$ with respect to the hyperplane $\Gamma_\lambda:=\{x\in\R^n: x_2=\tan\theta (x_1-\lambda)\}$, and the domain $D_\lambda:=\{x\in\R^n: 0<x_2<\tan\theta (x_1-\lambda)\}$. Then, the function $\psi_\lambda(x):=y_2(x)-y_2(\rho_\lambda(x))$ is negative in $D_\lambda$, for every $\lambda\in\R$. \end{lemma} \begin{proof} We set $\textbf{\em n}=(\cos(\theta+\frac{\pi}{2}), \sin(\theta+\frac{\pi}{2}),0,\ldots,0)\in\R^n$, and denote by $(p',q',\zeta)\in\R\times\R\times\R^{n-2}$ the image by $\rho_\lambda$ of a point $(p,q,\zeta)\in D_\lambda$, and by $D'_\lambda$ the set $\rho_\lambda(D_\lambda)$. It is obvious that $\psi_\lambda(x_1,0,\zeta)< 0$, $\forall x_1> \lambda$, $\forall \zeta\in\R^{n-2}$, and that $\psi_\lambda(x)= 0$, $\forall x\in \Gamma_\lambda$. We can also check that for $(p,q,\zeta)\in D_\lambda$, we have $p>p'$ and $q<q'$, as well as: \begin{subequations} \begin{equation}\label{eqa1}
y_2(p,q,\zeta)=|y(p,q,\zeta)|\frac{q}{\sqrt{q^2+|\zeta|^2}}, \end{equation} \begin{equation}\label{eqa2}
y_2(p',q',\zeta)=|y(p',q',\zeta)|\frac{q'}{\sqrt{(q')^2+|\zeta|^2}}, \end{equation} \begin{equation}\label{eqa3}
\frac{q}{\sqrt{q^2+|\zeta|^2}}\leq\frac{q'}{\sqrt{(q')^2+|\zeta|^2}}, \end{equation}
\begin{equation}\label{eqa4} \Delta \psi_\lambda(p,q,\zeta)-c(p,q,\zeta)\psi_\lambda(p,q,\zeta)\geq 0, \end{equation} \end{subequations} with
$$c(p,q,\zeta)=
p+2 (|y(p,q,\zeta)|^2+|y(p,q,\zeta)||y(p',q',\zeta)|+|y(p',q',\zeta)|^2).$$
Next, for each $\lambda\in\R$ we consider the statement \begin{equation}\label{statem} \psi_\lambda(p,q,\zeta) <0, \quad \forall (p,q,\zeta)\in D_\lambda. \end{equation}
We shall first establish Lemma \ref{movingplane} in the case where $\theta\in (0,\frac{\pi}{4})$. According to Lemma \ref{lll2}, \eqref{statem} is valid for each $\lambda> s_\textbf{\em n}$. Set $\lambda_0=\inf\{\lambda\in \R: \psi_\mu<0 \text{ holds in $ D_\mu$, for each $\mu\geq \lambda$} \}$. We will prove $\lambda_0=-\infty$. Assume instead $\lambda_0\in\R$. Then, there exist a sequence $\lambda_k<\lambda_0$ such that $\lim_{k\to\infty}\lambda_k=\lambda_0$, and a sequence $(p_k,q_k,\zeta_k)\in D_{\lambda_k}$, such that \begin{equation}\label{assume} y_2(p_k,q_k,\zeta_k)\geq y_2(p'_k,q'_k,\zeta_k), \forall k. \end{equation} According to Lemma \ref{lll2}, we have $p'_k\leq s_\textbf{\em n}$, thus the sequence $(p_k,q_k)\subset\R^2$ is bounded, since by assumption $\theta\in(0,\pi/4)$. Up to subsequence we may assume that $\lim_{k\to\infty}(p_k,q_k)=(p_0,q_0)$, with $p'_0\leq s_\textbf{\em n}$.
We first examine the case where up to subsequence $\lim_{k\to\infty}\zeta_k=\zeta_0\in\R^{n-2}$. Note that $(p_0,q_0,\zeta_0)\in\overline{D_{\lambda_0}}$. By definition of $\lambda_0$, we have $\psi_{\lambda_0}\leq 0$ in $D_{\lambda_0}$, and $\psi_{\lambda_0}(p_0,q_0,\zeta_0)=0$ i.e. $y_2(p_0,q_0,\zeta_0)=y_2(p'_0,q'_0,\zeta_0)$. Now we distinguish the following cases. If $(p_0,q_0,\zeta_0)\in D_{\lambda_0}$, the maximum principle implies that $\psi_{\lambda_0}\equiv 0$ in $D_{\lambda_0}$. Clearly, this situation is excluded, since $y_2$ is positive in the half-space $\{x_2>0\}$. On the other hand, the maximum principle also implies that $\frac{\partial \psi_{\lambda_0}}{\partial \textbf{\em n}}(p,q,\zeta)=2 \frac{\partial y_2}{\partial \textbf{\em n}}(p,q,\zeta)>0$, provided that $(p,q,\zeta)\in \Gamma_{\lambda_0}$ and $q>0$. Furthermore, the previous inequality still holds on the subspace $\{p=\lambda_0\}\cap\{q=0\}$, since $\frac{\partial y_{2}}{\partial x_2}(x_1,0,\zeta)>0$ and $\frac{\partial y_2}{\partial x_1}(x_1,0,\zeta)=0$ hold, for every $ x_1\in\R$, and $ \zeta\in\R^{n-2}$ (cf. the proof of Lemma \ref{lll2}). As a consequence,$(p_0,q_0,\zeta_0)$ cannot belong to $\Gamma_{\lambda_0}$. Finally, since the case where $p_0>\lambda_0$ and $q_0=0$ is ruled out (because $y_2$ is positive in the half-plane $\{x_2>0\}$), we have reached a contradiction.
On the other hand, when $\lim_{k\to\infty}|\zeta_k|=\infty$, we have in view of \eqref{eqa1} and \eqref{eqa2}: \begin{equation}
\frac{ y_2(p_k,q_k,\zeta_k) }{y_2(p'_k,q'_k,\zeta_k)}=\frac{|y(p_k,q_k,\zeta_k)|q_k\sqrt{(q'_k)^2+|\zeta_k|^2}}{|y(p'_k,q'_k,\zeta_k)|q'_k\sqrt{q^2_k+|\zeta_k|^2}}\sim\frac{|y(p_k,q_k,\zeta_k)|q_k}{|y(p'_k,q'_k,\zeta_k)|q'_k},\text{ as $k\to\infty$}. \end{equation}
In addition, it follows from Lemma \ref{ass22}, that $\lim_{k\to\infty}\frac{|y(p_k,q_k,\zeta_k)|}{|y(p'_k,q'_k,\zeta_k)|}=\frac{h(p_0)}{h(p'_0)}$. Thus, since $p_0\geq p'_0$, $h(p'_0)\geq h(p_0)$, and $q'_0\geq q_0$ hold, the assumption $y_2(p_k,q_k,\zeta_k)\geq y_2(p'_k,q'_k,\zeta_k)$ implies that $p_0=p'_0$ and $q_0=q'_0$. Then, we notice that, $$ y_2(p'_k,q'_k,\zeta_k)- y_2(p_k,q_k,\zeta_k)=\sqrt{(p_k-p'_k)^2+(q_k-q'_k)^2}\frac{\partial y_2}{\partial \textbf{\em n}}((p_k,q_k,\zeta_k)+t_k \textbf{\em n}),$$ with $t_k\in(0,\sqrt{(p_k-p'_k)^2+(q_k-q'_k)^2})$, and
$\frac{\partial y_2}{\partial \textbf{\em n}}(x)\geq \frac{x_2}{|z|}\big(-\sin\theta\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,|z|)+\cos\theta\frac{x_2}{|z|}\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,|z|)\big)$.
Setting $(p_k,q_k,\zeta_k)+t_k \textbf{\em n}=:(\tilde p_k,\tilde q_k,\tilde \zeta_k)$, we deduce again from Lemma \ref{ass22} that \begin{align*}
\frac{\partial y_2}{\partial \textbf{\em n}}(\tilde p_k,\tilde q_k,\tilde \zeta_k)\geq \frac{\tilde q_k}{\sqrt{\tilde q_k^2+|\tilde\zeta_k|^2}}(-h'(p_0)\sin\theta +o(1)). \end{align*} Therefore, we obtain $ y_2(p'_k,q'_k,\zeta_k)- y_2(p_k,q_k,\zeta_k)>0$, for $k$ large enough, which is a contradiction. \begin{figure}
\caption{The sets $A_\lambda$, $A'_\lambda$, and the subspace $\Gamma_\lambda$, in the cases where $\lambda<s_\textbf{\em n}$ and $\lambda>s_\textbf{\em n}$. }
\label{fig}
\end{figure}
Next, we establish Lemma \ref{movingplane} in the case where $\theta \in [\frac{\pi}{4},\frac{\pi}{2})$. When $\theta=\frac{\pi}{4}$, it is clear that
\eqref{statem} is valid for each $\lambda>s_\textbf{\em n}$. Otherwise, when $\theta \in (\frac{\pi}{4},\frac{\pi}{2})$, let $A'_\lambda:=\{(p',q',\zeta)\in D'_\lambda: p'\leq s_\textbf{\em n}\}$, and let $A_\lambda=\rho_\lambda(A'_\lambda)$. Our first claim is that $m:=\inf_{A'_{s_\textbf{\em n}+1}}|y| >0$. Indeed, proceeding as in the proof of Lemma \ref{lem44}, one can see that \[
\lim_{x \in A'_{s_\textbf{\em n}+1},x_1\to-\infty}\frac{\sqrt{2}}{\sqrt{-x_1}}|y(x)|=1. \]
In addition, according to Lemma \ref{ass22}, we obtain that $\inf\{|y(x)|: x\in A'_{s_\textbf{\em n}+1}, \, s_\textbf{\em n}-l\leq x_1\leq s_\textbf{\em n}\}>0$, for every constant $l>0$. Thus, $m>0$. On the other hand, we have $\lim_{\lambda\to\infty}\sup\{| y(x)|: x\in A_\lambda\}=0$, since $\lim_{\lambda\to\infty}\inf\{ x_1: x\in A_\lambda\}=0$. As a consequence when $\lambda\geq s_\textbf{\em n}+1$ is large enough, we have \begin{equation}
\frac{ y_2(p,q,\zeta) }{y_2(p',q',\zeta)}\leq \frac{|y(p,q,\zeta)|}{|y(p',q',\zeta)|}<1, \ \forall (p,q,\zeta)\in A_\lambda, \end{equation} and also $y_2(p',q',\zeta)<y_2(p,q,\zeta)$, $\forall (p,q,\zeta)\in D_\lambda\setminus A_\lambda$, by definition of $s_\textbf{\em n}$. This establishes that \eqref{statem} holds for $\lambda$ large enough. Then, defining $\lambda_0$ as previously, we assume by contradiction that $\lambda_0\in\R$, and there exist sequences $\lambda_k\to\lambda_0$ and $(p_k,q_k,\zeta_k)\in D_{\lambda_k}$ satisfying \eqref{assume}. We need to show that $(p_k,q_k)$ is bounded in $\R^2$.
Indeed, if $\lim_{k\to\infty}p_k=\infty$, then we also have $\lim_{k\to\infty}q'_k=\infty$, as well as $p'_k\leq s_\textbf{\em n}$, in view of \eqref{assume} and the definition of $s_n$. In particular, it follows from Lemma \ref{lem44} and Lemma \ref{ass22} (resp. from the bound $|y(x)|\leq h(x_1)$) that
$\liminf_{k\to\infty}|y(p'_k,q'_k,\zeta_k)|\geq h(s_\textbf{\em n})$ (resp. $\lim_{k\to\infty}|y(p_k,q_k,\zeta_k)|=0$). As a consequence, we obtain \begin{equation}
\frac{ y_2(p_k,q_k,\zeta_k) }{y_2(p'_k,q'_k,\zeta_k)}\leq \frac{|y(p_k,q_k,\zeta_k)|}{|y(p'_k,q'_k,\zeta_k)|}\to 0, \text{ as $k\to\infty$}, \end{equation} which contradicts \eqref{assume}. Now that the boundedness of the sequence $(p_k,q_k)$ is established, to complete the proof we reproduce the arguments detailed in the case where $\theta\in (0,\frac{\pi}{4})$.
\end{proof}
Lemma \ref{movingplane} implies that $\forall \theta \in (0,\frac{\pi}{2})$, $\forall \lambda \in \R$, and $(p,q,\zeta)\in \Gamma_{\lambda}$ with $q>0$, we have
$\frac{\partial \psi_{\lambda}}{\partial \textbf{\em n}}(p,q,\zeta)=2\frac{\partial y_2}{\partial \textbf{\em n}}(p,q,\zeta)>0$, where $\textbf{\em n}=(\cos(\theta+\frac{\pi}{2}), \sin(\theta+\frac{\pi}{2}),0,\ldots,0)$. It follows that $\frac{\partial y_2}{\partial x_1}(x)\leq 0$, and $\frac{\partial y_2}{\partial x_2}(x)\geq 0$, provided that $x_2\geq 0$. Moreover, in the half-space $x_2\geq 0$, $\frac{\partial y_2}{\partial x_1}$ and $\frac{\partial y_2}{\partial x_2}$ satisfy respectively $\Delta \frac{\partial y_2}{\partial x_1}\geq (x_1+6|y|^2)\frac{\partial y_2}{\partial x_1}$, and $\Delta \frac{\partial y_2}{\partial x_2}\leq (x_1+6|y|^2)\frac{\partial y_2}{\partial x_2}$, thus $\frac{\partial y_2}{\partial x_1}$ (resp. $\frac{\partial y_2}{\partial x_2}$) cannot vanish in the open half-space $x_2>0$, since otherwise we would obtain by the maximum principle $\frac{\partial y_2}{\partial x_1}\equiv 0$ (resp. $\frac{\partial y_2}{\partial x_2}\equiv 0$). These situations are excluded by Lemma \ref{lll1} and the fact that $y_2>0$ in the half-space $x_2>0$. Therefore we have proved that $\frac{\partial y_{\mathrm{rad}}}{\partial x_1}(x_1,\sigma)=\frac{\partial y_2}{\partial x_1}(x_1,\sigma,0,\ldots,0)<0$, and $\frac{\partial y_{\mathrm{rad}}}{\partial \sigma}(x_1,\sigma)=\frac{\partial y_2}{\partial x_2}(x_1,\sigma,0,\ldots,0)>0$, provided that $\sigma>0$.
Finally, we consider the rescaled map $\tilde y(t_1,\ldots,t_n)$ as in \eqref{ytilde}, with $z=0$, and proceed as in the proof of Lemma \ref{lem44}. Given a sequence $l_k\to -\infty$, one can see, that up to subsequence, $\tilde y_k(t_1,t_2,\ldots,t_n):=\tilde y(t_1+l_k,t_2,\ldots,t_n)$ converges in $C^1_{\mathrm{ loc}}(\R^n;\R^{n-1})$ to a solution $u(t_1,t_2,\ldots,t_n)$, $u:\R^n\to\R^{n-1}$, of $\Delta u=|u|^2-u$. Moreover, $u$ is by construction $O(n-1)$-equivariant with respect to $\tau:=(t_2,\ldots,t_n)$ (cf. \eqref{equivv}), and minimal for $O(n-1)$-equivariant perturbations. We also notice that \eqref{eqder3} and $y_{x_1}(x_1,r)\cdot \frac{r}{|r|}<0$, $\forall x_1\in\R$, $\forall r \in \R^{n-1}\setminus\{0\}$, imply that
$|t_1+l_k|^{\frac{2}{3}}\tilde y_{t_1}(t_1+l_k,t_2,\ldots,t_n)\cdot\frac{\tau}{|\tau|}+O(|t_1+l_k|^{-\frac{1}{3}})\leq 0$, $\forall t_1\in\R$, $\forall \tau\in\R^{n-1}\setminus\{0\}$. Passing to the limit as $k\to\infty$, we deduce that $u_{t_1}(t_1,\ldots,t_n)\cdot\frac{\tau}{|\tau|}\leq 0$, $\forall t_1\in\R$, $\forall \tau\in\R^{n-1}\setminus\{0\}$. As a consequence the limits $\lim_{t_1\to\pm\infty}u(t_1,t_2,\ldots,t_n)=:v^\pm(t_2,\ldots,t_n)$ exist, and one can see that $v^\pm:\R^{n-1}\to\R^{n-1}$ is an $O(n-1)$-equivariant solution of $\Delta v^\pm=|v^\pm|^2v^\pm-v^\pm$, which is minimal for $O(n-1)$-equivariant perturbations. That is, $v^\pm\equiv \eta$, where $\eta:\R^{n-1}\to\R^{n-1}$ is the standard vortex solution of the Ginzburg-Landau system \eqref{gl}. In addition, in view of the monotonicity of $u$ along the $t_1$ direction, we obtain that $u(t_1,\ldots,t_n)=\eta(t_2,\ldots,t_n)$, and since this limit is independent of the sequence $\{l_k\}$, we have established that \eqref{scale2} holds in the case where $z=0$. This completes the proof of Theorem \ref{corpain2}.
\section*{Acknowledgments}
The author would like to thank Micha{\l } Kowalczyk for several fruitful discusssions during a visit at the University of Chile. He was partially supported by the National Science Centre, Poland (Grant No. 2017/26/E/ST1/00817)
\end{document} | arXiv |
If Alpha Centauri A's solar system exactly mirrored our own, what would we be able to detect?
Suppose there was an exact replica of our solar system 4.4 ly away (people included). What would we be able to detect and with what telescope(s)? Which planets? Could we detect radio transmissions and/or any atmospheres?
I assume detection would be optimal if we were co-planar with the other star's ecliptic, so what would we see in the best- and worst-case (90°-view?) scenarios?
Post Script: a few months later I asked something like this during a von Karman lecture by Neil Turner.
Nick T
Nick TNick T
$\begingroup$ Related: astronomy.stackexchange.com/questions/1037/… $\endgroup$
$\begingroup$ The radio detection side of things is dealt with by astronomy.stackexchange.com/questions/8146/… $\endgroup$
This is a broad question and too broad for me to answer comprehensively. It should be broken down into doppler methods, transits and direct imaging; and that's before we get to questions of detecting Kuiper belts, radio emission etc.
I'll stick for the moment with what I know about detection of planets using the doppler wobble technique.
Doppler Technique
The reflex radial velocity semi-amplitude of a star for the case of a planet of mass $m_2$ orbiting a star of mass $m_1$, in an elliptical orbit with eccentricity $e$, and orbital period $P$ and with an orbital axis inclined at $i$ to the line of sight from Earth is: $$ \left( \frac{2\pi G}{P}\right)^{1/3}\frac{m_2 \sin i}{m_1^{2/3}} (1-e^2)^{-1/2}. $$ A (very) detailed derivation is given by Clubb (2008).
So I built myself a little spreadsheet and assumed that all the planets were seen optimally at $i=90^{\circ}$ (they could not all be seen optimally, but the smallest inclination would be about $i=83^{\circ}$ for Mercury, so it doesn't make too much difference) I'll also assume the mass of Alpha Cen A is about $M \simeq 1.1M_{\odot}$.
The results are
RV semi-amplitude (m/s)
Mercury $8.3\times 10^{-3}$
Venus $8.1\times 10^{-2}$
Earth $8.4\times 10^{-2}$
Mars $7.5\times 10^{-3}$
Jupiter $11.7$
Saturn $2.6$
Uranus $0.28$
Neptune $0.26$
The limits of what are possible are well illustrated by a planet around Alpha Cen B, claimed to be in a 3 day orbit and with a mass similar to the Earth (Dumusque et al. 2012, and see exoplanets.org). The radial velocity semi-amplitude detected here was $0.51\pm 0.04$ m/s, and some spectrographs, notably the HARPS instruments, are routinely delivering sub 1 m/s precision. Thus Jupiter and Saturn would be detectable, Uranus and Neptune are right on the edge of detectability (remember you can average over many RV observations), but the terrestrial planets would not be found (Earth detections would require precisions below 10 cm/s. Remember also that the weaker signals would have to be dug out from the larger signals due to the Jupiter- and Saturn-like planets.
However, there is a second limitation: to find a planet using the doppler method you need to observe for at least a significant fraction of the orbital period. Given that current m/s precisions have been available for only $\sim 5$ years, it is unlikely that Saturn would yet have been detected.
A picture that illustrates the situation can be obtained from the exoplanets.org website, to which I have added lines that approximate where RV semi-amplitudes would be for 10 m/s and 1 m/s precision (assuming the Alpha Cen A mass and circular orbits). I've marked on the Earth, Jupiter and Saturn. Note that few objects have been discovered below the 1 m/s line. Also note the lack of planets between the 1 and 10m/s lines with periods longer than a couple of years - the recent increase in sensitivity has yet to feed through to lower mass, longer period exoplanet discoveries.
In conclusion: only Jupiter would have been so far found by the doppler technique.
Transit techniques
I'll also add a few comments about the transit technique. Transit detection will only work if the exoplanets orbit such that they cross in front of the star. So high inclinations are mandatory. Someone who is better at spherical trigonometry should use the published data for the solar system to work out how many (and which) planets transit in some highly optimal orientation. Given that the planets have orbital inclinations with a scatter of a few degrees, then some straightforward trigonometry and a comparison with the solar radius, tells you that these orbits will generally not all transit for any particular viewing angle. Indeed a number of the Kepler-discovered multiple transit systems are much "flatter" than the solar system.
The Kepler satellite is/was capable of detecting very small transiting planets thanks to its very high photometric precision (the dip in flux is proportional to the square root of the exoplanet radius). The picture below, presented by the NASA Kepler team (slightly out of date now), shows that planetary candidates have been discovered that are down to the size of Mars. However these tend to be in short period orbits because a transit signal needs to be seen a number of times, and Kepler studies this patch of sky for about 2.5 years (when this plot was produced).
So from this point of view, possibly Venus would have been seen, but none of the other planets could be confirmed.
However, there is a wrinkle. Alpha Cen A is way too bright for these kinds of studies and way brighter than the Kepler stars. You would have to build a special instrument or telescope to look for transits around very bright stars. Some of this work has been done by ground-based surveys (mainly finding hot Jupiters). A new satellite called TESS (Transiting Exoplanet Survey Satellite, launched in April 2018) is a two year mission, focused on finding small planets (Earth-sized and bigger) around bright stars. However, most of its targets (including Alpha Cen) are only observed for a 1-2 months, so only the inner parts of their planetary systems will be probed.
ProfRobProfRob
$\begingroup$ As a quick update, the TESS launch has (not too surprisingly) slipped to 2018, but on the other hand it will be launched on a Falcon 9, so there's that. $\endgroup$
– Emilio Pisanty
First, I think Rob Jeffries answer is brilliant. I'll just add a minor points that might be worth mentioning.
What would we be able to detect and with what telescope(s)?
Alpha Centauri A is a binary star with Alpha Centauri B and they are close enough in size to have no stable L4 or L5, so anything that orbits either one of them would need to be either very close (Mercury distance perhaps Venus) or very far and very cold, much greater than Pluto distance, orbiting both stars like Proxima Centauri does.
If you put a Jupiter in it's solar orbit around A or B, the 3 body effect would almost certainly create a wildly unstable orbit for the planet that likely wouldn't last long, so, one answer to this question is that our solar-system type of orbit around A or B is impossible.
Could we detect radio transmissions and/or any atmospheres?
For now, our detection of atmosphere is very limited and only to large planets very close to their stars, but, the article says they're working on that with bigger telescopes on the way, so maybe in a few years we'll get something on that for habitable zone planets.
exoplanet atmosphere detection
On radio-waves and worth mentioning, visible light, I couldn't find a good article, but if an alien planet shoots a message towards us in a tight beam - then, I'm sure we could detect that, provided they shoot a big enough beam, But could we detect another earth with our current output? I don't think we're close to that kind of detection technology.
(and if I got any of that wrong, I welcome correction).
(I asked something like this during a von Karman lecture by Neil Turner)
Did he answer you? Did he say anything good?
userLTKuserLTK
$\begingroup$ Click the link! He basically said detecting Jupiter via radial velocity would be do-able, but slow (an orbit or two), and transit method would be very low probability. $\endgroup$
– Nick T
Neal Turner's answer from the "The Birth of Planets" von Karman Lecture
How would we detect planets around a far-flung, identical copy of our solar system? Would our planets need to be detected using the transit method?
On the whole, yes. Jupiter you could probably detect by the radial velocity method if you're willing to wait one orbit or maybe two to be sure, so 12 years for Jupiter to go around the sun.
The other planets would be really tough. If they transited you could detect them with technology similar to ours. You'd have to be lucky because our solar system is not compact like [others discovered by Kepler]; it's quite spread out. If you have a planet very close to its star, you have a decent chance, if it has a random orientation, that it'll be along your line of sight. If it's very far away there's many more possibilities for its orientation and there's a much smaller probability if things are random that you'll get it along your line of sight exactly.
So for someone to see our Jupiter from a nearby star is much less likely than for us to see a hot Jupiter. There's only a small number of aliens looking at our solar system and seeing it through transits right now.
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